vol.71, #8 Notices of the American Mathematical Society 9781470478001

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ISSN 0002-9920 (print) ISSN 1088-9477 (online)

of the American Mathematical Society September 2024

Volume 71, Number 8

The cover design is based on imagery from “Topological Photonics: A Mathematical Perspective,” page 994.

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September 2024 Cover Credit: The image used in the cover design appears in “Topological Photonics: A Mathematical Perspective,” p. 994, and is courtesy of Ross Parker.

A Word from... Stephan Ramon Garcia...................... 992

Washington Update: The AMS Marks Twenty

Feature: Topological Photonics: A Mathematical

Karen Saxe

Ross Parker and Alejandro Aceves

News: AMS Updates................................................... 1112

Feature: Clusters and Weaves................................... 1004 Mikhail Gorsky and José Simental

News: Mathematics People........................................ 1114

Perspective................................................................... 994

Feature: Multiscale Modeling of Viscoelastic

Fluids..........................................................................1015

Paula A. Vasquez

Years of Sending Mathematicians to Congress........ 1082

Classified Advertising................................................. 1117 New Books Offered by the AMS................................ 1118 Meetings & Conferences of the AMS.........................1125

Feature: Codimension One Foliations on

Projective Manifolds................................................. 1025

Jorge Vitório Pereira

Early Career: Building Community and Keeping

Momentum............................................................... 1032 Finding and Creating Community in Your Department........................................................... 1032 Alan Chang and Rachel Greenfeld

Transitioning as an Early-Career Mathematician...................................................... 1034 Rosemarie Bongers

Do Mathematics Every Day.................................. 1037 Daniel J. Thompson

Productivity and Time Management in Research................................................................ 1040 Steven Senger

How Does Your Daily Life Change When You Become the Graduate Coordinator?.................... 1042 Chun-Kit Lai

Dear Early Career.................................................. 1043 Memorial Tribute: Eli Goodman (1933–2021)

and Ricky Pollack (1935–2018)............................... 1044

János Pach, Micha Sharir, Noga Alon, and Andreas Holmsen Book Review: The Mathematician............................ 1054 Reviewed by Thomas Garrity Education: Freeman Hrabowski: Advocate for

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FROM THE AMS SECRETARY Calls for Nominations & Applications....................... 1084

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State of the AMS: Executive Director Report........... 1086 2024 Election.............................................................. 1089

Election Information............................................ 1089 Candidate Biographies......................................... 1091

2025 Election............................................................... 1109

Call for Suggestions............................................... 1109 Nominations by Petition...................................... 1110

INVITATIONS FROM THE AMS

Christian Anderson

What is... a Parking Function?.................................. 1062 J. Carlos Martínez Mori

Bookshelf................................................................... 1067 Communication: Label Bias: A Pervasive and

Invisibilized Problem............................................... 1069 Yunyi Li, Maria De-Arteaga, and Maytal Saar-Tsechansky Communication: Double-Anonymous Peer

Review in Mathematics: Implementation for American Mathematical Society Journals................ 1079 Dan Abramovich, Henry Cohn, David Futer, and Robert Harington

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Notices of the American Mathematical Society

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© Copyright 2024 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability. Opinions expressed in signed Notices articles are those of the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society.

A WORD FROM. . . Stephan Ramon Garcia

The opinions expressed here are not necessarily those of the Notices or the AMS. National Hispanic Heritage Month (September 15— October 15) has long been recognized by the Notices with an annual special issue. For the past several years, I have had the honor of writing the “A Word from. . . ” introductions for these wonderful issues. My 2023 piece, written toward the beginning of that calendar year, stated that “stunning new developments in artificial intelligence are changing the world around us.” Far from being prophetic, these words perhaps undersold the extent to which artificial intelligence, and its uses and misuses, have changed everything. The 2024 election season is just beginning to hit its stride as I write these words, with the major parties having just identified their presumptive nominees. Bad actors are flooding the internet with falsehoods nearly indistinguishable from reality. With AI-generated text, images, sound, and video polluting the public discourse, the existence of simple facts can no longer be taken for granted. Objective truth seems an old-fashioned notion these days. Closer to home, AI has forced instructors at all levels to confront how we embrace, or defend ourselves against, the AI revolution. While our colleagues in the humanities and social sciences are already grappling with computergenerated essays written at a disturbingly high level, generative AI has been somewhat slower to affect the advanced mathematics curriculum. While Wolfram Alpha and similar websites have long been able to answer standard algebra and calculus problems, complete with step-by-step derivations, it is now only a matter of time before sophisticated AI algorithms can convincingly answer proof-based questions in upper-level courses. What will we do then? Stephan Ramon Garcia is W. M. Keck Distinguished Service Professor and Chair of the Department of Mathematics and Statistics at Pomona College. He served as an associate editor of the Notices from 2019–2021. His email address is [email protected]. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti3005

992

How will we adapt to the new landscape? Will we embrace the challenge or shrink from it? The flip side of all this is that the AI revolution is partially the byproduct of advances in the mathematical sciences. After all, computers are fundamentally mathematical in nature: when it comes down to it, they are machines for performing mathematical computations. It’s hard to tell where things will stand a year from now, save that AI will have progressed in novel and unexpected directions. How will the AI revolution help or hurt our communities? Will algorithmic bias, which we have been warned about for some years now, combine with AI algorithms to disadvantage some at the expense of others? In this time of uncertainty, there is one thing that is certain: this year’s Hispanic Heritage issue contains some exciting articles that should interest mathematicians of all stripes. The issue kicks off with four terrific feature articles. First, Ross Parker and Alejandro Aceves provide a mathematical perspective on topological photonics. Then Mikhail Gorsky and Jos´e Simental introduce us to cluster algebras and weaves. Next Paula Vasquez tells us about multiscale modeling of viscoelastic fluids. In our last feature article, Jorge Pereira explains codimension one foliations on projective manifolds. In our Math-Education section, Christian Anderson celebrates Freeman Hrabowski’s mathematical advocacy and his contributions to diversity in STEM. Our “What is. . . ?” column was written by J. Carlos Mart´ınez Mori, who answers the question: “What is a parking function?” We also have two important Communications in this issue. First, Yunyi Li, Maria de-Arteaga, and Maytal Saar-Tschanskye warn us about label bias. Then Dan Abramovich, Henry Cohn, David Futer, and Robert Harington catch us up on the implementation of double anonymous peer review in the mathematics journals of the AMS.

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 71, NUMBER 8

Although I highlighted the articles in honor of Hispanic Heritage Month, there is a lot more exciting content in this issue as well. Now let’s get on to the 2024 Hispanic Heritage Month special issue of the Notices!

New from the AMS

Stephan Ramon Garcia Credits

Photo of Stephan Ramon Garcia is courtesy of Gizem Karaali.

An Introduction to Real Analysis A conversational yet rigorous course in undergraduate real analysis Yitzhak Katznelson, Stanford University, Stanford, CA, and Yonatan Katznelson, University of California, Santa Cruz, CA This book introduces students to the foundations of real analysis and the analytic way of thinking in a style that is both conversational and mathematically rigorous. Pure and Applied Undergraduate Texts, Volume: 65; 2024; 264 pages, Softcover; ISBN: 978-1-4704-74218; List US$89; AMS members US$68; MAA members US$80.10; Order code AMSTEXT/65

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SEPTEMBER 2024

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Topological Photonics: A Mathematical Perspective Ross Parker and Alejandro Aceves 1. Introduction Topological photonics is a framework that follows both condensed matter physics and topology. It refers to designing the guiding properties of the propagating medium (e.g., a photonic crystal or a waveguide lattice) in such a way that the transport of electromagnetic energy is realized in unique, robust, and sometimes unexpected ways. Consider a simple thought experiment: imagine first the two-dimensional wave equation on a square domain, and assume homogeneous Dirichlet boundary conditions. We know that the accessible modes extend in periodic form throughout the whole domain and, in time, waves can propagate in all directions. This behavior is in response to the inherent symmetries of the medium. Imagine instead that we engineer the medium in such a way that all the energy concentrates in the boundary of the medium and propagates in only one direction. (In the language of optics, this would be seen as inhibiting back reflection and making the bulk medium act like an insulator). Typically, in describing a photonic system, we refer to physical quantities such as frequency, wave vector, polarization, and dispersion. Instead, in the relatively new field of topological photonics, the term “topology” refers to a property of a photonic material that characterizes global behavior of the wavefunctions on their entire dispersion map; most importantly, this property takes quantized values. Think of this as characterizing the “genus” of an object, like a doughnut, with the “object” being described in wave vector space rather than in physical space. There are analogues in photonics to the topological fact that continuous deformations will not change the genus of an object. As an example, photonic topological insulators that are designed using artificial materials can support topologically nontrivial unidirectional states of light. These Ross Parker is a research staff member at the IDA Center for Communications Research, Princeton. His email address is [email protected]. Alejandro Aceves is a professor in the department of mathematics at Southern Methodist University. His email address is [email protected]. Communicated by Notices Associate Editor Reza Malek-Madani. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2998

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states are characterized by a particular “genus-like” number. Since this number is quantized, this unidirectional property will be robust to perturbations in the underlying photonic structure. Photonics research often parallels or aims at explaining phenomena in other physical contexts. Bose-Einstein condensation in condensed matter physics is governed by the Gross-Pitaevskii equation, which is identical to the nonlinear Schrödinger equation that governs intense laser beam propagation in a dielectric medium such as air. In the quantum realm, nontrivial states of two-dimensional matter (e.g., a periodic lattice of atoms) with broken timereversal symmetry can have the property that the bulk is an insulator while states (modes) exist that carry current along the sample edges without dissipation. The characteristic “genus-like” integer is called the Chern number (see section 3 for an example), which arises out of topological properties of the material’s band structure (see the discussion in section 2 and section 3 below). In photonic crystals, a periodic variation of the dielectric properties of the medium affects photons in the same manner as solids modulate electrons (with the caveat that photons are bosons, while electrons are fermions). The question is whether the topological features are replicated in the analogous photonic system. In two foundational papers by Haldane and Raghu [HR08, RH08], the authors highlight the photonics analogue to quantum properties. They demonstrate the ingredients necessary to create a “one-way waveguide” which exhibits properties similar to the Quantum Hall Effect. While the model in [HR08] has not been experimentally realized, it motivated further work by Wang, Chong, Joannopoulos, and Soljaˇci´c, in which they first predicted the existence of edge states in a magneto-optical crystal in the microwave regime [WCJS08] and then demonstrated these experimentally [WCJS09]. Experiments by Rechtsman et al. [RZP+ 13] found topological edge states without the need for an external magnetic field by using a photonic crystal comprising helical waveguides. Since then, the field of topological photonics has matured and continues to be very active, both in theory and experiments, as well as in the linear and nonlinear regimes. While we have briefly discussed its origins, it is not our

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VOLUME 71, NUMBER 8

Figure 1. Top: schematic of SSH model, unit cell (𝑎𝑛 , 𝑏𝑛 ) in dotted box. Coupling constant is 𝑡 1 within unit cell, and 𝑡 2 between unit cells. Bottom: reciprocal lattice, first BZ [−𝜋, 𝜋] shown in red.

purpose to give a detailed history of the field (for this purpose, we point the interested reader to the review articles [JS14, LCG+ 22]). Instead, we will focus our discussion on three prototypical examples: the one-dimensional SuSchrieffer-Heeger model, the two-dimensional Haldane model, and the model of a photonic Floquet topological insulator from [RZP+ 13]. We hope that this article highlights why this is a fertile area for mathematicians to explore and contribute to with their expertise.

2. SSH Model The Su-Schrieffer-Heeger (SSH) model [SSH79], is the simplest lattice model that exhibits topological features. It was devised to describe the electrical conductivity in a doped polyacetylene polymer chain. The lattice comprises repeating, two-node unit cells, where the couplings within and between unit cells are given by 𝑡1 and 𝑡2 , respectively (Figure 1, top). The optical analogue is a linear lattice of waveguides in which the nearest-neighbor couplings are staggered (this can be implemented, e.g., by altering the physical spacings between the fibers). Mathematically, the SSH model can be described by the discrete nonlinear Schrödinger equation 𝑖𝑎𝑛̇ + 𝑡1 𝑏𝑛 + 𝑡2 𝑏𝑛−1 + 𝛾|𝑎𝑛 |2 𝑎𝑛 = 0 𝑖𝑏𝑛̇ + 𝑡1 𝑎𝑛 + 𝑡2 𝑎𝑛+1 + 𝛾|𝑏𝑛 |2 𝑏𝑛 = 0,

(1)

where (𝑎𝑛 , 𝑏𝑛 ) is the 𝑛th unit cell, and 𝛾 > 0 is the strength of the cubic nonlinearity. (A rigorous mathematical derivation can be found in [AC22]). Our analysis follows that of [AOP16, Chapter 1] and [AC22, Section 8]. The topological features of the optical SSH model can be understood by studying the linear model (𝛾 = 0). As a first step, we look for plane wave solutions of the form 𝑎 𝑎 w(𝑘) = ( 𝑛 ) = ( ) 𝑒𝑖(𝑘𝑛−𝜆𝑡) , 𝑏𝑛 𝑏

(2)

where 𝜆 is the frequency and 𝑘 is the wavenumber. Equation (2) is periodic in 𝑘 with period 2𝜋, since w(𝑘+2𝑚𝜋) = w(𝑘) for any integer 𝑚. The points {2𝑚𝜋 ∶ 𝑚 ∈ ℤ} define another linear lattice, which is called the reciprocal lattice (Figure 1, bottom). The first Brillouin zone (BZ) is the set of points closer to the origin than any other point of the SEPTEMBER 2024

Figure 2. Left: band structure of the SSH model for 0 < 𝑡 2 < 𝑡 1 (top) and 0 < 𝑡 1 < 𝑡 2 (bottom). Right: circle in the complex plane traced counterclockwise by ℎ(𝑘) for 𝑘 ∈ [−𝜋, 𝜋].

reciprocal lattice, which in this case is the interval [−𝜋, 𝜋]. Due to the 2𝜋-periodicity, the BZ is topologically equivalent to the unit circle 𝑆1 . Substituting the ansatz (2) into (1) and simplifying, we obtain the 𝑘-dependent eigenvalue problem 𝐻(𝑘)v = −𝜆(𝑘)v, where v = (𝑎, 𝑏)⊤ , and 𝐻(𝑘) is the Hermitian matrix 0 ℎ(𝑘) 𝐻(𝑘) = ( ∗ ), ℎ (𝑘) 0

ℎ(𝑘) = 𝑡1 + 𝑡2 𝑒−𝑖𝑘 .

Since 𝐻(𝑘) is 2𝜋-periodic, we only need to consider 𝑘 ∈ [−𝜋, 𝜋], i.e., in the first BZ. We note that since we are posing the problem on the full integer lattice, 𝑘 can take any value in [−𝜋, 𝜋]. The eigenvalues of 𝐻(𝑘) are 𝜆(𝑘) = ±|ℎ(𝑘)| = √𝑡12 + 𝑡22 + 2𝑡1 𝑡2 cos 𝑘,

(3)

which is the dispersion relation 𝜆(𝑘)2 = |ℎ(𝑘)|2 relating the frequency 𝜆 and the wavenumber 𝑘. Each eigenvalue 𝜆(𝑘) is a continuous function of the wavenumber on the first BZ, and is called a band. Since 𝐻(𝑘) is a 2 × 2 matrix, the SSH model has two bands. All of the bands of the system form its band structure, which is illustrated in the left column of Figure 2. (These terms are borrowed from solid state physics, where the band structure describes the energy levels that electrons can occupy in a solid). When 𝑡1 ≠ 𝑡2 , there is a space between the upper and lower bands, which is known as a band gap. This band gap has size 2Δ, where Δ = |𝑡1 − 𝑡2 |. The band gap closes when 𝑡1 = 𝑡2 . Since the eigenvalues (3) are unchanged if 𝑡1 and 𝑡2 are exchanged, it appears at first glance that the cases 𝑡1 > 𝑡2 and 𝑡1 < 𝑡2 are identical, i.e., that the problem is symmetric about 𝑡1 = 𝑡2 . Interestingly, this is not the case. For a complete picture, we need to look at the eigenvectors of 𝐻(𝑘) as well.

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The normalized eigenvectors corresponding to the eigenvalues ±𝜆(𝑘) are given by v± (𝑘) =

̂ ±ℎ(𝑘) ), 1 √2 1

(

ℎ(𝑘) ̂ ℎ(𝑘) = . |ℎ(𝑘)|

(4)

̂ Since ℎ(𝑘) is a complex number of unit modulus whose argument is the same as that of ℎ(𝑘), we can write v± (𝑘) as ±𝑒𝑖𝜙(𝑘) ) ( 1 √2 𝑡 sin 𝑘 Im ℎ(𝑘) =− 2 . tan 𝜙 = 𝑡1 + 𝑡2 cos 𝑘 Re ℎ(𝑘)

v± (𝑘) =

1

(5)

−1 𝑡1 < 𝑡2 , 0 𝑡1 > 𝑡2

(6)

where a winding number of −1 represents a single clockwise trip around the origin. (The winding number is undefined if 𝑡1 = 𝑡2 ). The same topological information can be obtained in a different way by computing a quantity known as the Berry phase [Ber84] (also known as the Zak phase [Zak89] in 1D). Intuitively, the Berry phase is the phase angle accumulated by a complex vector, e.g., one of the eigenvectors v± (𝑘), around a closed contour in 𝑘-space. We will take a brief digression to discuss these concepts, following [AOP16, Chapter 2] and [Van18, Chapter 3], before computing them for the SSH model. Let v(𝑘) be a normalized eigenvector of 𝐻(𝑘). For any wavenumbers 𝑘1 and 𝑘2 , we define the relative phase between v(𝑘1 ) and v(𝑘2 ) by 𝛾12 = arg⟨v(𝑘2 ), v(𝑘1 )⟩,

(7)

where arg 𝑧 is the phase, or argument, of the complex number 𝑧. (We are using the Hermitian inner product ⟨u, v⟩ = u† v = ∑𝑗 𝑢𝑗∗ 𝑣𝑗 , where the dagger symbol denotes the conjugate transpose; the complex conjugation is placed on the first component to be consistent with the Dirac notation of quantum mechanics). The relative phase 𝛾12 satisfies the equation ⟨v(𝑘2 ), v(𝑘1 )⟩ 𝑒𝑖𝛾12 = . (8) |⟨v(𝑘2 ), v(𝑘1 )⟩| It is important to note that the eigenvector v(𝑘) is not unique. In particular, since it is a unit vector, it is specified only up to multiplication by a constant unit complex number 𝑒𝑖𝜃 . The transformation v(𝑘) ↦ 𝑒𝑖𝜃 v(𝑘) is 996

⟨𝑒𝑖𝜃2 v(𝑘2 ), 𝑒𝑖𝜃1 v(𝑘1 )⟩ = 𝑒𝑖(𝜃1 −𝜃2 ) ⟨v(𝑘2 ), v(𝑘1 )⟩, thus 𝛾12 ↦ 𝛾12 + (𝜃1 − 𝜃2 ). We wish to define the change of phase of v(𝑘) in such a way as to be gauge invariant. To do this, we take a sequence (𝑘1 , 𝑘2 , … , 𝑘𝑁 ) of 𝑁 points in 𝑘-space ordered in a loop. We then define the discrete Berry phase by 𝛾 = arg 𝑒𝑖(𝛾12 +𝛾23 +⋯+𝛾𝑁,1 )

As the wavenumber 𝑘 varies from −𝜋 to 𝜋 over the BZ, the complex number ℎ(𝑘) traces a clockwise circle in the complex plane with center (𝑡1 , 0) and radius 𝑡2 . This circle encloses the origin when 𝑡1 < 𝑡2 , but does not when 𝑡1 > 𝑡2 (Figure 2, right column). The topological invariant is the winding number of ℎ(𝑘), which is the number of times ℎ(𝑘) travels counterclockwise around the origin. We can see from Figure 2 that Indℎ (0) = {

called a gauge transformation. The relative phase 𝛾12 is not invariant under a gauge transformation, since if we take v(𝑘𝑗 ) ↦ 𝑒𝑖𝜃𝑗 v(𝑘𝑗 ), ⟨v(𝑘2 ), v(𝑘1 )⟩ transforms to

= arg (⟨v(𝑘2 ), v(𝑘1 )⟩⟨v(𝑘3 ), v(𝑘2 )⟩ ⋯ ⟨v(𝑘1 ), v(𝑘𝑁 )⟩) , which is the phase accumulated by v(𝑘) around the loop. Unlike the relative phases 𝛾𝑗𝑘 , the Berry phase is gauge invariant; if we take the gauge transformations v(𝑘𝑗 ) ↦ 𝑒𝜃𝑗 v(𝑘𝑗 ), the Berry phase transforms to 𝛾 + (𝜃1 − 𝜃2 ) + (𝜃2 − 𝜃3 ) + ⋯ + (𝜃𝑁 − 𝜃1 ), which is equal to 𝛾, since all of the 𝜃𝑗 cancel. We note that the Berry phase is only unique up to an integer multiple of 2𝜋 unless we take the principal value of the argument, i.e., restrict arg 𝑧 to (−𝜋, 𝜋]. We now move from discrete to continuous. In particular, we wish to compute the phase accumulated by v(𝑘) along a continuous, closed path. For small Δ𝑘, let Δ𝛾 be the relative phase accumulated between v(𝑘) and v(𝑘+Δ𝑘). Following (8), Δ𝛾 satisfies the equation 𝑒𝑖∆𝛾 =

⟨v(𝑘 + Δ𝑘), v(𝑘)⟩ . |⟨v(𝑘 + Δ𝑘), v(𝑘)⟩|

(9)

Since Δ𝑘 is small and v is a unit vector, the denominator in (9) is approximately 1, thus 𝑒𝑖∆𝛾 ≈ ⟨v(𝑘 + Δ𝑘), v(𝑘)⟩. Expanding both sides in a Taylor series to first order in Δ𝛾 and Δ𝑘 and simplifying, we find that Δ𝛾 is approximately given by 𝑑 v(𝑘)⟩ Δ𝑘. (10) Δ𝛾 ≈ 𝑖 ⟨v(𝑘), 𝑑𝑘 We define the Berry connection by 𝐴(𝑘) = 𝑖 ⟨v(𝑘),

𝑑 𝑑 v(𝑘)⟩ = 𝑖v(𝑘)† v(𝑘), 𝑑𝑘 𝑑𝑘

(11)

which is the coefficient of Δ𝑘 on the RHS of (10). We then define the Berry phase to be the integral of Berry connection around a closed contour 𝒞: 𝛾 = ∮ 𝐴(𝑘)𝑑𝑘.

(12)

𝒞

As in the discrete case, the Berry connection is not gauge invariant, while the Berry phase is invariant under gauge transformations (modulo integer multiples of 2𝜋).

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Figure 3. Eigenvalues of 𝐵 for 0 < 𝑡 2 < 𝑡 1 (top left), 𝑡 1 = 𝑡 2 (top right), and 0 < 𝑡 1 < 𝑡 2 (bottom left). Edge mode eigenvectors v = (a, b) for 0 < 𝑡 1 < 𝑡 2 (bottom right); a (solid blue line) and b (dotted orange line) are either in-phase (top) or out-of-phase (bottom). 𝑁 = 25 unit cells with Dirichlet boundary conditions.

Returning to the SSH model, we first compute the Berry connection using the eigenvector v+ (𝑘): 1 𝑖𝜙′ (𝑘)𝑒𝑖𝜙(𝑘) 𝑒𝑖𝜙(𝑘) ), )⟩ ( 0 1 √2 √2 1 = − 𝜙′ (𝑘). 2 We then compute the Berry phase by integrating the Berry connection from −𝜋 to 𝜋. This is a closed contour in the BZ since the endpoints of [−𝜋, 𝜋] correspond to the same point on the unit circle 𝑆1 . The Berry phase is 𝐴(𝑘) = 𝑖 ⟨

1

(

𝜋

𝜋 1 𝛾 = − ∫ 𝜙′ (𝑘)𝑑𝑘 = { 2 −𝜋 0

𝑡1 < 𝑡2 𝑡1 > 𝑡2 ,

which is the change in phase of the eigenvector v+ (𝑘) over the BZ. Since it is a constant multiple of (6), it conveys the same information as the winding number. The fundamental topological difference between the two asymmetric lattice configurations (𝑡1 < 𝑡2 and 𝑡1 > 𝑡2 ) becomes evident when we consider a finite lattice. Specifically, we take a lattice comprising 2𝑁 waveguides (𝑁 unit cells) with Dirichlet boundary conditions at the two ends (𝑏0 = 0 and 𝑎𝑁+1 = 0). For the linear system, solutions are standing waves of the form (a, b)⊤ 𝑒−𝑖𝜆𝑡 , where a = (𝑎1 , … , 𝑎𝑁 ) and b = (𝑏1 , … , 𝑏𝑁 ) represent the 𝑎 and 𝑏 sublattices of the system. Substituting this ansatz into (1) yields the eigenvalue problem 𝐵v = −𝜆v, where v = (a, b)⊤ ∈ ℂ2𝑁 and 𝐵 is the off-diagonal block matrix 𝐵=(

0 𝐵0⊤

𝐵0 ), 0

𝑡1 𝐵0 = (𝑡2

𝑡1 ). ⋱ ⋱

An intuitive understanding of the difference between the two cases can be gained by considering the two extreme SEPTEMBER 2024

configurations, where one of the coupling constants is set to 0. If 𝑡1 > 0 and 𝑡2 = 0, the lattice comprises 𝑁 independent dimers with internal coupling constant 𝑡1 . The eigenvalues of 𝐵 are ±𝑡1 , each with multiplicity 𝑁. On the other hand, if 𝑡1 = 0 and 𝑡2 > 0, the lattice instead comprises 𝑁 − 1 independent dimers (staggered from the ones in the previous case) with internal coupling constant 𝑡2 , as well as two unconnected nodes at the ends of the lattice. In addition to eigenvalues at ±𝑡2 , each with multiplicity 𝑁 −1, the matrix 𝐵 has two eigenvalues at 0. The eigenvectors corresponding to these zero eigenvalues are (1, 0, … , 0)⊤ and (0, … , 0, 1)⊤ . These are called edge modes, since they are localized at the ends of the lattice. Since 𝐵 is a 2𝑁 × 2𝑁 matrix, its spectrum is a discrete set of 2𝑁 eigenvalues, as opposed to the two continuous bands of eigenvalues found from the dispersion relation in the infinite lattice case. The eigenvalues of 𝐵 are shown in Figure 3 for 0 < 𝑡2 < 𝑡1 , 𝑡1 = 𝑡2 , and 0 < 𝑡1 < 𝑡2 . The two asymmetric configurations contain a “gap,” which closes when 𝑡1 = 𝑡2 . This eigenvalue gap is analogous to the band gap in the infinite lattice case. When 0 < 𝑡2 < 𝑡1 , there are no eigenvalues in this gap, and all of the eigenvectors are nonlocalized. When 0 < 𝑡1 < 𝑡2 , however, there are two eigenvalues close to (but not exactly at) 0 which lie within this gap (these eigenvalues approach 0 in the limit 𝑁 → ∞). As in the case where 𝑡1 = 0, these eigenvalues correspond to edge modes, since a and b are localized to the left and right edges of the lattice, respectively (Figure 3, bottom right). All of the remaining modes are nonlocalized. Finally, we briefly comment on what occurs when a cubic nonlinearity (𝛾 > 0) is present (see [MS21] for a more thorough treatment). Standing wave solutions of the form (a, b)⊤ 𝑒−𝑖𝜆𝑡 solve the equation 𝐵v + 𝛾v3 + 𝜆v = 0, where v = (𝑎1 , … , 𝑎𝑁 , 𝑏1 , … , 𝑏𝑁 ). Numerical continuation experiments show that the edge modes from the linear model persist for small 𝛾.

3. Haldane Model We now turn to a two-dimensional model. We start with a honeycomb lattice (Figure 4), which is constructed from a two-site unit cell, with sites labeled 𝑎 and 𝑏. These unit cells tile the plane periodically along the two primitive lat3 √3

tice vectors v1 = ( , 2

2

3

√3

2

2

) and v2 = ( , −

) to obtain a

hexagonal lattice. We note that the 𝑎-sites and 𝑏-sites form two offset, triangular sublattices. This structure is similar to that of the material graphene, which is a hexagonal lattice constructed entirely from carbon atoms. The spatial location of a unit cell is specified by the vector rn = 𝑚v1 +𝑛v2 , where n = (𝑚, 𝑛) ∈ ℤ2 . It is therefore natural to index the unit cells by the vector n; the locations of the lattice sites 𝑎n and 𝑏n in unit cell n are 𝑟n and 𝑟n + (1, 0),

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respectively. Each node in the honeycomb lattice is connected to its three nearest neighbors with coupling strength 𝑡1 > 0, which is a coupling between sublattices. The directions of the nearest-neighbor (NN) couplings are 1

√3

2

2

given by the vectors 𝛿1 = ( , −

1 √3

), 𝛿2 = ( , 2

2

), and

𝛿3 = −(𝛿1 + 𝛿2 ) = (−1, 0), which are depicted in Figure 4. The resulting linear model can be written as 𝑖𝑎ṅ + 𝑡1 ∑ 𝑏m = 0,

𝑖𝑏ṅ + 𝑡1 ∑ 𝑎m = 0,

⟨m⟩

⟨m⟩

(13)

where the angle brackets indicate that the sum is taken over nearest neighbors. The system (13) obeys time-reversal symmetry, i.e., is invariant under the transformation 𝑡 ↦ −𝑡, (𝑎n , 𝑏n ) ↦ (𝑎∗n , 𝑏∗n ). The Haldane model [Hal88] adds two more terms to the honeycomb model. We will see that this results in a band gap in the spectrum, similar to what occurs in the SSH model with unequal couplings. First, the Haldane model has an on-site energy term of magnitude 𝑡0 , which takes opposite signs on the 𝑎 and 𝑏 sublattices. In addition, there is an imaginary, next-nearest neighbor (NNN) coupling term with strength 𝑖𝑡2 . (In the original Haldane model, this coupling term has the complex strength 𝑡2 𝑒𝑖𝜙 ; we take 𝜙 = 𝜋/2 here for simplicity). Each node is coupled to its six next-nearest neighbors, which is a coupling within sublattices. These couplings are staggered so that there is no net flux into or out of a single lattice site. The directions of the next-nearest neighbor couplings are given ′ ′ ′ by the vectors 𝛿1 = v1 , 𝛿2 = −v2 , and 𝛿3 = v2 −v1 (Figure 4). Using this notation, the linear model for the Haldane lattice is

Figure 4. Schematic of the Haldane lattice. Rhombus is unit cell with sites 𝑎 and 𝑏. Primitive lattice vectors v1 and v2 . Nearest neighbor coupling vectors 𝛿1 , 𝛿2 , and 𝛿3 . ′ ′ ′ Next-nearest neighbor coupling vectors 𝛿1 , 𝛿2 , and 𝛿3 .

gion in k-space, since w(k) is periodic in k. Specifically, w(k + r∗ ) = w(k), where r∗ is called a reciprocal lattice vector. To determine the reciprocal lattice vectors, we note that w(k + r∗ ) = w(k) if and only if 𝑒𝑖k⋅rn = 𝑒𝑖(k+r

⟨m⟩

⟪m⟫

⟨m⟩

2𝜋 v∗𝑖 ⋅ v𝑗 = 2𝜋𝛿 𝑖𝑗 = { 0

⟪m⟫

𝑎n 𝑎 ) = ( ) 𝑒𝑖(k⋅rn −𝜆𝑡) , 𝑏n 𝑏

(15)

where k = (𝑘𝑥 , 𝑘𝑦 ) is the wave vector. As in the onedimensional case, we restrict ourselves to a bounded re998

∗ ⋅r

n

.

𝑖=𝑗 𝑖 ≠ 𝑗.

The points r∗ define another lattice, which is called the reciprocal lattice, and its periodicity is given by the primitive reciprocal lattice vectors v∗1 and v∗2 . For a twodimensional lattice, v∗1 and v∗2 can be computed in terms of the primitive lattice vectors v1 and v2 using the formulas 2𝜋𝑄v2 , v1 ⋅ 𝑄v2

v∗2 =

2𝜋𝑄v1 , v2 ⋅ 𝑄v1

For the honeycomb lattice, v∗1 =

0 𝑄=( 1 2𝜋

2𝜋

where the double angle brackets indicate that the sum is taken over next-nearest neighbors. The signs of the NNN couplings are indicated by the arrows in Figure 4, where outward and inward pointing arrows denote couplings of +𝑖𝑡2 and −𝑖𝑡2 , respectively. The arrangement of the arrows in two staggered, counterclockwise triangles ensures no net flux results from the NNN term. Time-reversal symmetry is broken when 𝑡2 ≠ 0, but is unaffected by the on-site term 𝑡0 . As in the SSH model, the first step is to compute the band structure, which is found by looking for plane wave solutions to (14) of the form w(k) = (

= 𝑒𝑖k⋅rn 𝑒𝑖r

This implies that 𝑒 𝑛 = 1, i.e., r∗ ⋅r𝑛 = 𝑚(r∗ ⋅v1 )+𝑛(r∗ ⋅v2 ) is an integer multiple of 2𝜋. To satisfy this criterion, we take r∗ = 𝑙1 v∗1 + 𝑙2 v∗2 for integers 𝑙1 and 𝑙2 , where

v∗1 = (14)

n

𝑖r∗ ⋅r

𝑖𝑎ṅ + 𝑡0 𝑎n + 𝑡1 ∑ 𝑏m ± 𝑖𝑡2 ∑ 𝑎m = 0 𝑖𝑏ṅ − 𝑡0 𝑏n + 𝑡1 ∑ 𝑎m ± 𝑖𝑡2 ∑ 𝑏m = 0,

∗ )⋅r

3

−1 ). 0

(1, √3) and v∗2 =

(1, −√3). The first BZ is the set of points closer to the 3 origin than any other point of the reciprocal lattice (outlined hexagon in Figure 5, top left). This is the Voronoi cell around the origin, which is a unit cell of the reciprocal lattice. Equivalently, the first BZ is the rhombus spanned by the reciprocal lattice vectors v∗1 and v∗1 . Since opposite sides of this rhombus are identified due to periodicity, the first BZ has the topology of a torus. Substituting the ansatz (15) into (14) and simplifying, we obtain the k-dependent eigenvalue problem 𝐻(k)v = −𝜆(k)v, where 𝐻(k) = (

𝑡0 − 𝑓(k) ℎ(k) ) ℎ∗ (k) −𝑡0 + 𝑓(k) ′

ℎ(k) = 𝑡1 ∑ 𝑒−𝑖k⋅𝛿𝑗 ,

𝑓(k) = 2𝑡2 ∑ sin (k ⋅ 𝛿𝑗 ) .

𝑗

𝑗 ∗

We note that ℎ(−k) = ℎ (k) and 𝑓(−k) = −𝑓(k). Since 𝐻(k) is periodic in k along the reciprocal lattice vectors, we

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Figure 5. Top left: reciprocal lattice for Haldane model (blue dots). Red arrows are primitive reciprocal lattice vectors v∗1 and v∗2 . Outlined hexagon is first BZ, which is equivalent to rhombus. Γ is the origin of the reciprocal lattice, 𝐾 and 𝐾 ′ are the two Dirac points, and 𝑀 is the midpoint of the Dirac points. Band structure of Haldane model for 𝑡 0 = 0, 𝑡 2 = 0 (top right), 𝑡 0 = 0, 𝑡 2 = 0.1 (bottom left), and 𝑡 0 = 0.3, 𝑡 2 = 0.1 (bottom right), following piecewise linear path Γ → 𝐾 ′ → 𝑀 → 𝐾 → Γ in BZ. 𝑡 1 = 1.

only need to compute the eigenvalues of 𝐻(k) over the first BZ. The eigenvalues of 𝐻(k) give us the two bands ±𝜆(k), where 𝜆(k) = √|ℎ(k)|2 + (𝑡0 − 𝑓(k))2 . 1

by 𝐾 =

3

(1,

√3

′

) and 𝐾 =

2𝜋 3

(1, −

1 √3

(17)

It follows from (17) that there is no band gap when 𝑡0 = 0 and 𝑡2 = 0 (Figure 5, top right). For 𝑡0 = 0 and 𝑡2 > 0, SEPTEMBER 2024

(19)

where 𝐷k v(k) is the Jacobian matrix for v(k). The Berry phase is the integral of the Berry connection around a closed contour 𝒞 in k-space: 𝛾 = ∮ A(k) ⋅ 𝑑k.

(20)

(21)

𝑆

). The Dirac points

𝜆(𝐾 ′ ) = ||𝑡0 − 3√3𝑡2 || .

A(k) = 𝑖v(k)† 𝐷k v(k),

𝛾 = ∫ Ω(k)𝑑𝑆,

𝐾 and 𝐾 ′ are not equivalent in the BZ, since they are not related by translation through reciprocal lattice vectors. (The remaining Dirac points are equivalent to either 𝐾 or 𝐾 ′ ). Near the Dirac points, the bands are called Dirac cones, since, to leading order, they are linear in k. At the Dirac points, 𝜆(𝐾) = ||𝑡0 + 3√3𝑡2 || ,

Analogous to equation (11), we define the Berry connection A = (𝐴𝑥 , 𝐴𝑦 ) by

Using Stokes’s theorem (i.e., Green’s theorem in two dimensions), we can write the Berry phase as

(

Plots of the band structure of the Haldane model for several parameter configurations are shown in Figure 5. (As is typically done, e.g., in [HR08, Figure 1], the bands are plotted following a piecewise linear path in the BZ). The eigenvalue 𝜆(k) has six minima, which are called Dirac points. These are located at the corners of the hexagonal BZ (Figure 5, top left). We label the two Dirac points with 𝑘𝑥 > 0 1

(18)

𝒞

±𝜆(k) + 𝑔(k) ) ℎ∗ (k) √2𝜆(k)(𝜆(k) ± 𝑔(k)) 𝑔(k) = 𝑡0 − 𝑓(k).

2𝜋

𝛾12 = arg⟨v(k2 ), v(k1 )⟩.

(16)

The corresponding normalized eigenvectors are v± (k) =

there is a band gap of size 6√3𝑡2 (Figure 5, bottom left). The Dirac cones at both 𝐾 and 𝐾 ′ touch when 𝑡2 = 0. For fixed 𝑡0 > 0 and 𝑡2 > 0, the behavior at the two Dirac points is no longer symmetric (Figure 5, bottom right). It follows from (17) that 𝜆(𝐾) > 0, thus the Dirac cones at 𝐾 can never touch. The Dirac cones at 𝐾 ′ , however, touch when 𝑡2 = 𝑡2∗ , where 𝑡2∗ = 𝑡0 /3√3. Therefore, there is a band gap for both 0 < 𝑡2 < 𝑡2∗ and 𝑡2 > 𝑡2∗ , and the band gap closes when 𝑡2 = 𝑡2∗ . The closure of the band gap corresponds to a topological transition, which we will discuss below. To understand what is occurring topologically, we will extend the concepts we discussed in section 2 from one to two dimensions (for a more rigorous treatment, see [AOP16] and [Van18]). Let v(k) be a normalized eigenvector of 𝐻(k). As in the one-dimensional case, we are interested in how the phase of v(k) changes along a closed path in the BZ. For any k1 and k2 , we define the relative phase between v(k1 ) and v(k2 ) by

where 𝑆 is the region in the plane enclosed by 𝒞, and Ω(k) = 𝜕𝑥 𝐴𝑦 − 𝜕𝑦 𝐴𝑥

(22)

is called the Berry potential. Evaluating the derivatives in (22) and using the equivalence of mixed partials, we can also write the Berry potential as Ω(k) = −2 Im ([𝜕𝑘𝑥 v(k)]† 𝜕𝑘𝑦 v(k)).

(23)

The topological quantity of interest is the integral of the Berry connection around the boundary of the first BZ. By the Chern theorem (see, for example, [Nak90], as well as the intuitive explanation below), this is an integer multiple of 2𝜋. We then define the Chern number of v(𝑘) by 𝐶=

1 1 ∮ A(k) ⋅ 𝑑k = ∫ Ω(k)𝑑𝑆, 2𝜋 𝜕BZ 2𝜋 BZ

(24)

which is an integer. We will use a numerical method to compute the Chern number [FHS05]. We first choose a mesh size 𝑁, and then

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discretize the first BZ (rhombus in Figure 5, top left) using the (𝑁 + 1)2 points 𝑚 ∗ 𝑛 ∗ v + v 𝑁 1 𝑁 2

k𝑚,𝑛 =

𝑚, 𝑛 = 0, 1, … , 𝑁.

The discretized grid comprises 𝑁 2 rhombus-shaped plaquettes 𝑃𝑚,𝑛 , with corners k𝑚,𝑛 , k𝑚+1,𝑛 , k𝑚+1,𝑛+1 , and ′

′

𝑚 ,𝑛 k𝑚,𝑛+1 . Let 𝛾𝑚,𝑛 be the relative phase between v(k𝑚,𝑛 ) and v(k𝑚′ ,𝑛′ ). Then the discrete Berry phase around the boundary of the BZ is defined by 𝑁−1

𝑁−1

𝑚+1,0 𝑁,𝑛+1 𝛾𝑁 = arg exp [𝑖( ∑ 𝛾𝑚,0 + ∑ 𝛾𝑁,𝑛 𝑚=0

𝑛=0

𝑁−1

+ ∑

(25)

𝑁−1 𝑚,𝑁 𝛾𝑚+1,𝑁

+ ∑

𝑚=0

0,𝑛 𝛾0,𝑛+1 )],

𝑛=0

where the subscript 𝑁 denotes the mesh size. Instead of computing this, which involves the sum of 4𝑁 gaugedependent relative phases, we will take the sum of the discrete Berry phases around each plaquette [AOP16, Chapter 2.1.3]. The discrete Berry phase around 𝑃𝑚,𝑛 is given by 𝑚+1,𝑛+1 𝑚+1,𝑛 + 𝛾𝑚+1,𝑛 𝐹𝑚𝑛 = arg exp [𝑖(𝛾𝑚,𝑛

(26)

𝑚,𝑛 𝑚,𝑛+1 + 𝛾𝑚,𝑛+1 + 𝛾𝑚+1,𝑛+1 )],

where we take the principal value of the argument, i.e., take 𝐹𝑚𝑛 ∈ (−𝜋, 𝜋]. Taking the product of the Berry phases 𝐹𝑚𝑛 for all 𝑁 2 plaquettes, 𝑁−1

𝑁−1

∏ 𝑒𝑖𝐹𝑚𝑛 = exp (𝑖 ∑ 𝐹𝑚𝑛 ) 𝑚,𝑛=0

𝑚,𝑛=0 𝑁−1

(27)

𝑚+1,𝑛 𝑚+1,𝑛+1 + 𝛾𝑚+1,𝑛 = exp (𝑖 ∑ [𝛾𝑚,𝑛 𝑚,𝑛=0

+

𝑚,𝑛+1 𝛾𝑚+1,𝑛+1

𝑚,𝑛 + 𝛾𝑚,𝑛+1 ]).

Consider an internal edge of the mesh connecting the points k𝑚,𝑛 and k𝑚′ ,𝑛′ . This edge appears in exactly two adjacent plaquettes, but in opposite orientations, which implies that the sum on the RHS of (27) contains the rel𝑚,𝑛 𝑚′ ,𝑛′ ative phases 𝛾𝑚,𝑛 and 𝛾𝑚 ′ ,𝑛′ , each exactly once. Since the ′

𝑁−1

(28)

𝑚,𝑛=0

from which it follows that the discrete Berry phase 𝛾𝑁 and 𝑁−1 the plaquette sum ∑𝑚,𝑛=0 𝐹𝑚𝑛 are equal, modulo an integer multiple of 2𝜋. 1000

Finally, we use the discrete Berry phase around the BZ to define the discrete Chern number 𝑁−1

𝐶𝑁 =

1 ∑ 𝐹 . 2𝜋 𝑚,𝑛=0 𝑚𝑛

(29)

Since the BZ is a torus, opposite boundaries of the BZ are identified. In particular, this means that opposite external edges of the mesh are equivalent, e.g., the edge between k0,𝑛 and k0,𝑛+1 is the same as that between k𝑁,𝑛 and k𝑁,𝑛+1 . Since the two members of each pair of equivalent external edges appear exactly once, but in opposite orientations, in the sum on the RHS of (27), the relative phase contributions from all external edges cancel as well. This implies that the exponent in (28) is 0, so that 𝛾𝑁 is an integer multiple of 2𝜋, and 𝐶𝑁 is an integer. We can think of the Chern number 𝐶 as the limit of 𝐶𝑁 as 𝑁 → ∞, which provides an intuitive explanation for why the Chern number is integer valued. Returning to the Haldane model, we compute the Chern numbers of the two bands using the above discretization with 𝑁 = 100. First, we consider the case when 𝑡0 = 0. When 𝑡2 = 0, the Chern numbers of both bands are 0, and when 𝑡2 > 0, the Chern numbers of the upper and lower bands are 1 and −1, respectively. When 𝑡0 > 0, the Chern number of the upper band is 𝐶={

′

𝑚 ,𝑛 Hermitian inner product is conjugate-symmetric, 𝛾𝑚,𝑛 = 𝑚,𝑛 −𝛾𝑚′ ,𝑛′ , thus the relative phase contributions from all internal edges cancel. This implies that the exponents on the RHS of equations (25) and (27) are the same, i.e.,

𝑒𝑖𝛾𝑁 = exp (𝑖 ∑ 𝐹𝑚𝑛 ) ,

Figure 6. Ribbon of Haldane lattice with armchair edges on top and bottom, infinite in horizontal direction. Unit cell enclosed in dotted lines.

0 0 < 𝑡2 < 𝑡2∗ 1 𝑡2 > 𝑡2∗ ,

and the Chern number of the lower band has the same magnitude but opposite sign. The Chern number changes from 0 to 1 when the band gap closes at 𝑡2 = 𝑡2∗ . This transition from the nontopological to the topological regime is analogous to what occurs in the SSH model. The fundamental difference between the nontopological and topological states can be most easily seen in a lattice which is finite in at least one dimension. As in the onedimensional case, imposition of a boundary will give rise to edge modes. The simplest example is a ribbon lattice,

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connect the upper and lower bands. A plot of the associated eigenvectors for these edge modes (Figure 7, bottom) shows that they are indeed localized to the bottom and top edges of the ribbon.

4. Photonic Floquet Topological Insulator

Figure 7. Band structure for ribbon of Haldane lattice with armchair edges. Top left: 𝑡 0 = 0.25 and 𝑡 2 = 0 (nontopological state). Top right: 𝑡 0 = 0.25 and 𝑡 2 = 0.2 (topological state). Blue and red lines are edge modes which connect the upper and lower bands. Band gap closes at 𝑡 2 = 𝑡2∗ = 0.0481. Bottom: eigenvector (a, b)⊤ corresponding to the two edge modes from top right for 𝑘 = 𝜋/4. 𝑁 = 40 and 𝑡 1 = 1 in all cases.

While the Haldane model has not been realized experimentally, it has motivated further theoretical and experimental work in topological photonics. In [RZP+ 13], Rechtsman et al. performed experiments with a photonic crystal array of coupled, helical waveguides arranged in a honeycomb lattice (as in the Haldane model). The helical waveguides induce temporal modulation of the photonic crystal, which breaks time-reversal symmetry and leads to topological states. We start with the following lattice model, which is a modification of (13) and can be derived from Maxwell’s equations (see [AC17, AC22]): 𝑖𝑎ṅ + 𝑡1 ∑ 𝑒𝑖𝑅A(𝑧)⋅(rm −rn ) 𝑏m = 0 ⟨m⟩

𝑖𝑏ṅ + 𝑡1 ∑ 𝑒𝑖𝑅A(𝑧)⋅(rm −rn ) 𝑎m = 0,

(31)

⟨m⟩

which is finite in one direction and infinite in the other. There are many possible configurations for the edges of the ribbon (see, for example, [LFR24]), and we note that the edge mode behavior and properties depend on this choice. We will use armchair edges for the top and bottom edges of the ribbon, which are illustrated in Figure 6. The unit cell of the ribbon (dotted square in Figure 6) comprises 𝑁 sites of each type, for a total of 2𝑁 sites. Solutions to the linear system are standing waves of the form (a, b)⊤ 𝑒𝑖(𝑘𝑛−𝜆𝑡) , where a = (𝑎1 , … , 𝑎𝑁 ) and b = (𝑏1 , … , 𝑏𝑁 ) represent the 𝑎 and 𝑏 sublattices of the system. The wavenumber 𝑘 runs over the first BZ, which is [−𝜋, 𝜋]. Substituting this ansatz into (14), we obtain the eigenvalue problem 𝐵v = −𝜆v, where v = (a, b)⊤ ∈ ℂ2𝑁 , 𝐵 is the 2𝑁 × 2𝑁 block matrix 𝐵=(

𝑡0 + 𝑖𝑡2 𝐵2 (𝑘) 𝑡1 𝐵1 (𝑘) ), 𝑡1 𝐵1† (𝑘) −𝑡0 + 𝑖𝑡2 𝐵2† (−𝑘)

(30)

𝐵1 (𝑘) is the 𝑁 × 𝑁 tridiagonal matrix whose rows alternate between (1, 1, 1) and (1, 𝑒𝑖𝑘 , 1), and 𝐵2 (𝑘) is the 𝑁 × 𝑁 pentadiagonal, skew-Hermitian matrix whose rows alternate between (1, −1 − 𝑒−𝑖𝑘 , 0, 1 + 𝑒−𝑖𝑘 , −1) and (1, −1 − 𝑒𝑖𝑘 , 0, 1 + 𝑒𝑖𝑘 , −1). When 0 < 𝑡2 < 𝑡2∗ , there is a band gap, but the system does not possess any edge modes (Figure 7, top left). This corresponds to a Chern number of 0 for both bands. As 𝑡2 is increased, the band gap closes at 𝑡2 = 𝑡2∗ , and then reopens for 𝑡2 > 𝑡2∗ . A topological transition occurs at 𝑡2 = 𝑡2∗ , which is concurrent with the band gap closure. For 𝑡2 > 𝑡2∗ , the Chern numbers of the two bands are ±1. The resulting topological state is characterized by the appearance of edge modes (blue and red lines in Figure 7, top right) which SEPTEMBER 2024

where the dot denotes differentiation with respect to 𝑧, A(𝑧) = (sin(Ω𝑧), − cos(Ω𝑧))⊤ , and the angle brackets indicate that the sum is taken over nearest neighbors. We point out that in the photonics setting, the paraxial direction 𝑧 is the “time-like” variable. The parameters 𝑅 and Ω represent the radius of the helical waveguide and its frequency of rotation, respectively. The unit cells (𝑎n , 𝑏n ) and position vectors rn are the same as in the Haldane model, and rm − rn is the displacement vector between two unit cells. Time-reversal symmetry is broken when 𝑅 ≠ 0, since making the transformation 𝑧 ↦ −𝑧, (𝑎n , 𝑏n ) ↦ (𝑎∗n , 𝑏∗n ) and taking complex conjugates takes 𝑅 ↦ −𝑅 in (31). We look for solutions of the form (

𝑎n (𝑧) 𝑎(𝑧) 𝑖k⋅rn , )=( )𝑒 𝑏n (𝑧) 𝑏(𝑧)

(32)

where the wave vector k ranges over the first BZ. Unlike the Haldane model, the system (31) is nonautonomous, i.e., it depends on 𝑧. As a consequence, the functions 𝑎(𝑧) and 𝑏(𝑧) will also depend on 𝑧. Substituting this ansatz into (31) and simplifying, we obtain the linear, k-dependent, nonautonomous ODE u(𝑧) ̇ = 𝑖𝐻(𝑧, k)u(𝑧),

(33)

where u = (𝑎, 𝑏)⊤ , 0 ℎ(𝑧, k) 𝐻(𝑧, k) = ( ∗ ) ℎ (𝑧, k) 0 ℎ(𝑧, k) = 𝑡1 ∑ 𝑒−𝑖A(𝑧)⋅𝛿𝑗 𝑒−𝑖k⋅𝛿𝑗 ,

(34)

𝑗

and the 𝛿𝑗 are the nearest-neighbor coupling vectors from Figure 4.

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Figure 8. Floquet bands for (33) for 𝑅 = 0 (left) and 𝑅 = 1 (right). Plot shows imaginary part of Floquet exponents over the first BZ. For both cases, 𝑡 1 = 1, Ω = 2𝜋.

Since the matrix 𝐻(𝑧, k) is periodic in 𝑧 with period 𝑇 = 2𝜋/Ω, it follows from Floquet theory (see, for example, [Chi06, Chapter 2.4]) that (33) has solutions of the form u(𝑧) = v(𝑧)𝑒𝜆𝑡 , where v(𝑧 + 𝑇) = v(𝑧), and 𝜆 is called a Floquet exponent. To compute the Floquet exponents, let Φ(𝑧) be the fundamental matrix solution for the system, so that u(𝑧) = Φ(𝑧)u0 is the unique solution to (33) with initial condition u0 . Let 𝑀 = Φ(𝑇), which is called the monodromy matrix. The Floquet multipliers 𝜇 are the unique eigenvalues of the monodromy matrix 𝑀, and the Floquet exponents, which are unique modulo 2𝜋𝑖/𝑇, are related to the Floquet multipliers by 𝜇 = 𝑒𝜆𝑇 . Since the two columns of the monodromy matrix are the unique solutions to (33) at 𝑧 = 𝑇 with initial conditions (1, 0)⊤ and (0, 1)⊤ , respectively, it is straightforward to compute the monodromy matrix numerically using a standard ODE solver. Once we have computed 𝑀, we can then calculate the Floquet multipliers and exponents numerically. Figure 8 shows the Floquet band structure of the system, which is obtained by plotting the imaginary part of the Floquet exponents as the wavenumber k varies over the first BZ (the Floquet exponents are purely imaginary). When 𝑅 = 0, there is no band gap, which is expected since the system reduces to the honeycomb model (13). In this case, the matrix 𝐻(𝑧, k) is constant in 𝑧, and the Floquet exponents are the eigenvalues of 𝑖𝐻. A band gap opens when 𝑅 ≠ 0, which can be seen in the right panel of Figure 8. This band gap is associated with a topological state, i.e., a nonzero Chern number. The Chern number for each Floquet band can be computed using the corresponding eigenvector of the monodromy matrix and the numerical method from the previous section. When 𝑅 = 0, the Chern numbers of both bands are 0. For 𝑅 ≠ 0 and frequency Ω = 2𝜋, the Chern numbers of the two bands are 1 and −1. The edge modes that are a consequence of the nontrivial Chern number are both predicted theoretically and demonstrated experimentally in [RZP+ 13].

5. Conclusions and Future Directions In this article, our hope is to illustrate by way of examples the rich mathematics underpinning the study of topological photonics. By confining photons with topological protection, we can generate structures which have applications 1002

in topological lasers, buffers, and other optical elements. The signature of these states (e.g., edge modes) is the presence of band gaps in the frequency versus wavenumber dispersion relation. While there have been many experimental demonstrations of these concepts, fully threedimensional topological photonic bandgaps have not been achieved to date, which is a very promising direction. Another exciting emerging field where mathematical modeling can play an important role is that of quantum topological photonics. In a quantum setting, the application of topological photonics to quantum optics could help to generate robust quantum light sources and protect photons from decoherence during photon propagation. References

[AC17] Mark J. Ablowitz and Justin T. Cole, Tight-binding methods for general longitudinally driven photonic lattices: Edge states and solitons, Phys. Rev. A 96 (2017), Paper No. 043868, DOI 10.1103/PhysRevA.96.043868. [AC22] Mark J. Ablowitz and Justin T. Cole, Nonlinear optical waveguide lattices: asymptotic analysis, solitons, and topological insulators, Phys. D 440 (2022), Paper No. 133440, DOI 10.1016/j.physd.2022.133440. MR4461678 [AOP16] J´anos K. Asboth, ´ L´aszlo´ Oroszl´any, and Andr´as P´alyi, A short course on topological insulators: Band structure and edge states in one and two dimensions, Lecture Notes in Physics, vol. 919, Springer, Cham, 2016, DOI 10.1007/9783-319-25607-8. MR3467967 [Ber84] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proc. Roy. Soc. London Ser. A 392 (1984), no. 1802, 45–57, DOI 10.1098/rspa.1984.0023. MR738926 [Chi06] Carmen Chicone, Ordinary differential equations with applications, 2nd ed., Texts in Applied Mathematics, vol. 34, Springer-Verlag, New York, 2006, DOI 10.1007/0-38735794-7. [FHS05] Takahiro Fukui, Yasuhiro Hatsugai, and Hiroshi Suzuki, Chern numbers in discretized Brillouin Zone: Efficient method of computing (spin) Hall conductances, J. Phys. Soc. Jpn. 74 (2005), no. 6, 1674–1677, DOI 10.1143/JPSJ.74.1674. [Hal88] F. D. M. Haldane, Model for a quantum Hall effect without Landau levels: Condensed-matter realization of the ”parity anomaly”, Phys. Rev. Lett. 61 (1988), 2015–2018, DOI 10.1103/PhysRevLett.61.2015. [HR08] F. D. M. Haldane and S. Raghu, Possible realization of directional optical waveguides in photonic crystals with broken time-reversal symmetry, Phys. Rev. Lett. 100 (2008), Paper No. 013904, DOI 10.1103/PhysRevLett.100.013904. [JS14] John D. Joannopoulos and Marin Soljaˇci´c, Topological photonics, Nature Photonics 8 (2014), 821–829, DOI 10.1038/nphoton.2014.248. [LCG+ 22] Zhihao Lan, Menglin L. N. Chen, Fei Gao, Shuang Zhang, and Wei E. I. Sha, A brief review of topological photonics in one, two, and three dimensions, Reviews in Physics 9 (2022), Paper No. 100076, DOI 10.1016/j.revip.2022.100076.

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NEW FROM THE [LFR24] J. L. Lado and J. Fern´andez-Rossier, Theory of edge states in graphene-like systems, Encyclopedia of condensed matter physics (second edition), 2024, pp. 350–360, DOI 10.1016/B978-0-323-90800-9.00207-9. [MS21] Y.-P. Ma and H. Susanto, Topological edge solitons and their stability in a nonlinear Su-Schrieffer-Heeger model, Phys. Rev. E 104 (2021), no. 5, Paper No. 054206, DOI 10.1103/physreve.104.054206. MR4349854 [Nak90] Mikio Nakahara, Geometry, topology and physics, Graduate Student Series in Physics, Adam Hilger, Ltd., Bristol, 1990, DOI 10.1887/0750306068. MR1065614 [RH08] Srinivas Raghu and Frederick Duncan Michael Haldane, Analogs of quantum-Hall-effect edge states in photonic crystals, Phys. Rev. A 78 (2008), no. 3, Paper No. 033834, DOI 10.1103/PhysRevA.78.033834. [RZP+ 13] Mikael C. Rechtsman, Julia M. Zeuner, Yonatan Plotnik, Yaakov Lumer, Daniel Podolsky, Felix Dreisow, Stefan Nolte, Mordechai Segev, and Alexander Szameit, Photonic Floquet topological insulators, Nature 496 (2013), no. 7444, 196–200, DOI 10.1038/nature12066. [SSH79] W. P. Su, J. R. Schrieffer, and A. J. Heeger, Solitons in polyacetylene, Phys. Rev. Lett. 42 (1979), 1698–1701, DOI 10.1103/PhysRevLett.42.1698. [Van18] David Vanderbilt, Berry phases in electronic structure theory: Electric polarization, orbital magnetization and topological insulators, Cambridge University Press, 2018, DOI 10.1017/9781316662205. [WCJS08] Zheng Wang, Y. D. Chong, John D. Joannopoulos, and Marin Soljaˇci´c, Reflection-free one-way edge modes in a gyromagnetic photonic crystal, Phys. Rev. Lett. 100 (2008), Paper No. 013905, DOI 10.1103/PhysRevLett.100.013905. [WCJS09] Zheng Wang, Yidong Chong, J. D. Joannopoulos, and Marin Soljaˇci´c, Observation of unidirectional backscattering-immune topological electromagnetic states, Nature 461 (2009), no. 7265, 772–775, DOI 10.1038/nature08293. [Zak89] J. Zak, Berry’s phase for energy bands in solids, Phys. Rev. Lett. 62 (1989), 2747–2750, DOI 10.1103/PhysRevLett.62.2747.

The Structure of Pro-Lie Groups Karl H. Hofmann, Technische Universität, Darmstadt, Germany; and Tulane University, New Orleans, US; and Sidney A. Morris, La Trobe University, Bundoora: Australia, and Federation University Australia, Ballarat, Australia A topological group is said to be almost connected if the quotient group of its connected components is compact. This book exposes a Lie theory of almost connected pro-Lie groups (and hence of almost connected locally compact groups) and illuminates the variety of ways in which their structure theory reduces to that of compact groups on the one hand and of finite dimensional Lie groups on the other. It is, therefore, a continuation of the authors’ monograph on the structure of compact groups (1998, 2006, 2014, 2020, 2023) and is an invaluable tool for researchers in topological groups, Lie theory, harmonic analysis, and representation theory. EMS Tracts in Mathematics, Volume 36; 2023; 840 pages; Hardcover; ISBN: 978-3-98547-048-8; List US$129; AMS members US$103.20; Order code EMSTM/36

Explore more titles at bookstore.ams.org/emstm Ross Parker

Alejandro Aceves

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

Credits

Figures 1–8 are courtesy of Ross Parker. Photo of Ross Parker is courtesy of Sarah Louise Parker. Photo of Alejandro Aceves is courtesy of Adriana Aceves.

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Clusters and Weaves Mikhail Gorsky and Jos´e Simental 1. Cluster Algebras

Thus, a cluster algebra is defined by two pieces of data:

Cluster algebras were defined by Sergey Fomin and Andrei Zelevinsky [9] in 2001, with the goal of providing a combinatorial framework for problems related to total positivity and canonical bases in Lie theory. Since then, various connections have been developed between the theory of cluster algebras and many areas of mathematics, including representation theory and categorification, integrable systems, mathematical physics, symplectic and algebraic geometry, and higher Teichmüller theory, to name just a few. The goal of this article is to introduce weaves as a technique to construct cluster algebra structures on the coordinate rings of certain algebraic varieties that appear naturally in the study of Legendrian links and in Lie theory, and which generalize many of the motivating examples of cluster algebras. 1.1. Cluster algebras. Cluster algebras are a special type of commutative algebras that enjoy nice algebraic and combinatorial properties. In contrast with the commutative algebras we encounter in a first algebra course, or even in an algebraic geometry class, cluster algebras are typically not explicitly given by generators and relations. Rather, cluster algebras are defined to be subalgebras of a field of rational functions ℱ ≅ ℂ(𝑥1 , … , 𝑥𝑟 ) specified by a set of generators 𝑆 ⊆ ℱ, known as the set of cluster variables. While the set 𝑆 may be infinite, it can be expressed as a union of distinguished finite subsets of the same cardinality 𝑟, which is the transcendence degree of the field ℱ. Each of these finite subsets is called a cluster. A key property of cluster algebras is that, in order to know all of the cluster variables, it is enough to know a single cluster and a certain combinatorial datum together with a rule, called mutation, for how to get all the other such pairs from it. Mikhail Gorsky is a postdoctoral researcher in the Department of Mathematics of the University of Vienna. His email address is mikhail.gorskii@univie .ac.at. His work received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001159). Jos´e Simental is an investigador asociado C at the Instituto de Matem´aticas of the Universidad Nacional Aut´onoma de M´exico. His email address is [email protected]. His work received support from CONAHCyT Project CF-2023-G-106. Communicated by Notices Associate Editor Han-Bom Moon. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti3000

1004

1. A finite set 𝐱 = {𝑥1 , … , 𝑥𝑟 }, called an initial cluster, consisting of algebraically independent elements that form a free generating set of the field ℱ. 2. A complementary piece of data given by a quiver (i.e., an oriented graph) 𝑄 without loops or oriented 2cycles, whose vertex set is identified with 𝐱. The pair Σ ∶= (𝐱, 𝑄) is known as an initial seed, and it determines a cluster algebra 𝒜(Σ): from the initial seed Σ all other clusters are obtained as follows. For 𝑘 = 1, … , 𝑟, define a new element 𝑥𝑘′ ∈ ℱ by the equation: 𝑥𝑘 𝑥𝑘′ = ∏ 𝑥𝑖 + ∏ 𝑥𝑗 , 𝑖→𝑘

(1)

𝑘→𝑗

where by 𝑖 → 𝑘 and 𝑘 → 𝑗 we mean arrows in the quiver 𝑄. The condition that 𝑄 has no loops is imposed so that 𝑥𝑘 does not appear on the right-hand side of (1), while the condition that 𝑄 has no 2-cycles is imposed so that no element 𝑥𝑖 divides the right-hand side of (1). Now we create a new cluster 𝜇𝑘 (𝐱) by replacing in 𝐱 the element 𝑥𝑘 by the element 𝑥𝑘′ , that is, 𝜇𝑘 (𝐱) = (𝐱 ⧵ {𝑥𝑘 }) ∪ {𝑥𝑘′ }. Note that this gives 𝑟 new clusters: 𝜇1 (𝐱), … , 𝜇𝑟 (𝐱). Each one of these new clusters is known as a mutation of the original cluster 𝐱. Now we want to mutate each of the clusters 𝜇1 (𝐱), … , 𝜇𝑟 (𝐱), and so on. However, we also need to update the quiver 𝑄 in a compatible way every time we mutate. This is given by the procedure of quiver mutation. Given a vertex 𝑘 of 𝑄, the mutation 𝜇𝑘 (𝑄) is a new quiver obtained from 𝑄 by the following three-step procedure: (i) For each pair of arrows 𝑗 → 𝑘 → 𝑖, insert a new arrow 𝑗 → 𝑖. (ii) Reverse all arrows incident with 𝑘. (iii) The previous two steps may have created 2-cycles. Remove a maximal collection of these. With this in hand, we start with an initial seed Σ = (𝐱, 𝑄); we create 𝑟 new seeds 𝜇𝑖 (Σ) = (𝜇𝑖 (𝐱), 𝜇𝑖 (𝑄)), 𝑖 = 1, … , 𝑟; from each one of these seeds we create more seeds, and so on. As an exercise, the reader may prove that mutation is involutive: 𝜇𝑘 (𝜇𝑘 (Σ)) = Σ. By definition, the cluster algebra 𝒜(𝐱, 𝑄) is the subalgebra of ℱ generated by all clusters that can be obtained from 𝐱 by iterated mutations. Note that, since mutation is involutive, the choice of an initial seed is not important: the cluster algebra associated with an arbitrary seed Σ′ which can be obtained from Σ by finitely many mutations coincides with 𝒜(𝐱, 𝑄).

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Variations. Before giving examples of cluster algebras, let us explain variations of the definition that are commonly used in the literature. First, given our initial cluster 𝐱, we may declare some elements of it to be unmutable, or frozen. We are not allowed to mutate at these elements. As a consequence, these elements will belong to every cluster. To reflect this, we declare the corresponding vertices of 𝑄 to be frozen; we are not allowed to mutate at these. A quiver 𝑄 with frozen vertices is also known as an ice quiver. Second, if 𝑥𝑘 is a frozen variable, it is common to adjoin the element 𝑥𝑘−1 to the cluster algebra 𝒜. In this case, we say that the frozen variable 𝑥𝑘 is invertible. Below, we indicate frozen variables by putting them in a box; invertible frozen variables will, in addition, be colored blue. Example 1.1. Consider the space of 2 × 2-matrices, whose algebra of polynomial functions is ℂ[𝑎11 , 𝑎12 , 𝑎21 , 𝑎22 ]. One such polynomial function is the determinant det = 𝑎11 𝑎22 − 𝑎12 𝑎21 . Note that 𝑎11 𝑎22 = det +𝑎12 𝑎21 . Interpreting this as an exchange relation of the form (1), we obtain that ℂ[𝑎11 , 𝑎12 , 𝑎21 , 𝑎22 ] is a cluster algebra with the following initial seed: 𝑎12

𝑎11

𝑎21

det

(ℂ× )4 → {𝑀 ∈ GL(2, ℂ) ∣ 𝑎11 , 𝑎12 , 𝑎21 ≠ 0}, 𝑥 (𝑥, 𝑦, 𝑧, 𝑤) ↦ ( 𝑧

1.2. The geometry of cluster algebras. From the definition, each cluster variable is a rational function in the initial cluster. The following is one of the key early results in the theory of cluster algebras. Theorem 1.2 (The Laurent phenomenon). Let 𝒜 be a cluster algebra, and let 𝐱 be a cluster of 𝒜. Then, each cluster variable of 𝒜 can be expressed as a Laurent polynomial in the variables of 𝐱. The Laurent phenomenon was proved by Fomin and Zelevinsky in [9]. One way to interpret this result is that, for every cluster 𝐱 = {𝑥1 , … , 𝑥𝑟 }, we have 𝒜 ⊆ ℂ[𝑥1±1 , … , 𝑥𝑟±1 ], where both algebras are considered inside our fixed field ℱ. Note that, since 𝑥1 , … , 𝑥𝑟 ∈ 𝒜, we actually have ℂ[𝑥1 , … , 𝑥𝑟 ] ⊆ 𝒜 ⊆ ℂ[𝑥1±1 , … , 𝑥𝑟±1 ], so if we adjoin to 𝒜 the inverse to the element 𝑥1 ⋯ 𝑥𝑟 , we have: 𝒜[(𝑥1 ⋯ 𝑥𝑟 )−1 ] = ℂ[𝑥1±1 , … , 𝑥𝑟±1 ].

(2)

The equation (2) has a neat algebro-geometric interpretation. To each finitely generated commutative algebra 𝒜 with no nonzero nilpotent elements, algebraic geometry associates a space Spec(𝒜), called an affine algebraic variety, whose algebra of regular functions to ℂ is precisely 𝒜. For

𝑦 ). 𝑥−1 (𝑤 + 𝑦𝑧)

The union of all the cluster tori is typically properly contained in Spec(𝒜). The Laurent phenomenon also suggests to consider the upper cluster algebra 𝒰 ∶=

Note that if we choose to invert the frozen variable det, we obtain the algebra of functions on the group GL(2, ℂ).

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example, if 𝒜 = ℂ[𝑥1 , … , 𝑥𝑛 ]/(𝑝1 , … , 𝑝 𝑘 ) then Spec(𝒜) consists of the points in ℂ𝑛 where the polynomials 𝑝1 , … , 𝑝 𝑘 all vanish. As another example, if 𝒜 = ℂ[𝑥1±1 , … , 𝑥𝑛±1 ] then Spec(𝒜) is (ℂ× )𝑛 . The condition that the algebra ℂ[Spec(𝒜)] is 𝒜 determines Spec(𝒜) uniquely: if we have an affine algebraic variety 𝑋 whose algebra of regular functions is 𝒜, then 𝑋 is isomorphic to Spec(𝒜). Then, (2) tells us that the set of 𝑝 ∈ Spec(𝒜) for which all 𝑥1 (𝑝), … , 𝑥𝑟 (𝑝) are nonzero, which is open in Spec(𝒜), is isomorphic to a torus (ℂ× )𝑟 . Thus, each cluster 𝐱 defines an open torus 𝕋𝐱 ⊆ Spec(𝒜), known as a cluster torus. Example. We go back to Example 1.1: Spec(𝒜) = Mat(2 × 2, ℂ). We have a cluster {𝑎11 , 𝑎12 , 𝑎21 , det}. Then, Theorem 1.2 tells us that the set of invertible (2 × 2)matrices with 𝑎11 , 𝑎12 , 𝑎21 ≠ 0 is isomorphic to a torus (ℂ× )4 . Indeed, one can check that the following map is an isomorphism:

⋂ 𝐱

ℂ[𝑥1±1 , … , 𝑥𝑟±1 ],

is a cluster

where the intersection is taken inside our fixed field ℱ. By Theorem 1.2, 𝒜 ⊆ 𝒰. In some cases, these two algebras coincide, but this does not always happen. As we will see next, from a geometric point of view it is more natural to consider the algebra 𝒰. Note that, by the same reasoning as with 𝒜, we have that every cluster 𝐱 defines a cluster torus 𝕋𝐱 ⊆ Spec(𝒰). 1.3. Constructing cluster structures. Given a commutative algebra 𝑅, how to decide whether it has an (upper) cluster algebra structure? Since cluster algebras are subalgebras of fields, a necessary condition on 𝑅 to have a cluster algebra structure is that it is an integral domain. Beyond that, this seems like a daunting task. We would need to: 1. Construct a set 𝑆 ⊆ 𝑅 of cluster variables. 2. Partition 𝑆 into clusters. 3. Find a mutation rule to move between the clusters. The Laurent phenomenon, Theorem 1.2, gives us a geometric way to move forward. If 𝑅 is to be an (upper) cluster algebra, then Spec(𝑅) must contain cluster tori. So a first task is to find candidates for these cluster tori. After finding these tori, we must find a system of coordinates on them: these will be the candidates for the cluster variables. When this is done, the most difficult part is to find a mutation rule that allows us to mutate every coordinate of a cluster torus which is not an invertible element of 𝑅, for we can define the invertible elements to be frozen variables. This

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is not an easy task. However, the following result tells us that we do not necessarily have to find all (possibly infinitely many!) cluster tori: we only need to find one of them and all of its possible one-step mutations. Lemma 1.3 (The starfish lemma [1]). Let 𝑅 be a finitely generated ℂ-algebra that is a normal domain with fraction field ℱ. Assume that we are given a seed Σ = (𝐱, 𝑄) with 𝐱 ⊆ ℱ such that: 1. The cluster 𝐱 consists of elements of 𝑅. 2. The noninvertible (in 𝑅) elements of 𝐱 are pairwise coprime in 𝑅. 3. For each noninvertible element 𝑥𝑘 ∈ 𝐱, mutation at the vertex 𝑘 replaces 𝑥𝑘 with 𝑥𝑘′ ∈ 𝑅, and (𝑥𝑘 , 𝑥𝑘′ ) are coprime in 𝑅. Then, the upper cluster algebra 𝒰 associated to Σ is contained in 𝑅. Let us say a few words on the assumptions of the theorem. The assumption that the algebra 𝑅 is normal is made so as to be able to use an algebraic version of Hartogs’s theorem from complex analysis, that says that a function on a normal algebraic variety that is regular outside of codimension 2 must be regular everywhere. This condition is satisfied if, for example, Spec(𝑅) is smooth. The coprimeness conditions on the variables are made so as to ensure that both the cluster 𝐱 and its one-step mutations generate the fraction field ℱ of 𝑅. As we will see below, the starfish lemma is a powerful tool when trying to find cluster algebra structures on a given commutative algebra. 1.4. Why? Let us mention a few reasons why you would be interested in constructing a cluster structure on a given commutative algebra. As mentioned above, one of Fomin and Zelevinsky’s original motivations was to provide an algebraic framework for the study of total positivity. The Mat(2 × 2, ℂ) example is a good illustration of this. Let us say that a 2 × 2-matrix 𝐴 is totally positive if all of its minors det +𝑎12 𝑎21 are positive real numbers. Since 𝑎22 = , to check 𝑎11

total positivity it is enough to verify that the elements of the cluster given in Example 1.1 give positive numbers: we do not have to check all minors. In general, the algebra of functions on Mat(𝑛 × 𝑛, ℂ) admits a cluster structure, with each cluster containing 𝑛2 elements. To check total positivity of a matrix it is sufficient to check that the elements of a single cluster, i.e., 𝑛2 functions, evaluate to positive numbers, which is more efficient than verifying that all the 2 𝑛 ∑𝑘=1 (𝑛) minors of the matrix are positive. 𝑘 More generally, given a cluster algebra 𝒜 one can define the totally positive space Spec(𝒜)+ to be the set of points 𝑝 ∈ Spec(𝒜) such that 𝑥(𝑝) > 0 for every cluster variable 𝑥. It is a highly nontrivial result that it is enough to verify that 𝑥(𝑝) > 0 for all elements in a single cluster, [13]. Thus, having a cluster structure not only allows us to define a 1006

notion of positivity, it also provides (many!) efficient tests for it. Another motivation of Fomin and Zelevisnky was the study of (semi)canonical bases in Lie theory. Much later, several families of bases for quite general (upper) cluster algebras 𝒜 (resp. 𝒰) were proved to exist in [13, 17]; see [18] for a survey. Each of these is a vector space basis of 𝒜 (resp. 𝒰) including all cluster variables; the basis constructed in [13] also has positive structure constants. In general, having a cluster structure on 𝑅 comes with a wealth of combinatorial properties that allow to explicitly study the geometry and topology of the variety Spec(𝑅).

2. Points in the Projective Line In this section, we find cluster structures on varieties defined from configurations of points. Let ℙ1 denote the complex projective space of dimension 1, that is, the set of lines passing through the origin in ℂ2 . We will use homogeneous coordinates and denote by [𝑎 ∶ 𝑏] ∈ ℙ1 the line passing through the origin and (𝑎, 𝑏) ≠ (0, 0). Thus, [𝑎 ∶ 𝑏] = [𝜆𝑎 ∶ 𝜆𝑏] for any 0 ≠ 𝜆 ∈ ℂ, i.e. we can simultaneously rescale the coordinates without changing the point in ℙ1 . We will denote 𝟎 = [1 ∶ 0], ∞ ∶= [0 ∶ 1] ∈ ℙ1 . Now let 𝑚 > 0 and consider the space 𝑋(𝑚) consisting of (𝑚 + 1)-tuples of elements of ℙ1 , (𝐱0 , … , 𝐱𝑚 ) ∈ (ℙ1 )𝑚+1 , satisfying: (i) 𝐱0 = 𝟎, 𝐱𝑚 = ∞. (ii) 𝐱𝑖 ≠ 𝐱𝑖+1 for all 𝑖 = 0, … , 𝑚 − 1. Thus, 𝑋(1) = {(𝟎, ∞)} is a single point. Also, 𝑋(2) = {(𝟎, 𝐱, ∞) ∈ (ℙ1 )3 ∣ 𝟎 ≠ 𝐱 ≠ ∞}, so that 𝑋(2) ≅ ℙ1 ⧵ {𝟎, ∞} ≅ ℂ× , the space of nonzero complex numbers. These are both algebraic varieties. Let us show that 𝑋(𝑚) is an algebraic variety for all 𝑚. Since we are required to have 𝐱1 ≠ [1 ∶ 0], we must have 𝐱1 = [𝑎1 ∶ 𝑏1 ] with 𝑏1 ≠ 0. Rescaling, we have that 𝐱1 = [𝑧1 ∶ 1] for a unique 𝑧1 ∈ ℂ. Now, since (𝑧1 , 1) and (−1, 0)1 form a basis of ℂ2 , 𝐱2 = [𝑎2 ∶ 𝑏2 ], where (𝑎2 , 𝑏2 ) = 𝜆1 (𝑧1 , 1) + 𝜆2 (−1, 0). Since 𝐱2 ≠ 𝐱1 , we have 𝜆2 ≠ 0. Rescaling again, we may assume 𝜆2 = 1 and, renaming 𝜆1 =∶ 𝑧2 , we have 𝐱2 = [𝑧1 𝑧2 − 1 ∶ 𝑧2 ]. Similarly, 𝐱3 is represented by an element in ℂ2 of the form 𝑧3 (𝑧1 𝑧2 − 1, 𝑧2 ) + (−𝑧1 , −1) for a unique 𝑧3 ∈ ℂ. Continuing recursively, we obtain a family of polynomials defined by 𝑝−1 ≡ 0, 𝑝0 ≡ 1, and 𝑝 𝑖 = 𝑧𝑖 𝑝 𝑖−1 (𝑧1 , … , 𝑧𝑖−1 ) − 𝑝 𝑖−2 (𝑧1 , … , 𝑧𝑖−2 )

(3)

so that 𝐱𝑖 = [𝑝 𝑖 (𝑧1 , … , 𝑧𝑖 ) ∶ 𝑝 𝑖−1 (𝑧2 , … , 𝑧𝑖 )]. The requirement that 𝐱𝑚 = ∞ now becomes 𝑝𝑚 (𝑧1 , … , 𝑧𝑚 ) = 0, since ∞ = [0 ∶ 1]. Thus, 𝑋(𝑚) ≅ {(𝑧1 , … , 𝑧𝑚 ) ∈ ℂ𝑚 ∣ 𝑝𝑚 (𝑧1 , … , 𝑧𝑚 ) = 0}. 1The choice of sign is made so that the matrix (𝑧1

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We see that 𝑋(𝑚) is an affine algebraic variety: it is given by the zeros of a polynomial equation. Note also that 𝐱𝑚 = ∞ implies that 𝐱𝑚−1 ≠ ∞ and, conversely, if 𝐱𝑚−1 ≠ ∞ we obtain an element of 𝑋(𝑚) by setting 𝐱𝑚 = ∞. We thus obtain an equivalent description of the variety 𝑋(𝑚):

torus 𝕋𝐱 we are forced to have 𝐱𝑖 ≠ ∞ for every 𝑖 = 0, … , 𝑚− 1, we can represent this torus in a similar way: 𝟎

𝐱1

…

𝐱𝑚−3 𝐱𝑚−2 𝐱𝑚−1

∞

𝑋(𝑚) ≅ {(𝑧1 , … , 𝑧𝑚−1 ) ∈ ℂ𝑚−1 ∣ 𝑝𝑚−1 (𝑧1 , … , 𝑧𝑚−1 ) ≠ 0}, which realizes 𝑋(𝑚) as an open subvariety of ℂ𝑚−1 . As an open subvariety of a ℂ𝑚−1 , 𝑋(𝑚) is smooth of dimension 𝑚 − 1 and ℂ[𝑋(𝑚)] ≅ ℂ[𝑧1 , … , 𝑧𝑚 ]/(𝑝𝑚 ) −1 ≅ ℂ[𝑧1 , … , 𝑧𝑚−1 ][𝑝𝑚−1 ]. (6)

The recursion (3) is very similar to the rule (1). To make this more explicit, let us rewrite (3) as 𝑧𝑖 𝑝 𝑖−1 = 𝑝 𝑖 + 𝑝 𝑖−2 .

(4)

So that 𝐱 ∶= {𝑝1 , … , 𝑝𝑚−1 } seems to be a cluster in ℂ[𝑋(𝑚)], with mutations given by 𝜇1 (𝐱) = {𝑧2 , 𝑝2 , … , 𝑝𝑚−1 }, 𝜇2 (𝐱) = {𝑝1 , 𝑧3 , … , 𝑝𝑚−1 } and 𝜇𝑖 (𝐱) = {𝑝1 , … , 𝑝 𝑖−1 , 𝑧𝑖+1 , 𝑝 𝑖+1 , … , 𝑝𝑚−1 }. Note that since 𝑝𝑚−1 is invertible, we may consider it frozen. In order to verify that this is the mutation rule for a cluster structure on ℂ[𝑋(𝑚)], we need to verify that there exists a quiver 𝑄 providing a mutation rule of the form (1) that coincides with (4). From (4) we can see that the neighbors of 𝑝 𝑖−1 must be 𝑝 𝑖 and 𝑝 𝑖−2 , so that 𝑄 exists and is as follows: 𝑝1

𝑝2

⋯

𝑝𝑚−2

𝑝𝑚−1

Since 𝑋(𝑚) is smooth, ℂ[𝑋(𝑚)] is normal. We are then in position to apply the starfish lemma 1.3 to conclude that the upper cluster algebra 𝒰(𝐱, 𝑄) is contained in ℂ[𝑋(𝑚)]. On the other hand, the algebra ℂ[𝑋(𝑚)] is generated by 𝑧1 , … , 𝑧𝑚−1 and the invertible variable 𝑝𝑚−1 . These all belong to 𝒜(𝐱, 𝑄). We then conclude that 𝒰(𝐱, 𝑄) ⊆ ℂ[𝑋(𝑚)] ⊆ 𝒜(𝐱, 𝑄) and thus these are all equalities. By virtue of it being a cluster variety, the variety 𝑋(𝑚) must contain a collection of open tori 𝕋𝐱 , one for each cluster 𝐱. Let us study these cluster tori, starting with the initial cluster 𝐱 = {𝑝1 , … , 𝑝𝑚−2 , 𝑝𝑚−1 }. By definition, 𝕋𝐱 consists of the points (𝑧1 , … , 𝑧𝑚−1 ) ∈ ℂ𝑚−1 such that 𝑝 𝑖 (𝑧1 , … , 𝑧𝑚−1 ) ≠ 0 for all 𝑖 = 1, … , 𝑚 − 1. Recalling that 𝐱𝑖 = [𝑝 𝑖 (𝑧1 , … , 𝑧𝑖 ) ∶ 𝑝 𝑖−1 (𝑧2 , … , 𝑧𝑖 )] and that ∞ = [0 ∶ 1], we obtain:

so that now every point 𝐱0 = 𝟎, 𝐱1 , … , 𝐱𝑚−1 is separated from 𝐱𝑚 = ∞ by a blue edge. Thus, this diagram represents the points (𝐱0 , … , 𝐱𝑚 ) ∈ 𝑋(𝑚) such that 𝐱𝑖 ≠ ∞ for all 𝑖 = 0, … , 𝑚 − 1, which is precisely the cluster torus 𝕋𝐱 . Let us now examine the mutation 𝐲 = 𝜇𝑚−2 (𝐱) = {𝑝1 , … , 𝑝𝑚−3 , 𝑧𝑚−1 , 𝑝𝑚−1 }. In the torus 𝕋𝐲 , we have 𝑝1 (𝑧) ≠ 0, … , 𝑝𝑚−3 (𝑧) ≠ 0, 𝑧𝑚−1 ≠ 0, 𝑝𝑚−1 ≠ 0. As before, the nonvanishing conditions on 𝑝’s mean that 𝐱1 ≠ ∞, … , 𝐱𝑚−3 ≠ ∞, 𝐱𝑚−1 ≠ ∞. What does the condition 𝑧𝑚−1 ≠ 0 mean? By definition, 𝐱𝑖 represents the line passing through the origin and a point 𝑥𝑖 ∈ ℂ2 . The points 𝑥𝑖 are defined recursively by 𝑥0 = (1, 0), 𝑥−1 = (0, 1) and 𝑥𝑖 = 𝑧𝑖 𝑥𝑖−1 − 𝑥𝑖−2 . Thus, 𝑥𝑚−1 = 𝑧𝑚−1 𝑥𝑚−2 − 𝑥𝑚−3 , and 𝑧𝑚−1 ≠ 0 if and only if the elements 𝑥𝑚−1 , 𝑥𝑚−3 ∈ ℂ2 are linearly independent, that is, if and only if 𝐱𝑚−1 ≠ 𝐱𝑚−3 . Thus, 𝕋𝐲 is given by 𝐱0 , … , 𝐱𝑚−3 , 𝐱𝑚−1 ≠ ∞,

𝐱𝑚−1 ≠ 𝐱𝑚−3 ,

and can be pictorially represented as follows: 𝟎

𝐱1

…

𝐱𝑚−3 𝐱𝑚−2 𝐱𝑚−1

∞

𝕋𝐱 = {(𝐱0 , … , 𝐱𝑚 ) ∈ 𝑋(𝑚) ∣ 𝐱𝑖 ≠ ∞, 𝑖 = 0, … , 𝑚 − 1}. (7)

Let us represent this torus pictorially. First, we represent the elements of 𝑋(𝑚) graphically, as follows: 𝟎

𝐱1

…

𝐱𝑚−3 𝐱𝑚−2 𝐱𝑚−1

∞ (5)

Here, two points in ℙ1 labeling adjacent regions separated by a blue edge are required to be different. Since in the SEPTEMBER 2024

In fact, every cluster torus in 𝑋(𝑚) can be pictorially represented by a diagram similar to those in (6) and (7). Thus, any cluster torus is defined by requiring some pairs of elements among 𝐱0 , … , 𝐱𝑚−1 , 𝐱𝑚 ∈ ℙ1 to be distinct.

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Conversely, every diagram like those in (6) and (7) (that is, a rooted binary tree with 𝑚 leaves) defines a cluster torus in 𝑋(𝑚). See Figure 1.

In fact, any flag arises from a matrix in this way. Different matrices may give rise to the same flag. Multiplying the columns by arbitrary nonzero scalars does not change the subspaces 𝐹𝐴𝑖 . Also, adding scalar multiples of the first (𝑖 − 1) columns to the 𝑖-th column does not change the space 𝐹𝐴𝑖 . Recalling that elementary column operations can be expressed by matrix multiplication on the right, we see that • 𝐹𝐴• = 𝐹𝐴𝑈

for every invertible upper triangular matrix 𝑈. In fact, if we let 𝐺 = GL(𝑛) be the group of invertible 𝑛 × 𝑛-matrices and 𝐵 = B(𝑛) its subgroup consisting of upper triangular matrices, the set of all flags in ℂ𝑛 can be identified with Figure 1. Graphical representations of the cluster tori inside 𝑋(4).

In this context, seed mutation has a clear combinatorial meaning. It corresponds to a local move of the following form: (8)

The varieties 𝑋(𝑚) are some of the simplest varieties whose coordinate algebra admits a cluster algebra structure. Moreover, in this case we have a complete understanding of the cluster tori, and there are only finitely many of them. The number of cluster tori in 𝑋(𝑚) is given by the Catalan number 𝐶𝑚−1 - an ubiquitous sequence in combinatorics. In the next section, we will generalize the variety 𝑋(𝑚) to braid varieties. This is a rather large class of varieties that appear naturally in many different mathematical contexts: geometric representations of Weyl groups, Legendrian link invariants, and homological algebra in a Lie theoretic context, to give just a few examples.

ℱ(𝑛) ∶= 𝐺/𝐵. The space ℱ(𝑛) is known as the flag variety, and it is a projective algebraic variety. It is not affine, but can be glued from affine varieties in a similar way to how ℙ𝑛 is glued from many copies of ℂ𝑛 . To see how ℱ(𝑛) generalizes the projective space ℙ1 , note that ℱ(2) consists of all chains of subspaces {0} ⊆ 𝐹 1 ⊆ ℂ2 . Since the first (that is, {0}) and last (ℂ2 ) spaces are always fixed, we see that ℱ(2) can be identified with the space of all lines in ℂ2 . This is precisely ℙ1 , i.e., ℱ(2) ≅ ℙ1 . Flag varieties appear naturally in representation theory: the flag variety ℱ(𝑛) can be used to geometrically construct all irreducible representations of the symmetric group 𝑆𝑛 . They also appear naturally in Schubert calculus, which is the study of incidence problems of (translations of) linear spaces, and can be interpreted as a study of the cohomology of ℱ(𝑛) and related varieties. (Anti)standard flags. Having generalized the projective space ℙ1 , we now find counterparts of the points 𝟎, ∞ ∈ ℙ1 . Let 𝑒 1 = (1, 0, … , 0), 𝑒 2 = (0, 1, 0, … , 0), … , 𝑒 𝑛 = (0, 0, … , 0, 1) ∈ ℂ𝑛 be the standard basis. • The standard flag 𝐹std is the flag defined by 𝑖 𝐹std = span(𝑒 1 , … , 𝑒 𝑖 ), • and the antistandard flag 𝐹ant is the flag defined by

3. Braid Varieties We generalize every ingredient appearing in the definition of 𝑋(𝑚). The flag variety. The projective space ℙ1 naturally generalizes to flag varieties. Let us fix a number 𝑛 > 0. A complete flag in ℂ𝑛 is a sequence of vector subspaces:

𝑖 𝐹ant = span(𝑒 𝑛 , … , 𝑒 𝑛−𝑖+1 ). 1 1 When 𝑛 = 2, we have that 𝐹std = span((1, 0)), and 𝐹ant = span((0, 1)). Thus, we see that in this case the standard flag is precisely 𝟎, while the antistandard flag is ∞. Relative position of flags. So far, we have generalized:

𝐹 • = ({0} = 𝐹 0 ⊆ 𝐹 1 ⊆ ⋯ ⊆ 𝐹 𝑛 = ℂ𝑛 )

• • ℙ1 ⇝ ℱ(𝑛), 𝟎 ⇝ 𝐹std , ∞ ⇝ 𝐹ant .

with the property that dim(𝐹 𝑖 ) = 𝑖 for all 𝑖 = 0, … , 𝑛. One way to think about a flag is as an invertible 𝑛 × 𝑛-matrix: if 𝐴 is an invertible 𝑛 × 𝑛-matrix, its columns 𝑣 1 , … , 𝑣 𝑛 ∈ ℂ𝑛 are linearly independent, so the space 𝐹𝐴𝑖 = span(𝑣 1 , … , 𝑣 𝑖 ) is 𝑖-dimensional and this gives rise to a flag:

It is tempting to generalize 𝑋(𝑚) by considering all tuples • of consecutively distinct flags that start with 𝐹std and end • with 𝐹ant . However, this variety would be too large to be well-behaved, and we will instead study pieces of its certain decomposition. For example, if 𝑚 = 2, the variety is ℱ(𝑛)⧵ • • {𝐹std , 𝐹ant }, that is not an affine algebraic variety when 𝑛 > 2.

𝐹𝐴• = ({0} ⊆ 𝐹𝐴1 ⊆ ⋯ ⊆ 𝐹𝐴𝑛 = ℂ𝑛 ). 1008

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Instead, we observe that if 𝐹 • , 𝐺 • ∈ ℱ(2) ≅ ℙ1 are different flags, then they are forced to be different in the first subspace: as 𝐹 0 = 𝐺0 = {0} and 𝐹 2 = 𝐺2 = ℂ2 , we have 𝐹 1 ≠ 𝐺 1 . Generalizing this, we will specify where two flags 𝐹 • , 𝐺 • ∈ ℱ(𝑛) are different. Definition 3.1. Let 𝐹 • , 𝐺 • ∈ ℱ(𝑛) be flags, and let 𝑖 = 1, … , 𝑛 − 1. We say that 𝐹 • , 𝐺 • are in position 𝑖 if 𝐹 𝑖 ≠ 𝐺 𝑖 , but 𝐹 𝑗 = 𝐺 𝑗 for all 𝑗 ≠ 𝑖. Let us classify the flags that are in position 1 with respect • to 𝐹std ∈ ℱ(3). Such a flag must satisfy 𝐹 2 = span(𝑒 1 , 𝑒 2 ). Since 𝐹 1 ⊆ 𝐹 2 , there must exist 𝜆1 , 𝜆2 ∈ ℂ such that 1 𝐹 1 = span(𝜆1 𝑒 1 + 𝜆2 𝑒 2 ). Since we are requiring 𝐹 1 ≠ 𝐹std = span(𝑒 1 ), we must have 𝜆2 ≠ 0. Rescaling, we may assume 𝜆2 = −1 and thus 𝐹 1 = span(𝑧𝑒 1 − 𝑒 2 ) for a unique 𝑧 ∈ ℂ. In fact, with a similar reasoning it is possible to parametrize all flags that are in position 𝑖 with respect to a given flag. In order to do this, we define a family of matrices 𝐵𝑖 (𝑧) ∈ GL(𝑛) depending on a parameter 𝑧 ∈ ℂ: 1 ⎛ ⋮ ⎜ 0 𝐵𝑖 (𝑧) = ⎜ ⎜0 ⎜⋮ ⎝0

⋯ ⋱ ⋯ ⋯ ⋱ ⋯

𝑧 −1 1 0

⋯ 0 ⎞ ⋱ ⋮ ⎟ ⋯ 0⎟ , ⋯ 0⎟ ⋱ ⋮⎟ ⋯ 1⎠

where the nonidentity block is in the 𝑖-th and (𝑖+1)-st rows and columns.2 Lemma 3.2. Let 𝐴 ∈ GL(𝑛) be an invertible matrix. The set of all flags which are in position 𝑖 with respect to the flag 𝐹𝐴• is given by • {𝐹𝐴𝐵 ∣ 𝑧 ∈ ℂ}. 𝑖 (𝑧) We warn the reader that, if 𝐹𝐴• = 𝐹𝐴•′ , it does not follow • that 𝐹𝐴𝐵 = 𝐹𝐴•′ 𝐵𝑖 (𝑧) for all 𝑧 ∈ ℂ. Rather, what is true 𝑖 (𝑧) is that for 𝑧 ∈ ℂ there exists a unique 𝑤 ∈ ℂ such that • 𝐹𝐴𝐵 = 𝐹𝐴•′ 𝐵𝑖 (𝑤) . In other words, the parametrization of 𝑖 (𝑧) the flags in position 𝑖 with respect to 𝐹𝐴• given by Lemma 3.2 depends on the chosen matrix 𝐴, and not just on the flag 𝐹𝐴• . Braid varieties. We are ready to generalize the varieties 𝑋(𝑚). Instead of one natural number 𝑚 > 0, the variety will depend on an 𝑚-tuple, or a word, 𝐣 = (𝑗1 , 𝑗2 , … , 𝑗𝑚 ) ∈ {1, … , 𝑛 − 1}𝑚 , that specifies how the flags change. Definition 3.3. Let 𝐣 = (𝑗1 , … , 𝑗𝑚 ) ∈ {1, … , 𝑛 − 1}𝑚 . The braid variety 𝑋(𝐣) is the variety of (𝑚 + 1)-tuples of flags in ℱ(𝑛), (𝐹0• , … , 𝐹𝑚• ) ∈ ℱ(𝑛)𝑚+1 , satisfying: • • (1) 𝐹0• = 𝐹std ; 𝐹𝑚• = 𝐹ant . • (2) For 1 ≤ 𝑖 ≤ 𝑚, 𝐹𝑖−1 and 𝐹𝑖• are in position 𝑗 𝑖 . 2The −1 sign is there just so that det 𝐵 (𝑧) = 1, which makes some formulas 𝑖 nicer.

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Note that, if 𝑛 = 2 and 𝐣 = (1, 1,⏟…⎵⎵ ,⏟ 1), then 𝑋(𝐣) ≅ 𝑋(𝑚), ⏟⎵⎵ 𝑚 times

so the varieties 𝑋(𝐣) do indeed generalize 𝑋(𝑚) from Section 2. We have the following two facts. (i) If 𝐣 = (𝑗1 , … , 𝑗 𝑘 , 𝑗 𝑘+1 , … , 𝑗𝑚 ) and 𝐣′ = (𝑗1 , … , 𝑗 𝑘+1 , 𝑗 𝑘 , … , 𝑗𝑚 ), then the varieties 𝑋(𝐢) and 𝑋(𝐢′ ) are isomorphic provided that |𝑗 𝑘 − 𝑗 𝑘+1 | > 1. (ii) If 𝐣 = (𝑗1 , … , 𝑗 𝑘 , 𝑗 𝑘 + 1, 𝑗 𝑘 , … , 𝑗𝑚 ) and 𝐣′ = (𝑗1 , … , 𝑗 𝑘 + 1, 𝑗 𝑘 , 𝑗 𝑘 +1, … , 𝑗𝑚 ), then the varieties 𝑋(𝐣) and 𝑋(𝐣′ ) are isomorphic. These isomorphisms mimick the relations in the positive braid monoid. This is the reason that 𝑋(𝐣) is called a braid variety. We refer the reader to [2, 4] for details on this. Just as 𝑋(𝑚) from Section 2, 𝑋(𝐣) is a smooth, affine algebraic variety. We verify that it is indeed an affine algebraic variety by showing that it is defined by the vanishing of several polynomials. For this, given an element (𝐹0• , … , 𝐹𝑚• ) ∈ 𝑋(𝐣), we choose particular matrices in GL(𝑛) representing the flags • • 𝐹0• , … , 𝐹𝑚• . Since 𝐹0• = 𝐹std , we have that 𝐹0• = 𝐹Id , where • Id ∈ GL(𝑛) is the 𝑛 × 𝑛-identity matrix. Since 𝐹1 is in position 𝑗1 with respect to 𝐹0• , thanks to Lemma 3.2 there exists a unique 𝑧1 ∈ ℂ such that • 𝐹1• = 𝐹Id𝐵 𝑗

1

(𝑧1 )

= 𝐹𝐵• 𝑗

1

(𝑧1 ) .

Since 𝐹2• is in position 𝑗2 with respect to 𝐹1• , there exists a unique 𝑧2 ∈ ℂ such that 𝐹2• = 𝐹𝐵• 𝑗

1

(𝑧1 )𝐵𝑗2 (𝑧2 ) .

Continuing like this, we obtain a unique element (𝑧1 , … , 𝑧𝑚 ) ∈ ℂ𝑚 such that 𝐹𝑘• = 𝐹𝐵• 𝑗

1

(𝑧1 )⋯𝐵𝑗 (𝑧𝑘 ) 𝑘

for each 𝑘 = 1, … , 𝑚. Now, the condition that • 𝐹𝑚• = 𝐹ant imposes conditions on the matrix 𝐵𝑗1 (𝑧1 )𝐵𝑗2 (𝑧2 ) ⋯ 𝐵𝑗𝑚 (𝑧𝑚 ), and we obtain that 𝑋(𝐣) is isomorphic to the collection of tuples (𝑧1 , … , 𝑧𝑚 ) ∈ ℂ𝑚 such that (𝐵𝑗1 (𝑧1 ) ⋯ 𝐵𝑗𝑚 (𝑧𝑚 ))𝑝,𝑞 = 0 if 𝑝 ≤ 𝑛 − 𝑞. Note that each entry of the matrix product 𝐵𝑗1 (𝑧1 ) ⋯ 𝐵𝑗𝑚 (𝑧𝑚 ) is a polynomial in ℂ[𝑧1 , … , 𝑧𝑚 ]. Thus, 𝑋(𝐣) is given by the vanishing of several polynomials, and so it is indeed an affine algebraic variety. The fact that 𝑋(𝐣) is smooth is more complicated this time; in general, 𝑋(𝐣) cannot be identified with an open set in ℂ𝑁 . For smoothness of 𝑋(𝐣) we refer the reader to [7]. Braid varieties generalize a wide class of algebraic varieties appearing in Lie theory, such as positroid and Richardson varieties [2, 10, 14, 15]. They also have incarnations in the study of Legendrian link invariants, such as the augmentation variety or the moduli space of microlocal rank 1 sheaves associated with certain Legendrian links [5, 12, 20].

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The fact that the varieties 𝑋(𝑚) admit cluster structures generalizes as follows. Theorem 3.4 ([4]). For any 𝐣 ∈ {1, … , 𝑛 − 1}𝑚 such that the braid variety 𝑋(𝐣) is nonempty, the coordinate ring ℂ[𝑋(𝐣)] admits a cluster algebra structure. Remark 3.5. Let us elaborate on the condition 𝑋(𝐣) ≠ ∅. If the word 𝐣 is not “complicated enough,” then there will be • • no space to get from 𝐹std to 𝐹ant according to the word 𝐣, and thus 𝑋(𝐣) will be empty. For example, if 𝐣 = (1, 1, 2) and 𝑛 = 3, then 𝑧1 𝑧2 − 1 𝐵1 (𝑧1 )𝐵1 (𝑧2 )𝐵2 (𝑧3 ) = ( 𝑧2 0

−𝑧1 𝑧3 −𝑧3 1

𝑧1 1). 0

Since we cannot have simultaneously 𝑧1 𝑧2 − 1 = 0 and 𝑧2 = 0, there are no triples (𝑧1 , 𝑧2 , 𝑧3 ) ∈ ℂ3 such that the flag associated to 𝐵1 (𝑧1 )𝐵1 (𝑧2 )𝐵2 (𝑧3 ) is the antistandard flag, and so 𝑋(1, 1, 2) = ∅. One way to fix this is to relax • the condition on the last flag being 𝐹ant , and require that it is a flag all whose subspaces are given by the vanishing of • some coordinates, and that is as transversal to 𝐹std as the word 𝐣 allows. See [4] for details. In what follows, we will use this more general definition of a braid variety so that 𝑋(𝐣) is always nonempty. These varieties still admit a cluster structure, and the construction we will explain applies to this case as well, with minimal modifications. The construction of an initial seed for the cluster algebra on ℂ[𝑋(𝐣)] is considerably more involved than its counterpart for 𝑋(𝑚) described in Section 2. However, a great deal of cluster tori in 𝑋(𝐣) are still readily accessible using a graphical calculus similar to the one used in Section 2. The calculus becomes more complicated because we need to take into account the isomorphisms (i) and (ii). The key object is a weave, introduced in [6], that we discuss in detail in the next section.

4. Weaves In order to motivate weaves, let us take a word 𝐣 = (𝑗1 , … , 𝑗ℓ ), and picture an element of the braid variety 𝑋(𝐣) as follows:

Here, the vertical lines are colored with colors 1, … , 𝑛−1. • The flag in the leftmost region is 𝐹std , and the flag in the • rightmost one is 𝐹ant . If two regions are separated by a line of color 𝑖, then the corresponding flags are in position 𝑖, that is, they differ precisely in the 𝑖-th subspace. 1010

Now suppose we have the following configuration:

where both vertical lines are of the same color 𝑖. Here, the flags 𝐹1• and 𝐹3• either differ in precisely the 𝑖-th subspace, or they are equal. It is natural to picture these two possibilities using the following diagrams, that we refer to as a trivalent vertex and a cup, respectively.

Note that the condition 𝐹1• ≠ 𝐹3• is open, and the complement 𝐹1• = 𝐹3• is closed. Demazure weaves, which we will now define, are designed to characterize open subsets of a braid variety, and so they only use trivalent vertices, never cups. A Demazure weave 𝔴 on 𝐣 = (𝑗1 , … , 𝑗ℓ ) is a graph in a rectangle 𝑅, such that edges of 𝔴 are colored 1, … , 𝑛 − 1 and vertices are of four types: 1. Univalent vertices, located only on the top and bottom sides of 𝑅. The edges adjacent to the vertices on the top side spell precisely 𝐣, and the word spelt by the colors of the edges adjacent to the vertices on the bottom side is so that no more trivalent vertices can be drawn starting from it, even after applying tetra- and hexavalent vertices described below. 2. Trivalent vertices, as pictured in Figure 2. 3. Hexavalent vertices, as pictured in Figure 2. 4. Tetravalent vertices, as pictured in Figure 2.

Figure 2. The types of vertices in the interior of the rectangle 𝑅 of the definition of a weave. Note that the edges adjacent to a trivalent vertex all have the same color; the edges adjacent to a tetravalent vertex are of two distant colors; and the edges adjacent to an hexavalent vertex are of neighboring colors.

See Figure 3 for an example. Given a Demazure weave 𝔴 on 𝐣, we consider the space 𝑋(𝔴) of all configurations of flags 𝐹𝐶• , one per connected component 𝐶 of 𝑅 ⧵ 𝔴, satisfying the following conditions. • The flag labeling the region bordering the left side of • the rectangle is 𝐹std . • The flag labeling the region bordering the right side of • the rectangle is 𝐹ant .3 3More precisely, this region should be labeled by the flag whose subspaces are • as 𝐣 allows; given by vanishing of coordinates and that is as transversal to 𝐹std see Remark 3.5.

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• If two regions are separated by an edge of color 𝑖, then the corresponding flags are in position 𝑖. Note that, by definition, for any element of 𝑋(𝔴), the flags labeling the regions bordering the top side of the rectangle give an element of 𝑋(𝐣). In fact, these flags propagate, using isomorphisms (i) and (ii), to fill the entire rectangle in a unique way, i.e., there is an embedding 𝑋(𝔴) ↪ 𝑋(𝐣). Not every collection of flags forming an element in 𝑋(𝐣) will propagate consistently through the entire diagram to give an element of 𝑋(𝔴): the trivalent vertices impose open conditions on the element of 𝑋(𝐣). See Figure 3. Note that, when 𝑛 = 2, we recover precisely the diagrams in Section 2 that give cluster tori in 𝑋(𝑚). More generally, the elements of 𝑋(𝐣) that propagate to give an element of 𝑋(𝔴) form a cluster torus 𝕋𝔴 . We remark that different weaves may determine the same tori: the map from Demazure weaves to cluster tori is not injective.

Figure 3. A weave on 𝐣 = (1, 1, 2, 2, 1, 1, 2, 2) and its flag moduli. All the flags are determined by 𝐹std , 𝐹1 , … , 𝐹7 and 𝐹ant . The shortest path from 𝐹std to 𝐹4 is indicated in orange; the one from 𝐹std to 𝐹7 is indicated in purple.

To get a seed from a cluster torus 𝕋𝔴 , we need to specify: (1) A collection of functions in ℂ[𝑋(𝐣)] whose nonvanishing locus is precisely the torus 𝕋𝔴 . (2) An ice quiver, 𝑄𝔴 , specifying the mutation rule. Both constructions are quite technical. We explain some ideas behind them and refer the reader to [4] for details. We take the weave in Figure 3 as a motivating example. Intuitively, a collection of flags (𝐹std , 𝐹1 , … , 𝐹7 , 𝐹ant ) belongs to 𝕋𝔴 if and only if each flag 𝐹1 , 𝐹2 , … , 𝐹7 is as far from the standard flag as possible. This can be made precise using matrix minors. For example, the flag 𝐹2 = 𝐹𝐵1 (𝑧1 )𝐵1 (𝑧2 ) is not equal to the standard flag if and only if the (2, 1) entry of the matrix 𝐵1 (𝑧1 )𝐵1 (𝑧2 ) is nonzero. Note that 2 = 𝑠1 (1), where 𝑠1 = (12) is a transposition in the symmetric group. Then, we can rephrase by saying that 𝐹2 is not equal to the standard flag if and only SEPTEMBER 2024

if the minor 𝑚1 = Δ𝑠1 {1},{1} (𝐵1 (𝑧1 )𝐵1 (𝑧2 )) is nonzero.4 Similarly, the flag 𝐹4 is as far as possible from 𝐹std if the minor 𝑚2 = Δ{2,3},{1,2} (𝐵1 (𝑧1 )𝐵2 (𝑧2 )𝐵3 (𝑧3 )𝐵4 (𝑧4 )) is nonzero. Note that {2, 3} = 𝑠1 𝑠2 {1, 2} where 𝑠2 = (23), and that the path joining the regions labeled with 𝐹std and 𝐹4 in Figure 3 that crosses the minimal number of edges in the weave 𝔴 crosses precisely an edge of color 1 followed by an edge of color 2. By looking at flags immediately to the right of each trivalent vertex, we are led to also consider the following minors: 𝑚3 = Δ{3},{1} (𝐵1 (𝑧1 ) ⋯ 𝐵1 (𝑧6 )), 𝑚4 = Δ{2,3},{1,2} (𝐵1 (𝑧1 ) ⋯ 𝐵1 (𝑧6 )𝐵2 (𝑧7 )), 𝑚5 = Δ{2,3},{1,2} (𝐵1 (𝑧1 ) ⋯ 𝐵1 (𝑧6 )𝐵2 (𝑧7 )𝐵2 (𝑧8 )). The torus 𝕋𝔴 is given by the nonvanishing of 𝑚1 , … , 𝑚5 . However, note that 𝑚1 = 𝑧2 , 𝑚2 = 𝑧2 𝑧4 , so that 𝑚1 and 𝑚2 are not coprime. Nevertheless, once we factor 𝑚𝑖 into irreducibles, there will be exactly one irreducible factor that does not appear in 𝑚𝑗 for 𝑗 < 𝑖. These irreducible factors are the cluster variables. For a general weave, the cluster variables will be irreducible factors of certain minors of partial products in 𝐵𝑖1 (𝑧1 ) ⋯ 𝐵𝑖ℓ (𝑧ℓ ), that measure how far from each other the corresponding flags are. To find the quiver 𝑄𝔴 we need to introduce one more ingredient: Lusztig cycles. Each trivalent vertex 𝑣 has an associated Lusztig cycle 𝛾𝑣 , that is an integer-valued function on the edges of the weave 𝔴. This function is completely determined by requiring that it is 0 in every edge that is not below the trivalent vertex 𝑣; that it is 1 on the southern edge of 𝑣; and that below 𝑣 it satisfies a tropical version of Lusztig’s relations between factorization coordinates [16]. We refer the reader to [4] for the most general case of these rules. When all the values of the Lusztig cycle are equal to 0 or 1, the Lusztig cycle propagates downward through the vertices of 𝔴 according to Figure 4. Figure 5 shows the weave of Figure 3 with all its Lusztig cycles drawn. Once the Lusztig cycles are computed, the quiver 𝑄𝔴 is obtained using an intersection pairing between these. In the setting of [6], where a weave is a combinatorial shadow of a surface in ℝ4 , Lusztig cycles represent homology cycles in this surface and the intersection pairing corrresponds to topological intersection. The intersection pairing can be computed combinatorially, by adding contributions at each 3- and 6-valent vertex. We again refer the reader to [4] for the most general case of this. In case all Lusztig cycles have weight 0 or 1, the contributions are as in Figure 6. Finally, frozen variables correspond to those trivalent vertices whose Lusztig cycle has a nonzero value at the southern boundary of the weave. The reader is invited to verify that the quiver associated to the weave in Figures 3 and 5 4Recall that if 𝐴 is an 𝑛 × 𝑛-matrix and 𝐼, 𝐽 ⊆ {1, … , 𝑛} are sets of the same size,

then the minor Δ𝐼,𝐽 (𝐴) is the determinant of the square submatrix obtained by considering only the rows indexed by 𝐼 and the columns indexed by 𝐽 of 𝐴.

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is as follows, where circles indicate mutable vertices and squares indicate frozen ones.

A more precise version of Theorem 3.4 is: Theorem 4.1. For any Demazure weave 𝔴 on a word 𝐣, the functions on ℂ[𝑋(𝐣)] obtained using irreducible factors of matrix minors, together with the quiver 𝑄𝔴 obtained using Lusztig cycles, give a cluster structure on ℂ[𝑋(𝐣)]. Moreover, any two weaves on 𝐣 give rise to the same cluster structure on ℂ[𝑋(𝐣)].

Figure 4. Lusztig propagation rules when all values are 0 or 1. An edge colored purple means that the value of the Lusztig cycle at that edge is 1; otherwise the value is 0.

Figure 6. Intersection pairing between Lusztig cycles at 6- and 3-valent vertices.

of the motivating examples of cluster algebras [1], while the case of positroid varieties is studied by P. Galashin and T. Lam in [10], and cluster structures on double BottSamelson varieties were studied in detail by D. Weng and L. Shen in [19]. That open Richardson varieties admit cluster structures was conjectured by B. Leclerc in [15], and proved using braid varieties and weaves in [4]. Braid varieties provide a unifying context to all the cluster structures mentioned above. Moreover, even in the cases when a cluster structure was previously known, weaves usually give combinatorial access to more clusters than earlier constructions; see, e.g., [3]. Finally, we mention some exciting recent developments related to braid varieties: independent constructions of cluster structures using a 3D analogue of Postnikov’s plabic graphs [11] and, for some braids and similar varieties, entirely via symplecto-geometric means [5]; a construction of cluster structures for braid varieties of other Lie types [4]; a construction of cluster Poisson structures on braid varieties and the proof of cluster duality for these, including the existence of 𝜗-bases [4, 8, 13]; and connections to Khovanov-Rozansky homology [2]. ACKNOWLEDGMENTS. We are grateful to the referees, O. Chugreeva, J. de Loera Ch´avez, and R. Manr´ıquez for their thoughtful suggestions.

References

Figure 5. The weave from Figure 3 with its Lusztig cycles indicated. There is one Lusztig cycle per trivalent vertex, and each Lusztig cycle only takes values 0 or 1.

Special cases of braid varieties include double Bruhat cells, double Bott-Samelson varieties, open positroid and Richardson varieties. The case of double Bruhat cells is one 1012

[1] Arkady Berenstein, Sergey Fomin, and Andrei Zelevinsky, Cluster algebras. III. Upper bounds and double Bruhat cells, Duke Math. J. 126 (2005), no. 1, 1–52, DOI 10.1215/S0012-7094-04-12611-9. MR2110627 [2] R. Casals, E. Gorsky, M. Gorsky, and J. Simental, Positroid links and braid varieties, arXiv:2105.13948. [3] M. Castronovo, M. Gorsky, J. Simental, and D. Speyer, Cluster deep loci and mirror symmetry, arXiv:2402.16970. [4] R. Casals, E. Gorsky, M. Gorsky, I. Le, J. Simental, and L. Shen, Cluster structures on braid varieties, arXiv:2207.11607. [5] Roger Casals and Daping Weng, Microlocal theory of Legendrian links and cluster algebras, Geom. Topol. 28 (2024), no. 2, 901–1000, DOI 10.2140/gt.2024.28.901. MR4718130 [6] Roger Casals and Eric Zaslow, Legendrian weaves: 𝑁-graph calculus, flag moduli and applications, Geom. Topol. 26 (2022), no. 8, 3589–3745, DOI 10.2140/gt.2022.26.3589. MR4562568

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[7] Laura Escobar, Brick manifolds and toric varieties of brick polytopes, Electron. J. Combin. 23 (2016), no. 2, Paper 2.25, 18, DOI 10.37236/5038. MR3512647 [8] Vladimir Fock and Alexander Goncharov, Moduli spaces of local systems and higher Teichmüller theory, Publ. ´ Math. Inst. Hautes Etudes Sci. 103 (2006), 1–211, DOI 10.1007/s10240-006-0039-4. MR2233852 [9] Sergey Fomin and Andrei Zelevinsky, Cluster algebras. I. Foundations, J. Amer. Math. Soc. 15 (2002), no. 2, 497–529, DOI 10.1090/S0894-0347-01-00385-X. MR1887642 [10] Pavel Galashin and Thomas Lam, Positroid varieties and cluster algebras (English, with English and French sum´ Norm. Sup´er. (4) 56 (2023), no. 3, maries), Ann. Sci. Ec. 859–884, DOI 10.24033/asens.2545. MR4650160 [11] P. Galashin, T. Lam, M. Sherman–Bennett, and D. Speyer, Braid variety cluster structures, I: 3D plabic graphs, arXiv:2210.04778. [12] Honghao Gao, Linhui Shen, and Daping Weng, Augmentations, Fillings, and Clusters, Geom. Funct. Anal. 34 (2024), no. 3, 798–867, DOI 10.1007/s00039-024-00673y. MR4743511 [13] Mark Gross, Paul Hacking, Sean Keel, and Maxim Kontsevich, Canonical bases for cluster algebras, J. Amer. Math. Soc. 31 (2018), no. 2, 497–608, DOI 10.1090/jams/890. MR3758151 [14] Allen Knutson, Thomas Lam, and David E. Speyer, Positroid varieties: juggling and geometry, Compos. Math. 149 (2013), no. 10, 1710–1752, DOI 10.1112/S0010437X13007240. MR3123307 [15] B. Leclerc, Cluster structures on strata of flag varieties, Adv. Math. 300 (2016), 190–228, DOI 10.1016/j.aim.2016.03.018. MR3534832 [16] G. Lusztig, Total positivity in reductive groups, Lie theory and geometry, Progr. Math., vol. 123, Birkhäuser Boston, Boston, MA, 1994, pp. 531–568, DOI 10.1007/978-1-46120261-5_20. MR1327548 [17] Fan Qin, Bases for upper cluster algebras and tropical points, J. Eur. Math. Soc. (JEMS) 26 (2024), no. 4, 1255–1312, DOI 10.4171/jems/1308. MR4721032 [18] Fan Qin, Cluster algebras and their bases, Representations of algebras and related structures, EMS Ser. Congr. Rep., EMS Press, Berlin, [2023] ©2023, pp. 335–369. MR4693645 [19] Linhui Shen and Daping Weng, Cluster structures on double Bott-Samelson cells, Forum Math. Sigma 9 (2021), Paper No. e66, 89, DOI 10.1017/fms.2021.59. MR4321011 [20] Vivek Shende, David Treumann, and Eric Zaslow, Legendrian knots and constructible sheaves, Invent. Math. 207 (2017), no. 3, 1031–1133, DOI 10.1007/s00222-016-06815. MR3608288

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Mikhail Gorsky

Jose´ Simental

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All figures are courtesy of the authors. Photo of Mikhail Gorsky is courtesy of Eugene Gorsky. Photo of Jos´e Simental is courtesy of Adolfo Arroyo-Rabasa.

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Multiscale Modeling of Viscoelastic Fluids Paula A. Vasquez 1. Introduction Although you may be unfamiliar with the term viscoelasticity, viscoelastic fluids (VEFs) are ubiquitous in our daily lives. The shampoo you used this morning, the salad dressing you ate yesterday, and about every biological fluid in your body; all of them are VEFs. As their name implies, viscoelastic materials exhibit both viscous and elastic behaviors. Viscous behavior is related to a fluid’s resistance to flow. The higher the viscosity, the more the fluid resists motion. Honey, for example, has a high viscosity, while water has a low viscosity. Conversely, elasticity pertains to reversible deformations, such as the snapback of a rubber band. Under applied deformations, VEFs display an instantaneous pure elastic response, followed by a time-dependent mechanical response and energy dissipation, characteristic of viscous liquids. The differences from one VEF to another comes from the relative timescales of these elastic and viscous responses. This “duality” in the behavior of VEFs plays a critical role in their applications. For example, paints can be thin enough to be applied with a brush, yet thick enough to stay on the wall. And although mayonnaise appears semisolid in a jar, it can be easily spread on bread. The study of VEFs falls within the field of rheology. Rheology investigates how materials deform or flow during and after a load is applied. Measuring rheological properties is pertinent to all materials, from liquids such as water, polymers, and protein solutions to semisolids such as gels and creams and to solid polymers such as resins. Within rheology, at the most basic level, fluids can be divided into Newtonian and non-Newtonian according to their response to flow. From a modeling point of view, all Newtonian fluids are described by the well-known Navier–Stokes equations [Bat99]. This set of equations works well on systems in which the flow does not alter the dynamics of indiPaula A. Vasquez is an associate professor of mathematics at the University of South Carolina. Her email address is [email protected]. Communicated by Notices Associate Editor Reza Malek-Madani. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti3001

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vidual constituents. In contrast, for non-Newtonian fluids, applied fields can alter the local microstructure. Hence, there is no single set of equations that can comprehensively describe all non-Newtonian materials. What these fluids have in common is that their properties emerge from the collective behavior of many microstructural components. The field of VEFs offers many opportunities for mathematical exploration, particularly in the development of new constitutive models and numerical techniques. The absence of a unified equation for VEFs and their wide range of applications in industrial and biological processes has resulted in extensive research activity in this field. However, there are still many issues that need to be addressed. On the numerical side, key challenges involve the loss of accuracy and convergence of numerical methods due to nonlinearities in the constitutive equations. Another challenge is the change of type of the partial differential equations (PDEs), which sometimes leads to a loss of wellposedness. Additionally, in certain cases, fluid flows result in rapid changes of the solutions in specific regions, making it necessary to implement adaptive mesh techniques. For reviews on this area, the reader is referred to [OP02, Keu04, AOP21]. Fractional calculus is another rapidly growing area in the field of VEFs. It provides a more detailed understanding of the memory effect through the use of fractional derivatives [Mai22]. The behavior of VEFs falls on a spectrum between that of fully elastic solids and fully viscous fluids. Fractional models provide a unified framework to understand this entire range by varying the order of the fractional derivative. This allows fractional models to achieve comparable accuracy to classical models, but with fewer parameters, making data fitting less complex. In addition, the field of VEFs offers numerous opportunities for the mathematical analysis of existing constitutive equations. To understand the dynamics of VEFs, it is crucial to evaluate the stability of these systems and understand how the interplay between viscous and elastic properties influences their behavior. However, there is still much to be explored regarding how different flow conditions affect the solutions of these models. Only a limited subset of these equations has been thoroughly studied in

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Figure 1. VEFs exist at different scales because their macroscopic responses depend on the dynamics of their microstructural components.

This coupling of micro and macro scales is particularly crucial for some materials. For example, one might like to study biochemical changes in the mucus network and their impact on the mucociliary clearance process, or perhaps investigate how changes in the viscoelastic properties of the cytoplasm affect cellular function. In these situations, modeling platforms that consider dynamics at the microscale are better suited. To model the microstructure explicitly, we consider a VEF to be composed of two major components: a solvent and some dynamic network at the microscale. We consider the system to be a continuum. This means it can be described by classical mechanics and its state under a deformation is determined by the fundamental hydrodynamic fields of density, momenta, and energy, [Bat99, BCAH87], Conservation of mass:

terms of their existence and uniqueness [RT21]. Investigating the connections between VEFs constitutive models and broader concepts in dynamical systems could uncover previously unexplored complexities. Perturbation analysis is another valuable field commonly used to gain insight and address unresolved issues pertaining to VEFs. Singular perturbation methods are especially well-suited for examining VEFs because they can effectively manage the wide range of temporal and spatial scales inherent in the dynamics of these materials. By analyzing extreme cases of high and low elasticity, we can gain valuable insights into the underlying dynamics, circumventing the need to solve complex systems of equations. This review aims to introduce the reader to basic concepts related to VEFs, focusing mainly on constitutive modeling. We note that each class of VEFs has unique properties, posing different mathematical challenges. Providing a complete overview of every class of VEFs and their mathematical representations is beyond the scope of this introductory review. Instead, we discuss some commonalities and focus on a specific class of VEFs. When suitable, the reader will be referred to more in-depth reviews and textbooks.

Conservation of momentum:

𝐷𝜌 + 𝜌𝛁 ⋅ 𝐮 = 0, 𝐷𝑡 𝐷 (𝜌 𝐮) = 𝛁 ⋅ 𝝈 + 𝜌𝐟b , 𝐷𝑡

𝐷𝑒 = 𝝈 ∶ 𝛁𝐮 + 𝛁 ⋅ (𝜅𝛁𝑇) . 𝐷𝑡 Here 𝜌 represents the material density, 𝐮 the velocity field, 𝝈 the Cauchy stress tensor, 𝐟b body forces, 𝑒 the internal energy per unit mass, 𝜅 the thermal conductivity and 𝑇 the temperature. And, the material derivative is defined as, Conservation of energy:

𝜌

𝐷 (⋅) 𝜕 (⋅) = + 𝐮 ⋅ 𝛁 (⋅) . 𝐷𝑡 𝜕𝑡 If the flow is incompressible (𝜌 = constant), isothermal (𝑇 = constant), and in the absence of body forces (𝐟b = 0), the conservation equations become, 𝛁 ⋅ 𝐮 = 0, (1a) 𝐷𝐮 𝜌 = 𝛁 ⋅ 𝝈. (1b) 𝐷𝑡 The Cauchy stress tensor can be decomposed into isotropic and extra stress components, 𝝈 = −𝑝 𝜹 + 𝝉,

2. Multiscale Modeling of VEFs

where 𝜹 is the identity tensor. At equilibrium, the isotropic component is the thermodynamic pressure, 𝑝, while the extra stress tensor, 𝝉, vanishes [Gra18]. Under these considerations, the resulting conservation equations are,

From a mathematical perspective, modeling challenges of VEFs arise from the need to describe the dynamics resulting from the complex interactions among microscopic constituents and how such interactions dictate material properties and functions at the macroscale (Fig. 1). In short, there is a need for continuous communication across multiple scales of time and space. Ideally, a constitutive equation for VEFs should provide sufficient insight into the microscopic changes that lead to a given macroscopic response.

𝛁 ⋅ 𝐮 = 0, (2a) 𝐷𝐮 𝜌 = −𝛁𝑝 + 𝛁 ⋅ 𝝉. (2b) 𝐷𝑡 For a viscous or Newtonian fluid, the stress is directly ⊺ proportional to the strain rate, 𝜸 ̇ = 𝛁𝐮 + (𝛁𝐮) , so that 𝝉 = 𝜂 𝜸.̇ In this case 𝝉 is known as a viscous stress and 𝜂 is the fluid’s viscosity. We note that in some references, 1 the strain rate tensor is defined as 𝐃 = 𝜸,̇ and 𝝉 = 2 𝜂 𝐃. 2 Here we follow the notation proposed in [BCAH87] and

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use 𝜸.̇ The resulting conservation of momentum equation is known as the incompressible Navier–Stokes equation, 𝜌

𝐷𝐮 = −𝛁𝑝 + 𝜂𝚫𝐮. 𝐷𝑡

(3)

Inspection of (3) shows that differences between materials will only affect the value of 𝜂, but not the functional form of the governing equations. Then, by varying the viscosity constant, the same numerical algorithms can describe different materials, as long as they behave as Newtonian fluids. Coming back to (2a)–(2b), in the case of VEFs there is a need to introduce an extra term in stress, 𝝉𝑝 , which accounts for the contributions from the microstructure, 𝝉 = 𝜂 𝜸+𝝉 ̇ 𝑝 . In our discussion of (3) we noted that for Newtonian fluids there is a single, general constitutive equation capable of describing different Newtonian materials, so that (3) suffices to describe a large class of fluids. In contrast, there is not a single all-encompassing constitutive equation for viscoelastic materials. The details of 𝝉𝑝 , particularly the constitutive equation that defines it, will differ depending on the material or the specific flow conditions. Thus, the formulation of constitutive equations describing different viscoelastic materials is a prolific area of research [BCAH87, Lar99]. Broadly speaking, we understand relatively well the connection between VEFs and external flow fields: the dynamics of the underlying microstructure directly control the behavior of a VEF under these deformation fields. Therefore, when dealing with constitutive equations for VEFs, challenges arise at three main levels: • At the level of their derivation, since different materials require different mathematical descriptions of their microstructure; • At the level of their numerical simulation, since the resulting equations are of different class and each has its unique numerical challenges; • At the level of their mathematical treatment, since researchers have only established the existence and uniqueness of the solution of VEF constitutive equations in only a few cases (See for example [Ren85, LM00, RT21]). The mathematical representation of the microstructure, the length scale at which this representation will be rendered, and the numerical methods used to solve the resulting constitutive equations will all depend on the particulars of the fluid and the chosen mathematical description. Accordingly, many studies have tackled one aspect or another and even combinations of them. However, a full description is out of the scope of this review. Moving forward, we focus our attention on one type of models which originates from kinetic theory and a specific category of VEFs: polymeric fluids. SEPTEMBER 2024

2.1. Coarse-grained representation of the microstructure. To better understand how we can develop mathematical models of VEFs based on representations of the microstructure, here we will focus on polymeric fluids. In these fluids, polymer chains compose the microstructure. The underlying dynamics driving the material’s response to deformation result from both individual chain configurations and interchain dynamics. Among others, these include coiling and uncoiling processes, hindered motion due to physical entanglements, hydrodynamic effects caused by the presence of other molecules, and in some cases, physical cross-linking between the polymer chains. Within the context of kinetic models and polymeric fluids, various approaches have developed to describe the coupling between macroscopic responses and microstructure dynamics. One family of models is the so-called bead-spring models. These models use molecular coarsegraining to describe the behavior of polymer chains represented as beads connected by massless springs. These models are based on molecular physics by considering the interaction between individual polymer chains and the surrounding fluid. The extra stress arising from the polymer molecules, i.e., the microstructure, depends strongly on their spatial configuration, the most important features being their orientation and their extension. The simplest of these models considers only two beads and it is known as the elastic dumbbell model. The configuration of each dumbbell is fully specified by its stretching and orientation. Although it is widely recognized that a dumbbell is too simple to be able to describe any complicated dynamics in polymeric molecules, it is also well known that stretching and orientation alone suffice to give a qualitative description of steady-state rheological properties and flows with slow characteristic timescales [BCAH87]. Accordingly, these models had been extensively used to develop “an elementary but broad understanding of the relation between macro-molecular motions and rheological phenomena” [BCAH87]. 2.2. Dumbbell models. To model a given microstructure this class of models uses a coarse-grained approximation at the mesoscale consisting of noninteracting elastic dumbbells. The VEF system will be described by the dynamics of these dumbbells in a solvent. The solvent is assumed to be an incompressible Newtonian fluid of viscosity 𝜂𝑠 . The configuration of the dumbbell is described by the end-toend connector vector 𝐐 = 𝐫2 − 𝐫1 and the center-of-mass 1 vector 𝐫𝑐 = (𝐫1 + 𝐫2 ). 2

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tion of 𝐐, 4𝑘𝐵 𝑇 2 ⊺ 𝑑𝐐𝑡 = (𝛁𝐮) ⋅ 𝐐𝑡 − 𝐅 (𝐐𝑡 ) + 𝑑𝐖𝑡 . 𝜁 √ 𝜁

(6)

The only remaining task is to establish the form of the spring law, denoted as 𝐅(𝐐), where 𝐅1 = −𝐅2 = 𝐅. 2.2.1. Hookean dumbbells. For Hookean dumbbells 𝐅(𝐐) = 𝐻𝐐, where 𝐻 is the spring constant. With this, (6) becomes, To keep it short, this review only discusses homogeneous flows. This means that the velocity gradient is assumed to be the same everywhere. We refer the reader to [BAB91] for a discussion on how to introduce spatial variations into the dumbbell equations. To capture the dynamics of each dumbbell, we start from a balance of forces at the inertia-less limit. That is we use Newton’s second law, 𝑚𝐚⃗ = ∑ 𝐅, but assume the mass is negligible [BCAH87, Ött96], ⊺

0 = −𝜁 [𝑑𝐫𝑖 (𝑡) + (𝛁𝐮) ⋅ 𝐫𝑖 (𝑡)] − 𝐅⏟ 𝑖 (𝑡) ⏟⎵⎵⎵⎵⎵⎵⎵⏟⎵⎵⎵⎵⎵⎵⎵⏟ Drag force

+ √4𝑘𝐵 𝑇 𝜁 𝑑𝐖 𝑖 (𝑡) ⏟⎵⎵⎵⎵⏟⎵⎵⎵⎵⏟

Spring force

𝑖 = 1, 2.

(4)

Thermal noise

Here 𝜁 is the drag coefficient, 𝐮 is the fluid velocity, 𝐅(𝑡) denotes a functional form of the spring force, and 𝑘𝐵 𝑇 is the thermal energy. 𝐖(𝑡) denotes a Wiener process where each component of the vector 𝐖(𝑡) is a random number drawn from a normal distribution with zero mean and variance equal to 1. Equation (4) shows that changes in the configuration of the dumbbell, namely its orientation and extension, are the result of three competing forces. The drag force, imposed by the solvent molecules onto the beads, has the tendency of aligning the dumbbells with the macroscopic flow. The thermal or Brownian force tends to randomize their configuration. And, the spring force tends to bring both beads together, which counteracts the stretching effects of the drag and thermal forces. Rearranging (4), gives the following two stochastic differential equations (SDEs) describing the evolution of each bead’s position, ⊺

𝐅1 (𝑡) 4𝑘𝐵 𝑇 + 𝑑𝐖1 (𝑡), 𝜁 √ 𝜁

(5a)

⊺

𝐅2 (𝑡) 4𝑘𝐵 𝑇 + 𝑑𝐖2 (𝑡). 𝜁 √ 𝜁

(5b)

𝑑 𝐫1 (𝑡) = (𝛁𝐮) ⋅ 𝐫1 (𝑡) − 𝑑 𝐫2 (𝑡) = (𝛁𝐮) ⋅ 𝐫2 (𝑡) −

Since the spatial location of the dumbbells is not relevant under the homogeneous flow assumption, the only variable of interest is the end-to-end vector, 𝐐. By subtracting (5a) from (5b), we obtain the SDE describing the evolu1018

⊺

𝑑𝐐𝑡 = (𝛁𝐮) ⋅ 𝐐𝑡 −

4𝑘𝐵 𝑇 2𝐻 𝐐 + 𝑑𝐖𝑡 . 𝜁 𝑡 √ 𝜁

(7)

For convenience, we will make these equations nondimensional. To couple these equations with the conservation equations, we will scale the time using the macroscopic timescale, 𝑡∗ . In addition, we introduce the following characteristic microscopic time and length scales, respectively, 𝜁 𝑘 𝑇 𝜆= , 𝐿𝑚 = √ 𝐵 . 4𝐻 𝐻 The nondimensional variables are then given by, 𝐻 𝐐̃ = 𝐐 ⋅ ( ), 𝑘 √ 𝐵𝑇

𝑡̃ =

𝑡 . 𝑡∗

Scaling (7) and dropping the tildes gives, 1 1 𝐐𝑡 + √ 𝑑𝐖𝑡 , (8) 2 𝐷𝑒 𝐷𝑒 where the nondimensional group 𝐷𝑒 = 𝜆/𝑡∗ is the socalled Deborah number. Since it is the ratio of micro-tomacro characteristic timescales, this dimensionless group compares how long it takes for a material to adapt to deformations relative to the process’s characteristic timescale. Note that, we could have chosen a different macroscopic timescale, namely, 𝐿/𝑈, where 𝐿 and 𝑈 are, respectively, characteristic macroscopic length and velocity. Here, the resulting nondimensional group, 𝑊𝑖 = 𝜆 𝑈/𝐿, is called the Weissenberg number, and it represents the ratio of elastic to viscous forces. For many applications 𝐷𝑒 = 𝑊𝑖 and it is very common to confuse these two nondimensional groups. To better understand the difference between 𝐷𝑒 and 𝑊𝑖, we recommend reading [Poo12]. 2.2.2. FENE dumbbells. The linear spring law used in the Hookean dumbbell model is unphysical, since it allows the end-to-end vector, 𝐐, to stretch without limit. One modification of this law uses finitely extensible nonlinear elastic (FENE) springs laws. FENE-type models are derived by introducing Warner’s force law [BCAH87], 𝐻𝐐 𝐹 (𝐐) = , (9) 2 1 − (𝑄/𝑄max ) ⊺

𝑑𝐐𝑡 = (𝛁𝐮) ⋅ 𝐐𝑡 −

2

where 𝑄2 = |𝐐| = 𝐐⋅𝐐 and 𝑄max is the maximum allowed extension of the dumbbell.

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Introducing this spring law into (6) and nondimensionalizing as before gives ⊺

𝑑𝐐𝑡 = (𝛁𝐮) ⋅ 𝐐𝑡 − +

1 √𝐷𝑒

𝑑𝐖𝑡 ,

𝐐𝑡 1 ( ) 2𝐷𝑒 1 − 𝑄2 /𝑏 (10)

where 𝑏 = 𝐻 𝑄2max /(𝑘𝐵 𝑇). Finally, we note that, in this dumbbell formulation, we only considered the most basic dynamics. Still, dumbbell models can integrate other physical assumptions. For instance, certain models account for a nonisotropic drag [BD83] while others include breaking and reforming dynamics [VMC07]. Before we move forward, we note that dumbbell models are not the only, nor the most prominent, class of models used to describe polymeric systems. We have chosen this particular class of models for specific reasons. Among them, the Hookean dumbbell model, although physically unrealistic, stands out as the only model in its class with an exact closure [BCAH87, BAB91]. This allows for a more coherent explanation of the transition through the different length scales that are discussed in this review. We acknowledge that dumbbell models have their limitations, especially when dealing with complex flows. These drawbacks primarily stem from their oversimplification of several key aspects in polymer physics. For instance, they do not account for entanglements, excluded volume effects, nor internal chain dynamics. For a deeper understanding of more physically relevant models, we highly recommend the book by Doi and Edwards [DEE88]. For readers interested in the analytical and numerical solutions to SDEsbased models, the book by Öttinger is another excellent starting point [Ött96]. 2.3. Fokker–Planck representation. In the previous section, we discussed how SDEs can describe the evolution of the end-to-end vector 𝐐𝑡 . This implies that 𝐐𝑡 is a stochastic or random process. To fully capture the system’s dynamics, it is necessary to solve thousands of SDEs. Now, if instead of “following” individual realizations of this process, i.e., solving (8) or (10), we decide to “follow” ensembles of realizations, we would need a different set of equations. This new set of equations should describe the same system, but instead of using stochastic variables, it uses deterministic variables that fluctuate because of stochasticity. To accomplish this we use the probability density function (PDF), 𝜓 (𝐐, 𝑡), which describes the probability of finding dumbbells with configurations in the interval (𝐐, 𝐐 + 𝑑𝐐) at time 𝑡. Risken’s book [Ris84] excellently explains the differences between these two representations, which we summarize in Fig. 2. Let Ψ (𝐫, 𝐐, 𝑡) represent the configuration number density function, so that the number density of dumbbells SEPTEMBER 2024

Figure 2. Levels of description of a system using Langevin and Fokker–Planck Equations. Figure adapted from [Ris84].

with end-to-end vector 𝐐 and center of mass at position 𝐫, at time 𝑡 is given by, 𝑛 (𝐫, 𝑡) = ∫ Ψ 𝑑𝐐. In the homogeneous case the spatial dependence can be neglected, so that Ψ = 𝑛 𝜓 (𝐐, 𝑡). When dealing with PDFs, one can do an expansion with similar flavor as the Taylor expansion taught in calculus. This expansion is known as the Kramers–Moyal expansion [Ris84]. If the expansion is truncated after the second term, the resulting equation is called a Fokker–Planck equation, also known as the forward Kolmogorov equation. A brief summary of how Brownian dynamics can be described by Langevin and their corresponding Fokker–Planck equations is given in [MDV20], but for a more comprehensive treatment, see [Ris84]. The general form of a Fokker–Planck equation on the variable 𝐐 is 𝜕𝜓 𝜕 =− [𝐴 (𝐐, 𝑡) 𝜓 (𝐐, 𝑡)] 𝜕𝑡 𝜕𝐐 1 𝜕2 + [𝐁 (𝐐, 𝑡) 𝐁⊺ (𝐐, 𝑡) 𝜓 (𝐐, 𝑡)] , 2 𝜕𝐐2

(11)

which corresponds to the Langevin equation, 𝑑𝐐𝑡 = 𝐴 (𝐐𝑡 , 𝑡) + 𝐁 (𝐐𝑡 , 𝑡) 𝑑𝐖𝑡 .

(12)

Thus, we can use (12) together with (8) or (10) to obtain the Fokker–Planck equations corresponding to the Hookean and FENE models. Hookean dumbbells 𝜕𝜓 𝜕 1 ⊺ =− 𝐐) 𝜓] [((𝛁𝐮) ⋅ 𝐐 − 𝜕𝑡 𝜕𝐐 2𝐷𝑒 1 𝜕2 𝜓 + 𝐷𝑒 𝜕𝐐2

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(13)

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FENE dumbbells 𝜕𝜓 𝜕 ⊺ =− [((𝛁𝐮) ⋅ 𝐐 𝜕𝑡 𝜕𝐐 1 𝐐 1 𝜕2 𝜓 − . (14) ( )) 𝜓] + 2 2𝐷𝑒 1 − 𝑄 /𝑏 𝐷𝑒 𝜕𝐐2 Just like it was the case for the Langevin equations, we can relate each term in the Fokker–Planck equations with their physical counterpart. Since the derivatives are on 𝐐, this means these terms depend on the dumbbell’s configuration. In (13)–(14) the first term on the right-hand side describes transport by the macroscopic flow. The second term relates to the spring force, whose effect is to concentrate the distribution function about 𝐐 = 0. The last term represents diffusion in configuration space; its effect is to “spread out” the distribution function. Just as with the previous section, we will avoid going into details about solution strategies for these Fokker– Planck equations in order to maintain brevity and accessibility in this review. However, the book by Risken [Ris84] is a good starting point for those interested in this subject. 2.4. Macroscopic representation. So far, we have described two ways in which we can capture the dynamics of a microstructure comprising chains represented by elastic dumbbells. The first representation considers the equation of motion of individual dumbbells given by SDEs. The second representation considers ensembles of dumbbells and gives the evolution equation for their PDFs as partial differential equations (PDEs). The next logical step is to discuss how to couple these dynamics with the macroscopic flow field described by the conservation of mass and momentum equations. As discussed in previous sections, the equations for the conservation of mass and momentum of an isothermal, incompressible, viscoelastic fluid, in the absence of external forces, are given by 𝛁 ⋅ 𝐮 = 0, (15a) 𝐷𝐮 𝜌 = −𝛁𝑝 + 𝜂𝑠 𝚫𝐮 + 𝛁 ⋅ 𝝉𝑝 . (15b) 𝐷𝑡 Here we have used the notation 𝜂𝑠 to denote the viscosity of the Newtonian component of the VEF, i.e., the solvent. We will scale this equations as, 𝝉𝑝 𝑝 𝑡 𝐱 𝐮 𝑡 ̃ = ∗ , 𝐱̃ = , 𝑝̃ = , 𝐮̃ = , 𝝉𝑝̃ = , 𝑡 𝑈𝑡∗ 𝑈 𝑛𝑘𝐵 𝑇 𝜌𝑈 2 where 𝑡∗ and 𝑈 are characteristic time and velocity, 𝑛 is the dumbbells’ number density, and 𝑘𝐵 𝑇 is the thermal energy. Dropping the tildes, gives 𝛁 ⋅ 𝐮 = 0,

(16a)

𝛽 1−𝛽 𝐷𝐮 = −𝛁𝑝 + 𝚫𝐮 + 𝛁 ⋅ 𝝉𝑝 . (16b) 𝐷𝑡 𝑅𝑒 𝑅𝑒 𝐷𝑒 Here, 𝛽 = 𝜂𝑠 /𝜂0 is the ratio of the solvent to the total viscosity, where 𝜂0 = 𝜂𝑠 + 𝜂𝑝 is the zero-shear rate viscosity 1020

Figure 3. Components of the stress tensor, where 𝜏𝑖𝑗 are stresses resulting from forces in the 𝑖-direction, acting in the face of the volume with normal vector in the 𝑗-direction. In general, 𝝉 is a symmetric tensor; see proof for symmetry in Appendix A1 of [OP02].

of the fluid, with 𝜂𝑝 = 𝑛𝑘𝐵 𝑇 𝜆 being the polymer contribution to the viscosity. And, 𝑅𝑒 = 𝜌𝑈 2 𝑡∗ /𝜂0 is the Reynolds number, which is the ratio of inertial to viscous forces. Together with a constitutive equation for 𝝉𝑝 , (16) give the mathematical description, in time and space, of the resulting flow field of a VEF. However, these equations are applicable only when considering the fluid as a continuous medium. This means that we need to find an expression for 𝝉𝑝 that can “translate” the dynamics at the microscopic and mesoscopic levels, described in previous sections, to the continuum level. As dumbbells move about the solvent fluid, there is a drag force imposed on the beads by the fluid’s velocity, 𝐮. This drag on the beads causes an extra stress on the solvent, which in turns changes its velocity. This exchange between the dumbbells and the fluid depends on the momentum transferred between the beads in each dumbbell, which is modulated by the connector vector 𝐐. Because of this dependence on 𝐐, momentum transfer perpendicular to the flow exerts additional viscous forces, i.e., resistance to flow. While momentum transfer parallel to the flow gives elastic properties to the fluid. In order to understand how single dumbbells contribute to the stress, let’s start by examining the significance of each entry in the stress tensor. In three-dimensional space, the stress tensor is a 3 × 3 matrix. If we consider a piece of fluid as a cube, then the 𝑖, 𝑗 entry corresponds to the stress resulting from a force in the 𝑖 ̂ direction imposed in the face with a normal vector in the 𝑗 ̂ direction; see Fig. 3. The contribution to the stress from a single dumbbell is then given by Kramer’s relation [BCAH87, Ött96], 𝒯 = 𝐻𝑓(𝐐) ⊗ 𝐐 − 𝑘𝐵 𝑇𝜹, where 𝐹(𝐐) = 𝐻𝑓(𝐐) is the spring force, 𝜹 the identity matrix, and ⊗ indicates the tensor product of vectors. And, we can connect 𝐐, at the microscopic or mesoscopic scales, to the macroscopic stress using ensemble averages. In the configuration space represented by 𝐐, the ensemble

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average of any function 𝑔(𝐐) is given by

2.4.1. Hookean dumbbells. In this case 𝑓(𝐐) = 𝐐, so that

⟨𝑔(𝐐)⟩ = ∫ 𝑔(𝐐) Ψ(𝐫, 𝐐, 𝑡) 𝑑𝐐.

(17)

We use (17) to find the stress resulting from the collective dynamics all of dumbbells as 𝝉𝑝 = ⟨𝒯⟩ = 𝑛 ∫ 𝐻𝑓(𝐐) 𝐐 𝜓 𝑑𝐐 − 𝑛𝑘𝐵 𝑇𝜹, = 𝑛𝐻 ⟨𝑓(𝐐) 𝐐⟩ − 𝑛𝑘𝐵 𝑇𝜹, where 𝑛 ⟨𝐹(𝐐) 𝐐⟩ is the contribution from the tension on the spring with spring law 𝐹(𝐐), and 𝑛𝑘𝐵 𝑇 capture effects due to Brownian motion. • For Hookean springs, 𝑓(𝐐) = 𝐐 and we obtain, 𝝉𝑝 = ⟨𝐐𝐐⟩ − 𝜹.

(18) 2

• For FENE springs 𝑓(𝐐) = 𝐐/(1 − (𝑄/𝑄max ) ), so that, 𝝉𝑝 = ⟨

𝐐𝐐 ⟩ − 𝜹. 1 − 𝑄2 /𝑏

(19)

Here, we used the same nondimensionalization as before. To find a constitutive equation for 𝝉𝑝 , we start with the general form of the Fokker–Planck equation for elastic dumbbells, 𝑓(𝐐) 𝜕𝜓 𝜕 1 𝜕2 𝜓 ⊺ =− . [((𝛁𝐮) ⋅ 𝐐 − ) 𝜓] + 𝜕𝑡 𝜕𝐐 𝐷𝑒 𝐷𝑒 𝜕𝐐2 Recall that in this representation we assume the spring force is of the form 𝐅(𝐐) = 𝐻𝑓(𝐐). To find an expression for ⟨𝐐𝐐⟩, we multiply (20) throughout by 𝐐𝐐, and integrate over the configuration space. We use the divergence theorem and the fact that 𝜓 → ∞ as 𝐐 → ∞ to obtain [BCAH87] 𝜕 ⟨𝐐𝐐⟩ ⊺ = (𝛁𝐮) ⋅ ⟨𝐐𝐐⟩ + ⟨𝐐𝐐⟩ ⋅ (𝛁𝐮) 𝜕𝑡 1 1 − 𝜹. ⟨𝐐𝑓(𝐐)⟩ + 𝐷𝑒 𝐷𝑒

1 1 𝜹. ⟨𝐐𝐐⟩ + 𝐷𝑒 𝐷𝑒

Using (18) we obtain the constitutive equation for the extra stress tensor of Hookean dumbbells, 𝐷𝑒 𝝉𝑝,(1) + 𝝉𝑝 = 𝐷𝑒 𝜸.̇

𝜕 (⋅) ⊺ − (𝛁𝐮) ⋅ (⋅) − (⋅) ⋅ (𝛁𝐮) , 𝜕𝑡

This is the well-known Upper Convected Maxwell (UCM) model [BCAH87]. We should emphasize that the UCM model provides an exact closure to the Hookean dumbbell model because (21) can be directly obtained from (8). As we will see below, this is not the case for the FENE dumbbells. Finally, if instead of considering a constitutive equation for only 𝝉𝑝 , we consider the total extra stress, 𝝉 = 𝜂𝑠 𝚫𝐮+𝝉𝑝 , the constitutive equation for 𝝉 is known as the Oldroyd-B Model. For a discussion of the mathematical considerations and challenges arising from the description of VEF using this model, see [RT21]. 2.4.2. FENE dumbbells. For FENE dumbbells, (20) gives, ⟨𝐐𝐐⟩(1) = −

1 𝐐𝐐 1 𝜹. ⟨ ⟩+ 𝐷𝑒 1 − 𝑄2 /𝑏 𝐷𝑒

Because of the nonlinear term, it is not possible to obtain a close-form constitutive equation of 𝝉𝑝 for FENE dumbbells. Instead, several closures have been suggested to allow the formulation of macroscopic constitutive equations. Here we will discuss the so-called Peterlin approximation, which results in the FENE-P model [BCAH87]. Other FENE closures are discussed in [DLY05]. Peterlin proposed a separate average of the numerator and denominator of the spring law [BCAH87], 1 1 ⟨𝐐𝐐⟩ + 𝜹, 𝐷𝑒 1 − ⟨𝑄2 ⟩ /𝑏 𝐷𝑒 ⟨𝐐𝐐⟩ 𝝉𝑝 = − 𝜹. 1 − ⟨𝑄2 ⟩ /𝑏

⟨𝐐𝐐⟩(1) = −

we arrive at, ⟨𝐐𝐐⟩(1)

1 1 =− 𝜹. ⟨𝐐𝑓(𝐐)⟩ + 𝐷𝑒 𝐷𝑒

𝐀≡𝑑 (20)

In the next sections we show how we can use (20) to formulate constitutive equations for 𝝉𝑝 corresponding the Hookean and FENE dumbbells. SEPTEMBER 2024

(21)

(22) (23)

In this way, instead of restricting the length of individual dumbbells to be less than 𝑄max , the Peterlin’s approximation relaxes the restriction where only the average dumbbell length needs to be less than the prescribed maximum extension. Individual dumbbell lengths can thus exceed 𝑄max as long as the average stays within bounds. For convenience, we define a nondimensional configuration tensor, 𝐀, as

If we define the upper convected derivative as, (⋅)(1) =

⟨𝐐𝐐⟩(1) = −

⟨𝐐𝐐⟩ , ⟨𝑄2 ⟩0

(24)

where ⟨𝑄2 ⟩0 is the mean-square end-to-end spring length at equilibrium (absence of flow) and 𝑑 = 3 is the dimensionality [BCAH87]. Note that if we scale the end-to-end vector as before, we have a nondimensional conformation

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1021

3. Conclusions

Figure 4. Solutions in simple shear flow for macroscopic models (solid lines) compared to stochastic simulations (dashed lines). For simple shear flow, the velocity is ⊺ prescribed as 𝐮 = [𝛾0̇ 𝑦, 0, 0] . Defining 𝐷𝑒 = 𝜆 𝛾0̇ gives the ⊺ nondimensional velocity as 𝐮 = [𝑦, 0, 0] . (A) Hookean dumbbell vs. UCM model. (B) FENE dumbbell vs. FENE-P model.

tensor 𝐀 = 3 ⟨𝐐𝐐⟩. The constitutive equation for 𝐀 in three-dimensions is then found as, 𝝉𝑝 =

𝐀 − 𝜹. 1 − trace (𝐀) /(3𝑏)

𝐷𝑒𝐀(1) +

𝐀 = 𝜹. 1 − trace (𝐀) /(3𝑏)

(25a) (25b)

To elucidate the trade-offs involved in the Peterlin approximation, in Fig. 4 we show solutions of (8), (10), (21), and (25) under simple shear flow. We compare the macroscopic closures given by the UCM and FENE-P models with their corresponding stochastic counterparts: the Hookean and FENE models. Since the UCM model is an exact closure of the Hookean dumbbell model, solutions to (8) and (21) will always agree with each other, as shown in Fig. 4(A). On the other hand, the approximation used in the FENE-P formulation leads to deviation between solutions of (10) and (25). Fig. 4(B) shows that these differences are more noticeable at higher deformation rates, when dumbbells are close to their maximum extension. As mentioned above, FENE-P is not the only closure proposed for the FENE model. The mathematical analysis of various closures for the FENE model remains an active area of research, involving the exploration of different approximations and/or higher order moments, e.g., [DLY05].

1022

This review aims to introduce the reader to the fundamental aspects of mathematical modeling of viscoelastic fluids (VEFs). The most important factor being that the underlying microstructure of VEFs is what determines their properties at the macroscale. This microstructure can comprise polymer molecules, colloidal particles, emulsion drops, etc. The common characteristic is that these structures are larger than the solvent molecules and, as a result, bring about additional stresses to the system. This results in two components of the extra stress, one from the Newtonian component (solvent) and the other from the microstructure. We discussed three levels of description used in modeling this microstructure. The first description focuses on the evolution of individual dumbbells using Langevin-type SDEs. The second description is concerned with the evolution of the PDF of the dumbbell’s configuration through Fokker–Planck equations. The third description provides information at the macroscopic level using PDEs. Among the different representations, macroscopic constitutive models offer a higher level of computational feasibility, enabling us to find solutions in complex flows or geometries. However, by using these models, one must make a compromise regarding how accurately we can describe the underlying molecular physics. Similarly, although computationally expensive, Langevin or Fokker– Planck descriptions are more amenable to incorporating additional degrees of freedom. This allows us to develop models that better capture the intricacies of physical processes. Hence, in finding the appropriate level of description from a mathematical standpoint, it is important to strike a balance between the complexity of molecular information and the computational costs involved. References

[AOP21] M. A. Alves, P. J. Oliveira, and F. T. Pinho, Numerical methods for viscoelastic fluid flows, Annual Review of Fluid Mechanics 53 (2021), 509–541. [BAB91] Aparna V. Bhave, Robert C. Armstrong, and Robert A. Brown, Kinetic theory and rheology of dilute, nonhomogeneous polymer solutions, The Journal of Chemical Physics 95 (1991), no. 4, 2988–3000. [Bat99] G. K. Batchelor, An introduction to fluid dynamics, Second paperback edition, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1999. MR1744638 [BCAH87] Robert Byron Bird, Charles F. Curtiss, Robert C. Armstrong, and Ole Hassager, Dynamics of polymeric liquids, volume 2: Kinetic theory, Wiley, 1987. [BD83] R. Byron Bird and J. R. DeAguiar, An encapsulted dumbbell model for concentrated polymer solutions and melts i. theoretical development and constitutive equation, Journal of nonNewtonian Fluid Mechanics 13 (1983), no. 2, 149–160.

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[DEE88] Masao Doi, Sam F. Edwards, and Samuel Frederick Edwards, The theory of polymer dynamics, Vol. 73, Oxford University Press, 1988. [DLY05] Qiang Du, Chun Liu, and Peng Yu, FENE dumbbell model and its several linear and nonlinear closure approximations, Multiscale Model. Simul. 4 (2005), no. 3, 709–731, DOI 10.1137/040612038. MR2203938 [Gra18] Michael D. Graham, Microhydrodynamics, Brownian motion, and complex fluids, Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge, 2018, DOI 10.1017/9781139175876. MR3837153 [Keu04] Roland Keunings, Micro-macro methods for the multiscale simulation of viscoelastic flow using molecular models of kinetic theory, Rheology Reviews 2004 (2004), 67–98. [Lar99] Ronald G. Larson, The structure and rheology of complex fluids, Oxford University Press, 1999. [LM00] P. L. Lions and N. Masmoudi, Global solutions for some Oldroyd models of non-Newtonian flows, Chinese Ann. Math. Ser. B 21 (2000), no. 2, 131–146, DOI 10.1142/S0252959900000170. MR1763488 [Mai22] Francesco Mainardi, Fractional calculus and waves in linear viscoelasticity—an introduction to mathematical models, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2022. Second edition [of 2676137], DOI 10.1142/p926. MR4485803 [MDV20] Andrei Medved, Riley Davis, and Paula A Vasquez, Understanding fluid dynamics from Langevin and Fokker– Planck equations, Fluids 5 (2020), no. 1, 40. [OP02] R. G. Owens and T. N. Phillips, Computational rheology, Imperial College Press, London, 2002, DOI 10.1142/9781860949425. MR1906885 [Ött96] Hans Christian Öttinger, Stochastic processes in polymeric fluids, Springer-Verlag, Berlin, 1996. Tools and examples for developing simulation algorithms, DOI 10.1007/978-3-642-58290-5. MR1383323 [Poo12] R. J. Poole, The Deborah and Weissenberg numbers, Rheol. Bull 53 (2012), no. 2, 32–39. [Ren85] M. Renardy, Existence of slow steady flows of viscoelastic fluids with differential constitutive equations, Z. Angew. Math. Mech. 65 (1985), no. 9, 449–451, DOI 10.1002/zamm.19850650919. MR814684 [Ris84] H. Risken, The Fokker-Planck equation, Springer Series in Synergetics, vol. 18, Springer-Verlag, Berlin, 1984. Methods of solution and applications, DOI 10.1007/978-3-64296807-5. MR749386 [RT21] Michael Renardy and Becca Thomases, A mathematician’s perspective on the Oldroyd B model: progress and future challenges, J. Non-Newton. Fluid Mech. 293 (2021), Paper No. 104573, 12, DOI 10.1016/j.jnnfm.2021.104573. MR4266315 [VMC07] Paula A. Vasquez, Gareth H. McKinley, and L. Pamela Cook, A network scission model for wormlike micellar solutions: I. model formulation and viscometric flow predictions, Journal of non-Newtonian Fluid Mechanics 144 (2007), no. 2-3, 122–139.

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Paula A. Vasquez Credits

All figures are courtesy of the author. Photo of Paula A. Vasquez is courtesy of the University of South Carolina.

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Codimension One Foliations on Projective Manifolds Jorge Vit´orio Pereira In simple terms, a smooth foliation ℱ on a manifold 𝑋 is a decomposition of 𝑋 into a disjoint union of immersed smooth subvarieties. Although the origins of the subject trace back to the end of the 19th century, the definition of a smooth foliation only appeared in the mid-20th century, in the work of Georges Reeb. Foliations naturally appear in various mathematical disciplines, subjecting them to different perspectives. For some, they are topological objects, others emphasize their dynamical nature. Here, our focus lies on seeing foliations as algebro-geometric objects. In particular, we will restrict ourselves to the discussion of holomorphic foliations, with an emphasis on codimension one holomorphic foliations on projective manifolds.

1. Foliations A smooth holomorphic foliation ℱ of codimension 𝑞 on a complex manifold 𝑋 is defined by a collection of holomorphic submersions {𝜑𝑖 ∶ 𝑈 𝑖 → 𝑉 𝑖 ⊂ ℂ𝑞 }, where the sets 𝑈 𝑖 form an open covering of 𝑋, and whenever the domains of two submersions, say 𝑈 𝑖 and 𝑈 𝑗 , have a nonempty intersection, there exists a holomorphic transition function 𝜑𝑖𝑗 ∶ 𝜑𝑖 (𝑈 𝑖 ∩ 𝑈 𝑗 ) → 𝜑𝑗 (𝑈 𝑖 ∩ 𝑈 𝑗 ) satisfying 𝜑𝑖 |𝑈 ∩𝑈 = 𝜑𝑗 |𝑈 ∩𝑈 ∘ 𝜑𝑖𝑗 . 𝑖

𝑗

𝑖

𝑗

We will use “smooth foliation” rather than “smooth holomorphic foliation” throughout the text to avoid wordiness. A smooth foliation defines on 𝑋 an equivalence relation, namely the smallest equivalence relation which identifies points on the same connected components of fibers of the submersions 𝜑𝑖 . Its equivalence classes are the leaves of ℱ which are immersed submanifolds of 𝑋. In practice, a smooth foliation is rarely presented through a collection of submersions as above. A classical theorem by Frobenius allows to recover a smooth foliation by means of the holomorphic vector fields tangent to the Jorge Vit´orio Pereira is a researcher at IMPA. His email address is jvp@impa .br.

Communicated by Notices Associate Editor Han-Bom Moon. For permission to reprint this article, please contact: [email protected].

level sets of the submersion 𝜑𝑖 , or by means of the holomorphic 1-forms vanishing on the level sets of the 𝜑𝑖 . If ℱ is a smooth foliation on 𝑋 then the holomorphic vector fields tangent to it define a subbundle 𝑇ℱ of the tangent bundle of 𝑋. The Lie bracket of any two local sections of 𝑇ℱ defined on the same open subset of 𝑋 is also a local section of 𝑇ℱ . Reciprocally, the Frobenius theorem guarantees that any subbundle 𝐸 of 𝑇𝑋 with set of local sections closed under Lie bracket is the tangent bundle of a (uniquely defined) smooth foliation ℱ. When this is the case, we say that the subbundle 𝐸 is involutive. Dually, the holomorphic 1-forms which are zero when restricted (that is, pulled-back) to the leaves of a smooth 1 foliation ℱ define a subbundle 𝑁ℱ∗ of Ω𝑋 called the conormal bundle of ℱ. When ℱ has codimension one, our main case of interest, any section 𝜔 of 𝑁ℱ∗ over a open set 𝑈 𝑖 where ℱ is defined by a submersion 𝜑𝑖 ∶ 𝑈 𝑖 → 𝑉 𝑖 ⊂ ℂ is of the form 𝜔 = 𝑓𝑑𝜑𝑖 for some holomorphic function 𝑓. It follows that 𝜔 ∧ 𝑑𝜔 = 0. Reciprocally, a dual formulation of the Frobenius theorem, implies that for any rank 1 such that the differential of any one subbundle 𝐸 of Ω𝑋 germ of section 𝜔 of 𝐸 satisfies 𝜔 ∧ 𝑑𝜔 = 0 is the conormal bundle of a codimension one smooth foliation ℱ. When this is the case, we say that the subbundle 𝐸 is integrable. For example, the 1-form 𝛼 = 𝑑𝑧 + 𝑥𝑑𝑦 − 𝑦𝑑𝑥 is not a section of the conormal bundle of any foliation on ℂ3 since 𝛼 ∧ 𝑑𝛼 = 2𝑑𝑥 ∧ 𝑑𝑦 ∧ 𝑑𝑧 vanishes nowhere. In contrast, the 1-form 𝛽 = 𝑑𝑧 + 2𝑥𝑧𝑑𝑥 + 2𝑦𝑧𝑑𝑦 satisfies 𝛽 ∧ 𝑑𝛽 = 0 hence generates the conormal bundle of a foliation on ℂ3 . There is an analogue formulation of the Frobenius theorem for subbundles of 1 Ω𝑋 of arbitrary rank: a sub1 bundle 𝐸 ⊂ Ω𝑋 is the conormal bundle of a smooth foliation (in other words, is integrable) if, and only if, the differential of any secFigure 1. Real picture of some of the leaves of the foliation on tion 𝜔 of 𝐸 defined on a sufficiently small open subset the affine 3-space defined by the 1-form 𝑑𝑧 + 2𝑥𝑧𝑑𝑥 + 2𝑦𝑧𝑑𝑦.

DOI: https://doi.org/10.1090/noti2999

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1025

𝑈 can be written as 𝑞

𝑑𝜔 = ∑ 𝜂𝑖 ∧ 𝜔𝑖 , 𝑖=1

where 𝜔𝑖 are sections of 𝐸 and 𝜂𝑖 are 1-forms, all defined on 𝑈. 1.1. Algebraic foliations. If 𝑋 is a smooth algebraic variety then it makes sense to consider algebraic subbundles of 𝑇𝑋 which are closed under Lie bracket, as well as 1 algebraic subbundles of Ω𝑋 which are integrable. One obtains equivalent definitions of what we call algebraic foliations. Notice that we are not imposing algebraicity of the holomorphic submersions defining ℱ nor of the leaves of ℱ. In general, they are not algebraic. For instance, if 𝑋 = ℂ3 and we take ℱ as the smooth foliation defined by the subbundle 𝐸 of Ω1ℂ3 generated by the 1-form 𝛽 = 𝑑𝑧 + 2𝑥𝑧𝑑𝑥 + 2𝑦𝑧𝑑𝑦 considered above (clearly a polynomial 1-form, hence 𝐸 is algebraic), then ℱ is defined by the submersion 𝜑(𝑥, 𝑦, 𝑧) = 𝑧 exp(𝑥2 + 𝑦2 ) which clearly has nonalgebraic level sets, except for 𝜑−1 (0); see Figure 1. 1.2. Singular foliations. The existence of a smooth foliation on a compact complex manifold imposes strong restrictions on the manifold. For instance, the only smooth hypersurfaces on projective spaces admitting smooth foliations are the quadrics on the three-dimensional projective space1 (hence isomorphic to ℙ1 × ℙ1 ). Moreover, there are only two smooth foliations on ℙ1 × ℙ1 . Of course, those are the ones defined by the fibers of the two natural projections to ℙ1 . The sparsity of examples of smooth foliations on compact/projective manifolds, invites the consideration of singular foliations. To avoid delving into the technical details of the definition of this concept in general, we restrict ourselves to the codimension one case. A singular codimension one foliation on a complex manifold 𝑋 is defined by a 1 (𝑈 𝑖 )}, where collection of holomorphic 1-forms {𝜔𝑖 ∈ Ω𝑋 the open subsets 𝑈 𝑖 form an open covering of 𝑋, such that for every 𝑖 the 1-form 𝜔𝑖 has zero locus of codimension at least two and satisfies the Frobenius integrability condition 𝜔𝑖 ∧𝑑𝜔𝑖 = 0 and, whenever 𝑈 𝑖 ∩𝑈 𝑗 ≠ ∅, the holomorphic 1-forms 𝜔𝑖 |𝑈 ∩𝑈 and 𝜔𝑗 |𝑈 ∩𝑈 differ by multiplication 𝑖

𝑗

𝑖

𝑗

by a nowhere vanishing holomorphic function. The singular set sing(ℱ) of ℱ is, by definition, the union of the zero sets of the 1-forms 𝜔𝑖 . When the ambient is an algebraic manifold 𝑋, one can adopt an alternative definition. If one considers the equivalence relation on the set of nonzero rational 1-forms identifying two rational 1-forms when they differ by multipli-

cation by a nonzero rational function then a codimension one foliation on 𝑋 can be defined as an equivalence class of 1-forms such that some (and hence any) representative 𝜔 satisfies 𝜔 ∧ 𝑑𝜔 = 0. From now on, we will use the term foliation to refer to a singular holomorphic foliation. 1.3. First examples. 1.3.1. Algebraically integrable foliations. If 𝑓 ∶ 𝑋 99K ℙ1 is a nonconstant meromorphic map, then the differential of 𝑑𝑓 defines a foliation on 𝑋. Reciprocally, an argument of Darboux, refined first by Jouanolou and then by Ghys, see [Ghy00] and references therein, guarantees that a Figure 2. Algebraically integrable foliation on 𝑋 = ℂ2 codimension one foliation defined by the rational map on a compact manifold 𝑋 𝑓(𝑥, 𝑦) = (𝑥2 − 1)/(𝑦2 − 1). leaving invariant infinitely many distinct compact hypersurfaces2 is defined by (the differential of) a nonconstant meromorphic function 𝑓 ∶ 𝑋 99K ℙ1 . Codimension one foliations of this form are called algebraically integrable foliations. 1.3.2. Foliations on surfaces. When the ambient manifold 𝑋 has dimension two and is algebraic, it is a trivial matter to construct foliations, since there are many nonequivalent rational 1-forms, all of which, due to dimensional reasons, satisfying the integrability condition 𝜔 ∧ 𝑑𝜔 = 0. A sufficiently general rational 1-form on 𝑋 defines a foliation without invariant projective curves. Moreover, there are foliations for which every leaf is dense in the Euclidean topology. There are also foliations for which every leaf is Zariski dense but not dense in the Euclidean topology. Not much is known about the topological behavior of leaves of arbitrary foliations on surfaces. For instance, the minimal set problem for foliations on ℙ2 [CLNS88]—is it true that the topological closure of every leaf of every foliation of ℙ2 contains a singularity of the foliation?—remains unsolved up to date. 1.3.3. Closed rational 1-forms. Another class of “obvious” examples of codimension one foliations on projective manifolds are the foliations defined by closed rational/meromorphic 1-forms. Locally, a closed meromorphic 1-form is written as 𝑘

∑ 𝜆𝑖

1For the experts. When the hypersurface has dimension two, the statement is a

consequence of the classification of smooth foliations on compact complex surfaces by Brunella. For higher-dimensional manifolds, the statement is a consequence of Bott’s vanishing principle combined with the fact that hypersurfaces of dimension at least three have cyclic Picard group.

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𝑖=1

𝑑𝑓𝑖 + 𝑑ℎ, 𝑓𝑖

2A hypersurface 𝐻 is invariant by a codimension one foliation ℱ on a manifold

𝑋 if the restriction of 𝐻 − sing(ℱ) is a union of leaves of ℱ |𝑋−sing(ℱ) .

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where 𝑓𝑖 are holomorphic functions, 𝜆𝑖 are complex numbers (the residues of the 1-form), and ℎ is a meromorphic function, see [Per23]. On projective manifolds, or more generally compact Kähler manifolds, basic Hodge theory allow us to write any closed rational 1-form as a sum of a closed rational 1-form with simple poles, known as logarithmic 1-forms, and a closed rational 1-form without residues (the so-called 1forms of second type, which are locally exact). Reciprocally, as was observed by Andr´e Weil, given any formal 𝑘 sum 𝑅 = ∑𝑖=1 𝜆𝑖 𝐻𝑖 of hypersurfaces with complex coefficients, there exists a closed logarithmic 1-form with that sum as its residue if, and only if, the homology class of 𝑅 is 𝐻2𝑛−2 (𝑋, ℂ) is equal to zero. An important property of foliations defined by closed rational 1-forms is that they admit nontrivial rational infinitesimal symmetries, that is rational vector fields not everywhere tangent to the foliation and whose local flow (wherever defined) sends leaves of ℱ to leaves of ℱ. If 𝜔 is a closed rational 1-form defining a foliation ℱ then any rational vector 𝑣 satisfying 𝜔(𝑣) ∈ ℂ∗ is a nontrivial rational infinitesimal symmetry. Reciprocally, if a codimension one foliation ℱ admits a nontrivial rational infinitesimal symmetry generically transverse to it then ℱ is defined by a closed rational 1-form. In general, foliations do not admit nontrivial global rational/meromorphic infinitesimal symmetries which are generically transverse to them. 1.3.4. Pull-backs under rational maps. Given a singular codimension one foliation ℱ on a projective manifold 𝑋 and a rational map 𝑓 ∶ 𝑌 99K 𝑋 then, if the image of 𝑓 is not contained in a leaf of ℱ, we obtain a pull-back foliation 𝑓∗ ℱ on 𝑌 defined by the pull-back of a representative 1-form 𝜔 defining ℱ with polar and zero divisor not contained in the image of 𝑓.

2. Foliations on Projective Spaces Let ℱ be a codimension one foliation on ℙ𝑛 defined by a rational 1-form 𝜔. Let us denote by 𝑁 = (𝜔)∞ − (𝜔)0 the divisor (formal sum of hypersurfaces with integer coefficients) of poles minus the divisor of zeros of 𝜔. If 𝑖 ∶ ℙ1 → ℙ𝑛 is a sufficiently general linear embedding then 𝑖∗ 𝜔 is a ra1 Figure 3. Degree two foliation tional 1-form on ℙ . Beon ℙ2 defined by the level sets sides the zeros and poles of a degree three polynomial. coming from 𝑁, the 1-form The blue line is tangent to the 𝑖∗ 𝜔 acquires new zeros corfoliation at the marked points. responding to the tangencies of 𝑖(ℙ1 ) with the folia-

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tion. The number of these tangencies (counted with the appropriate multiplicities) is, by definition, the degree of ℱ. Since the divisor of poles minus zeros of any 1-form on ℙ1 has degree 2, the degree of ℱ is exactly the degree of 𝑁 minus 2. Starting with a rational 1-form on ℙ𝑛 , we can lift it to 𝑛+1 ℂ as a rational homogeneous 1-form. After multiplying such 1-form by the defining (homogeneous) equation of the divisor −𝑁, we obtain a polynomial 1-form 𝑛

Ω = ∑ 𝐴𝑖 𝑑𝑥𝑖 , 𝑖=0

where 𝐴0 , … , 𝐴𝑛 are homogeneous polynomials in ℂ[𝑥0 , … , 𝑥𝑛 ] without a common factor. If the original 1form satisfied the Frobenius integrability condition, then the same holds true for Ω, that is Ω ∧ 𝑑Ω = 0. Moreover, Ω satisfies the descent condition 𝑛

Ω(𝑅) = ∑ 𝑥𝑖 𝐴𝑖 = 0, 𝑖=0 𝑛 ∑𝑖=0

where 𝑅 = 𝑥𝑖 𝜕𝑥𝑖 is the radial, or Euler, vector field. Reciprocally, starting with a polynomial 1-form Ω as above, we can set 𝑥0 = 1, to obtain a polynomial 1-form on ℂ𝑛 , or rather a rational 1-form on ℙ𝑛 without codimension one zeros and polar divisor equal to (𝑑 + 2)𝐻 where 𝐻 is the hyperplane {𝑥0 = 0}. A moment of reflection reveals that the degree of the foliation ℱ defined by Ω equals deg 𝐴𝑖 − 1. 2.1. Space of foliations. Let 𝑉𝑛,𝑑 be the vector space3 formed by (𝑛 + 1)-uples (𝐴0 , … , 𝐴𝑛 ) of polynomials of degree 𝑑 + 1 in ℂ[𝑥0 , … , 𝑥𝑛 ] satisfying the equation ∑ 𝑥𝑖 𝐴𝑖 = 0. The subset Fol𝑑 (ℙ𝑛 ) ⊂ ℙ𝑉𝑛,𝑑 formed by equivalence classes of (𝑛 + 1)-uples (𝐴0 , … , 𝐴𝑛 ) for which the 1-form 𝑛 𝜔 = ∑𝑖=0 𝐴𝑖 𝑑𝑥𝑖 satisfies the Frobenius integrability condition 𝜔 ∧ 𝑑𝜔 = 0 and has zeros of codimension at least two is a locally closed set defined by a collection of quadratic equations on the coefficients of the polynomials 𝐴𝑖 . Problem 2.1. Describe the irreducible components of the locally closed subset Fol𝑑 (ℙ𝑛 ) defined by the Frobenius integrability condition. A variant of the classical Euler formula for homogeneous polynomials 𝑖𝑅 𝑑𝐹 = deg(𝐹)𝐹, implies that 𝑖𝑅 𝑑Ω is proportional to Ω whenever 𝑑Ω ∧ 𝑑Ω = 0 and 𝑖𝑅 Ω = 0. It follows that for 𝑛 = 2, the problem is trivial as the integrability condition Ω ∧ 𝑑Ω is always satisfied by homogeneous 1-forms on ℂ3 annihilated by the Euler vector field. Therefore, for every 𝑑 ≥ 0, Fol𝑑 (ℙ2 ) has only one irreducible component. 3The vector space 𝑉

𝑛,𝑑

is nothing but 𝐻 0 (ℙ𝑛 , Ω1ℙ𝑛 (𝑑 + 2)).

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For 𝑛 ≥ 3, the problem is highly nontrivial, even hopeless in its full generality, but spurred a lot of activity. To have an idea of the complexity of the problem at hand, a simple dimension count reveals that 𝑑+4 𝑑+5 dim 𝑉3,𝑑 = 4( )−( ), 3 3 and that the Frobenius integrability condition on 𝑉3,𝑑 translates into (2𝑑+3) quadratic equations on the coeffi3 cients of the polynomials 𝐴𝑖 . In degree zero, the variety Fol0 (ℙ𝑛 ) is smooth, irreducible, and isomorphic to the Grassmanian of lines on the dual ℙ𝑛 . Moreover, its points correspond to foliations on ℙ𝑛 defined the level sets of linear projections ℙ𝑛 99K ℙ1 . It is unclear who were the first mathematicians to observe this fact. In degree one, when 𝑛 ≥ 3, the determination of the irreducible components of Fol1 (ℙ𝑛 ) dates back to the 1970s and is due to Jouanolou [Jou79]. Inspired by earlier works of Jacobi and Darboux, Jouanolou proves that Fol1 (ℙ𝑛 ) has exactly two irreducible components. One of them parameterizes foliations which are pull-backs of foliations on ℙ2 under linear projections ℙ𝑛 99K ℙ2 , while the other parameterizes foliations defined by pencil of hypersurfaces generated by a quadric hypersurface and a double hyperplane. The determination of the irreducible components of Fol2 (ℙ𝑛 ), again assuming that 𝑛 ≥ 3, had to wait almost 20 years. It is considerably more involved and is the subject of a celebrated paper by Cerveau and Lins Neto [CLN96]. Theorem 2.2. For every 𝑛 ≥ 3, the set Fol2 (ℙ𝑛 ) has exactly 6 irreducible components. One of the irreducible components of Fol2 (ℙ𝑛 ) parameterizes linear pull-back foliations. The other irreducible components parameterize foliations defined by logarithmic 1-forms (one for each partition of 4 with at least two summands). The two irreducible components corresponding to partitions with exactly two summands (4 = 2 + 2 = 1 + 3) parameterize, respectively, pencils of quadrics and pencils of cubics containing a hyperplane with multiplicity three. The general foliation parameterized by any of the two other components corresponding to partitions with at least three summands is not algebraically integrable. The sixth irreducible component (the so-called exceptional component) has general member corresponding to a foliation defined by the linear pull-back under a projection ℙ𝑛 99K ℙ3 of the foliation on ℙ3 = ℙℂ≤3 [𝑡] defined by the action of affine group 𝑡 ↦ 𝜆𝑡 + 𝜇, 𝜆 ∈ ℂ∗ , 𝜇 ∈ ℂ, on polynomials of degree at most three. 1028

Figure 4. On the left, an isolated tangency of a leaf of foliation with a tangent hyperplane. On the right, the induced foliation at the tangent hyperplane. The central point corresponds to the tangency point.

Proof of Theorem 2.2 for 𝑛 = 3 (sketch). Start with a codimension one foliation ℱ on ℙ3 and consider the map from ℙ3 −sing(ℱ) to ℙ̌ 3 which sends a point 𝑝 to the hyperplane tangent to ℱ at 𝑝, the so-called Gauss map of ℱ. If the Gauss map of ℱ is not dominant, then classical results on the geometry of (germs of) surfaces in ℙ3 imply that every leaf of ℱ is either a cone over a curve or the tangential surface of a curve. With this information at hand, one verifies that a foliation with a nondominant Gauss map is either a linear pull-back of a foliation on ℙ2 or its leaves form a one-parameter family of algebraic cones with vertices moving along a line. The first case spans an irreducible component of Fol𝑑 (ℙ3 ) for any degree 𝑑, while degree two foliations in the second case are contained in the irreducible component parameterizing pencils of quadrics. When the Gauss map of ℱ is dominant, the restriction of ℱ to a general ℙ2 is a foliation admitting a singular point (corresponding to the tangency of a smooth point of ℱ with the ℙ2 in question) where the foliation is locally defined by a the differential 𝑑𝑓 of a holomorphic function of the form 𝑓(𝑥, 𝑦) = 𝑥2 + 𝑦2 + ℎ.𝑜.𝑡.; see Figure 4. In the terminology of the theory of ordinary differential equations, such singularities are called centers. A classical result by Dulac published in 1908, describes, rather precisely, quadratic vector field on ℂ2 admitting a center. In general, degree two foliations on ℙ2 are not defined by quadratic vector fields on an affine chart. This is the case if, and only if, the line at infinity is invariant. Computer-aided calculations implies that for degree two foliations on ℙ2 the existence of a center singularity automatically implies the existence of an invariant line not passing through it. After moving such invariant line to infinity, one is in position to use Dulac’s classification in order to classify foliations on ℙ3 with dominant Gauss map. □ Arriving at this point, it is natural to wonder why not pursue the same strategy in order to classify degree three foliations on ℙ3 . Unfortunately, it is very unlikely that

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the strategy outlined above could work in degree three. The main obstruction is the lack of an analogue of Dulac’s classification in degree three or higher. The problem of classifying foliations on ℙ2 with a center (the so-called Poincar´e center problem) is a well-studied and acknowledgedly difficult problem, probably harder than Problem 2.1. The equations defining the center variety are considerably harder to obtain than the integrability equations for codimension one foliations. For instance, back in 2007, Van Bothmer tried to write down, with the help of a computer, the equations for the center problem in degree three without success as explained in the Introduction of [GvB07], to which we refer for further information on the Poincar´e center problem.

3. Intrinsic Meaning of Low Degree Instead of seeking for a generalization of Cerveau-Lins Neto classification to higher degrees, one may try to understand, in more intrinsic terms, what are the factors imposing the constrained behavior of low-degree foliations on projective spaces. The birational geometry of varieties provides a nice conceptual framework to approach this question. The work of Mori and others from the 1980s and its subsequent developments taught us that much of the geometry of projective manifolds is governed by how the linebundle of differential forms of top degree, the so-called canonical bundle, intersects curves. In analogy with the case of projective manifolds, one is naturally lead to define the canonical bundle of a singular foliation as the line-bundle of holomorphic forms along the leaves of ℱ of degree dim ℱ. More formally, one defines the canonical bundle of a foliation as the dual of the determinant of its tangent sheaf. 3.1. Negative canonical bundle. The study of algebraic foliations having negative canonical bundle was initiated by Miyaoka in a landmark paper [Miy87] that explores the idea that negativity of the canonical bundle of a foliation along a sufficiently general curve should be related to the possibility of deforming the curve along the foliation. In rough terms, a deformation of a curve along a foliation is a map 𝑖 ∶ 𝑇 × 𝐶 → 𝑋 from the product of a connected parameter space 𝑇 with the curve 𝐶 to 𝑋 such that for some point 𝑡0 ∈ 𝑇, 𝑖(𝑡0 , ⋅) is the inclusion of 𝐶 and for any given point 𝑥 ∈ 𝐶, the image of 𝑇 ×{𝑥} under 𝑖 is completely contained in the leaf of ℱ passing through 𝑖(𝑡0 , 𝑥). In general, the negativity of the canonical line-bundle of ℱ alone is not sufficient to produce such deformations. Nevertheless, when working with algebraic foliations one may collect the coefficients of (finitely many) defining equations of the foliations in a finitely generated ℤ-algebra 𝑅, reduce everything in sight modulo a maximal prime ideal of 𝑅, and obtain arithmetic shadows in positive characteristic of the

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foliation and of the curve. Then one is in place to explore particular features of foliations and morphisms defined in positive characteristic. It turns out that after composing the inclusion of 𝐶 in 𝑋 with a sufficiently high power of the Frobenius morphism, the sought for deformations are shown to exist in positive characteristic and are used to produce rational curves tangent to the leaves of the foliation through a variant of Mori’s bend-and-break argument. In conclusion, one deduces that the foliation is uniruled, that is, through a general point of 𝑋 passes a rational curve everywhere tangent to the foliation. Miyaoka’s paper was not motivated by internal questions in foliation theory, but instead by the study of projective manifolds. His result is an important ingredient on the proof of the so-called abundance conjecture for projective 3-folds. It was later extended, with different proofs, by Bogomolov-McQuillan, Bost, and Campana-Paun. These lead to a cone theorem for foliations by curves [BM16] and to the uniruledness of foliations with nonpseudoeffective canonical bundle [CP19]. In a related vein, Araujo-Druel-Kovacs ´ [ADK08], motivated by a question of Beauville pertaining to the study of symplectic singularities, classified foliations with canonical bundle as negative as possible: foliations of dimension 𝑟 such that the canonical bundle is the 𝑟-th power of the dual of an ample line-bundle. It turns out that these are precisely the degree zero foliations (of arbitrary codimension) on projective spaces. 3.2. Trivial canonical bundle. Codimension one foliations of degree two on ℙ3 are examples of foliations with trivial canonical bundle. Touzet established in [Tou08] a classification of smooth codimension one foliations with trivial canonical class. He builds on previous work on the universal covering of compact Kähler manifolds with decomposable tangent bundle [BPT06] as well as results on the geometry of Ricci flat complete Kähler manifolds. The classification is rather precise: a smooth codimension one foliation with trivial canonical class is, after an e´ tale covering, the product of compact Kähler manifold with trivial canonical class (seen as a foliation with only one leaf) and a compact Kähler manifold admitting a codimension one locally free action of an abelian Lie algebra (including the trivial case of the action of the zero dimension Lie algebra on an arbitrary compact Riemann surface). Later, Touzet’s classification was extended in [LPT18] to describe singular codimension one foliation with trivial canonical class. Theorem 3.1. Let ℱ be a codimension one foliation with trivial canonical bundle on a projective manifold 𝑋. If ℱ is not uniruled then, after a finite e´ tale covering, (𝑋, ℱ) is the product of a compact Kähler manifold with trivial canonical class and a codimension one foliation induced by a codimension one action of an abelian Lie algebra.

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Although the statement is essentially the same as in the smooth case, it is noteworthy that the algebraicity of 𝑋 (and hence of ℱ) is included as one of the assumptions. It is conceivable that the same result holds for compact Kähler manifolds, but its proof involves techniques specific to algebraic manifolds. Firstly, it is observed, building on complex-analytic arguments due to Demailly [Dem02] and using the Boucksom-Demailly-Paun-Peternell uniruledness criterion (available only to projective manifolds), that when 𝑋 is not uniruled, then ℱ is automatically smooth. Next, when 𝑋 is uniruled, one studies the deformation of rational curves along ℱ to establish that the transverse dynamics of ℱ is rather constrained. More specifically, one proves that ℱ is a transversely projective foliation (see §4.1 below for a definition of this concept). In parallel, one studies the reduction to positive characteristic of ℱ. As observed in [CLNL+ 07], it is relatively easy to produce infinitesimal transverse symmetries in positive characteristics. Although these symmetries cannot be lifted back to characteristic zero in general, when the canonical bundle is trivial this can be done in most cases. This is used to show that ℱ is either defined by a closed rational 1-form (perhaps after a ramified covering) or the singularities of ℱ are rather constrained (admit local holomorphic first integrals). By combining the outcomes of both arguments, one concludes that ℱ is always defined by a closed rational 1form without codimension one zeros. From there, the conclusion follows from a study of the Albanese morphism of 𝑋. The study of deformations of rational curves along codimension one foliations was further developed in [LPT20] and eventually lead to a (partial) classification of the irreducible components of the space of degree three foliations on ℙ𝑛 , 𝑛 ≥ 3, see [dCLP22]. Likewise, the reduction to positive characteristic of codimension one foliations on projective manifolds is investigated in [MP23] and applied to the detection of previously unknown irreducible components of the space of foliations.

4. Global Structure The structure results presented so far imply that foliations with negative or trivial canonical bundles are either uniruled, hence pull-backs of foliations on lower-dimension manifolds, or (finite quotients of) foliations defined by closed rational 1-forms. Below we present a different class of codimension one foliations which, in general, behave differently. 4.1. Riccati and transversely projective foliations. Let 𝐸 be the total space of ℙ1 -bundle over a complex manifold 𝑋. A Riccati foliation on 𝐸 is a codimension one foliations which is completely transverse to a general fiber of the ℙ1 -

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bundle. The Riemann-Hilbert correspondence [Del70] allows one to build Riccati foliations from representations of the fundamental group of the complement of hypersurfaces on complex manifolds to GL(2, ℂ) producing a wealth of examples with rich dynamical behavior. A foliation on a complex manifold 𝑋 is transversely projective if there exists a ℙ1 -bundle over 𝑋 with total space 𝐸, a Riccati foliation ℛ on 𝐸 and a rational section 𝜎 ∶ 𝑋 99K 𝐸 of 𝐸 such that ℱ = 𝜎∗ ℛ. The class of transversely projective foliations includes, as particular cases, the algebraically integrable Figure 5. Numerical approximation of the real trace foliations as well as the foliof some of the leaves of a ations defined by closed raRiccati foliation on 𝔻 × ℙ1 . The tional 1-forms. The study of light gray vertical lines transversely projective foliarepresent open subsets of the 1 fibers of the ℙ -bundle. In the tions, akin to the study of Riccati foliations, is intrinpicture, the foliation is transversal to all fibers except sically related to the study one. Over it one can observe of representations of the the formation of a singularity, fundamental group of comthe point of confluence of the plex manifolds to the auleaves in the bottom half of the tomorphism group of ℙ1 . picture. For more about transversely projective foliations on projective manifolds, the reader is invited to consult [LP07] for a detailed discussion about the concept, and [LPT16] for a description of their global structure,. 4.2. Cerveau-Lins Neto conjecture. All the known examples of codimension one foliations on projective manifolds are either pull-backs under rational maps of foliations on surfaces or are transversely projective. Conjecture 4.1. Let ℱ be a codimension one singular holomorphic foliation on a projective manifold 𝑋. If ℱ is not the pull-back under a rational map of a foliation on a projective surface then ℱ is a transversely projective foliation. The validity of this conjecture would impose strong restrictions on the holonomy representation of algebraic leaves of codimension one foliations. When ℱ is the pullback of a foliation on a surface, the holonomy representation of any algebraic leaf would factor (up to finite index) through a representation of the fundamental group of an algebraic curve. If instead the foliation is transversely projective then the holonomy representation of any algebraic leaf would have solvable image. The main result of [CLPT19] provides evidence toward Conjecture 4.1 as it implies that any representation of the fundamental group

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of a quasi-projective manifold to the group of germs of diffeomorphisms of (ℂ, 0) satisfies this dichotomy. Further evidence is provided by the main results of [CLNL+ 07]. References

[ADK08] Carolina Araujo, St´ephane Druel, and S´andor J. Kov´acs, Cohomological characterizations of projective spaces and hyperquadrics, Invent. Math. 174 (2008), no. 2, 233– 253, DOI 10.1007/s00222-008-0130-1. MR2439607 [BM16] Fedor Bogomolov and Michael McQuillan, Rational curves on foliated varieties, Foliation theory in algebraic geometry, Simons Symp., Springer, Cham, 2016, pp. 21–51. MR3644242 [BPT06] Marco Brunella, Jorge Vitorio ´ Pereira, and Fr´ed´eric Touzet, Kähler manifolds with split tangent bundle (English, with English and French summaries), Bull. Soc. Math. France 134 (2006), no. 2, 241–252, DOI 10.24033/bsmf.2507. MR2233706 [CLNS88] C. Camacho, A. Lins Neto, and P. Sad, Minimal sets ´ of foliations on complex projective spaces, Inst. Hautes Etudes Sci. Publ. Math. 68 (1988), 187–203 (1989). MR1001454 [CP19] Fr´ed´eric Campana and Mihai P˘aun, Foliations with positive slopes and birational stability of orbifold cotangent bun´ dles, Publ. Math. Inst. Hautes Etudes Sci. 129 (2019), 1–49, DOI 10.1007/s10240-019-00105-w. MR3949026 [CLN96] D. Cerveau and A. Lins Neto, Irreducible components of the space of holomorphic foliations of degree two in 𝐂P(𝑛), 𝑛 ≥ 3, Ann. of Math. (2) 143 (1996), no. 3, 577–612, DOI 10.2307/2118537. MR1394970 [CLNL+ 07] Dominique Cerveau, Alcides Lins-Neto, Frank Loray, Jorge Vitorio ´ Pereira, and Fr´ed´eric Touzet, Complex codimension one singular foliations and Godbillon-Vey sequences (English, with English and Russian summaries), Mosc. Math. J. 7 (2007), no. 1, 21–54, 166, DOI 10.17323/1609-4514-2007-7-1-21-54. MR2324555 [CLPT19] Benoît Claudon, Frank Loray, Jorge Vitorio ´ Pereira, and Fr´ed´eric Touzet, Holonomy representation of quasiprojective leaves of codimension one foliations, Publ. Mat. 63 (2019), no. 1, 295–305, DOI 10.5565/PUBLMAT6311910. MR3908795 [dCLP22] Raphael Constant da Costa, Ruben Lizarbe, and Jorge Vitorio ´ Pereira, Codimension one foliations of degree three on projective spaces, Bull. Sci. Math. 174 (2022), Paper No. 103092, 39, DOI 10.1016/j.bulsci.2021.103092. MR4354288 ´ [Del70] Pierre Deligne, Equations diff´erentielles à points singuliers r´eguliers (French), Lecture Notes in Mathematics, Vol. 163, Springer-Verlag, Berlin-New York, 1970. MR417174 [Dem02] Jean-Pierre Demailly, On the Frobenius integrability of certain holomorphic 𝑝-forms, Complex geometry (Göttingen, 2000), Springer, Berlin, 2002, pp. 93–98. MR1922099 ´ [Ghy00] Etienne Ghys, À propos d’un th´eorème de J.-P. Jouanolou concernant les feuilles ferm´ees des feuilletages holomorphes (French, with English summary), Rend. Circ. Mat. Palermo (2) 49 (2000), no. 1, 175–180, DOI 10.1007/BF02904228. MR1753461

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[GvB07] Hans-Christian Graf von Bothmer, Experimental results for the Poincar´e center problem, NoDEA Nonlinear Differential Equations Appl. 14 (2007), no. 5-6, 671–698, DOI 10.1007/s00030-007-5036-x. MR2374205 ´ [Jou79] J. P. Jouanolou, Equations de Pfaff alg´ebriques (French), Lecture Notes in Mathematics, vol. 708, Springer, Berlin, 1979. MR537038 [LPT16] Frank Loray, Jorge Vitorio ´ Pereira, and Fr´ed´eric Touzet, Representations of quasi-projective groups, flat connections and transversely projective foliations (English, with ´ polytech. Math. 3 English and French summaries), J. Ec. (2016), 263–308, DOI 10.5802/jep.34. MR3522824 [LPT18] Frank Loray, Jorge Vitorio ´ Pereira, and Fr´ed´eric Touzet, Singular foliations with trivial canonical class, Invent. Math. 213 (2018), no. 3, 1327–1380, DOI 10.1007/s00222-018-0806-0. MR3842065 [LPT20] Frank Loray, Jorge Vitorio Pereira, and Fr´ed´eric Touzet, Deformation of rational curves along foliations, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 21 (2020), 1315–1331, DOI 10.2422/2036-2145.201712_010. MR4288634 [LP07] Frank Loray and Jorge Vitorio ´ Pereira, Transversely projective foliations on surfaces: existence of minimal form and prescription of monodromy, Internat. J. Math. 18 (2007), no. 6, 723–747, DOI 10.1142/S0129167X07004278. MR2337401 [MP23] W. Mendson and J. V. Pereira, Codimension one foliations in positive characteristic, Journal of the Institute of Mathematics of Jussieu (2023), 1–46. [Miy87] Yoichi Miyaoka, Deformations of a morphism along a foliation and applications, Algebraic geometry, Bowdoin, 1985 (Brunswick, Maine, 1985), Proc. Sympos. Pure Math., vol. 46, Amer. Math. Soc., Providence, RI, 1987, pp. 245–268, DOI 10.1090/pspum/046.1/927960. MR927960 [Per23] J. V. Pereira, Closed meromorphic 1-forms, 2023. [Tou08] Fr´ed´eric Touzet, Feuilletages holomorphes de codimension un dont la classe canonique est triviale (French, with Eng´ Norm. Sup´er. lish and French summaries), Ann. Sci. Ec. (4) 41 (2008), no. 4, 655–668, DOI 10.24033/asens.2078. MR2489636

´ Jorge Vitorio Pereira Credits

Figures 1–5 are courtesy of Jorge Vitorio ´ Pereira. Author photo is courtesy of Dayse Haime Pastore.

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EARLY CAREER The Early Career Section offers information and suggestions for graduate students, job seekers, early career academics of all types, and those who mentor them. Krystal Taylor and Ben Jaye serve as the editors of this section. Next month’s theme will be Publishing and Presenting Mathematics.

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Building Community and Keeping Momentum Finding and Creating Community in Your Department Alan Chang and Rachel Greenfeld Having a network of people in your circle can benefit your career and well-being. A sense of community can make people happier and therefore more productive. Academically, community can impact the quality of one’s research, for example, through gaining exposure to different fields and creating collaborations between people working in different areas, as well as merging perspectives of different inFor permission to reprint this article, please contact: [email protected].

Alan Chang is an assistant professor of mathematics at the Washington University in St. Louis. His email address is [email protected]. Rachel Greenfeld is an assistant professor of mathematics at Northwestern University. Her email address is [email protected]. DOI: https://doi.org/10.1090/noti3009

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dividuals. Mathematicians at every stage of their career can contribute to the community and atmosphere in their department. While it takes some effort and initiative to build relationships, the rewards are well worth the effort. In this article, we share our experience and advice on finding and creating community. Tea time, common room, lunches. There are various ways to create opportunities for people in the department to interact and get to know each other. For instance, if your department already has a weekly or daily tea time, go to those! You might be surprised by how meaningful attending them can be. Often we get to meet new people during tea time and get exposed to other areas of research while talking with them. The atmosphere at tea is usually very friendly and welcoming and gives one a sense of belonging. If your department does not have a regular tea time, you can ask about organizing one. There is a good chance that your colleagues will be willing to help set one up. In addition to tea time, one can use lunchtime to interact with colleagues. How about inviting a colleague to have lunch together in the common room? Once, when I was assigned to teach a class I hadn’t taught before, I found who taught it the previous term and asked if they wanted to have lunch together sometime. They happily agreed, and this ended up being a very fun and productive lunch. My colleague shared several useful tips regarding the class I was going to teach and we talked about other things as well. We had many more lunches together afterwards! Asking for teaching advice is just one of many excuses to invite a colleague for lunch. There are many more ways one can take advantage of the common room in the department to interact with others. In some places, such as Princeton University, professors and TAs even hold their office hours at the common room, and often people feel comfortable participating in the discussions that emerge although they are not necessarily taking or teaching the class. In general, don’t be afraid to speak up, introduce yourself, and play an active role in initiating social events. You never know when something you have to say might positively influence or inspire another person. Research seminars and colloquia and their dinners. Another way to meet people in the department is to attend seminars. I recommend regularly attending at least the research seminar closest to your field, even if some of the topics seem far from your research. In addition to being

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Early Career exposed to research, this gives you the chance to see your colleagues regularly, and can help you feel like part of a research group. At many places, there will be a seminar lunch or dinner before/after the talk. These are great opportunities to get to know people in the department. If you regularly attend the seminar dinner for your research area, then you’ll get to know people by seeing them often. At my department, everyone is welcome to attend the dinner. Sometimes younger members of the department are shy and need some encouragement—I do my best to encourage people to attend. Many departments have colloquiums, which are intended for a broader audience. In contrast to seminar dinners, colloquium dinners are a great way to meet people working in other fields. As a postdoc, I made the mistake of not attending colloquium dinners until my very last semester. Once I started attending, I had meaningful conversations with department members, and I made some good friends whom I only wished that I had more time to get to know. Working seminars / “What is. . . ?” seminars. While research seminars often have outside speakers, it is also good to have a way for people in the department to give presentations to each other. Such seminars with internal speakers are an excellent way to encourage the creation of a community. There are many ways these kinds of seminars can be structured, and we give some examples below. Last semester at WashU, the analysis group had a weekly working seminar with no particular topic—each week, someone gave a presentation on a paper they were reading. At the end of each meeting, we had everyone in the room go around and share what they’ve been working on. This created accountability and opened up a way for us to learn about what others are doing. This semester, based on the suggestion of graduate students, we are trying a different format: we chose a textbook to work through, and people are taking turns giving presentations every week. Another type of seminar that encourages the creation of a community is a “What is. . . ?” seminar. At IAS, we have such a weekly seminar. The idea of the seminar is for a speaker to explain a basic concept in their field to a general audience. One of the main goals of the seminar is for the people of the department—graduate students, postdocs and faculty—to get to know each other through their mathematics. Such a seminar has the potential to take one’s research in new directions by making new techniques and concepts available. You might end up collaborating with a colleague who works in a different field and build new bridges in mathematics! Math chats (Q&A) with faculty members. The WashU math department runs an in-person “math chats” series, which was started by graduate students. The series allows

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students to get to know the professors on an academic and personal level, and also to learn from faculty members’ experiences and perspectives. Each meeting features one professor, who gives a short introduction, and then spends the rest of the time answering graduate students’ questions. When I participated as a professor, the students asked me questions about many things—about my research, my grad school experiences, my hobbies, my favorite Pok´emon, advice about collaboration, etc. In turn, I asked the students many questions as well, so we all got to know each other better. Group chats. A way to easily reach your peers is to create a group chat (WhatsApp, Discord, Slack, etc.). For example, if you are a grad student, you could create one for all grad students, or just your year. I have used these groups to organize social events or just to share memes. Side note: I’ve found group chats to also work very well in small conferences, e.g., AMS sectional meetings. Messages sent in these groups have ranged from “Any plans for dinner tonight?” and “[X] and I are hanging out in the hotel lounge. Come join us!” to “Does anyone know how to use an iron? I need to iron my shirt.” Conclusion. To conclude, there are many ways to find a community in your department and initiate activities that would encourage social interactions among people in your department. Some departments already have various interactive events set up; if yours does not, whether you are a student, postdoc, or faculty, don’t be afraid to take the initiative and organize one of the activities mentioned above, or anything else you have in mind. It only takes one person to make a meaningful change in the department.

Alan Chang

Rachel Greenfeld

Credits

Photo of Alan Chang is courtesy of Washington University in St. Louis/Sean Garcia. Photo of Rachel Greenfeld is courtesy of Dan Kamoda, Institute of Advanced Studies.

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Early Career

Transitioning as an Early-Career Mathematician Rosemarie Bongers I am a mathematician. I majored in mathematics and physics in college, where I fell in love with the abstraction and patterns that gave me a sense of universal truth. I found my area of geometric measure theory while I was in graduate school, and I continued to learn harmonic analysis as a postdoctoral lecturer. I discovered a passion for education, curriculum development, and pedagogical research around this time, leading to a teaching postdoc and eventually a teaching professorship that I currently hold. I am also a transgender woman who came to this personal truth about herself at the age of 29. For me, being transgender is a source of joy and the deepest happiness that I’ve felt; it’s also a source of struggle, challenge, discrimination, and emotional labor. This identity and awareness permeate every aspect of my life, my career, and my living in American society in 2024. My goal here is to explain some of the surprising things I’ve learned through my transition and to discuss how these discoveries intersect with me being an early-career mathematician. Although every transgender identity is unique, we are every bit as diverse and complicated as non-trans folks. I hope that what I write here will resonate with other genderqueer people. I also hope to give a bit of insight into the trans experience, and help spark conversations about how to support the trans community in a time when the community is the focus of political debate and frequently subjected to extreme discrimination and legal restrictions. 0.1. Terminology. As with any mathematical paper, it’s important to get the key definitions correct. I am a transgender woman who was assigned male at birth. This means that when I was born, someone made a choice to check the little “M” box on my birth certificate and kicked into operation a whole set of legal and social mechanisms—all without asking me for my thoughts. For most people, this is okay and these folks are called cisgender, or just cis; all it means is that your internal sense of self matches how society treats you. If you’ve never really thought about these things or had a complaint with your assignation, then that’s pretty typical. On the other hand, my internal sense of identity is female, regardless of what decisions were made more than 30 years ago. This means that I am more at peace, more fulfilled, and just plain happier living as a woman. Rosemarie Bongers (she/they) is an assistant teaching professor at UC Merced. Her email address is [email protected]. DOI: https://doi.org/10.1090/noti3006

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Some important notes need to be mentioned here. First of all, gender is different from sexuality. Gender is about our internal sense of self, whereas sexuality is about who we want to have relationships with. Many transgender people are straight, gay or lesbian, asexual, or on any variety of spectra of desire for relationships. Secondly, gender itself is a broad spectrum that incorporates all sorts of expression (both along the traditional male/female binary as well as directions of being agender). Thirdly, there is a huge amount of diversity even in terms of transgender identities: after finding out that one identifies as being trans, some people dress differently; some change their pronouns (e.g., from he/him/his to she/her/hers or they/them/theirs); some use medical interventions such as hormone replacement therapy or genderaffirming surgery; some do nothing. In any case, the spectrum of trans identities is every bit as rich, complicated, and messy as cis identities—so it’s always dangerous to assume that you know a person’s thoughts or feelings just from knowing their gender identity or which pronouns they use. Likewise, it’s important to discuss pronouns and lived names. Pronouns are not the same as gender; many people use multiple sets of pronouns (such as he/they) and this may differ from a stereotypically masculine or feminine presentation. Likewise, many trans people change their name, and it’s common to use it as a lived name for a long time before it is legally changed (if ever! Remember that not everyone does this). I do not refer to these as preferred pronouns or names because they truly are lived; if a trans person tells you their name or pronouns, listen to them and use them. Being misgendered or deadnamed (i.e., unwillingly referred to by your previous name) can be profoundly insulting and painful experiences that erase our identities and tell us that our truths do not matter. When we make mistakes, it’s frequently best to address them, apologize for them, and move past them. Finally, to emphasize: every transgender life and experience is different. I speak for myself and myself alone; there are many ways in which I experience privilege (e.g., as a white, able-bodied, native English speaker in academia) and many ways in which I do not. I do not speak for all transgender people, and there are so many trans authors—especially those from indigenous, BIPOC, and two-spirit communities—whose work should be read, discussed, and elevated. 0.2. My story: the joy of being trans. Although I didn’t have the terminology to describe it then, many of my oldest memories revolve around wanting to be a girl—as if it was something that I could aspire to. Due to the social expectations and religious pressures in my childhood, I chose to suppress this. To act “normal” and to fit in as best as I could. But suppressing the complicated parts of your

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Early Career life also mean that you don’t get to fully experience the happy, exciting, or awe-inspiring parts either; it just emotionally deadens you. I also didn’t know that being gay, or queer, or transgender was something that was okay; as with many others who grow up in a religious or socially isolated home, I didn’t know any out queer people until I was late into graduate school. My feelings about being out of place and of trying to play-act as a man pushed me into deep depression and anxiety, and I didn’t even know what was wrong. I compare this to the feeling of a left-handed person who was forced to write with their right hand; something feels wrong, and you don’t quite get how everything comes so easily to other people, and you’re never quite as good at it as the others. It took the isolation of the pandemic (between online teaching, not seeing family or friends for months, and the constant background of fear) to finally break me and force me to face the deepest source of my unease in the world. I came out first to myself, then to my partner, then to some of my closest friends. Work and family followed shortly after that and I began to transition. I am incredibly grateful that my partner is one of the kindest, most caring people that I know; her support kept me alive through some incredibly dark times. I am also proud of the community of friends that I was a part of; without them, I could never have started to live my authentic truth. As I initially came out, I was also deeply fortunate to work in a department where many of my colleagues were queer and the culture was open, validating, and supportive. There was never any doubt that I would be anything less than an equal, valued, and trusted colleague and friend. At the time, I didn’t really know any other transgender mathematicians; but believing that at least within my department I would be validated and loved made all the difference. A few months later, I began the process of medical transitioning. As I grow through this experience, I am so deeply grateful that I could start the process. I have no regrets about my transition, save that I did not come out sooner. As people have started to naturally see me the way I am on the inside, I feel calmer and increasingly at peace. There are so many moments of joy that derive from this and from the simple feeling of relaxation in my body. In many ways, I am now “fully” socially transitioned. I dress as I wish, I act as I wish, I speak as I wish. My name has been legally changed. For the first time in my life, I am unapologetically me. There is still a moment of joy every time someone calls me “Rose” or “Rosemarie.” There is still a moment of happiness in seeing myself in the mirror, being what I dreamed of. Many cis commentators reduce being transgender to dysphoria, that feeling of the distress from a mismatch between body and mind. But the counterpart to that is euphoria; the deep fulfillment

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and happiness and love that I feel when I just get to be me in the deepest ways possible. Moving through society as my true self helps me to realize just how much joy there is in life; to paraphrase the video essayist Abigail Thorn, living in my body is finally a source of relaxation and peace. This is not to say that my life is problem-free; there are many challenges, difficulties, and emotional labors that I face due to my femininity and identity. But I believe that it is important to center joy as a core part of the transgender experience—because it is. 0.3. On coming out. Many portrayals of coming out focus on it as a singular moment in a queer person’s life—the moment where they share their identity with the world; but this is far less complicated than the actual experience. For many queer people, myself included, it is a years-long process involving dozens of emotional conversations and moments of uncertainty. This is doubly true for trans folks, especially when there are changes in lived names or pronouns—as there were for me. The inconsistency between lived name and legal name can forcibly out a person; simply having to show ID or fill out an application on MathJobs can lead to this; and we have no control over it. My coming out process started small. I consider myself to be the first person I came out to—because it was a struggle even to admit to myself that I was different. I then came out to some of my closest friends, then to my coworkers, then to my family, and then more broadly. Each conversation is exhilarating and terrifying at the same time because you don’t truly know how the person you’re telling will handle you sharing your deepest truth. These are some of the times in my life when I have felt the most vulnerable: in that moment between saying the words and hearing the response. On the other hand, when people have responded with joy (one friend in particular screamed in happiness!), I had some of the highest moments of my life until now. But coming out is a huge challenge. Most people are not transgender, and many people still hold strong prejudices. There is genuine fear in coming out in the wrong context; transgender people are routinely fired, discriminated against, shunned or disowned, or physically attacked due to their identities. In the current political environment where transgender people are the “demon” a major political party fights against, this moment is especially fraught. In many states, it remains legal to discriminate against trans people in housing and employment and transgender people face high rates of poverty and marginalization throughout the United States. This is also a time when there are dozens or hundreds of new laws that focus on restricting trans rights and which work to marginalize or erase the community.

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Early Career 0.4. Transitioning on the job market. I was applying for jobs during the height of my social transition. In the space between when I submitted my applications for faculty positions and the time that I interviewed for them, I changed the name and pronouns that I use. I was also experimenting with new ways of dressing, starting speech therapy, and learning so much about being myself. It came at a time of incredible emotional labor but also happiness at being able to follow my dreams. I chose to center my transgender identity within my application materials. I talked about how being trans had shaped my perspective in my teaching statement and my diversity/equity/inclusion statement. I included the pronouns I was using at the time on my CV. It would have been impossible to read my documents without knowing that this was a core part of me that shaped my experiences. I was incredibly nervous about doing this; in a sense, this was the first time I was coming out to a broader mathematical community outside my home department, and I didn’t know how people would take it. I thought that if I centered it, I could at least screen out a transphobic department or university without putting much labor into the process. Happily, most of my experiences were very positive. In the job I currently hold, I felt so incredibly supported all the way through the interview process. The committee never called me the wrong name or made me feel like an other. The department chair offered to connect me to queer faculty on campus who had experience living in the area. They made efforts to make me feel welcome and safe and that being transgender was an asset and not a mark against my application. I feel beyond fortunate to have the job that I currently hold, and my department’s commitment to me has been apparent throughout my first year here. My department and university clearly work to support people with marginalized identities, and I have found a loving and accepting queer community that help me to work in the spaces that I truly care about. Unfortunately, this is not the case in every department. As I met search committees, it became painfully clear to me that for many departments a commitment to diversity and equity are merely lip service or words on a departmental webpage. In one on-campus interview, I was repeatedly misgendered by the search committee to their faculty. While this may seem like a small word choice or slip-up, they are reflections of how important these issues are to a department. The indications that a department and a university give can really show the difference between paying lip service to DEI issues and being honestly committed to doing the hard work of creating an affirming and safe environment. 0.5. Transitioning in the classroom. Before I transitioned, I had already been teaching for nearly a decade as a graduate TA and as a postdoc. I fit exactly the stereotype

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of a slightly absent-minded mathematician—down to the plaid collared shirt and cargo pants. This also meant that students immediately viewed me as competent and intelligent, even when I was only a graduate student; it also meant that they were unlikely to talk to me as a human being, to come to me for support, or to show me who they truly were. Now I am visibly queer in the classroom; I make a point of briefly coming out on the first day of class and to build community around the idea that everyone’s whole self is welcome. It is important that students can see people of all gender identities at the front of a math classroom, because there are students of all gender identities in a math classroom. One of my proudest teaching moments was at the end of a semester where a nonbinary student told me how happy they were to be in my class, because they had never seen someone like them as a teacher. Visibility and representation matter. Teaching as a woman is a vastly different experience for me than teaching as a man was, and the labor of teaching has become so different and in many ways has increased. Students connect with me much more now, and are more willing to talk to me about the challenges that they face. This leads to deeper connections and more impactful teaching, but the unobserved labor of supporting students through crises or anxiety can have terrible impacts on faculty mental health and burnout. On a negative note, students are far more willing to openly doubt my competence in a field I’ve taught for years. I face sexist or demeaning comments routinely; in a surprise to me, very few of these have to do with transitioning. The vast majority come simply from being a woman in a mathematics classroom. 0.6. The silent labor of being transgender: thoughts for our allies. While transitioning has been the greatest source of joy in my life, being trans involves a lot of labor that does not fall on cisgender colleagues. Depending on the steps that a person takes in their transition, we may face dozens or hundreds of hours of legal and administrative work for a name change. Some of us have frequent medical appointments to manage hormone levels. Some go through hundreds of hours of painful hair removal, or months-long recoveries from procedures. While being deadnamed or misgendered might take a single word on the part of a colleague, it can have a strong emotional impact of reminding you how you are other and how your true self is not seen, tolerated, or loved. A political group demonstrating on campus might be a small distraction for a cisgender professor, but when such a group is actively working to criminalize or erase transgender identities, it is impossible to ignore. Even routine conference travel can become challenging due to harassment during an ID check and extra searches from the TSA. Many of us who are

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Early Career transgender women also face the burden of simply being a woman in academia, and have to deal with our opinions being automatically less valuable and learn what it’s like to be constantly spoken over. In any case, we face a naturally discriminatory environment where our concerns are not viewed as important, by default. Because transgender people are a minority in most academic spaces (and most spaces within society, period), it is incumbent upon our cisgender allies to learn how to support our community. Building an inclusive department or inclusive pedagogical practice isn’t just a political goal; it has genuine impacts on people’s lives. It creates a space where we can all contribute, all bring our knowledge and creativity, and all explore the beauty of mathematics. As such, I wanted to share some thoughts for mathematicians about how you can help: • Learn about transgender identities by reading and hearing their stories. Trans identities are beautiful and diverse and have been a part of humanity since the beginning. And do remember that every transgender person is different and none of us can speak for all of us. • Remember that we are more than our gender identities; we are just as complex, fascinating, and messy as anyone else. Don’t reduce us to just our trans-ness, and don’t expect every trans person to educate you on trans issues. • Calling someone the wrong name or wrong pronoun is something that unfortunately happens regardless of best intentions; if you do it, make a quick, sincere, and direct apology and move on. Don’t use it as an opportunity to bring up every moment that you’ve misgendered a trans person. Better yet, ask the person how they want it to be handled. • If you see someone being called the wrong name or pronoun, address it. It can be exhausting for that labor to always fall on the queer folks in the community. This goes for pretty much any kind of discriminatory behavior. • Remember that while intent matters, so does impact. Something might feel minor to you but be emotionally devastating to another person. Be willing to do the work to address this. • Intentionally create welcoming spaces where trans people don’t have to worry if they’ll be safe or comfortable. Work with your campus’s LGBTQ+ center or its analogue to figure out how your department can be better. • Think about intersectionality. Although our struggles are different, many of the things that the community can do to support women, racially marginalized groups, or disabled colleagues, are the exact same things that the community can do to support transgenSEPTEMBER 2024

der people. Inclusive practices are often exactly the same as good practices. • The responsibility to address transphobia belongs to all of us. If you aren’t sure how you can contribute, then read, learn, and discuss these issues. 0.7. Conclusion. I hope that there are two takeaways that you’ll keep: that being transgender can be a deeply joyful, satisfying, and fulfilling experience in a beautiful life. And that there is an incredible emotional labor that comes with being trans or actually transitioning. Our transgender colleagues deserve the support of their friends, their departments, and the mathematical community at large. I know that I am incredibly fortunate; I have had the privilege to work within departments that are affirming and am surrounded by a community that loves me as I am. Not every transgender person is so fortunate, and we, as the mathematical community, must do the hard work to bring in everyone.

Rosemarie Bongers Credits

Photo of Rosemarie Bongers is courtesy of Rosemarie Bongers.

Do Mathematics Every Day Daniel J. Thompson A fact of life in our profession is that we often spend extended periods of time facing two opposite challenges. The first challenge many of us face is multimonth periods of time spent with extremely full schedules with teaching commitments, committees, seminars, and the whirlwind of activity of the academic year. This poses obvious challenges to our research productivity and schedule. The second challenge, which may receive less attention, is long stretches of time when our schedules are completely open, and here the challenge is for us to find ways to organize our time efficiently and be productive. While the second situation may be considered “optimal” for research and getting Daniel J. Thompson is a professor of mathematics at The Ohio State University. His email address is [email protected]. DOI: https://doi.org/10.1090/noti3008

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Early Career a lot done, it certainly comes with its own set of potential pitfalls. In this article, I want to share some thoughts on how to keep our mathematical fires burning consistently, and also how to balance an ambitious research schedule against the need to recharge and to be mindful of our wellbeing. Semester-pocalyse. We often face stretches of time when we have far more on our plates than we can reasonably do. This might be for a stretch of weeks, a couple of months, or even a whole semester. If you are reading this as an early career person, I’m sure you can relate, but unfortunately this only gets worse with more seniority. So what can you do to keep your mathematics moving forward when overloaded with myriad other duties? My advice is deceptively simple. Do mathematics every day. By this, I mean to consciously set aside time for mathematics every day, and to do this no matter what else is going on (barring a major crisis or something really exceptional going on). I used to do this intuitively—I’ve always enjoyed working in coffee shops, and even when I’ve been really busy I would often relocate to a coffee shop for an hour and work on mathematics—the change of scenery would free me from feeling the need to reply to emails and remove the chance of interruptions. My caffeine dependency thus led to me doing consistent daily research even during the busiest of times! In recent years, I’ve carved out research time more intentionally and systematically. This came from a breakthrough in my bass-playing. I’ve been a dedicated bassist since I was a teenager, but my playing had been languishing due to a lack of practice for several years (correlating with starting a tenure-track position and having kids). What used to be possible for me in terms of finding practice time was no longer possible. I rebooted my playing during the pandemic, taking an online course on the Discover Double Bass platform with jazz bass maestro John Goldsby, who had this to say about practice: “Practicing 15 minutes a day, every day, is better for your development than two hours of hectic bass playing every weekend.” This was a lightbulb moment for me. For years, I’d felt that I was too busy to practice. The idea that I could make progress by doing 15 minutes of serious practice each day was a revelation. What kind of crisis would have to be going on that I couldn’t find 15 minutes in my day? It would have to be quite a crisis if I couldn’t find just 15 minutes! Another aspect of this is that 15 minutes is the floor, not the ceiling—if the going is good, you just keep on going. Over the course of a year or two, this approach worked wonders for my playing, and now I’m swinging my way through jazz standards and burning my way through the bluegrass songbook. I got to thinking about how to apply this to mathematics research.

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A common mistake that we make in research is the same one I was making with my music practice. It’s a common trap to feel like we need a substantial block of uninterrupted time to get in the zone and make progress. For research, maybe we are waiting for a free day or even a free week so we can really get in the zone. Those are certainly optimal conditions, and those big blocks of time are great to have when they’re available. However, it’s important to let go of the idea that you need to wait for optimal conditions to do research. Being able to make progress in nonoptimal conditions is the way to produce research consistently over the long-term. My advice to you during busy periods is to decide how much research time each day can be a realistic floor for your research time. Is it one hour? Is it half an hour? Maybe even carving out two hours each day would be realistic depending on your schedule. The point is to figure out what can realistically be done EVERY workday based on your schedule for the semester, and to stick to it as religiously as possible. This will keep your research moving along, and when a block of uninterrupted time does open up, you are ready to make the most of it. Otherwise, when that uninterrupted block of time comes up, you risk needing most of the time just to get back up to speed on your projects, and there may be little time remaining to make further progress. Let your subconscious do some of the work. A common misconception is to think that we’re only working when we’re sitting in front of the computer, or reading a paper, and actively concentrating. In fact, the subconscious often does a lot of the heavy-lifting. The “active work” of concentrating on reading a paper, or working through a calculation, is sometimes most useful as a process of getting the ideas into our head. We may not know how to solve a problem when we’re consciously trying to do it. When we step away from a problem we’ve concentrated hard on, and come back the next day, often our subconscious has been working away in the meantime, and suddenly we know what to do next. This is another benefit of the ‘Do math every day’ approach. Each day that you learn some mathematics, even for a relatively modest amount of time, your subconscious will continue to work away on it while you are teaching your classes or busy with other duties. When not to do mathematics every day. Burnout happens in our profession. We’re running a marathon, not a sprint, and it’s important to have the tools to keep on going indefinitely. My previous advice to do math every day aims to stimulate your subconscious to keep on working on math even while you’re doing other tasks. Sometimes you just need to rest and to let your mind be quiet. Without rest, stress becomes cumulative and can lead to chronic anxiety issues. The classic recommendation by

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Early Career psychologists is to give yourself the following periods of downtime [1]: One hour per day, One day per week, One week out of every twelve to sixteen weeks. Taking a full day away from research each week is certainly important. It’s probably difficult for most people to take off a full week every quarter; I haven’t been doing that. However, that is the classic evidence-based advice from psychologists, and maybe we would all benefit from it when possible. Taking at least a couple of week-long breaks in the year seems realistic and in the spirit of this advice. During these times, you want to rest your brain as completely as possible so you can come back fresh and strong when you return to work. Balancing semester duties. When things are really busy, it is an unfortunate fact of university life that other tasks squeeze our research time. Our teaching and service work have hard deadlines that push these duties to the top of our priority list. To counteract this, it’s good to create some firm deadlines in our research too. Regular meetings with collaborators are a great way to provide this, since it’s natural for us to want to have some progress to report at the meeting. Giving a talk on your research, or an expository talk on an adjacent subject can be another great way to generate momentum. Deadlines like these can guide how we use our research time and help us use it efficiently. While adding research deadlines is useful for direction and motivation, it can be freeing to balance this by occasionally going “rogue” and thinking about some interesting mathematics even if it is not directly useful for a current project. For example, learning what a colleague is working on or spending a little time with an interesting paper that jumped out at you from the daily arXiv listing is a great way to stay productive when you need a break from your main research projects and deadline-oriented work. If you’re lucky, you might even find some inspiration for a great new project. If service has taken over your schedule to the point where research feels almost impossible, then I would advise looking at what you can let go of in the future. Personally, I always try to complete the service assignments I committed to as best I can, even if I find out the hard way that I took on more than I should. However, if I have taken on too much, I try to make a change for the following year. Most things we sign up for, committee assignments for example, are for a fixed amount of time. If you find yourself doing an unreasonable amount of service during the semester, then in most cases it is absolutely reasonable to let some things go the following year to preserve more of your time for research. Alternatively, if you anticipate having an unusually demanding service load, and you’re unwilling or unable to let anything go, you can request to teach courses you’ve taught before to reduce time spent on teaching preparation.

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Wide open spaces—summer and sabbatticals. Being productive on research when we are NOT busy with teaching, committees, etc., is an underrated challenge. Many of us oscillate between the extremes of our schedules being completely full and mostly empty as semesters begin and end, and it can be a challenge to adapt. We often begin our summers, or a semester without teaching duties, with high hopes for extraordinary research productivity. Indeed, these are the times when one can really make a lot of research progress, but these periods can also be overwhelming, particularly when paired with lofty selfexpectations. The “do math every day” mantra still applies, but now the challenge is to use your research time well and also to be mindful about staying healthy. I recommend creating a research “master plan” for your summer or semester. Not only should you create a list of the broad projects that you would like to work on, but you should write down a couple of pages worth of specific research tasks, e.g., “understand paper X; polish the proof of Lemma 3.7; work on the introduction of paper Y; work on the main technical result needed for project Z,” and research-related tasks, e.g., “schedule regular meetings with collaborator A; email expert B asking them a question.” If your to-do list solely consists of “do great math research today,” that will likely be overwhelming, and some days it will be unclear where to start. We often lose some perspective and mental clarity if we’ve just spent a couple of days working on a technical lemma, or doing a MathSciNet deep dive into the literature on a topic we’re learning. When we hit a wall on writing up a technical lemma, or trying to absorb all the information in a literature search, it can be valuable to change gears. That is where your to-do list of specific tasks becomes useful to fall back on. If you hit a wall with one task, or you’re unsure how to proceed, just consult your list and pick whatever task looks good to you to work on that day. Some people go further and schedule their days into blocks of time for specific tasks. If that style works for you, then great. Personally, I prefer to leave my time unstructured, but to use my list of specific tasks to guide me and keep me on track and productive. It’s also important to keep up with non-math activities to help you stay mentally sharp and healthy. This could be exercise, music, social activities, or just getting outside. This is a good idea for overall well-being, but it’s also good for the mathematics itself. Judiciously chosen time spent on non-math activities helps our subconscious organize and process all the hard work we do. Let me finish with a quote from legendary record producer Rick Rubin [2]: Take the example of an album. If you’re a musician struggling with ten songs, narrow your focus to two. When we make the task more manageable and focused,

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Early Career a change occurs. . . . Going from two to three is easier than going from zero to two. And if you happen to get stuck on three, then skip it and get four and five done. Complete as many elements of the project as you can without getting hung up. . . Often the knowledge we gain from finishing the other pieces becomes a key to overcoming earlier obstacles. References

[1] Edmund J. Bourne. The Anxiety and Phobia Workbook, Seventh Edition. New Harbinger Publications, 2020. [2] Rick Rubin. The Creative Act: A Way of Being. Penguin, 2023.

Daniel J. Thompson Credits

Photo of Daniel J. Thompson is courtesy of Neal Havener.

Productivity and Time Management in Research Steven Senger I’m relaxing in a coffee shop overlooking Ngo.c Kh´anh Lake in Hanoi, sipping the last of my ba.c xiu, an absurdly delicious coconut cream coffee drink, as a summer-long research program at the Vietnam Institute for the Advanced Study of Mathematics draws to a close. I’m reflecting a bit on how I got here, and how I can keep producing enough mathematics to hopefully get asked back someday. I came into our trade by accident, as many of us do. One of my college buddies happened to be a mathematician, and because I was an electrical engineer by training, I figured I should be able to at least understand what some of the big ideas were. Even though math had never been my focus before, I registered for a graduate program in engineering and applied mathematics at the same school where I had finished my undergrad without really thinking much about Steven Senger is an associate professor at Missouri State University. His email address is [email protected]. DOI: https://doi.org/10.1090/noti3011

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what that would mean for my future; I really just enjoyed school and wanted to keep learning. Despite feeling like I was a few steps behind in my mathematical foundation, I always thought I had some useful physical intuition because of my engineering background, particularly when it came to Fourier analytic techniques. Partially due to an all too common blend of pride and selfdoubt, when I started graduate school, I spent more energy on traveling around to play music and go rock climbing than on studying. While I did fine in my coursework, I proceeded to get soundly outclassed by my fellow students, who seemed to cut through research roadblocks by applying ideas from this or that paper that I’d never heard of. “So the support isn’t compact? That’s fine, we’ll just consider this Radon-Nikodym derivative. Oh, you’re worried about some pesky symmetries? Just get a bound using the orbit-stabilizer theorem.” In hindsight, this is all standard, but when I was just getting started, I was completely overwhelmed by the relative ease with which my compatriots navigated the same challenges that kept me up all hours of the night. I am extremely fortunate to have a world-class advisor, Alex Iosevich, who was careful to meet me where I was in my research journey, and instead of chastising me for not studying hard enough, he merely gave me objectively reasonable research problems, and when I couldn’t solve them, I worked out for myself that I needed to decide what was important and prioritize accordingly. I played fewer shows and spent less time climbing but I still kept a balance. Slowly, the tricky calculations began to seem routine, and I got a better sense for which barriers were just better left for another day. So instead of a series of brilliant epiphanies, I got a feel for how to make slow and steady progress on a number of fronts and increase my personal chances of producing a steady stream of research ideas. However, during my postdoc, I learned that this focusing has limits. I pretty much ditched my hobbies entirely, and put everything into mathematics. Somehow, even though I was dedicating more time and energy than ever to research, all of my projects seemed to grind to a halt. So I worked even harder, applied for grants, designed and taught a well-received graduate topics course, and ended up developing a severe anxiety disorder. After a particularly nasty attack I passed out in my office and took a ride in an ambulance. I think it’s funny that after working jobs in skydiving, rock climbing, music, and even being a bouncer for a while, my most dangerous professional hazard was in mathematics. I am so grateful to have a solid network of people to support me, and while I still have issues, I have learned to not work myself so hard all the time. With these adventures, I found good approximate upper and lower bounds for how hard I can work and still

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Early Career produce. However, as soon as I got a handle on life after graduate school, the game changed again with my tenuretrack appointment in 2014. I am very lucky that I got my top choice, Missouri State University. It’s a state school, close to my family, with research programs similar to my own. The thing is, once you’re a faculty member, the committee meetings and teaching preparation balloon up without warning. Okay, I heard plenty of warnings to guard my time, keep my pedagogical expectations reasonable, and not to let too many committees hop on my plate, but I was an alacritous new member of the team, and wanted to prove my worth! Luckily, my mentors both within the department and beyond were careful not to let me bite off more than I could chew. After a few semesters, I began to appreciate their wisdom more personally. Of course, teaching and service tasks took precious time and energy. One of the pitfalls for the budding researcher is working hard for a day or a week, or whatever, and feeling so accomplished with their hard work on teaching, service, and just surviving, that research can seem to be a luxury task for another day. The thing I’ve noticed is that the longer breaks I take between serious research pushes, the harder it is for me to get moving forward again. It’s almost like exercising. If you skip your exercise routine too many times, it gets much harder to get it moving again. So my two-pronged approach to research praxis is consistency and variety. For me personally, I really need to think about research for a good chunk of time at least once a week, or I am likely to lose whatever framework for a problem I had in my head, as well as atrophy my creative reasoning abilities. During busier parts of the semester, this may be as simple as rereading notes from previous weeks, or going back through the deltas and epsilons of a current manuscript. A while back, my friend Paul Baginski suggested that we meet up once a week over Zoom to discuss our (very different!) mathematics research programs. Even though he and I don’t collaborate on manuscripts, I feel a responsibility to have something new to tell him each week. This gives me a deadline for my research, which keeps me motivated during the week. Teaching, service, and other activities tend to have deliverables with regular deadlines: grading student quizzes within one week, finishing the report for the ad hoc subcommittee, or even submitting an abstract for an upcoming talk. For better or worse, the deadlines in research are often flexible, and the expectations can get nebulous at best. As a result, they can sometimes get lost in the shuffle. How many times have you heard, “I’m still thinking about that,” or, “This week I’ve really been trying to read,” from a collaborator in a research meeting? Meeting regularly with Paul forces me to have concrete things to say each week, even if the concrete thing I say is, “I failed to get research done this week.” When I have to acknowl-

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edge it out loud, it reminds me to prioritize it in the week to come. While the idea of regularly attacking a problem makes sense in an abstract, maybe even romantic way, the reality of slamming your head into the same expression every time you sit down to do research can be punishing at best. So I try to make sure that I always have a curated selection of active projects, with difficulty/feasibility ranging from, “simply turn the crank and the student gets a good start on a paper,” through, “this is fun, but I don’t know how soon we’ll finish,” all the way to seriously difficult problems that I mostly study for fun. My goal is to make sure I have a moving pool of ideas that I can use as vehicles to shepherd students through their first projects, but that I can occasionally spin off into serious research papers. Then I separately have hard conjectures that I can try new ideas on after working through some of the more standard projects. The hope is that these feed and support one another, so that simpler cases come out of hard problems to provide projects for students, and the surprising pieces that come out of student projects can give me new angles of attack on harder problems. This way, I can have a fairly steady stream of papers, but I never feel like I’m just beating the same dead horse. To put this in perspective, here is a sketch of some of my current research program. At the top, I have a few insights on the celebrated Erdos ˝ unit distance problem (How often can the most common distance occur in a large finite point set in the plane?) that I haven’t yet seen written down anywhere, so I can always entertain myself by pushing on those ideas. I certainly think about it every day, but I maybe push on it in earnest once a week. Now, this particular problem hasn’t seen concrete progress since the eighties, so I don’t put all my eggs in that basket. There are a number of related problems that have seen some traction recently, and I see these as a safer investment of my time and energy. For example, instead of a single distance determined by a pair of points, we could consider multiple distances determined by 𝑘-tuples of points. This is still challenging, but not so hopeless, so I can publish papers on problems like these with my colleagues. If we get started proving things, and it seems like we’ll mostly be applying standard techniques to get results, then I’ll invite some students along for the ride, and I can put more energy into setting up an environment where they can read some papers, learn some things, and prove some of the theorems. Of course, when I do have time to push hard, I like to go to a coffee shop, put on headphones, and melt into a problem. The background bustle of the shop keeps me from getting distracted by silence, and any music that I can’t sing along to will prevent my ears from latching onto any passing conversations. Everyone has different preferences, but

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Early Career I think it is important to experiment and get a sense for where you can be the most effective for each task. For example, I can proofread a draft of a manuscript just about anywhere, like while resting between sets at the climbing gym, but I have a hard time coming up with new ideas unless I am really insulated from distractions. When I don’t have the best conditions for hard thinking, but want to be creative, I like to type up notes. Even if a particular idea didn’t work with one problem, I can sometimes recast that idea successfully in a different context, and that’s much easier to see when you’ve clearly elucidated it in a beautifully typeset document. I also think that many creative pursuits follow natural inhale/exhale cycles. Sometimes I’m taking more ideas in, either by reading papers that seem interesting on arXiv, trying to brush up on this or that topic that I haven’t seen in a while, or even just refereeing papers. I find that I don’t write as much or have as many new ideas during these times, but I perceive that they give me the fuel for the opposite end of the cycle, which is when I’m grinding out the last few lemmas on a manuscript that should have been submitted weeks ago, trying to pull together a new big project with some collaborators, or just playing around with undergrads and finding connections that I hadn’t suspected. I find that these times last longer and have more output that I am proud of directly after a longer, harder input phase. So I don’t get particularly disturbed by a week or two without a concrete breakthrough, assuming that I have managed to digest or resurrect some significant ideas. I guess what I’ve come to believe after almost twenty years of working on research is that I can’t force success, but I can get a sense for what conditions make me more likely to make new connections and communicate them effectively. It is really like art in that I won’t necessarily know when something wonderful is about to come out, but I know nothing will come out if I don’t at least show up and consistently push on a variety of ideas.

Steven Senger Credits

Photo of Steven Senger is courtesy of Riley McCullough.

How Does Your Daily Life Change When You Become the Graduate Coordinator? Chun-Kit Lai When I got tenured, I was recommended to become a graduate coordinator in my University (San Francisco State University, SFSU). This has been by far the most satisfying departmental service I have ever done. My life has become much busier after taking on this role in addition to doing research and teaching, but the experience and the outcome have been rewarding and unforgettable. The ultimate degree that SFSU provides is a master’s in mathematics. Every year, we attract a diverse pool of applications from different ethnicities, geographic locations, and ages (even retired people apply to our graduate program!). The first job of the graduate coordinator is answering all the questions raised by these applicants. To save time in replying to these emails, it is important to make sure the department website is informative and attractive. The next major task is reviewing these applications after our first priority deadline (the deadline that we will start reviewing our first pool of applications, but we will still accept applications after this deadline), which could take a couple of days. All these applicants have different expectations and qualifications, but a common theme is that they have a passionate heart toward mathematics and mathematics teaching. They expect our program to fill their hearts. When I see them accept the offer and later come to my office in person, I regard them as joining our family. We will teach them advanced mathematics, but indeed their presence also changes how we deliver our lectures and shape our curriculum. Another major task as graduate coordinator is to help them choose the right courses and find an advisor for their thesis. There are set courses our students can choose from and there are also flexible electives for our students to take. Whenever a student comes for advice, it is important to listen carefully to what they need. In particular, I will try to gain some insight into what their mathematical interest is. I then direct them to my colleagues whose interests overlap with theirs. It is also my job to make sure that they are on track to finish all required courses. Graduate students in SFSU normally stay for only two years, some three years. They are like a close family doing homework together, talking about their research projects together, having fun together, and finally graduating together. After their graduation, some continue on to PhD Chun-Kit Lai is an associate professor at San Francisco State University. His email address is [email protected]. DOI: https://doi.org/10.1090/noti3010

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Early Career programs, some become lecturers in community colleges, some work in industry and some continue to enjoy their retirement. We may or may not have time to get together again, but our students all leave with a satisfying memory of their life at SFSU. I have signed more than 50 graduation forms in my five years of service. Most of our students come in with minimal undergraduate mathematics background, but they graduate with a thesis on a cuttingedge research topic. Seeing their growth and their success is the most fulfilling experience in this position of graduate coordinator.

Chun-Kit Lai Credits

Photo of Chun-Kit Lai is courtesy of Sin Yee Chau.

Dear Early Career I had been working on a paper for some time, and a colleague gave me a few ideas and suggestions which have improved the paper quite a bit. Should I offer coauthorship? —Confused author Dear Confused author, Firstly, congrats on the interesting developments in your research! Your question is delicate, and there is no one-size-fits-all answer, but I can say that in my experience being generous with offering coauthorship has been a good decision in the longer term. However, one should only do this when appropriate.a Essentially my guideline is that if the results and their proofs would be present without the discussions with your colleague, and the discussions have streamlined the presentation, then including an acknowledgement might be more appropriate.b However, if the discussions have led to significant changes to the main results or the structure of the mathematics, then it would be beneficial for you to have a conversation with the colleague as to whether they would like to be included in the paper as a coauthor. If you have been working on the paper for a while, this can be a difficult decision to arrive at, as you SEPTEMBER 2024

may feel like the effort level has not been equitable. It is beneficial to have colleagues who are generous with their ideas, and you do not know how much effort they have made in arriving at their suggestions, and so thinking in this manner can be not an accurate reflection of “work done.” In my experience, observations of colleagues which at the time felt like they came out of nothing, were, in fact, modifications of ideas they had attempted to execute in different contexts several times previously. Their observations were really the result of thousands of hours of work. Assuming the observations can be separated from the text without disturbing the flow of the exposition, a common alternative is to keep the authorship of the paper as it is, but to include the observations in a coauthored appendix. This is particularly suitable if 1. the conversations have led to applications of your main results that you did not envisage, but have not modified the primary results themselves, or 2. if you need a modification of a “standard” technical tool from another area, and the colleague has helped you with this due to their expertise in that area. Finally, I can also attest to the fact that if you offer an important tool for people to use in a paper, only to not be invited to be a coauthor, then you can be a bit weary of communicating with them going forward. Consequently, if you value the potential collaborative future with this colleague, then being generous with offering coauthorship at this stage can be hugely beneficial for you. —Early Career editors

Have a question that you think would fit into our Dear Early Career column? Submit it to Taylor [email protected] or [email protected] with the subject Early Career. DOI: https://doi.org/10.1090/noti3007 a The AMS Ethics Guidelines, which broach coauthorship briefly, may be found here: https://www.ams.org/about-us /governance/policy-statements/sec-ethics. b Being generous with acknowledgments is a good policy. If someone has helped you with any technical aspect of the paper (be it mathematics, references, or typesetting), then acknowledging their assistance is important to show that you valued their contribution. Additionally, if you are giving a seminar talk and you arrive at a particular moment when someone’s guidance helped, you should feel free to give them a shout-out.

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Eli Goodman (1933–2021) and Ricky Pollack (1935–2018) J´anos Pach, Micha Sharir, Noga Alon, and Andreas Holmsen Twin Primes: Eli Goodman and Ricky Pollack J´anos Pach In my eyes, Eli (Jacob) Goodman and Ricky Pollack were inseparable. Exactly when and where they first met was a matter of discussion between them. Was it in the alcoves of City College, playing chess or in the NYU library, listening to recordings of classical music? Eli’s father was a well-known secular Jewish scholar, who published extensively in Yiddish and English. Ricky was one of the “red diaper babies,” his parents were Communists, constantly harassed by the authorities. Both of them were passionately interested in mathematics, in music, and in literature. Both of them played the piano, Eli at a semiprofessional level. After attending the Bat Mitzvah of one of Eli’s daughters and listening to the band Klezmatics, Ricky started taking clarinet lessons from David Krakauer. He would not travel anywhere without his clarinet. Eli went even further: he got a degree in composition and he cofounded the New York Composers Circle. His pieces were performed by leading musicians and were recorded. Both of them had brilliant supervisors, but they did not have an easy start in mathematics. Eli’s supervisor Communicated by Notices Associate Editor Emilie Purvine. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2997 J´anos Pach is a professor of mathematics at HUN-REN R´enyi Institute of Math´ ematics, Budapest, Hungary and Ecole Polytechnique F´ed´erale de Lausanne, Lausanne, Switzerland. His email address is [email protected].

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Figure 1. Eli and Ricky, the main organizers of the 2008 Discrete Geometry Conference in Oberwolfach, Germany.

was Heisuke Hironaka, who later got the Fields Medal for groundbreaking discoveries in algebraic geometry. For many years, Eli worked relentlessly on a conjecture that turned out to be false. His favorite teacher at NYU was Harold N. Shapiro, a number theorist, who collaborated with Paul Erdos ˝ and Richard Bellman, and loomed large in mathematical circles. Ricky became his student. They spent a lot of time together. It was easy to learn from Shapiro, but difficult to shine next to him. By the midseventies, both Eli and Ricky were ready to venture into a new field that they felt was their own. They were lucky: during their sabbaticals they hit on roughly the same subject and shortly afterward they found out about their common interest. At McGill University, Montreal, Willy Moser told Ricky about the happy ending problem of Erdos, ˝ Esther Klein, and George Szekeres, and he almost instantly got obsessed with it. Is it true that any set of 2𝑛−2 + 1 points in general position in the plane has 𝑛

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elements that form the vertex set of a convex 𝑛-gon? If yes, this bound would be best possible. We still do not know the answer to this question, but a few years ago Andrew Suk showed that (2 + 𝑜(1))𝑛 points always suffice. Eli had been searching for a simple geometric question that can be approached by encoding the underlying configurations and translating the problem into a purely combinatorial one. The happy ending problem appeared to be a perfect candidate. They started to explore these ideas by introducing (rediscovering) the notions of order types and allowable sequences. They did not get any closer to the proof of the Erdos-Szekeres ˝ conjecture, but the approach quickly yielded fruit. They proved Grünbaum’s conjecture that every set of eight pseudolines (two-way infinite curves, any pair of which cross precisely once) is stretchable [7]. They continued to make progress related to a number of other questions raised in Branko Grünbaum’s classic 1972 treatise, Arrangements and Spreads [14]. However, the most elegant early application of the method of allowable sequences was found by Peter Ungar [18], a legendary problem solver of Hungarian origin. He proved that any set of 𝑛 points in the plane, not all of which are collinear, determine at least 𝑛 distinct directions, provided that 𝑛 is even. His argument was reproduced by Aigner and Ziegler in their popular volume Proofs from THE BOOK. Erdos ˝ often said as a joke that God kept a book with only the most elegant mathematical arguments, and he rarely allows anyone to have a glance into it. In 1980, Ricky and Eli started a geometry seminar at Courant Institute (NYU) which was attended by faculty and students of many universities from the Greater New York area, including Rutgers, Princeton, Columbia, CUNY, Stony Brook, and Pace University, and by researchers from Bell Labs, AT&T, and IBM. Over the years, when passing through the Big Apple, almost all important figures working in combinatorics, discrete geometry, computational geometry, or convexity gave a talk in this seminar. The abstracts of these talks were widely circulated. In the preinternet era, anyone following these announcements had a pretty good overview of the most exciting new developments in our field. It was in this seminar that Peter Ungar learned about allowable sequences, which enabled him to prove his above-mentioned theorem on directions, originally conjectured by Scott. Twenty-five years later, Rom Pinchasi, Micha Sharir, and I managed to settle Scott’s problem in three dimensions [15]. All three of us attended the meetings of this seminar for years. I also had the privilege of co-organizing it in the first decade of the 21st century. In the beginning, it was not clear whether such a seminar would ever fly. As Joe Malkevitch recalls, Ricky doubted if anyone would show up if they “put out a shingle.” Yet people did come, and they came in ever grow-

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Figure 2. Janos ´ Pach, Ricky, and Eli at the International Conference on Intuitive Geometry, Siofok, ´ Hungary, 1985.

ing numbers. Why? It is hard to deny that the charismatic personalities of Ricky and Eli played a big role in this. They picked the right speakers and fascinating topics, and they had a good nose for significant new developments in the subject. Sometimes they were wrong, especially Ricky, who easily fell in love with a new problem. But this only added to the thrill of novelty and discovery. The topics covered in the seminar have opened up new avenues of research for most participants, including many established senior mathematicians and computer scientists. A touch of luck has also contributed to the remarkable success of the seminar. The early 1980s witnessed an explosive surge in computing power, which resulted in a sustained appreciation for algorithmic techniques, an appreciation that has only grown stronger over time. NYU had a Robotics Lab, codirected by Jack Schwartz and Micha Sharir, who laid the mathematical foundations for motion planning. As recognition of their work, in 1986, they were invited to speak at the International Congress of Mathematicians in Berkeley. Many practical questions such as the so-called piano movers’ problem, visibility and ray-shooting problems raised deep questions about arrangements of points, lines, curves, convex sets, and other geometric objects in Euclidean spaces. It turned out that some closely related questions, with deep ties to number theory, functional analysis, discrete geometry, and information theory, had been investigated before by Gauss, Hilbert, Minkowski, Fejes Toth, ´ Rogers, Conway, Erdos, ˝ Lov´asz, Spencer, Szemer´edi, Trotter, and others. Many of their results proved to be applicable in the design of efficient geometric algorithms. A new field of computational geometry was born. It was also popularized by Ron Graham and Frances Yao’s concise and elegantly written survey, titled “A whirlwind tour of computational geometry,” published in the American Mathematical Monthly. The number of graduates in computer science far surpassed the number of mathematics graduates. The new generation of computer scientists were equally well-versed in discrete mathematics and geometry as their counterparts in mathematics. They were familiar with the happy ending problem and the probabilistic method (or the

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random sampling technique, as it was called in computer science), they learned about the Szemer´edi-Trotter theorem on the number of incidences between points and lines and Lov´asz’s theorem on halving lines. They not only knew about these results, but soon improved on them! In particular, in a seminal paper presented at the 2nd Annual Symposium on Computational Geometry in 1986, David Haussler and Emo Welzl borrowed a technique for set-systems of bounded Vapnik-Chervonenkis dimension, and applied it to a wide range of questions in geometry. This paved the way for a series of new discoveries, including far-reaching extensions of the Szemer´edi-Trotter theorem and a substantial improvement to Lov´asz’s upper bound on the number of halving lines (by Clarkson, Edelsbrunner, Guibas, Sharir, and Welzl, and by Dey, all of whom are computer scientists!) The relationship between discrete geometry and computational geometry proved to be mutually beneficial and resulted in remarkable breakthroughs on both sides. It was perhaps Ron Graham, who first suggested that the time was ripe to launch a journal devoted entirely to discrete geometry. He must have talked to some publishers, as in 1984, Cambridge University Press, Wiley, and Springer-Verlag all expressed interest in such a project. Ricky and Eli negotiated with all of them. Forty years ago, scientific publishing was a completely different business than what it is today. Almost all mathematics editors held PhDs. They were deeply embedded in the mathematics community, they had a good sense of scholarly quality and commercial value. Striking a balance between the two was, of course, crucial, but their primary goal was the advancement of science. Finally, the project was embraced by the late Walter Kaufmann-Bühler from Springer, about whom Ricky and Eli always spoke with the greatest admiration. Confident that the marriage of the classical subject of discrete geometry and the newly emerging field of computational geometry would be fruitful and long-lasting, they named the new journal Discrete & Computational Geometry (DCG). Through their seminar, the organization of numerous conferences, and their groundbreaking mathematical and editorial work, Ricky and Eli nurtured a thriving community that kept the subject and the journal as fresh and vibrant as ever. After all these years, it still fills me with pride to have contributed two papers to the inaugural issue of Discrete & Computational Geometry in 1986 and to have had the honor of serving for several years, alongside Ricky and Eli, as coeditor-in-chief of the journal they founded. Ricky and Eli made many important discoveries in discrete and computational geometry, convexity, geometric transversal theory, and real algebraic geometry. (Some of their achievements will be mentioned below, by Micha Sharir, Noga Alon, and Andreas Holmsen.) However, they

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Figure 3. Participants of the first Computational Geometry Conference at Bellairs Institute of McGill University, Holetown, St. James, Barbados, in 1986. Micha Sharir in the middle, Ricky above him, in red shirt, at the top.

always considered their most important legacy to be the creation of the journal and a large, friendly community of researchers around it. In their own ways, both of them were fundamentally social creatures, for whom mathematics, as Ricky’s son, Danny, once put it, was an “intensely social enterprise.” They were born in New York, and until the very last years of their lives, they both lived in New York. They could not imagine moving anywhere else. New York was their natural, cultural, and mathematical habitat. Shortly before his death, Eli completed his excellent novel, which has just appeared in print [5]. The protagonist is a professor of mathematics from New York City, who mysteriously disappears from his Manhattan apartment. From a short article published in the Times, one can learn that He was last seen there three weeks ago at a party in his honor, but failed to show up for his classes the following Monday. A police spokesman indicated that no signs of disturbance were found in his apartment and that no correspondence has turned up that might indicate his whereabouts. Eli and Ricky have disappeared from the New York scene, but their huge social and professional footprints are destined to linger for generations to come.

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Figure 4. Participants of the Monte Verità Conference on Discrete and Computational Geometry, in Ascona, Switzerland, 1999. In the middle, Ricky, wearing a red striped shirt. Eli is the 4th from the right.

A Tribute to Ricky Pollack and Eli Goodman Micha Sharir Ricky Pollack and Eli Goodman were, in many aspects, the founding fathers of discrete and computational geometry, as a thriving, active, and mainly interactive research area. Before turning to their scientific achievements, I would like to highlight two aspects of their leadership and influence on the field, which were not emphasized in the previous contribution. A. The Discrete and Computational Geometry Conference series. Computational Geometry was a young field in 1986 when Eli and Ricky launched the journal DCG. Discrete Geometry had been around for several decades, but bringing the two fields together was to a large extent the work of Eli and Ricky. Alongside the journal, they simultaneously launched a conference called Discrete and Computational Geometry, in Santa Cruz, CA, which had a huge impact in defining and directing the field, as a common discipline. It was so successful that they continued the tradition, by organizing two follow-up conferences, one at Mt. Holyoke, MA (Discrete and Computational Geometry Ten Years Later, 1996), and one at Snowbird, UT (Discrete and Computational Geometry Twenty Years Later, 2006). A fourth conference, Discrete and Computational Geometry Thirty Years Later, was organized (by others) at Monte Verita, Switzerland, in 2016. B. The books. Each of them contributed a major book in this area. Eli (with Joe O’Rourke) edited the Handbook of Discrete and Computational Geometry [6], a monumental 1500-page collection of papers surveying all aspects of the field. It later mushroomed into an even bigger collection (1937 pages), with Csaba Toth ´ as a third editor. Micha Sharir is a professor emeritus of computer science at Tel Aviv University, Tel Aviv, Israel. His email address is [email protected].

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Ricky was a coauthor, with Saugata Basu and MarieFran¸coise Roy, of another extremely influential book, Algorithms in Real Algebraic Geometry [2], which I will discuss later. The major networking ventures undertaken by Eli and Ricky had an enormous effect on the community, as did their technical contributions to discrete and computational geometry. They spanned many topics, including order types and allowable sequences, Helly-type results, algorithms in real algebraic geometry, and a variety of applications in computational geometry. A common thread in many of their works is the use of topological considerations in the analysis of structures in discrete geometry. Most notably, they looked for topological generalizations of standard concepts, such as pseudolines instead of lines, topological planes, and more. Perhaps the first example of such studies is their groundbreaking work on allowable sequences, which I will mention shortly. In the remainder of this note I would like to combine a brief review of some of the major achievements of Eli and Ricky with illustrations of how these have influenced my own research. I divide the discussion into four themes. 1. Allowable sequences and order types. Take a set 𝑃 of 𝑛 points in the plane, and project it onto a line ℓ. In general, we get a sequence 𝑃∗ of 𝑛 distinct points on ℓ. As we rotate ℓ, the sequence does not change combinatorially, except at certain critical orientations of ℓ, at which a block of consecutive elements of 𝑃 ∗ , or several blocks simultaneously, collapse into points and then are reversed. This evolution of 𝑃 ∗ as ℓ rotates is called an allowable sequence; see [11]. Many interesting properties of 𝑃 can be deciphered out of its allowable sequence, but the most intriguing question is whether a given allowable sequence (a sequence that obeys the evolution rule given above) is realizable, that is, whether it comes out of an actual set 𝑃 of 𝑛 points. The answer is that most sequences are not realizable, and that deciding whether a given sequence is realizable is PSPACE-complete (a computational complexity term, meaning, roughly, “intractable”). This however was not known at the time when they were working on the problem, and they were very excited about finding an effective solution to the decidability problem. As a matter of fact, when I first met Ricky in 1982, he enthusiastically gave me a “research announcement” (as these things were called those days) where he and Eli obtained an effective (albeit, sadly, wrong) solution. Let me switch to the related topic of order types, which generalizes these concepts to higher dimensions. For example, in the plane, the order type of a set 𝑃 of 𝑛 points specifies the orientation (left turn, right turn, or straight) of every ordered triple of points of 𝑃. Order types,

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introduced by Eli and Ricky, can be regarded as a concise purely discrete way of representing the “essence” of point configurations or, dually, of arrangements of lines (in the plane) or of hyperplanes (in higher dimensions). In fact, in this dual setting, order types can naturally be defined for arrangements of more general curves and surfaces, most notably for arrangements of pseudolines and pseudohyperplanes. Again, the question of the realizability of order types was a major topic of study. It was another manifestation of their interest in how basic discrete and combinatorial properties can be studied in a purely topological context, that had motivated many of their joint works. Let me mention a fairly fresh result (Comput. Geom., 2023), which is based on Eli and Ricky’s pioneering work on order types. We studied subquadratic algorithms for some 3Sum-hard geometric problems in the algebraic decision tree model. These are problems that are at least as hard as the 3Sum problem: determine whether a set of 𝑛 real numbers has a triple that sums to zero. These problems include collinearity testing, i.e., determining whether a set of 𝑛 points in the plane has a collinear triple. This problem is not known to have a subquadratic solution in the standard real-RAM model, and our work gave such a solution, for a restricted kind of collinearity testing (also known to be 3Sum-hard), in the algebraic decision tree model, where we only count algebraic sign tests involving the input data. We worked in the dual plane, where we have a set 𝐿 of 𝑛 lines (or other curves), and we want to preprocess the arrangement 𝒜(𝐿) for fast point location. This is of course a well known problem, which can be solved with 𝑂(𝑛2 ) storage and 𝑂(log 𝑛) query time, using line sweep and persistent search trees. However, to achieve this performance, one needs, among other things, to sort the vertices of 𝒜(𝐿) by their 𝑥-order, and each comparison in this sorting involves four input lines, two for each of the two vertices that are being compared. For our application, we wanted to obtain algebraic comparisons involving the input data that depend on a smaller number of elements, and for that it was crucial to reduce the number of lines involved in a comparison. The theory of order types was the tool that we needed. The order type information gives us the order of the vertices of 𝒜(𝐿) along each line of 𝐿, and each comparison that these sortings perform involves only three lines. This seemingly unimportant difference was crucial in improving the running time of our algorithm. This is just one, personal application, among many others, of the beautiful theory of order types; see [11]. 2. 𝑘-sets. Eli and Ricky’s papers on this topic have opened up a rich area of research on 𝑘-sets in configurations of points, and of levels in arrangements of curves and surfaces, in the plane and in higher dimensions. A 𝑘-set of a 1048

Figure 5. Ricky, Eli, and Peter McMullen at the AMS-IMS-SIAM Summer Research Conference on Discrete and Computational Geometry, Santa Cruz, CA, July 1986.

set 𝑃 of 𝑛 points in the plane, say, is a subset of size 𝑘 that can be cut off its complement by a half-plane. In a dual setting, the 𝑘-level in an arrangement of a set 𝐿 of 𝑛 lines (or other curves) in the plane, say, is the set of all vertices and edges of the arrangement 𝒜(𝐿) of 𝐿 that have exactly 𝑘 lines below them. What Eli and Ricky showed was that the number of at-most-𝑘-sets, namely the overall number of 𝑗-sets, for 𝑗 = 0, 1, … , 𝑘, is 𝑂(𝑛𝑘), which is an asymptotic worst-case tight bound; another proof with a tighter bound was given later by Alon and Gyori. ˝ As it turned out, this notion plays a crucial role in the analysis of randomized algorithms in computational geometry, and in many other computational and combinatorial problems in geometry; such as the celebrated probabilistic analysis technique of Clarkson and Shor. 3. Hadwiger-type theorems and geometric permutations. Hadwiger’s theorem gives a necessary condition for a finite collection of pairwise disjoint convex sets in the plane to have a line transversal (i.e., a line that crosses of all of them): If there is a linear ordering of the sets such that every triple of sets is met by a directed line in the corresponding order, then the entire collection has a line transversal. In a remarkable work, Eli and Ricky extended this result to arbitrary dimensions, giving a condition for the existence of a hyperplane transversal in terms of the multidimensional order type of the input sets, replacing the one-dimensional sorted order. Eli and Ricky were also interested in line transversals in higher dimensions; see [13]. A line transversal to any collection of disjoint convex sets meets all the sets in a given order or its reverse, depending on the direction of the transversal. This pair of orders (permutations) is called a geometric permutation. The study of geometric permutations, mainly to derive upper and lower bounds on the number of such permutations, took off from their pioneering work, and I have been involved in some of these

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studies. There are still many open challenges to better understand the structure of geometric permutations. 4. Algorithms in real algebraic geometry. A real semialgebraic set is a region in ℝ𝑑 that is defined by a Boolean combination of a finite number of polynomial equalities and inequalities. Given such a set 𝐴, how can we process it algorithmically? How do we determine whether 𝐴 is empty? Compute its connected components? Find a point in each connected component? These and many other basic problems in Computational Real Algebraic Geometry have been studied by Ricky, together with his colleague MarieFran¸coise Roy and his student Saugata Basu. This work culminated in their monumental book Algorithms in Real Algebraic Geometry [2], which has become, in a sense, the bible of this area, containing all the basic tools, techniques, and algorithms in computational real algebra and algebraic geometry. It is remarkable that the book is publicly available via open access, as demanded by the authors. I would like to finish by mentioning some of my joint work with Eli and Ricky, most of which are only with Ricky. An exception is a work with both of them on the space of hyperplane transversals to a family of 𝑛 separated and strictly convex sets in ℝ𝑑 , where we show that the maximum combinatorial complexity of this space is Θ (𝑛⌊𝑑/2⌋ ). A main feature of the analysis, mostly contributed by Eli and Ricky, is the analysis of the topology of the space of common tangents, of a special kind, of a collection of such sets. It is yet another manifestation of their interest in studying topological aspects of discrete geometry. Of my work with Ricky, I would like to mention one on counting and cutting cycles of lines in space. This was a notoriously difficult problem, which was solved much later, where the goal was to break the lines in a set of 𝑛 lines in ℝ3 into the smallest (or, at least, a small) number of pieces, so as to eliminate all the depth cycles between them. The newly derived upper bound is close to 𝑛3/2 , which is nearly tight in the worst case. However, back when the paper with Ricky appeared, only very partial results were known. The paper was accompanied by a paper of Pach, Pollack and Welzl, in which they showed that a 4 × 4 pattern of lines in space cannot be completely weaving, namely that it is impossible for each line to alternate between passing above and below the lines in the other set in order. To experiment with this finding, they went out to buy some toy sticks to physically test how they can weave. Without thinking too much, they naturally bought 4 × 4 = 16 sticks. . . . Several other works with Ricky are on quasi-planar graphs, on arrangements of Jordan arcs with three intersections per pair, and various problems on simple polygons. All this goes to show that Ricky and Eli were very curious

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and open-minded, and were interested in basically everything. Working with them was fun and very inspiring. The community at large, and I personally, are still reeling from the loss of both of them, and sorely miss their leadership and great science, not to mention friendship.

Eli Goodman, Ricky Pollack, and Semivarieties Noga Alon The friendship and collaboration between Eli Goodman and Ricky Pollack has been rare and productive, spanning decades of joint work and including the foundation of a leading journal and the organization of meetings and an active research seminar. Their joint papers stimulated a considerable amount of follow-up work. In this section, we focus on one of their beautiful contributions and describe some of its many subsequent developments. The basic idea appears in a remarkable short note [9], where Goodman and Pollack observed that a theorem of Milnor in real algebraic geometry can be used to provide an elegant nearly tight asymptotic estimate for the number of (simplicial) polytopes with 𝑛 vertices in ℝ𝑑 . Their approach paved the way to a significant amount of additional results in combinatorics, discrete and computational geometry, and related areas, obtained by applying powerful tools from real algebraic geometry. It is natural to speculate that the background of Goodman in algebraic geometry demonstrated by his influential early work in the subject [3] helped in the early development of this approach. Below I describe the background to Goodman and Pollack’s work on convex polytopes, followed by their results, and two research directions inspired by their work. 1. Connected components and sign patterns. There are several results that provide upper bounds for the number of connected components of real varieties or semivarieties. Following such estimates by Ole˘ınik–Petrovski (1949), Milnor (1964), and Thom (1965), Warren [19] proved that if 𝑚 ≥ ℓ ≥ 2, and {𝑃𝑖 (𝑥1 , … , 𝑥ℓ ), 1 ≤ 𝑖 ≤ 𝑚} is a set of 𝑚 polynomials of degree at most 𝑘 in ℓ real variables, then the number of connected components of the semivariety 𝑉 = {(𝑥1 , … , 𝑥ℓ ) ∈ ℝℓ , 𝑃𝑖 (𝑥1 , … 𝑥ℓ ) ≠ 0 for all 1 ≤ 𝑖 ≤ 𝑚} is at most (4𝑒𝑘𝑚/ℓ)ℓ . Noga Alon is a professor of mathematics at Princeton University, Princeton, New Jersey, and Tel Aviv University, Tel Aviv, Israel. His email address is [email protected].

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For each point 𝑥 = (𝑥1 , 𝑥2 , … 𝑥ℓ ) ∈ 𝑉, the sign pattern of the polynomials 𝑃𝑖 at the point 𝑥 is the vector (sign(𝑃1 (𝑥)), … , sign(𝑃𝑚 (𝑥))) ∈ {−1, 1}𝑚 . Let 𝑠(𝑃1 , … , 𝑃𝑚 ) denote the total number of distinct sign patterns of the polynomials 𝑃𝑖 , as 𝑥 ranges over all points of 𝑉. Since the sign of each polynomial cannot change in any connected component of 𝑉, it follows that for 𝑚, 𝑘, and ℓ as above, 𝑠(𝑃1 , … , 𝑃𝑚 ) ≤ (4𝑒𝑘𝑚/ℓ)ℓ . In applications it is sometimes desirable to bound the number of sign patterns (sign(𝑃1 (𝑥)), … , sign(𝑃𝑚 (𝑥)) ∈ {−1, 0, 1}𝑚 , where here 𝑥 ranges over all points of ℝℓ (including those in which some of the polynomials 𝑃𝑖 vanish). It is not difficult to show (see [1]) that the result of Warren implies that this number does not exceed (8𝑒𝑘𝑚/ℓ)ℓ . 2. Counting polytopes and configurations. Let 𝑐(𝑛, 𝑑) denote the number of (combinatorial types of) 𝑑polytopes on 𝑛 labeled vertices and let 𝑐𝑠 (𝑛, 𝑑) denote the number of simplicial 𝑑-polytopes on 𝑛 labeled vertices (that is, polytopes in which all facets are simplices). The problem of determining or estimating these two functions (especially for 3-polytopes) has been the subject of much effort and frustration of nineteenth-century geometers as described, for example, in the book Convex Polytopes by Grünbaum. Despite these efforts, for any 𝑑 ≥ 4 (and large 𝑛) the best known upper bound for both 𝑐𝑠 (𝑛, 𝑑) and 𝑐(𝑛, 𝑑) has been exponential in 𝑛⌊𝑑/2⌋ log 𝑛. This estimate follows from the upper bound theorem for convex polytopes. The remarkable result of Goodman and Pollack [8, 9] improved it dramatically to a bound exponential in 𝑑2 𝑛 log 𝑛. They started by bounding the number of order types of configurations of 𝑛 (labeled) points in ℝ𝑑 , defined in what follows. If (𝑝0 , 𝑝1 , … , 𝑝𝑑 ) is a sequence of 𝑑 +1 points in ℝ𝑑 , with 𝑝𝑖 = (𝑥𝑖1 , … , 𝑥𝑖𝑑 ) for each 𝑖, we say they have a positive orientation if the determinant of the matrix (𝑥𝑖𝑗 )0≤𝑖,𝑗≤𝑑 where 𝑥𝑖0 = 1 for each 𝑖, is positive. If the determinant is negative they have a negative orientation, and if the determinant is 0 they lie on a common hyperplane. The order type of a configuration 𝐶 of 𝑛 labeled points 𝑝1 , 𝑝2 , … , 𝑝𝑛 in ℝ𝑑 is a function 𝑤 from the set of all (𝑑 + 1)-subsets of [𝑛] = {1, 2, … , 𝑛} to {0, ±1}, where for 𝑆 = {𝑖0 , 𝑖1 , … , 𝑖𝑑 } with 1 ≤ 𝑖0 < 𝑖1 < … < 𝑖𝑑 ≤ 𝑛, 𝑤(𝑆) is +1, −1 or 0 according to the orientation of the points 𝑝𝑖0 , … , 𝑝𝑖𝑑 . Let 𝑡(𝑛, 𝑑) denote the number of distinct order types of configurations of 𝑛 labeled points in ℝ𝑑 . Note that 𝑡(𝑛, 𝑑) is the number of sign patterns of ( 𝑛 ) polynomials of 𝑑+1 degree 𝑑 in the 𝑑𝑛 real variables (𝑥𝑖1 , … , 𝑥𝑖𝑑 ), 𝑖 = 1, … , 𝑛, which are the coordinates of the points. The polynomials are just the determinants det(𝑥𝑖𝑘 𝑗 ), 0 ≤ 𝑘, 𝑗 ≤ 𝑑, where 𝑥𝑖𝑘 0 = 1 for all 𝑘 and 1 ≤ 𝑖0 < 𝑖1 < ⋯ < 𝑖𝑑 ≤ 𝑛. Therefore, the estimate of Warren (and its slight exten-

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sion for the total number of sign patterns) shows that 2 𝑡(𝑛, 𝑑) ≤ 𝑛(1+𝑜(1))𝑑 𝑛 . This immediately supplies a similar bound for the number 𝑐(𝑛, 𝑑) of 𝑑-polytopes on 𝑛 points. Indeed, the order type of a configuration that spans ℝ𝑑 determines which sets of its points lie on supporting hyperplanes of its convex hull. Hence, the order type of a configuration on a set of 𝑛 points in ℝ𝑑 which is the set of vertices of a convex polytope 𝑃 determines its facets and its complete combinatorial type. 3. Signrank. The sign-pattern of an 𝑚 by 𝑛 real matrix 𝐴 with nonzero entries (𝑎𝑖𝑗 )1≤𝑖≤𝑚,1≤𝑗≤𝑛 is an 𝑚 by 𝑛 matrix 𝑍(𝐴) = (𝑧𝑖𝑗 ) of 1, −1 entries where 𝑧𝑖𝑗 = sign 𝑎𝑖𝑗 . For an 𝑚 by 𝑛 matrix 𝑍 of 1, −1 entries, let 𝑟(𝑍) denote the minimum possible rank of a matrix 𝐴 such that 𝑍(𝐴) = 𝑍. Define 𝑟(𝑛, 𝑚) = max{𝑟(𝑍) ∶ 𝑍 is an 𝑚 by 𝑛 matrix over {1, −1}}. The problem of determining or estimating 𝑟(𝑛, 𝑚), and in particular 𝑟(𝑛, 𝑛), was raised by Paturi and Simon in the early 80s, motivated by the study of the so-called unbounded-error probabilistic communication complexity of a Boolean function of 𝑛 + 𝑛 bits. Alon, Frankl, and Rödl (cf. [1]) proved in 1985 that 𝑛 𝑛 ≤ 𝑟(𝑛, 𝑛) ≤ + 3√𝑛 16 2 2 and that if 𝑚/𝑛 → ∞ and (log2 𝑚)/𝑛 → 0 then 1 𝑟(𝑛, 𝑚) = ( + 𝑜(1)) 𝑛. 2 The lower bounds in both estimates are derived from the estimate of Warren by a simple counting argument. 4. Semialgebraic properties. A graph property is any family of graphs ℱ closed under isomorphism. Such a family is called semialgebraic if every vertex is a point in a real space of bounded dimension, and the adjacency of two vertices is determined by the signs of a finite set of bounded degree polynomials in the coordinates of the corresponding points. This can be extended to hypergraphs, but for simplicity we focus here on the case of graphs. Natural special cases of such properties are intersection graphs of simple geometric objects, like segments or disks in the plane, boxes in ℝ3 and more. The speed of a family ℱ is the function 𝑓(𝑛) = |ℱ𝑛 |, where ℱ𝑛 is the set of all graphs with 𝑛 vertices in the family. The results about the number of sign patterns of polynomials described here imply that the speed 𝑓(𝑛) of any semialgebraic family of graphs satisfies 𝑓(𝑛) ≤ 2𝑐𝑛 log 𝑛 , where 𝑐 = 𝑐(ℱ) is a constant that depends on the dimension and the degrees of the polynomials in the definition of the property. Examples can be found in [1] and the references therein. A result of Sauermann [17] shows that under mild conditions the estimate obtained for the constant 𝑐 = 𝑐(ℱ) by applying Warren’s theorem is tight. Besides their modest speed functions, it turns out

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that semialgebraic graph properties are simpler than general families of graphs in many respects. The study of their Ramsey properties and the investigation of additional extremal questions for such families received a considerable amount of attention in the last decade. It will surely keep being the subject of future research, like other topics initiated by the work of Eli Goodman and Ricky Pollack.

Eli Goodman, Ricky Pollack, and Geometric Transversals Andreas Holmsen One branch of combinatorial geometry in which the work of Eli and Ricky had a tremendous impact is what we call geometric transversal theory. This line of research, an offshoot of Helly’s theorem, was initiated in the 1930s by Vincensini and Santalo, ´ and explored further in the 50s and 60s by a number of prominent geometers such as Grünbaum, Hadwiger, Klee, and Danzer. The famous survey, “Helly’s theorem and its relatives,” gives a detailed account of the state of affairs in 1963, and motivated further study throughout the 70’s and 80’s. Eli and Ricky’s 1988 paper “Hadwiger’s transversal theorem in higher dimensions” stands as one of the milestones of geometric transversal theory. Their beautiful result related the (at the time) novel notion of order types to the classical study of geometric transversals, and paved the way for research directions in discrete and computational geometry which bear fruit to this day. The study of geometric transversals originated with Helly’s theorem, which asserts that for a family of at least 𝑑 + 1 compact convex sets in ℝ𝑑 , if every 𝑑 + 1 members can be intersected by a point, then the entire family can be intersected by a point. Can a similar theorem be true if the property “intersected by a point” is replaced by “intersected by a line,” or a plane, or more generally a 𝑘dimensional affine flat? This was the problem, posed by Vincensini in 1935, that initiated the study of geometric transversals, but it did not take long before Santalo´ realized that no such “Helly-type” theorem can exist for 𝑘-flats when 𝑘 > 0. While this situation may seem somewhat discouraging, it did not prevent further study of geometric transversals. Indeed, Santalo´ showed that if we restrict ourselves to families of axis parAndreas Holmsen is a professor of mathematics in the Department of Mathematical Sciences, KAIST, Daejeon, South Korea, and is a member of the Discrete Mathematics Group, Institute for Basic Science (IBS), Daejeon, South Korea. His email address is [email protected].

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1 2

4 3

Figure 6. The line transversal induces the ordering 1 ≺ 2 ≺ 3 ≺ 4.

allel boxes in ℝ𝑑 , then the “Helly number” for line transversals becomes 2𝑑−1 (2𝑑 − 1), and for hyperplane transversals it becomes 2𝑑−1 (𝑑 + 1). In fact, there has been extensive work on geometric transversals which investigates Hellytype theorems under various restrictions on the geometric shapes of the sets in the family. Rather than focusing on the geometric shape of the sets, Hadwiger’s approach has a more combinatorial flavor. Suppose a finite family of pairwise disjoint convex sets admits a line transversal. By orienting this line, it induces an ordering on the family, namely the order in which it meets the sets. In particular, any three members of the family are met by a line which is consistent with the given ordering. (See Figure 6.) What Hadwiger showed is that, in the plane, this obvious necessary condition is also sufficient: Theorem. A finite family 𝐹 of pairwise disjoint convex sets in the plane admits a line transversal if and only if there exists a linear ordering of ℱ such that every three members of ℱ are met by a line consistent with the ordering. By the mid 1980’s, Eli and Ricky had been investigating order types of point configurations in ℝ𝑑 for several years, when it dawned on them that the linear ordering in Hadwiger’s transversal theorem was simply a one-dimensional order type. They noticed that by making a bijection between a finite set and a point configuration in ℝ𝑘 , the order type of the point configuration induces what they called a 𝑘-ordering of the set, which for 𝑘 = 1 is precisely a linear ordering. This meant that they had just the right tool to generalize Hadwiger’s transversal theorem to higher dimensions! All that was missing was the right analog of pairwise disjointness, and the natural condition they found was to define a family of at least 𝑘 + 1 convex sets in ℝ𝑑 to be (𝑘 − 1)-separated if no 𝑘 + 1 of them admit a (𝑘 − 1)transversal. In particular, being 0-separated means that no two members have a 0-transversal, i.e., the members are pairwise disjoint. A consequence of the definition is the following: If a (𝑘 − 1)-separated family of convex sets in ℝ𝑑 admits a 𝑘-transversal, then by choosing one point

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from each set within the given 𝑘-transversal, we obtain a point configuration in ℝ𝑘 , and the order type of this configuration is independent of the choice of points. In this way, a 𝑘-transversal naturally induces a 𝑘-ordering of the family. These observations led Eli and Ricky to their celebrated generalization of Hadwiger’s theorem: Theorem ([10]). A finite (𝑑 −2)-separated family ℱ of convex sets in ℝ𝑑 admits a hyperplane transversal if and only if there is a (𝑑 − 1)-ordering of ℱ such that every 𝑑 + 1 members of ℱ are met by a hyperplane consistent with the (𝑑 − 1)-ordering. Another important development was made a couple years later by Wenger, a PhD student of Ricky’s, who was able to remove the disjointness assumption from Hadwiger’s transversal theorem. This requires a clarification of what it means for a line to intersect a family of convex sets consistently with an ordering, since such an ordering may no longer be uniquely determined. Wenger showed that it suffices for some choice of points to be consistent with the ordering, and shortly after, Pollack and Wenger extended this to higher dimensions as well. By an elegant proof from “the book,” which combines order types, combinatorial convexity, and the Borsuk–Ulam theorem, they proved what we now call the Goodman–Pollack–Wenger theorem: Theorem ([16]). A finite family ℱ of convex sets in ℝ𝑑 admits a hyperplane transversal if and only if for some 𝑘, 0 ≤ 𝑘 < 𝑑, there is a 𝑘-ordering of ℱ such that every 𝑘 + 2 of the sets are met by some 𝑘-flat consistent with the 𝑘-ordering. Over the years there have been a number of further generalizations and extensions of Eli and Ricky’s breakthrough result. Some of the highlights include: ▹ Anderson and Wenger (1996): Replaces the 𝑘ordering by the more general concept of an acyclic oriented matroid. ▹ Arocha et al. (2003): Shows that not only do we get a single hyperplane transversal, but in fact “many” of them, captured by what they call a virtual 𝑘-transversal. ▹ Arocha et al. (2008): Gives a colorful version of Hadwiger’s transversal theorem in the spirit of the B´ar´any– Lov´asz “colorful Helly theorem.” ▹ Cheong et al. (2023): Proves the colorful version of the Goodman–Pollack–Wenger theorem conjectured by Arocha et al. ▹ McGinnis (2023): Establishes an analog of the Goodman–Pollack–Wenger theorem for hyperplane transversal in ℂ𝑑 . For nearly two decades, Eli and Ricky (with various collaborators) continued working on geometric transversals, exploring Helly-type theorems, the topological structure 1052

and combinatorial complexity of the space of transversals, and convexity on the affine Grassmannian. Their survey [13] joint with Wenger, documents the explosion of work in geometric transversal theory that had taken place in the years following their breakthrough papers on the generalizations of Hadwiger’s transversal theorem. In the paper “Foundations of a theory of convexity on affine Grassmann manifolds,” Eli and Ricky asked whether there is a convex hull operator, conv𝑘 (⋅), on the space of 𝑘dimensional affine flats in ℝ𝑑 , which naturally extends the standard convex hull operator for points, and satisfies general properties such as: monotonicity, idempotence, antiexchange, and affine invariance. (For precise definitions see [12] or the expository article [4].) Their solution was as natural as the question: Fix an integer 0 ≤ 𝑘 < 𝑑. For a set ℒ of 𝑘-flats in ℝ𝑑 , define its dual, ℒ∗ , to be the family of all convex (point) sets which meets every flat in ℒ. For a family ℱ of convex (point) sets, define its dual, ℱ ∗ , to be the set of all 𝑘-transversals to the family ℱ. Now define the convex hull of a set ℒ of 𝑘-flats in ℝ𝑑 to be its double dual, that is, conv𝑘 ℒ = ℒ∗∗ . It turns out that this notion of convexity satisfies the four properties stated above, and indeed, when restricted to the case 𝑘 = 0, it reduces to the standard convexity. For 𝑘 > 0 a rich theory emerges which is closely tied to central questions in geometric transversal theory, and their paper explores some interesting examples ranging from rulings on a hyperboloid to certain Schubert varieties. In fact, many sophisticated constructions and counterexamples in geometric transversal theory can be traced back to this convexity structure. Today geometric transversal theory is an active area of research. In the last decade, we have witnessed an emergence of new and exciting directions motivated by recent trends in discrete and computational geometry, as well as developments on research problems dating back to Eli and Ricky’s seminal work in the area. References

[1] Noga Alon, Tools from higher algebra, Handbook of combinatorics, Vol. 1, 2, Elsevier Sci. B. V., Amsterdam, 1995, pp. 1749–1783. MR1373688 [2] Saugata Basu, Richard Pollack, and Marie-Fran¸coise Roy, Algorithms in Real Algebraic Geometry, 2nd ed., Algorithms and Computation in Mathematics, vol. 10, Springer-Verlag, Berlin, 2006. MR2248869 [3] Jacob E. Goodman, Affine open subsets of algebraic varieties and ample divisors, Ann. of Math. (2) 89 (1969), 160–183, DOI 10.2307/1970814. MR242843 [4] Jacob E. Goodman, When is a set of lines in space convex?, Notices Amer. Math. Soc. 45 (1998), no. 2, 222–232. MR1601812

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[5] Jacob E. Goodman, The Mathematician, FriesenPress, Altona, Canada, 2023. [6] Jacob E. Goodman and Joseph O’Rourke (eds.), Handbook of Discrete and Computational Geometry, 2nd ed., Discrete Mathematics and its Applications (Boca Raton), Chapman & Hall/CRC, Boca Raton, FL, 2004, DOI 10.1201/9781420035315. MR2082993 [7] Jacob E. Goodman and Richard Pollack, Proof of Grünbaum’s conjecture on the stretchability of certain arrangements of pseudolines, J. Combin. Theory Ser. A 29 (1980), no. 3, 385–390, DOI 10.1016/0097-3165(80)90038-2. MR600606 [8] Jacob E. Goodman and Richard Pollack, Upper bounds for configurations and polytopes in 𝐑𝑑 , Discrete Comput. Geom. 1 (1986), no. 3, 219–227, DOI 10.1007/BF02187696. MR861891 [9] Jacob E. Goodman and Richard Pollack, There are asymptotically far fewer polytopes than we thought, Bull. Amer. Math. Soc. (N.S.) 14 (1986), no. 1, 127–129, DOI 10.1090/S0273-0979-1986-15415-7. MR818067 [10] Jacob E. Goodman and Richard Pollack, Hadwiger’s transversal theorem in higher dimensions, J. Amer. Math. Soc. 1 (1988), no. 2, 301–309, DOI 10.2307/1990918. MR928260 [11] Jacob E. Goodman and Richard Pollack, Allowable sequences and order types in discrete and computational geometry, New Trends in Discrete and Computational Geometry, Algorithms Combin., vol. 10, Springer, Berlin, 1993, pp. 103– 134, DOI 10.1007/978-3-642-58043-7_6. MR1228041 [12] Jacob E. Goodman and Richard Pollack, Foundations of a theory of convexity on affine Grassmann manifolds, Mathematika 42 (1995), no. 2, 305–328, DOI 10.1112/S0025579300014613. MR1376730 [13] Jacob E. Goodman, Richard Pollack, and Rephael Wenger, Geometric transversal theory, New Trends in Discrete and Computational Geometry, Algorithms Combin., vol. 10, Springer, Berlin, 1993, pp. 163–198, DOI 10.1007/978-3-642-58043-7_8. MR1228043 [14] Branko Grünbaum, Arrangements and Spreads, Conference Board of the Mathematical Sciences Regional Conference Series in Mathematics, No. 10, American Mathematical Society, Providence, RI, 1972. MR307027 [15] J´anos Pach, Rom Pinchasi, and Micha Sharir, Solution of Scott’s problem on the number of directions determined by a point set in 3-space, Discrete Comput. Geom. 38 (2007), no. 2, 399–441, DOI 10.1007/s00454-007-1344-5. MR2343314 [16] Richard Pollack and Rephael Wenger, Necessary and sufficient conditions for hyperplane transversals, Combinatorica 10 (1990), no. 3, 307–311, DOI 10.1007/BF02122783. MR1092546 [17] Lisa Sauermann, On the speed of algebraically defined graph classes, Adv. Math. 380 (2021), Paper No. 107593, 55, DOI 10.1016/j.aim.2021.107593. MR4205109 [18] Peter Ungar, 2𝑁 noncollinear points determine at least 2𝑁 directions, J. Combin. Theory Ser. A 33 (1982), no. 3, 343– 347, DOI 10.1016/0097-3165(82)90045-0. MR676751

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[19] Hugh E. Warren, Lower bounds for approximation by nonlinear manifolds, Trans. Amer. Math. Soc. 133 (1968), 167– 178, DOI 10.2307/1994937. MR226281

Janos ´ Pach

Micha Sharir

Noga Alon

Andreas Holmsen

Credits

Figure 1 is courtesy of Archives of the Mathematisches Forschungsinstitut Oberwolfach. Figure 2, Figure 3, Figure 5, and photo of J´anos Pach are courtesy of J´anos Pach. Figure 4 is courtesy of Emo Welzl. Figure 6 and photo of Andreas Holmsen are courtesy of Andreas Holmsen. Photo of Micha Sharir is courtesy of Micha Sharir. Photo of Noga Alon is courtesy of Noga Alon.

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BOOK REVIEW

The Mathematician Reviewed by Thomas Garrity The Mathematician FriesenPress, 2023, 276 pp.

By Jacob E. Goodman One summer day in 2008, a younger colleague of mine from the Williams Chemistry Department looked somewhat frazzled. When asked why, she explained that her tenure packet was due the following Friday, and that she was desperately trying to finish two more papers, so that on her vita the papers could be listed as “submitted.” This made perfect sense to me. Only in December 2008 when the good news came that this colleague did indeed get tenure (a decision that I considered a no-brainer), did I briefly rethink our summer conversation. First, there is no way that having two more papers listed as submitted would push anyone over the tenure bar. But also, what if this colleague had only submitted these papers a few days later, after the tenure packet was due, but still listed them as submitted? Who would have ever known? As far as cutting corners go, it would not have been a huge deal. But at the time, my chemistry colleague never even considered this as an option. Suppose, though, we were not in the summer of 2008 in Massachusetts but instead in 1937 Germany, and the young academic was not a chemist but a young mathematician who is socially awkward and self-centered with a Jewish sounding name (and possibly with actual but hidden Thomas Garrity is the Webster Atwell Class of 1921 Professor of Mathematics at Williams College. His email address is [email protected]. Communicated by Notices Book Review Editor Emily Olson. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti3002

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Jewish roots). What would be mere “cutting corners” today could have had life and death consequences then. We are in the world of this novel. This novel is an historical mystery. Like most mysteries, part of the charm is not the mystery itself but the mileu in which the mystery occurs. The author, the late Jacob Goodman (see the memorial tribute in this issue), was most definitely a serious research mathematician, and hence this mystery is set in the world of academic mathematics. More specifically, the mystery shifts in the first half between the Columbia University math department of 1967 and German academe of the 1930s while the second half shifts between the math worlds of the 1930s and 1990s. The mystery is about the sudden disappearance in 1967 of Claus Eisenstadt, in the evening after a small congratulatory party in honor of his receiving a type of “lifetime achievement” award from his long-term school Columbia. We quickly learn that Eisenstadt arrived at Columbia as a Jewish refugee from Germany during the early years of World War II, but before Pearl Harbor. His reputation stemmed from two papers written right before his departure from Germany, papers that were critical in the development of algebraic topology. These papers were significantly different then his few earlier papers, which were simply not of the same caliber. In the ensuing years, Eisenstadt publishes a few more significant papers, usually with a coauthor, but overall his career does not live up to the promise hinted at in the two topology papers. We also quickly learn that Eisenstadt is quite eccentric, and not the lovable, charming type. In fact, in the very first paragraph of the novel, at the small party in his honor, we see him smashing the camera of a young journalism student who had the audacity of both trying to take his picture and to interview him about his award for the student paper. And the very next day, Eisenstadt was gone. As would be expected, this type of behavior leads to intense interest and gossip in the Columbia math department. In particular, Judy Carter, a second-year graduate

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Book Review student, becomes almost obsessed (within reason) with understanding who Eisenstadt really was; where he went; and why he vanished. Judy to a large extent represents a typical mathematician. She was the star math student as an undergraduate at a small school in the Midwest. On arrival at graduate school, she quickly finds herself at home in the cultural and intellectual hub of the Upper West Side of Manhattan around Columbia. By her second year, she starts to hear the siren song of the Grothendieck revolution in algebraic geometry. In the second part of the book, when we shift from 1967 to the midnineties, we see her as a successful midcareer research mathematician, long tenured at City University, with the standard sort of informal network of colleagues spread throughout the world. And it is suggested her success in mathematics is linked to certain obsessive tendencies, as she still wants to crack the mystery of Eisenstadt. We also get to see the mathematical journey of Eisenstadt in 1930s Germany, from his undergraduate days to what would now be called postdoc years. There are small intense seminars and interactions with “big” names such as Felix Hausdorff, Emil Artin (his thesis advisor), and Emmy Noether. There was a continual emphasis by his mentors and professors on the creation of new ideas, leading to understandable pressure on Eisenstadt to show that not only could he synthesize existing mathematics (even at a high level) but he could also discover new mathematics. This pressure is still present today. In fact, I seem to recall the graduate school insult (biting because of how true it often was) that someone was merely a “good student,” with the implication of course that they could learn but not create mathematics. There also appear other easily recognizable mathematical character types. In the 1967 world, there is a RussianJewish tenured math professor at Columbia. He leads a charmingly intense and decent intellectual life in New York City, which differs drastically from the life he would have led in the USSR had his family not escaped when he was a child. We are also introduced to a graduate student who flits from math topic to math topic and eventually decides to leave the PhD program. All of us had friends like this in graduate school. Unlike most though, this person eventually becomes a private detective (this is a mystery novel after all). There are other issues that naturally arise. For example, most readers of the Notices spend their lives in a math department full of interesting people, some of whom are quite eccentric. (Unlike the fictional Eisenstadt, almost all of the math eccentrics that I know are of the endearingly quirky type). But do any of us really know any of our colleagues? At the last Williams math department meeting, after reading this novel for the second time, I found myself looking at the other members of the department, SEPTEMBER 2024

many of whom I have known for decades, and wondering how many of them had dark secrets hidden from public view. (To be clear, I quickly concluded that the answer was “none”). I will include here a small warning that the novel contains a few explicit sex scenes that do not really fit with the rest of the book. These few scenes could also potentially get an American high school teacher in trouble if the book is recommended to nonadult students. The novel overall is a great read, one which will hold your interest from the first paragraph’s camera smashing incident to the exciting culmination at the end. I expect your nonmath friends will enjoy it. But Notices readers will enjoy it even more, seeing how the mystery unfolds in the world that we know so well.

Thomas Garrity Credits

Book cover is courtesy of Naomi Goodman. Photo of Thomas Garrity is courtesy of L. Pedersen.

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Freeman Hrabowski: Advocate for Mathematics and STEM Visionary Christian Anderson Introduction Science, Technology, Engineering, and Mathematics (STEM) education has been identified as a key factor to the United States’ national security (Athanasia and Cota, 2022). The country’s ability to produce STEM literate citizens and STEM professionals has received national attention over the past 7 to 10 decades. On the national landscape, Dr. Freeman Hrabowski has emerged as a STEM visionary, leading STEM education programs and producing a diverse cadre of STEM professionals. After a notable tenure as the President of University of Maryland Baltimore County (UMBC), Dr. Hrabowski created a blueprint and method to recruit, educate, and graduate STEM professionals, many of whom will go on to earn PhDs. The full impact of the legacy of Dr. Freeman Hrabowski is yet to be seen. His impact on the STEM community is still being measured as he continues to promote excellence in STEM education and advocates for diversity and representation. As a child growing up in Birmingham, Alabama, he advocated for African Americans’ rights in the face of unfair treatment in the Jim Crow South. As an adult, he became an advocate for diverse college students with academic promise to gain access to a rigorous and quality education that focused on STEM. Christian Anderson is an associate professor of mathematics education at Morgan State University. His email address is christian.anderson@morgan .edu. Communicated by Notices Associate Editor Asamoah Nkwanta. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti3016

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As we continue to celebrate the accomplishments of Dr. Hrabowski, we should acknowledge the unique role that the subject of mathematics played in his life and how his love for mathematics shaped his approach to creating innovative STEM programming, most notably, the Meyerhoff Scholars Program. The Meyerhoff Scholars Program is an undergraduate scholarship program at the University of Maryland Baltimore County (UMBC) which is designed to increase diversity among future leaders in STEM who have a desire to pursue a PhD or combined MD/PhD in a STEM field of study. To date, the Meyerhoff Scholars Program has approximately 1400 alumni with over 300 students currently enrolled in graduate and professional programs across the country. Moreover, there are Meyerhoff alumni who are faculty at prestigious universities (e.g., Harvard) or working as research scientists at top government agencies such as the National Institute of Allergy and Infectious Diseases and the National Institute of Health (NIH). Most notably, Meyerhoff Alumna, Kizzmekia Corbett, served as the leader of the team that developed Moderna’s COVID vaccine. UMBC leads the nation among predominately white institutions (PWI) who produce Black undergraduates that go on to earn PhDs in the natural sciences, life sciences, engineering, computer science, and mathematics. As we know, mathematics serves as the gateway to STEM fields. Unfortunately, mathematics is often viewed as a difficult subject and is often identified as the reason students leave STEM programs (Li & Schoenfeld, 2019). In addition, in many cases, mathematics, is taught in a way that does not promote creativity or invite diverse populations. Dr. Hrabowski’s love for mathematics and insights about the teaching of mathematics enabled him to train and inspire countless numbers of scientists,

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engineers, and other STEM professionals. Based on his own experience learning mathematics, he believes that mathematics should be taught in the context of the students who are learning it. Moreover, Hrabowski envisioned a program where learning takes place collaboratively. In his model, students are assigned to cohorts and taught how to learn and work together. This approach to learning has been discussed within scholarly literature. McCallum (2018) identified a similar approach to teaching mathematics known as making-sense, where the manner in which the students experience mathematics is just as important as their acquisition of the mathematical content. Hrabowski’s approach to STEM education with the Meyerhoff program created opportunities for students of color and females to engage in rigorous STEM coursework. Elements of the making-sense approach are evident in the Meyerhoff program. Like the approaches described by Schoenfeld et al. (2018), Hrabowski inspires and ignites his students’ love of mathematics and science to ascend to new heights of scholarship and research. Since mathematics is the gateway to advanced coursework in K-12 that leads to matriculation as STEM majors in college, the mathematics community should reexamine the way the subject is taught. Delvin (2000) argues that mathematics has four (4) different faces: 1) computation, formal reasoning, and problem solving, 2) a way of knowing, 3) a creative medium, and 4) applications. In a series of interviews, I met with several alumni of the Meyerhof Scholars Program to discuss their experiences in the program and to reflect upon the impact that Dr. Hrabowski had on their lives both personally and professionally. During my interview with Dr. Hrabowski, he shared his experiences as a student of mathematics, and he discussed how those experiences shaped his approach when designing STEM programs, consulting with STEM based organizations, and serving as a mentor to STEM professionals across the country.

An Advocate for Mathematics Early exposure to mathematics—“You learn to do, by doing”. In recent years, out of school learning experiences have been linked to increased student achievement (NeherAsylbekov and Wagner, 2022). Providing students with opportunities to link content knowledge with real world experiences outside of school allows students to deepen their conceptual understanding of the concepts being taught and increases their self-confidence to engage in the content. This was the case with Dr. Hrabowski. His approach to learning mathematics was rooted in a mathematics-rich home environment that was created by his parents.

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At an early age, Freeman Hrabowski’s love for mathematics was fostered and developed by routines and structures established in his home. While he was academically gifted, he developed a work ethic that deepened his love and appreciation for mathematics. His mother, an English teacher who was inspired by the New Math movement of the late 1950’s, believed that strong reading comprehension skills led to an ability to solve mathematical word problems. As a child, Hrabowski enjoyed solving problems that his parents presented to him on a regular basis. Hrabowski explains, “I found that the more problems I did, the more I understood concepts and the more fun I was having. That was happening with the math. . . .” Additionally, he recalls one of his mother’s fundamental beliefs when it came to learning mathematics, “you learn to do, by doing.” This mindset of engagement and repetition served as his mental foundation as he progressed through his educational journey. His continuous engagement with content (mainly with mathematics) increased his confidence as a student. As he entered new educational spaces, he was confident in his ability to engage with new learning. This approach was useful as he entered college at the age of fifteen. Mathematics as a passport to new experiences. Lived experiences have a direct impact on the development of a person’s belief system (Pajares, 1992). This is the case with Dr. Hrabowski. His varied experiences as a child and as a student, at all educational levels, shaped his beliefs about the teaching and learning of mathematics (and other STEM content areas). Experiencing mathematics in different educational settings by different types of people from various racial and cultural backgrounds, strengthened his belief that mathematics was universal and was accessible to all who worked hard to understand it. As Hrabowski matriculated through his elementary and secondary education, he excelled in mathematics, distinguishing himself from his peers. This led to opportunities for external learning experiences that allowed him to see mathematics in action and experience it at a high level. He spent summers participating in National Science Foundation (NSF) sponsored programs at various college campuses around the country. This provided him with the opportunity to see mathematicians engaging in rigorous and complex mathematical tasks. These experiences were pivotal for him because he had limited exposure to STEM professionals of color at his home school in Birmingham. Being educated in the Jim Crow South, Hrabowski attended segregated schools. He notes that the only Black scientist that he heard of as a child was George Washington Carver. By contrast, in the NSF summer programs, he had the opportunity to see

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scientists of color. He recalls, “I saw for the first-time Black PhDs in STEM, and that’s when I started thinking about the possibility of a PhD and of teaching math.” In addition to exposure to rigorous mathematics and diverse scientists and mathematicians, these summer programs exposed him to students of different races. Until then, Hrabowski never shared a classroom with white students. He recalls, “. . . another summer they sent me to Massachusetts to see what it would be like to be in class with white children.” Irrespective of the race of the teachers or the students, Hrabowski’s love of mathematics eclipsed the environments that he excelled in. Self-identity in the mathematics space. The concept of diversity, equity, and inclusion (DEI) has emerged on the social landscape as a mechanism to ensure that varying racial, cultural, and gender identity perspectives are represented in different facets of society, including the STEM spaces (NSF, 2023). The ability of someone to enter new spaces with confidence in their purpose and qualifications to be in those spaces is rooted in their self-identity. This was the case with Dr. Hrabowski. His early success with mathematics combined with his close family relations and his civil rights activism gave him a strong self-identity that was rooted in confidence in his own mathematics ability. This was needed as he matriculated through his graduate programs. Although Hrabowski was academically strong, he was aware of the need to advocate for himself to ensure his success in the classroom. Establishing positive relationships with his professors was a key factor in his academic success during his undergraduate years. Hrabowski attended Hampton Institute (now Hampton University) a Historically Black College / University (HBCU) in Virginia. As an undergraduate, Hrabowski recalls positive intellectual relationships with two of his African American female professors. One of them, he fondly remembered, “. . . she was such a fascinating intellectual and brought abstract algebra to life. And I enjoyed the proofs with her, and she prepared me well. She really did.” These positive experiences would increase Hrabowski’s self-confidence as he continued his matriculation through higher education. As a graduate student, Hrabowski attended a predominantly white university where his environment and experiences were somewhat different from his undergraduate experience. As an undergraduate, he was welcomed into his classes by his professors and fellow students. As a graduate student, he established himself as a serious and capable student as he developed relationships with his professors. He remembers, “. . . so when I went to grad school there was nobody black in my classes, no black professors.” He soon realized that his level of self-advocacy had to increase to make himself visible in the mathematics space. 1058

“I did quite well in the math, but nobody would work with me. Nobody would work with me, and in the first classes, when the white male professor would come in each class, the professor would look right at me, the only black in the class, and say, this is topology, or this is set theory. The idea was, You’re probably in the wrong class. And so by the third time, I said, this is ‘numerical analysis, right?’ But he came in, so let him know I’m supposed to be in here. . . .and I tell you that because I needed to let people know I knew where I was, and why I was there. . . .what I learned from the experience, though, was since the students, all male or white. . . .I needed to be aggressive in getting help from faculty members. I did well, but I had to, really, and they were not. They just kept saying, ‘Go work with the students’, and I said, ‘they won’t work with me’. And so, I did work with the students, and I did better than most. But I tell you that because by the time I finished my master’s, I had nobody to talk to. So, I did this combination of higher education administration and statistics in the social sciences for my PhD.” Dr. Hrabowski’s social isolation did not have a negative impact on his ability to be academically successful. It did, however, provide him with ideas and insight regarding the elements of a successful STEM education program that would educate and train a diverse group of scholars. Supporting and developing a positive self-identity was a key element that he identified. The creation of a support system that contained academic and social support became a foundational element in the STEM education programs that he developed.

STEM Visionary The under-representation of students of color and those students who identify as female in the STEM fields has been well documented in the scholarly literature (NSF, 2023). Hrabowski’s ability to navigate the mathematics space as a Black man is remarkable and has served as the impetus for his approach to teaching and promoting unique STEM educational programming. When he reflected on his experiences as a graduate student, Hrabowski shared, “. . . I tell you that because it shaped my thinking about preparing people for careers in STEM, and one of the points was collaboration and group work. As I began my career at UMBC, I knew I wanted to spend my life producing PhDs of all races with an emphasis on the students of color, but of all races, and women and men.” The ability to create a system that produces a diverse group of STEM scholars was not an easy task. Identifying

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students with academic promise was only the beginning. Creating a university infrastructure to support these students was a necessary component as well. The Meyerhoff Scholars program was the embodiment of Dr. Hrabowski’s vision for STEM education.

Developing the Model Dr. Hrabowski’s establishment of the Meyerhoff Scholars Program provided him with the mechanism to develop and refine his approach to producing a diverse group of PhDs in STEM fields. Using his own personal and professional experiences as a college administrator, the Meyerhof Scholars Program was centered around the following concepts: 1. Building a strong community among the students. 2. Creating a culture of high expectations for the students and faculty. 3. Understanding that it takes a scientist, mathematician, or engineer to produce a scientist, mathematician, or an engineer. Building a strong community among students. The concept of collaboration was a recurring theme that emerged during Dr. Hrabowski’s experiences as a student. The ability to learn by working with others is one of the key pieces of the Meyerhoff program. Providing students with the opportunity to authentically engage in rigorous STEM content with collaboration embedded in the curriculum created a sense of comradery within the program. To build a strong sense of community, Hrabowski elicited feedback from program alumni for strategies to strengthen relationships among the students. As a result, the summer induction program implemented a radical approach to encouraging the students to get to know one another. This approach required students to limit their use of their cell phones during the week. Hrabowski recalls making this decision with the assistance of his senior students from the Meyerhoff program. He stated: “. . . It was the students as they graduated from the program. We asked them one summer, what could we do to make the program better? And they said, If you want the students to get to know each other better, take the phones away during the week, so they have to depend on each other rather than on their high school friends and their mothers, and we did it. And it did make a difference.” As a result, students in the program learned to communicate and collaborate with one another as they matriculated through the program. This tactic also helped students to develop personal and professional networks that took them beyond the classroom. SEPTEMBER 2024

Creating a culture of high expectations for the students and faculty. High expectations have been a core value for Dr. Hrabowski his entire life, and he promoted that as he established the Meyerhoff Scholars program. He believed that engaging students in rigorous content and authentic research was key to their academic development. Moreover, Dr. Hrabowski created an environment where faculty were expected to engage their students in all phases of the research process. As a result, many graduates of the Meyerhoff program have gone on to become major participants in current STEM research. Understanding that it takes a scientist, mathematician, or engineer to produce a scientist, mathematician, or an engineer. Dr. Hrabowski believed that scientists, researchers, engineers, and other STEM industry professionals had insights into unique and practical applications of the concepts that were taught in classes. To this end, the Meyerhoffs were exposed formally and informally to STEM industry professionals throughout their matriculation in the program. Exposure to these professionals served as inspiration and motivation to the students in the Meyerhoff program.

Reflections from Meyerhoff Scholars Alumni Dr. Hrabowski’s model for the Meyerhoff Scholars Program provided the framework for many of the scholars to be successful in the program. I had an opportunity to interview several Meyerhoffs past and present to learn about the impact of the program on their personal and professional lives. During our discussion, three themes emerged as the Meyerhoffs shared their experiences while in the program. Theme 1: Developing and fostering relationships. As the Meyerhoffs reflected on their time in the program, they recalled the positive messages that they received from the program’s faculty and staff, especially the encouragement that they received from Dr. Hrabowski himself. Although, he was the President of the university, he was accessible to the students attending the university and to the students in the Meyerhoff program. The ability to develop meaningful relationships helped to create the culture of the Meyerhoff program, like the relationships that Dr. Hrabowski formed when he was an undergraduate at Hampton University. He created those same types of relationships with the students in the Meyerhoff program. A program alumnus remembers one of their first interactions with Dr. Hrabowski. “He was so nice and energetic! He asked me my name and where I was from, and what I planned to major in. It was like he really was trying to get know me. . . From that point on, every time he saw me on campus, he called me by name and he checked in with me.”

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Another Alumnus reflects on their relationship with Dr. Hrabowski. “Dr. Hrabowski really took a chance and believed in me. When I originally applied, I didn’t get in because my SAT scores weren’t that good. When I met with him, he told me that I would have to work hard, and if I did, I would be really successful. . . and I did! I went on to earn my MD/PhD.” In addition to developing relationships with several Meyerhoff scholars himself, Dr. Hrabowski created a culture that fostered relationships among the scholars themselves. Many of the program alumni were able to develop lifelong friendships with their fellow classmates. They reflected on their first time meeting one another at the summer orientation for the Meyerhoff program. The orientation was an intense 4–6-week program where the new cohort of scholars took classes that emphasized study skills, teamwork, collaboration, and group accountability. The program was structured for the new scholars to “buy-in” to the tenants of the program. One of the program scholars remembers her first week in the summer program. “I was really nervous when I first got there because I didn’t really know anyone. I recognized a couple of people that I met at one of the earlier receptions for admitted students, but that’s it. Dr. Hrabowski did a good job making us feel comfortable, and he shared his vision of us and the success that awaits us. He made me (us) feel really special. . . . Yeah, my roommate from freshman year, we were bridesmaids in each other’s weddings.” During their time on campus, the Meyerhoffs operated on campus as a pseudo-family. While many of them were active in other campus organizations and clubs, they participated in several academic and social activities as a group. They attended class together, they studied together, and they socialized around academics together as well. Theme 2: A culture of academic excellence. The comradery developed among the Meyerhoffs created a culture of achievement in the program as well. Academic excellence was an expectation. While competition existed within the program, the concepts of teamwork and shared success prevailed. One Meyerhoff explained her experiences in the program in this way. “We all had to do well. When we were in the summer program, everybody was responsible for everyone’s success. If one person didn’t understand a concept, it was the responsibility of the group to help them learn it. No one was allowed to be left behind.” One of the goals/expectations of the Meyerhoff program is that graduates would go on to graduate school. As a result, grade point averages (GPA) were of great 1060

importance to the students. This heightened emphasis on GPAs resulted in an increased focus on their academic performance. Meyerhoffs were encouraged to reach out and establish relationships with their professors to strengthen and build academic networks on campus just as Dr. Hrabowski had done as a student. In many cases, this resulted in opportunities for the Meyerhoffs (and other students) to engage in authentic research with their professors. These research opportunities allowed the students to deepen their understanding of the content that they were learning in class, which in turn strengthened the culture of the Meyerhoff Program.

Reflections for the Mathematics Community and the Future of STEM Diversity Mathematics can serve as an access point for advanced academic studies, and it can, unfortunately, act as a gatekeeper for those advanced academic studies as well. The academic career of Dr. Hrabowski is an example. One crowning achievement is the establishment of the Freeman Hrabowski Scholars Program sponsored by the Howard Hughes Medical Institute (HHMI) in 2022. The Hrabowski Scholars acts as an extension of the Meyerhoff Scholars program because its goal is to support early career faculty. HHMI (2022) describes the selection as this, “Faculty members are selected for their potential to become leaders in their fields and to create diverse and inclusive lab environments in which everyone can thrive”. Selected scholars receive a salary and a research budget that includes equipment for a 5-year period with an opportunity to renew for a second 5-year period. As I concluded my conversation with Dr. Hrabowski, I asked him if he had any parting recommendations for the mathematics community. He shared the following thought, “I want the mathematicians to be the ambassadors for the discipline, and to inspire others to keep learning. . . . We as mathematicians must be those who excite and inspire others. We have the capacity to do that, and we must do it with pride. Let’s do it with pride! That’s my message. We must speak with pride about mathematics.” Dr. Hrabowski’s legacy is still unfolding. His impact on STEM education will not be measured by the countless awards that he has received and will continue to receive. It will be measured by the countless numbers of STEM professionals that he has mentored, inspired, and produced as well as the manner in which these scientists contribute to their respective fields of study.

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References

[1] G. Athanasia and J. Cota, The U.S. Should Strengthen STEM Education to Remain Globally Competitive, Perspectives on Education, CSIS, 2022. [2] J. Li and A. H. Schoenfeld, Problematizing teaching and learning mathematics as “given” in STEM education, International Journal of STEM Education (2019), https://doi .org/10.1186/s40594-019-0197-9. [3] W. McCallum, Sense-making and making sense, 2018, https://blogs.ams.org/matheducation/2018. [4] National Center for Science and Engineering Statistics (NCSES), Diversity in STEM: Women, Minorities, and Persons with Disabilities, Special Report NSF 23-315, National Science, Alexandria, VA, 2023. [5] S. Neher-Asylbekov and I. Wagner, Effects of Out-of-School STEM Learning Environments on Student Interest: A Critical Systematic Literature Review, Journal of STEM Education Research 6 (2022), no. 1, 1–44. [6] M. F. Pajares, Teachers’ Beliefs and educational research: Cleaning up a messy construct, Review of Educational Research 62 (1992), no. 3, 307–332.

[7] V. Richardson, The role of attitudes and beliefs in learning to teach, in J. Sikula (Ed.), Handbook of Research on Teacher Education, Simon and Shuster, New York, NY, 1996.

Christian Anderson Credits

Photo of Christian Anderson is courtesy of Christian Anderson.

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WHAT IS. . .

a Parking Function? J. Carlos Mart´ınez Mori Consider a one-way street with 𝑛 ∈ {1, 2, …} numbered parking spots, denoted 𝑠1 , 𝑠2 , … , 𝑠𝑛 . A sequence of 𝑛 cars enters the street one at a time, each with a preferred spot. Upon its arrival, car 1 ≤ 𝑖 ≤ 𝑛, denoted 𝑐𝑖 , drives to its preferred spot 𝑥𝑖 ∈ [𝑛] ≔ {1, 2, … , 𝑛}. If spot 𝑥𝑖 is unoccupied, car 𝑐𝑖 is lucky and parks there. Otherwise, car 𝑐𝑖 displaces further down the street until it finds the first unoccupied spot in which to park, if such a spot exists. If no such spot exists, car 𝑐𝑖 reneges the search process unable to park. Let 𝐱 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ [𝑛]𝑛 be the 𝑛-tuple encoding the cars’ parking preferences. If all cars are able to park, then 𝐱 is said to be a parking function of length 𝑛. Parking functions were first implicitly studied by Pyke [Pyk59] in his study of Poisson processes and later on by Konheim and Weiss [KW66] in their study of hashing with linear probing. Parking functions can be seen as functions in the sense that the “parking experiment” narrated above assigns, to each 𝑛-tuple of parking preferences, a single 𝑛-tuple encoding its parking outcome. If the 𝑛-tuple of preferences is indeed a parking function, its outcome can be treated as a permutation of [𝑛] written in one-line notation. For example, as depicted in Figure 1, (1, 3, 1, 1) is a parking function of length 4 with outcome (1, 3, 2, 4). Classical enumerative results about parking functions foreshadow their rich mathematical structure. There are (𝑛 + 1)𝑛−1

(1)

J. Carlos Mart´ınez Mori is a Schmidt Science Fellow and a President’s Postdoctoral Fellow at the Georgia Institute of Technology. His email address is jcmm @gatech.edu. The arXiv version of this article includes a Spanish translation: https:// arxiv.org/abs/2404.15372. Communicated by Notices Associate Editor Emilie Purvine. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti3004

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Figure 1. The parking function (1, 3, 1, 1) has outcome (1, 3, 2, 4). Cars 𝑐1 and 𝑐2 are lucky and park in their preferred spots 𝑠1 and 𝑠3 , whereas cars 𝑐3 and 𝑐4 displace further down the street before ultimately parking in spots 𝑠2 and 𝑠4 , respectively.

parking functions of length 𝑛 ≥ 1 (OEIS A000272)—this is Cayley’s formula [Cay89] for the number of labeled trees on 𝑛+1 vertices. This count was established independently by Pyke [Pyk59] and Konheim and Weiss [KW66], and has been recovered bijectively by numerous authors; refer to Yan [Yan15, Section 13.2] and the references therein. An elegant proof of (1), due to Pollak as credited by Riordan [Rio69], is as follows. Consider a circular oneway street with 𝑛 + 1 numbered parking spots, denoted 𝑠1 , 𝑠2 , … , 𝑠𝑛 , 𝑠𝑛+1 . Say, it is a roundabout with no exits and a single entry between spots 𝑠𝑛+1 and 𝑠1 . There are 𝑛 cars attempting to park, so that each 𝑛-tuple 𝐱 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ [𝑛 + 1]𝑛 encodes a possibility for the cars’ parking preferences. How many of these 𝑛-tuples are in fact parking functions of length 𝑛? Since there are 𝑛 + 1 spots but only 𝑛 cars, there will always be one spot left unoccupied; which exact spot this is depends on the preferences. Note that the preference tuples form a group 𝐺 under component-wise addition modulo 𝑛 + 1, and consider the subgroup 𝐻 ≤ 𝐺 generated by the all-ones preference tuple (1, 1, … , 1). As depicted in Figure 2, the cosets 𝐱𝐻 = {𝐱 + ℎ ∶ ℎ ∈ 𝐻} for 𝐱 ∈ 𝐺 form equivalence classes of order 𝑛 + 1, each of which contains exactly one parking function, namely

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What is. . .

Figure 2. Pollak’s argument for (1) using the example in Figure 1. The tuples (1, 3, 1, 1), (5, 2, 5, 5), (4, 1, 4, 4), (3, 5, 3, 3), and (2, 4, 2, 2) form an equivalence class of order 5. Their outcomes are depicted as layers overlaying a circular one-way road with spots 𝑠1 , 𝑠2 , … , 𝑠5 . The only parking function in this class is (1, 3, 1, 1) because it is the only tuple that leaves spot 𝑠5 unoccupied in its outcome, as depicted in the outermost layer of the illustration.

Figure 3. The Dyck path 𝑁𝑁𝑁𝐸𝐸𝑁𝐸𝐸 of length 8, illustrated with bold dashed red lines, maps to the weakly increasing parking function 𝐱′ = (𝑥1′ , 𝑥2′ , 𝑥3′ , 𝑥4′ ) = (1, 1, 1, 3) of length 4. For example, the fourth 𝑁 step in the path is the sixth step overall, as illustrated with a solid bold red line. There are a total of two 𝐸 steps before it, so 𝑥4′ = 1 + 2 = 3.

Parking functions are also closely related to the Catalan numbers (OEIS A000108), which are given by the recurrence relation 𝑛

𝐶𝑛 = ∑ 𝐶𝑖−1 ⋅ 𝐶𝑛−𝑖

(3)

𝑖=1

whichever tuple leaves spot 𝑛 + 1 unoccupied in its outcome. Therefore, there are (𝑛 + 1)𝑛 /(𝑛 + 1) = (𝑛 + 1)𝑛−1 parking functions of length 𝑛. Now, consider an arbitrary 𝑛-tuple 𝐱 = (𝑥1 , 𝑥2 , … , 𝑥𝑛 ) ∈ [𝑛]𝑛 . Can one tell whether it is a parking function of length 𝑛 without explicitly considering its parking outcome? A classical inequality-based characterization of parking functions allows precisely this. Let 𝐱′ = (𝑥1′ , 𝑥2′ , … , 𝑥𝑛′ ) be the weakly increasing rearrangement of 𝐱. Then, the following holds: 𝐱 is a parking function of length 𝑛

⟺

𝑥𝑖′ ≤ 𝑖 for all 1 ≤ 𝑖 ≤ (2) 𝑛.

This characterization can be verified as follows. If there exists some 𝑖 for which 𝑥𝑖′ > 𝑖, then at least 𝑛 − 𝑖 + 1 cars attempt to park in the last 𝑛−𝑖 spots 𝑠𝑖+1 , 𝑠𝑖+2 , … , 𝑠𝑛 . This is more cars than spots available, so at least one of these cars will be unable to park. Conversely, if some car is unable to park, then there exists an earliest spot 𝑠𝑖 left unoccupied, which can only be the case if 𝑥𝑖′ > 𝑖. Note that (2) has the following remarkable implication: parking functions are invariant under the action of the symmetric group 𝔖𝑛 , which permutes their subscripts. In particular, the set of parking functions is obtained from the rearrangements of weakly increasing parking functions. For example, (1, 3, 1, 1), (1, 1, 3, 1), (1, 1, 1, 3), and (3, 1, 1, 1) are all parking functions of length 4, each with a different parking outcome, whereas no rearrangement of (1, 3, 3, 4) can possibly be a parking function of length 4. SEPTEMBER 2024

for 𝑛 ≥ 1 with 𝐶0 = 1. In particular, the set of weakly increasing parking functions of length 𝑛 is enumerated by (3). To verify this, suppose you construct a weakly increasing parking function of length 𝑛, denoted 𝐱′ = (𝑥1′ , 𝑥2′ , … , 𝑥𝑛′ ) ∈ [𝑛]𝑛 , with your choice of 1 ≤ 𝑖 ≤ 𝑛 for the greatest index satisfying 𝑥𝑖′ = 𝑖. For any such choice of ′ 𝑖, the (𝑖 − 1)-tuple (𝑥1′ , 𝑥2′ , … , 𝑥𝑖−1 ) must be a parking func′ ′ −𝑖+ − 𝑖 + 1, 𝑥𝑖+2 tion of length 𝑖 − 1, the (𝑛 − 𝑖)-tuple (𝑥𝑖+1 ′ 1, … , 𝑥𝑛 − 𝑖 + 1) must be a parking function of length 𝑛 − 𝑖, and by induction there are 𝐶𝑖−1 ⋅ 𝐶𝑛−𝑖 possibilities. As a consequence, weakly increasing parking functions are in bijection with the wide variety of Catalan objects; refer to Stanley [Sta15] for a survey. For example, weakly increasing parking functions of length 𝑛 are in bijection with Dyck paths of length 2𝑛; these are lattice paths from (0, 0) to (𝑛, 𝑛) using only north steps (1, 0) and east steps (0, 1), denoted “N” and “E” respectively, and which do not cross below the main diagonal. Armstrong, Loehr, and Warrington [ALW16, Section 2.2] describe the following bijection: given a Dyck path of length 2𝑛, obtain a weakly increasing parking function 𝐱′ = (𝑥1′ , 𝑥2′ , … , 𝑥𝑛′ ) of length 𝑛 by letting 𝑥𝑖′ be one plus the total number of E steps appearing before the 𝑖th N step for all 1 ≤ 𝑖 ≤ 𝑛. Figure 3 illustrates this construction using the weakly increasing parking function (1, 1, 1, 3); the weakly increasing rearrangement of the example in Figure 1. Note that if 𝐱′ = (𝑥1′ , 𝑥2′ , … , 𝑥𝑛′ ) ∈ [𝑛]𝑛 is a weakly increasing parking function, then the 𝑖th car 𝑐𝑖 (with preference 𝑥𝑖′ ) parks in the 𝑖th spot 𝑠𝑖 . Therefore, the total

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What is. . . displacement of a weakly increasing parking function 𝐱′ is given by 𝑛

∑ 𝑖 − 𝑥𝑖′ = 𝑖=1

𝑛

𝑛(𝑛 + 1) − ∑ 𝑥𝑖′ . 2 𝑖=1

(4)

This statistic has further combinatorial interpretations. For example, the number of full squares between a Dyck path and the main diagonal, which in the case of Figure 3 is 4, is the same as the total displacement of its corresponding weakly increasing parking function. Lastly, note that the dependency of (4) on 𝐱′ reduces to the summation of its terms. This implies that the total displacement of a parking function is also preserved under rearrangements! Despite first appearing in the literature more than six decades ago, the combinatorics of parking functions remains a vibrant area of research. Recent work ranges from the study of discrete statistics such as displacement [KY23, EHKMM23] or the number of lucky cars [GS04, SV24, SY23], the introduction of new variants and/or generalizations on the classical “parking experiment” rule (refer to Carlson et al. [CCH+ 21] for an accessible tour of endless possibilities in the style of “choose your own adventure”), polyhedral aspects [AW22, HLVM24], and surprising connections to seemingly unrelated objects [AAH+ 23, HKMM24], to mention just a few. In recent work, my collaborators and I use a subset of parking functions we call unit Fubini rankings, and in particular their outcome map, to characterize and enumerate the Boolean intervals of rank 𝑘 in the weak order poset of 𝔖𝑛 [EHKMM24]. Figure 4 illustrates our construction. References

[AAH+ 23] Yasmin Aguillon, Dylan Alvarenga, Pamela E. Harris, Surya Kotapati, J. Carlos Mart´ınez Mori, Casandra D. Monroe, Zia Saylor, Camelle Tieu, and Dwight Anderson Williams II, On parking functions and the tower of Hanoi, Amer. Math. Monthly 130 (2023), no. 7, 618–624, DOI 10.1080/00029890.2023.2206311. MR4623327 [AW22] Aruzhan Amanbayeva and Danielle Wang, The convex hull of parking functions of length 𝑛, Enumer. Comb. Appl. 2 (2022), no. 2, Paper No. S2R10, 10, DOI 10.54550/eca2022v2s2r10. MR4459984 [ALW16] Drew Armstrong, Nicholas A. Loehr, and Gregory S. Warrington, Rational parking functions and Catalan numbers, Ann. Comb. 20 (2016), no. 1, 21–58, DOI 10.1007/s00026015-0293-6. MR3461934 [CCH+ 21] Joshua Carlson, Alex Christensen, Pamela E. Harris, Zakiya Jones, and Andr´es Ramos Rodr´ıguez, Parking functions: choose your own adventure, College Math. J. 52 (2021), no. 4, 254–264, DOI 10.1080/07468342.2021.1943115. MR4309295 [Cay89] Arthur Cayley, A theorem on trees, Quarterly Journal of Pure and Applied Mathematics 23 (1889), 376–378. [EHKMM23] Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Mart´ınez Mori, Cost-sharing in parking games, arXiv preprint arXiv:2309.12265 (2023). 1064

Figure 4. Weak order poset of 𝔖6 with a Boolean interval of rank 3 highlighted, with minimal element 341625 and maximal element 436152 written in one-line notation. The outcome of (3, 5, 1, 2, 6, 4), a parking function of length 6, is (3, 4, 1, 6, 2, 5) and corresponds to the minimal element of the interval. The outcome of (3, 5, 1, 1, 5, 3), another parking function of length 6, is again (3, 4, 1, 6, 2, 5) and corresponds to the minimal element of the interval. In fact, (3, 5, 1, 1, 5, 3) is a “unit Fubini ranking with 3 distinct ranks” and corresponds to a Boolean interval of rank 6 − 3 = 3 (i.e., a cube), whereas (3, 5, 1, 2, 6, 4) is a “unit Fubini ranking with 6 distinct ranks” and corresponds to a Boolean interval of rank 6 − 6 = 0 (i.e., a node). Refer to Elder et al. [EHKMM24] for details.

[EHKMM24] Jennifer Elder, Pamela E. Harris, Jan Kretschmann, and J. Carlos Mart´ınez Mori, Parking functions, fubini rankings, and Boolean intervals in the weak order of 𝔖𝑛 , To appear in Journal of Combinatorics (2024). [GS04] Ira M. Gessel and Seunghyun Seo, A refinement of Cayley’s formula for trees, Electron. J. Combin. 11 (2004/06), no. 2, Research Paper 27, 23, DOI 10.37236/1884. MR2224940 [HLVM24] Mitsuki Hanada, John Lentfer, and Andr´es R. Vindas-Mel´endez, Generalized Parking Function Polytopes, Ann. Comb. 28 (2024), no. 2, 575–613, DOI 10.1007/s00026-023-00671-1. MR4747488 [HKMM24] Pamela E. Harris, Jan Kretschmann, and J. Carlos Mart´ınez Mori, Lucky Cars and the Quicksort Algorithm, Amer. Math. Monthly 131 (2024), no. 5, 417–423, DOI 10.1080/00029890.2024.2309103. MR4739576

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[KY23] Richard Kenyon and Mei Yin, Parking functions: from combinatorics to probability, Methodol. Comput. Appl. Probab. 25 (2023), no. 1, Paper No. 32, 30, DOI 10.1007/s11009-023-10022-5. MR4549917 [KW66] Alan G. Konheim and Benjamin Weiss, An occupancy discipline and applications, SIAM Journal on Applied Mathematics 14 (1966), no. 6, 1266–1274. [Pyk59] Ronald Pyke, The supremum and infimum of the Poisson process, Ann. Math. Statist. 30 (1959), 568–576, DOI 10.1214/aoms/1177706269. MR107315 [Rio69] John Riordan, Ballots and trees, J. Combinatorial Theory 6 (1969), 408–411. MR234843 [SV24] Anton´ın Slav´ık and Marie Vestenick´a, Lucky cars: expected values and generating functions, Amer. Math. Monthly 131 (2024), no. 4, 343–348, DOI 10.1080/00029890.2023.2293616. MR4723560 [Sta15] Richard P. Stanley, Catalan numbers, Cambridge University Press, New York, 2015, DOI 10.1017/CBO9781139871495. MR3467982 [SY23] Richard P. Stanley and Mei Yin, Some enumerative properties of parking functions, arXiv preprint arXiv:2306.08681 (2023). [Yan15] Catherine H. Yan, Parking functions, Handbook of enumerative combinatorics, Discrete Math. Appl. (Boca Raton), CRC Press, Boca Raton, FL, 2015, pp. 835–893. MR3409354

J. Carlos Mart´ınez Mori Credits

All figures are courtesy of J. Carlos Mart´ınez Mori. Author photo is courtesy of Nicole Semp´ertegui.

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BOOKSHELF New and Noteworthy Titles on our Bookshelf September 2024

follow their dreams. Ultimately, this documentary helped me reflect on what it means to be a mathematician while standing out from the stereotypical role of what is expected of mathematicians. The director has indicated there will be a sequel, which I hope will continue the stories started in Forging Resilience and expand upon some of the important themes of belonging and progress.

Directed by George Csicsery In February 2024, the Notices published the article “Making Journeys of Black Mathematicians,” which highlighted the recently released documentary film. The article, by Csicsery himself, outlined the motivation for the film and the process with which the film was made. Here, I plan to highlight aspects of the film that I hope will entice you to seek it out. There is a lot to learn from hearing the stories and journeys of others. While we all face uncertainty and difficulties, there is no doubt that Black mathematicians have faced unique challenges. The stories presented in the documentary span from the civil rights movement of the 1960s and integration to the current lack of representation among mathematics teachers across the United States. The interviews feature a wide range of mathematicians, including students, postdocs, and professors, both employed and retired. The film also discusses the important role HBCUs (Historically Black Colleges and Universities) have played in holding their students to a high standard and the power of community within the National Academy of Mathematicians (NAM). The film can either be watched through an individual rental or an institutional purchase. I screened the film at my institution, and my undergraduate students in attendance offered a variety of reactions. One student in computer science declared a mathematics minor the next day. Future teachers remarked on the importance of having high expectations for all students and how crucial it is for every student to hear they have capabilities and should This Bookshelf was prepared by Notices Associate Editor Emily J. Olson. Appearance of a book in the Notices Bookshelf does not represent an endorsement by the Notices or by the AMS. Suggestions can be sent to [email protected]. DOI: https://doi.org/10.1090/noti2995

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The Waltz of Reason Basic Books, 2023, 448 pp.

By Karl Sigmund

Cover is courtesy of Basic Books.

Cover is courtesy of George Csicsery/Zala Films.

Journeys of Black Mathematicians: Forging Resilience

Some of the greatest thinkers of all time have pondered deep and often insightful questions which are both mathematical and philosophical in nature. The quintessential example might be “What is a number?” Karl Sigmund, the author of this new book, states that many mathematicians and philosophers might reply to such a query by remarking that “what is?” questions make little sense. This book is a dance between math and philosophy, and when one seems to take the lead on a question over the other throughout history, we see “that the two fields have wonderful ways of stimulating and often surprising each other.” The text takes the reader on a tour through concepts and historical activities and figures. It contains pictures of mathematicians and philosophers and illustrations that describe mathematical concepts and proofs, but not many equations. The book is divided into four parts, which focus on number, algorithm, axiom, and proof; chance, probability, and the continuum; morality, economics, social contracts, politics, and law; and language and understanding. It is not exclusively chronological, as it occasionally needs to refer to a person or event in earlier times to explain a more recent topic of inquiry. The book includes many historical figures from both mathematics and philosophy and also indicates that sometimes it is hard to tell the difference between the two. The Waltz of Reason is not a technical book, but a book to be enjoyed by anyone interested in the interplay between math and philosophy.

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Label Bias: A Pervasive and Invisibilized Problem Yunyi Li, Maria De-Arteaga, and Maytal Saar-Tsechansky As machine learning (ML) systems are increasingly employed to automate various aspects of our lives and to assist in human decision-making, it is imperative that the accompanying issues of fairness, equity, and ethics concerning these AI artifacts are thoughtfully considered and appropriately addressed. While there has been a notable surge in research focused on the development of fairnessaware algorithms in recent years, a majority of these endeavors typically rely on a fundamental assumption: the training labels utilized to train these systems are accurate and unbiased. This assumption, while convenient, does not hold true in many critical practical scenarios. In this article, we argue that overlooking label bias while developing algorithms to address inductive bias runs the risk of invisibilizing and exacerbating existing societal biases. We do so by first presenting the concept of label bias and reviewing the literature that characterizes or conceptually discusses label bias in important domains including healthcare, predictive policing, and content moderation. Subsequently, we explain how confusion matrixbased measures of “fairness” used to mitigate inductive bias can overlook label bias. We then outline several noteworthy risks that may stem from this limitation. We conclude by proposing paths forward, emphasizing the imporYunyi Li is a PhD candidate at the University of Texas at Austin. Her email address is [email protected]. Maria De-Arteaga is an assistant professor at the University of Texas at Austin. Her email address is [email protected]. Maytal Saar-Tsechansky is a professor at the University of Texas at Austin. Her email address is [email protected]. Communicated by Notices Associate Editor Richard Levine. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti2941

SEPTEMBER 2024

tance of clearly discerning the types of bias that fairnessaware algorithms aim to address, and underscoring the need for developing algorithms that mitigate label bias.

1. Pervasiveness of Label Bias With advancements in AI technology and its ability to process vast amounts of data, ML systems are increasingly relied upon to make accurate predictions and assist in decision-making. This ability to process large amounts of data allows ML systems to uncover patterns in the data that link various features to outcomes, a task that was traditionally performed by human decision-makers. We refer to the outcome or the target variable’s value that ML systems are trained to predict as labels. Human-generated labels can be collected by recording historical human decisions, or by actively asking experts or crowd-sourced annotators to label the data as instructed. Labels used to train ML models may also correspond to observed outcomes that do not directly correspond to a human decision. For example, when hiring a salesperson, the observed outcome may be sales volume; and when developing algorithms for healthcare, there may be a medical outcome of interest. Regardless of the origin of the label, given a set of input features and label pairs, supervised ML methods aspire to induce models that offer consistent predictions for instances with unknown outcomes and achieve good performance over the population. This process is commonly known as “induction” in the ML domain. As ML and AI become ubiquitous in our daily life, researchers started to notice that AI technology might cause social harm. In 2013, Sweeney [Swe13] found racial bias in the targeting of online advertisements concerning arrest records. This bias was associated with names that are culturally belonging to Black or white individuals. The research highlighted how individuals with names that are

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perceived as Black-sounding may be more likely to encounter online ads suggesting they have arrest records, even if they do not [Swe13]. The evidence presented in this work emphasized the need to design technologies with consideration of societal consequences, particularly those related to structural racism. Researchers have found evidence of bias across multiple other ML technologies. For example, investigations into facial analysis software revealed that commercial facial classification algorithms exhibit lower accuracy when it comes to darker-skinned women [KBK+ 12, BG18]. Such findings, while centered on gender classification, have raised concerns regarding related technologies, such as facial analysis tools used by law enforcement. Law enforcement agencies utilize facial recognition technology to match suspects’ photos with mugshots and driver’s license images [Gar16]. As of 2016, it is estimated that over 117 million American adults— nearly half of the population—have their photos in facial recognition networks used by the police [Gar16]. As a result of biases in facial recognition, there are concerns that the deployment of such tools may disproportionately harm marginalized communities, including African Americans [Gar16]. Algorithmic bias can arise from various sources. One significant factor is the representation issue. For example, there might be an underrepresentation of minority groups in the training data. Underrepresentation occurs when certain demographic groups are not adequately represented or are disproportionately scarce in the dataset. The bias arises because ML algorithms learn patterns from the data they are trained on. If the training data is not diverse enough and lacks representation from minority groups, the algorithm may not learn the correct relationship between predictive covariates (features) and labels for those groups. Instead, the relationships it learns become dominated by the patterns it observed for the majority group(s). The landmark “Gender Shades” research [BG18] described how gender classification systems can be biased due to the representation issue in the training data. If a gender classification algorithm is trained primarily on images of lighter-skinned individuals, it is likely to only perform well in identifying and classifying faces of people with similar skin tones [BG18]. Another example of representation issue is the selective labels problem [LKL+ 17]1, which refers to a situation in which the observed labels in a dataset are not assigned randomly or uniformly across all instances but are influenced by the choices or decisions of human decision-makers [LKL+ 17]. In other words, the process of labeling data is selective and depends on certain con1We note that while this is sometimes referred to as a label bias problem, it is in

nature a bias caused by a representation issue, which is different from the label bias we will introduce in this article.

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ditions or criteria set by decision-makers. For example, if a model is trained to decide whom to grant bail, the available training data includes information about individuals who were granted bail in the past, but it may not include data about those who were not [LKL+ 17]. Another critical source of algorithmic bias, which we focus on in this paper, is label bias. Let 𝑌 ̃ denote an observed label used to train an ML system. Such a label is usually readily accessible in a set of training examples, but it may not always correspond to the ground truth or gold standard of interest, which we denote 𝑌 ∗ . Since we are concerned with bias affecting demographic subgroups, let 𝐺 be a variable denoting group membership. While, in general, 𝐺 does not need to be binary, we assume it is binary for simplicity in our explanation. Label bias refers to a “systematic disparity between the ground truth labels intended to train an AI system and the observed labels, such that the relationship underlying the mismatch differs across groups” [LDAST22]. The way the relationship differs may take different forms. For example, [LDAST22] proposes measuring gaps across different groups in the accuracy of observed labels 𝑌 ̃ with respect to latent ground truth labels 𝑌 ∗ . Under [LDAST22]’s definition, label bias occurs if there is a group of relevance 𝑔0 ∈ 𝐺 such that, 𝑃(𝑌 ̃ = 𝑌 ∗ |𝐺 = 𝑔0 ) ≠ 𝑃(𝑌 ̃ = 𝑌 ∗ |𝐺 ≠ 𝑔0 ). Naturally, other measures may be more relevant to determine label bias in different contexts. It may be the case that 𝑃(𝑌 ̃ = 𝑌 ∗ |𝐺 = 𝑔0 ) = 𝑃(𝑌 ̃ = 𝑌 ∗ |𝐺 ≠ 𝑔0 ), but the types of errors of observed labels 𝑌 ̃ with respect to latent ground truth labels 𝑌 ∗ are different for different groups. There are two types of errors that could occur: false positive error ( when 𝑌 ̃ = 1, 𝑌 ∗ = 0) and false negative error (when 𝑌 ̃ = 0, 𝑌 ∗ = 1). For example, let 𝐺 correspond to gender and 𝑔0 to women, then one may say that 𝑌 ̃ exhibits label bias with respect to 𝑌 ∗ if either or both of the following equations are satisfied, no matter what the accuracy rates are 𝑃(𝑌 ̃ = 1, 𝑌 ∗ = 0|𝐺 = 𝑔0 ) ≠ 𝑃(𝑌 ̃ = 1, 𝑌 ∗ = 0|𝐺 ≠ 𝑔0 ) 𝑃(𝑌 ̃ = 0, 𝑌 ∗ = 1|𝐺 = 𝑔0 ) ≠ 𝑃(𝑌 ̃ = 0, 𝑌 ∗ = 1|𝐺 ≠ 𝑔0 ). It is crucial to highlight that under this definition, identifying the existence of label bias requires the availability of ground truth labels for the training data used to train a supervised learning system. As a result, the presence of label bias relies on the specific predictive task that the supervised learning system aims to accomplish. Moreover, since the ground truth label is not available in many important and consequential contexts, it is not always possible to directly perform this assessment. Having formally introduced the notion of label bias, we now provide a review of label bias and its implications

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within three prominent domains: healthcare, predictive policing, and content moderation. 1.1. Label bias in healthcare. In 2019, Obermeyer et al. [OPVM19] examined racial disparities in an algorithm used to prioritize patients for high-risk care management programs. The algorithm was designed to identify patients with complex health needs who could benefit from such programs. However, researchers found that the predictive algorithm, trained to predict healthcare spending but used to forecast patients’ health risk scores, exhibited algorithmic bias. It consistently underestimated the severity of health needs for Black patients compared to their white counterparts. The racial bias in this widely used algorithm for distributing healthcare resources meant that Black patients with equivalent risk levels assigned by the algorithm tended to have more severe health conditions compared to white patients who received similar scores. Consequently, the algorithm disproportionately prioritized healthier white patients over more ill Black patients. The primary issue identified in Obermeyer et al.’s study [OPVM19] is label bias, wherein healthcare costs were utilized as the proxy label for health needs, yielding a systematically inaccurate measure of the underlying healthcare needs of patients. Historical inequities in the US have led to Black patients incurring lower costs than white patients with similar healthcare needs. Consequently, the algorithm favored healthier white patients over sicker Black patients, resulting in a significant reduction in the number of Black patients identified for health management programs. This serves as an excellent example of one of the sources of label bias referred to as the construct gap, wherein a mismatch occurs between the theoretical construct of interest (health risks in the example above) and the observed label used to train an ML system (healthcare spending) [LDAST22]. This discrepancy often arises due to the practical accessibility of one over the other. For instance, in the healthcare example that we outlined above, financial incentives lead to detailed data collection for insurance claims, which are then repurposed for other ML tasks [LDAST22]. Besides, the complexity of high-level objectives sometimes necessitates the use of proxies, posing potential risks of label bias in ML systems, as certain outcomes may be more evenly distributed in the population than others [PB19]. 1.2. Label bias in predictive policing. Predictive policing algorithms, which are increasingly used by law enforcement for proactive crime prevention, rely on data analysis and ML to predict potential criminal activities. Lum and Isaac (2016) [LI16] found that predictive policing of drug

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crimes leads to a disproportionate focus on historically over-policed communities. One way of understanding the bias in police-recorded data is label bias, which arises due to different error types for different neighborhoods, with false negatives more prevalent in higher-income or white communities and false positives more prevalent in lowerincome or Black communities. Such predictive policing algorithms trained using data that exhibit label bias may perpetuate and even amplify existing biases in the data, leading to ineffective or discriminatory practices. Specifically, informed by the algorithm’s predictions, police departments may focus their attention on areas predicted to be high-risk, leading to more arrests in those neighborhoods. These new arrests then feed back into the predictive policing algorithm, reinforcing the belief that those areas are high-crime zones, perpetuating a harmful feedback loop [LI16]. Moreover, while companies developing such systems have often claimed that their predictive policing systems rely on victim reports, which they argue are not influenced by biased police-recorded data, new research has shown that this alternative source of data is also prone to label bias. Akpinar et al. [ADAC21] highlights the existence of differential victim crime reporting rates across neighborhoods, and shows that this disparity can result in the misallocation of policing resources when using predictive models [ADAC21]. 1.3. Label bias in content moderation. Content moderation algorithms used by social media platforms, such as Twitter and Facebook, are shaping the online content we consume every day. However, algorithmic bias has been observed in these systems. For example, biases affecting posts written in African American English (AAE) have been studied, and researchers have revealed that AAE tweets are flagged as offensive by automated hate speech detection algorithms at a rate up to two times higher than other tweets [SCG+ 19]. The presence of racial bias in these systems is attributed to label bias: Sap et. al (2019) found unexpected correlations between surface markers of AfricanAmerican English and ratings of toxicity in widely used hate speech datasets used to train hate speech detection algorithms. As the training labels systematically contain more false positives with respect to ground truth for AAE posts, the automatic hate speech algorithms trained using such data are more likely to falsely identify an AAE tweet as containing hate speech. This bias in the algorithm’s performance can have detrimental consequences, further marginalizing historically disadvantaged communities. The misidentification of hate speech from minority communities may result in the unjust suppression of legitimate speech, and it may hinder efforts to address genuine issues faced by these

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Figure 1. Invisibilization of label bias in measures of “fairness” is exemplified through three confusion matrices: (a), which compares observed labels 𝑌 ̃ and algorithmic predictions 𝑌 ̂ ; (b), which contrasts ground truth labels 𝑌 ∗ and algorithmic predictions 𝑌 ̂ ; and (c), which compares the ground truth labels 𝑌 ∗ and observed labels 𝑌 ̃ . When label bias exists, resulting in a higher false positive rate for group 𝑔0 as illustrated in figure (c), the current fairness measures relying on confusion matrix (a) would fail to effectively capture the concerning error rate disparity evident in the confusion matrix (b), which was our intended matrix for fairness measures.

communities. This can be understood as a participatory injustice [NDAF22], as it makes it harder for some communities to share their perspectives and participate in conversations. Thus, the lack of fairness and accuracy in hate speech detection algorithms not only erodes user trust in the platforms but also raises serious concerns about issues of justice and who is able to make use of and participate in social media platforms. The label bias in datasets used to train hate speech detection algorithms often arises due to human labeling bias [LDAST22]. Identifying hate speech is a challenging task, as it heavily relies on the specific context and dialect. For instance, certain derogatory terms and phrases may be reappropriated by the communities that these terms have historically targeted, which means that a term may hold different meanings depending on the identity of the person using it, while remaining harmful and offensive when used by outsiders. In the example above, when collecting training labels from crowd-sourced annotators, who are mostly non-AAE speakers and mostly not sensitive to dialect, may mistakenly label an AAE post as containing hate speech when it is not [SCG+ 19]. Bias in human labeling is a widespread phenomenon, regardless of whether the labels are gathered from crowds or domain experts. Human cognitive biases and social stereotypes can creep into the labels collected from human annotators in a subtle way. These biases, when combined with the underrepresentation of annotators from minority groups, can result in labeled data that mirrors soci-

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etal prejudices and stereotypes. For example, Davani et al. [DAKD23] investigates the impact of social stereotypes on hate speech detection, revealing that aggregated annotations in curated datasets reflect normative stereotypes, and these biases contribute to systematic errors in hate speech classifiers.

2. Invisibilization of Label Bias in Measures of “Fairness” The risks posed by algorithmic bias have led to a surge in research on algorithmic fairness, which aims to mitigate algorithmic bias and enable the development of ML systems that do not exhibit disparate performance across groups. Increasingly, these methodologies are being adopted in practice[BBB+ 21] and thereby potentially impact humans, organizations, and societies. A cornerstone of algorithmic fairness work is the development of measures of fairness–quantitative metrics that seek to measure predictive bias. Such measures are then used to diagnose algorithmic bias, as well as to mitigate it, which often involves enforcing constraints that ensure a predictive algorithm meets a given performance based on a fairness measure. Perhaps the most popular family of algorithmic fairness measures are those that measure group fairness, which assesses the disparities in a metric of interest across groups [MPB+ 21]. The simplest group fairness measure is demographic parity, which measures gaps in selection rates across groups. For instance, in the context of healthcare needs, achieving

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demographic parity corresponds to the same rate of patients being predicted to have complex health needs across racial groups. While demographic parity has the advantage of not relying on observed labels, and thus is unaffected by issues of label bias, its shortcomings quickly become evident: what happens if health needs are different across groups? For this reason, equalizing measures that stem from the confusion matrix, which reflects erroneous predictions, is frequently preferred [MPB+ 21]. The confusion matrix, also referred to as the error matrix, is central to ML and algorithmic fairness. It stands as a foundational element for assessing the performance of predictions generated by ML classification models. This matrix, typically presented as a two-by-two arrangement (for a binary classification task), facilitates a comprehensive evaluation of the model’s predictive accuracy and potential biases. Essentially, the confusion matrix enables the inspection of the types of errors that a predictive model incurs, and serves as a comparative framework between the observed labels and the predictions generated by an algorithm, as depicted in Figure 1 (a). For each group, the matrix discerns between various types of accuracies and errors, systematically categorizing them into distinct categories: 1.

2.

3.

4.

True Positives (TP): Instances accurately predicted as positives by the algorithm, congruent with the observed labels. True Negatives (TN): Instances correctly predicted as negatives by the algorithm, congruent with the observed labels. False Positives (FP): Instances that are observed negatives, but are erroneously classified as positives by the algorithm. False Negatives (FN): Instances that are observed positives, but are ierroneously classified as negatives by the algorithm.

Most fairness measures depend on the confusion matrix. For example, a commonly used fairness measure, “equalized opportunity”, seeks to match the true positive rate across groups [HPS16]. Similarly, “equalized odds” matches both the true positive rate and the false positive rate across groups [HPS16]. Different types of errors reflected by the confusion matrix can have different implications from a fairness perspective. For example, a highprofile investigation on racial bias in recidivism prediction systems showed that a commercial algorithm used to assist judges in bail decisions was more likely to incur false negative errors for white defendants while incurring false positive errors for Black defendants [ALMK16]. This implies a systematic underestimation of recidivism risk for a specific group (eg. white defendants) and a systematic overestimation of recidivism risk for another group (eg. Black defenSEPTEMBER 2024

dants). Such disparities serve to uncover the potential bias inherent in the algorithmic decision-making process. In an ideal scenario, measures of algorithmic bias should compare algorithmic predictions with ground truth labels 𝑌 ∗ , resulting in a confusion matrix demonstrated in Figure 1 (b). Yet, the pervasiveness of label bias in many important domains often introduces a systematical disparity between the utilized observed labels for evaluation and the actual ground truth labels (please refer the definition of label bias in Section 1), resulting in a confusion matrix as depicted in Figure 1 (c). In the presence of label bias, fairness measures relying on observed labels, as depicted in confusion matrix 1 (a) can be misleading. To illustrate the risks of overlooking label bias and relying on observed labels in the measure of “fairness,” let us consider the context of algorithmic hate speech detection. Let 𝑔0 denote the group of posts written in AAE and 𝑔1 represent posts written in non-AAE (eg. Standard American English), with the task being the identification of hate speech within a post. In scenarios involving crowdsourced annotators, posts written in AAE are more likely to be incorrectly labeled as containing hate speech than posts written in non-AAE, leading to an elevated False Positive Rate (FPR) for 𝑔0 in Figure 1 (c). Consequently, the algorithm may learn this biased pattern and be more prone to erroneously flag AAE posts as containing hate speech (as seen in Figure 1 (b)). However, a challenge in assessing algorithmic bias emerges: both the algorithmic prediction 𝑌 ̂ and the biased annotation 𝑌 ̃ exhibit a high rate of false positives for group 𝑔0 . If the algorithm exhibits high accuracy, both algorithmic predictions and biased annotations incorrectly flag the same set of non-hateful instances as containing hate speech. Consequently, evaluating algorithmic prediction 𝑌 ̂ based on observed label 𝑌 ̃ might fail to identify the concerning pattern of a higher FPR for group 𝑔0 . The interplay among observed label 𝑌 ̃ , algorithmic prediction 𝑌 ̂ , and ground truth label 𝑌 ∗ is illustrated in Figure 2. At its core, the invisibilization of label bias stems from a conflation of inductive bias, which refers to bias that is introduced during training an ML algorithm, and the broader phenomenon of algorithmic bias. When addressing the concern of perpetuating and amplifying societal biases (algorithmic bias), it is important to acknowledge that such bias may be encoded in the observed label, and thus measures that assume the observed label to be accurate and unbiased are ill-posed to tackle the problem at hand. More precisely, consider breaking down a standard machine learning pipeline into three successive phases: 1) problem formulation and data preparation; 2) learning/training and evaluation; and 3) deployment. The potential for algorithmic bias to emerge spans all these stages. However,

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Figure 2. The interplay among observed label 𝑌 ̃ , algorithmic prediction 𝑌 ̂ , and ground truth label 𝑌 ∗ .

most of the proposed algorithms that aim to mitigate inductive bias pertain exclusively to bias arising during the learning phase, by assuming no bias exists in the other two phases. Label bias, which appears at the problem formulation and data preparation phase, signifies a data-oriented challenge that can cause harmful downstream effects for every following stage. If we inappropriately apply strategies intended to alleviate inductive bias in fields where label bias is a core issue, the efficacy of mitigating algorithmic bias might be compromised or might even be counterproductive.

3. Risks of Overlooking Label Bias In this section, we discuss that in scenarios when label bias exists, naive implementation of algorithms aimed at reducing inductive bias and data representation issues may not be fully effective, and in fact, could potentially have adverse outcomes. Specifically, failing to account for label bias may result in the incorrect automated identification of disadvantaged groups, misguide model selection, and mislead data collection. Even in an ideal scenario with perfect induction accuracy and comprehensive representation of the entire population in training data, the induced model can still learn biased patterns from the label bias. Rather than being a true reflection of the decision rule, this perfect-performing induction relying on the biased label may serve as a vehicle for the propagation of the societal biases intricately embedded into the label. 3.1. Misidentify disadvantaged group. Some studies suggest automatically detecting disadvantaged groups through algorithmic fairness metrics assessment [AAK+ 20, AAT22]. These studies then focus on addressing bias by targeting improvements in algorithm performance specifically for the disadvantaged group, as dynam-

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ically indicated by disparities in error rates. For instance, Anahideh et al. [AAT22] rely on the identification of instances that contribute most effectively to reducing disparities in a given metric. This approach first identifies the disadvantaged group based on a fairness metric and then seeks to enhance model performance for the group with the worst model performance. However, approaches rooted in data-driven analysis to discern the disadvantaged group often overlook contextual elements and historical injustices. This oversight becomes particularly concerning when historical biases lead to a biased labeling process. In such cases, there exists a potential for incorrectly identifying the marginalized group, which can lead to counterproductive efforts in bias mitigation [LDAST22]. 3.2. Misguide model selection. In addition to the potential consequences of misidentifying the disadvantaged group, an equally significant concern associated with neglecting label bias is the potential misguidance in the process of model selection. Consider a scenario in which multiple ML models are being evaluated, each vying for selection as the best-performing model. These algorithms exhibit comparable levels of accuracy in their predictions, while fairness metrics play a critical role in this decisionmaking process. These fairness metrics are used to gauge the performance of each algorithm in terms of bias mitigation. If label bias exists but is disregarded, the entire process of model selection becomes vulnerable to distortion. Label bias can significantly impact the apparent performance of algorithms, potentially favoring the majority group, and further marginalizing the already disadvantaged group. Selecting the top-performing model through evaluations reliant on the biased label can inadvertently

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favor models replicating the same errors present in the biased training label [LDAST22]. 3.3. Mislead data collection. A recent body of research proposes to address algorithmic bias through data collection. Early research in algorithmic fairness often attributed biased predictions to representation issues of the training dataset. While this is an important problem that leads to algorithmic bias, as discussed in Section 1, it is only one of many potential sources of bias. Due to the substantial expenses associated with data labeling, active learning learning techniques are commonly employed to facilitate the data collection process. These techniques can be tailored to identify the most useful training instances for label acquisition, which are subsequently used to train machine learning models. The design of appropriate heuristics, often in the form of utility functions, depends on the specific objectives of the final predictive model. Recent work has proposed fairness-aware algorithms for data collection [AAT22, AAK+ 20]. For instance, Anahideh et al. (2022) [AAT22] propose “fair active learning” (FAL), an active learning algorithm that proposes a utility function for selecting prospective training instances that aims to improve fairness metrics in addition to overall model accuracy. FAL selects instances for labeling based on both uncertainty-based Shannon entropy and expected improvement in fairness. In their experiments, FAL demonstrated a significant reduction in model bias while maintaining accuracy, as evaluated using the observed label [AAT22]. The advancement of fairness-aware active learning methodologies for data acquisition introduces a promising and dynamic strategy. Nevertheless, overlooking potential label bias can reduce the efficacy of these methods, and in some cases, may even exacerbate the existing bias they aim to mitigate. Li et al. (2022) [LDAST22] showed that incorporating active learning techniques, such as FAL, for additional data collection can exacerbate bias. The reason behind this is intuitive: if the label used for calculating the utility scores exhibit bias, the utility score becomes a biased measure. Furthermore, acquiring more instances with label bias may reinforce the model’s reliance on biased patterns rather than mitigating the intended bias.

4. Conclusion Label bias is a pervasive issue in the domain of machine learning. We have presented concrete examples where algorithmic bias emerges due to label bias, particularly in important domains such as healthcare, predictive policing, and content moderation. However, significant efforts to address algorithmic bias still overlook the presence of label bias and rely on “fairness” measures that assume the observed label perfectly aligns with the ground truth la-

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bel. When approaches designed to mitigate inductive bias are used in contexts where label bias is a core issue, inadvertent consequences may transpire. These consequences include misidentifying disadvantaged groups, misguiding model selection, and misleading data collection. We highlight the importance of distinctly identifying the various forms of bias that fairness-aware algorithms endeavor to mitigate, and underscore the necessity of developing algorithms that explicitly aim to address label bias. 4.1. Paths forward. While awareness of label bias in ML systems continues to grow, existing methods aimed at mitigating this bias are still inadequate. This highlights the need for further research and development in this area. We outline multiple paths forward to tackle the problem. A sociotechnical perspective. In some cases, label bias arises due to the difficulty of precisely collecting the ground truth labels for training an ML system, which closely relates to a significant challenge: problem formulation. Problem formulation in the context of ML refers to the process of defining and shaping a specific question or task that can be addressed using computational techniques and predictive models. It involves translating high-level objectives, goals, or real-world challenges into well-defined and tractable problems that can be answered using ML algorithms [PB19]. The target variable selected for prediction is central to problem formulation. A problem formulation process that considers the historical inequalities and societal biases in the choice of target variable can help alleviate the algorithmic bias and associated consequences [PB19]. For example, in the context of predicting health needs, researchers were able to mitigate algorithmic bias by changing the target label: rather than predicting health spending, they created a more holistic variable based on cost and health information [OPVM19]. Crucially, algorithmic audits must also adopt a sociotechnical perspective. Solely focusing on fairness measures without considering if the label used to create these measures may contain bias risks invisibilizing harms. Improving measurement. Throughout this article, we have focused on group fairness measures. This choice was made given the popularity of such measures, which is arguably due to the feasibility of applying them in practice. However, alternative measures of fairness have been proposed. For example, some causal notions of fairness aim to assess how predictions would change under different counterfactuals [MPB+ 21]. Fundamental issues on the validity of certain counterfactuals have been raised, and practical issues have largely prevented such measures from being used in practice, since we rarely have knowledge of the causal relationship between different covariates. However, many

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of these notions do not necessitate the absence of label bias. Further research on alternative measures of fairness may thus enable better assessments of algorithmic bias that stems from label bias. Improving mitigation. Moreover, for bias in human-generated labels, identifying the set of instances that have been mislabeled in the training dataset can be an effective way of preventing the propagation of societal bias through ML systems [LDAS23]. For instance, Li et al. (2023) [LDAS23] propose a pruning method—Decoupled Confident Learning (DeCoLe)—designed to mitigate label bias. DeCoLe identifies instances likely to be mislabeled based on groupand class-conditional label uncertainty and prunes them to create a training dataset with less bias. DeCoLe focuses on addressing a particular form of label bias, where errors in the label are conditioned on class and group. There is a need for additional algorithmic approaches to tackle different forms of label bias. In conclusion, addressing label bias in machine learning requires a multifaceted approach. Firstly, a proactive strategy involves avoiding label bias from the start by carefully formulating the machine learning task. This necessitates a keen awareness of potential biases and their implications throughout the problem definition stage. Secondly, when considering bias or fairness mitigation methods, it is crucial to critically evaluate the assumptions underlying these techniques. Many existing methods assume label accuracy, which may lead to suboptimal outcomes in real-world scenarios where mislabeling is prevalent. By being mindful of these assumptions, researchers and practitioners can make more informed choices in implementing mitigation strategies. Finally, to advance the state-ofthe-art, there is a need for the development of novel and improved bias measures as well as label mitigation methods. This involves exploring innovative approaches that go beyond existing paradigms and contribute to the ongoing evolution of techniques for ensuring fairness and reducing bias in machine learning systems. ACKNOWLEDGMENT. This research was supported in part by NIH grant R01NS124642 and by Good Systems, a UT Austin Grand Challenge to develop responsible AI technologies. References

[AAK+ 20] Jacob Abernethy, Pranjal Awasthi, Matthäus Kleindessner, Jamie Morgenstern, Chris Russell, and Jie Zhang, Active sampling for min-max fairness, arXiv preprint arXiv:2006.06879 (2020). [AAT22] Hadis Anahideh, Abolfazl Asudeh, and Saravanan Thirumuruganathan, Fair active learning, Expert Systems with Applications 199 (2022), 116981. 1076

[ADAC21] Nil-Jana Akpinar, Maria De-Arteaga, and Alexandra Chouldechova, The effect of differential victim crime reporting on predictive policing systems, Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, 2021, pp. 838–849. [ALMK16] Julia Angwin, Jeff Larson, Surya Mattu, and Lauren Kirchner, Machine bias (2016). [BBB+ 21] Chlo´e Bakalar, Renata Barreto, Stevie Bergman, Miranda Bogen, Bobbie Chern, Sam Corbett-Davies, Melissa Hall, Isabel Kloumann, Michelle Lam, Joaquin Quinonero ˜ Candela, and others, Fairness on the ground: Applying algorithmic fairness approaches to production systems, arXiv preprint arXiv:2103.06172 (2021). [BG18] Joy Buolamwini and Timnit Gebru, Gender shades: Intersectional accuracy disparities in commercial gender classification, Conference on fairness, accountability and transparency, 2018, pp. 77–91. [DAKD23] Aida Mostafazadeh Davani, Mohammad Atari, Brendan Kennedy, and Morteza Dehghani, Hate speech classifiers learn normative social stereotypes, Transactions of the Association for Computational Linguistics 11 (2023), 300– 319. [Gar16] Clare Garvie, The perpetual line-up: Unregulated police face recognition in america, Georgetown Law, Center on Privacy & Technology, 2016. [HPS16] Moritz Hardt, Eric Price, and Nati Srebro, Equality of opportunity in supervised learning, Advances in neural information processing systems 29 (2016). [KBK+ 12] Brendan F Klare, Mark J Burge, Joshua C Klontz, Richard W Vorder Bruegge, and Anil K Jain, Face recognition performance: Role of demographic information, IEEE Transactions on Information Forensics and Security 7 (2012), no. 6, 1789–1801. [LDAS23] Yunyi Li, Maria De-Arteaga, and Maytal SaarTsechansky, Mitigating label bias via decoupled confident learning, arXiv preprint arXiv:2307.08945 (2023). [LDAST22] Yunyi Li, Maria De-Arteaga, and Maytal SaarTsechansky, When more data lead us astray: Active data acquisition in the presence of label bias, Proceedings of the AAAI Conference on Human Computation and Crowdsourcing, 2022, pp. 133–146. [LI16] Kristian Lum and William Isaac, To predict and serve?, Significance 13 (2016), no. 5, 14–19. [LKL+ 17] Himabindu Lakkaraju, Jon Kleinberg, Jure Leskovec, Jens Ludwig, and Sendhil Mullainathan, The selective labels problem: Evaluating algorithmic predictions in the presence of unobservables, Proceedings of the 23rd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, 2017, pp. 275–284. [MPB+ 21] Shira Mitchell, Eric Potash, Solon Barocas, Alexander D’Amour, and Kristian Lum, Algorithmic fairness: choices, assumptions, and definitions, Annu. Rev. Stat. Appl. 8 (2021), 141–163, DOI 10.1146/annurev-statistics-042720-125902. MR4243544

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[NDAF22] Terrence Neumann, Maria De-Arteaga, and Sina Fazelpour, Justice in misinformation detection systems: An analysis of algorithms, stakeholders, and potential harms, Proceedings of the 2022 ACM Conference on Fairness, Accountability, and Transparency, 2022, pp. 1504–1515. [OPVM19] Ziad Obermeyer, Brian Powers, Christine Vogeli, and Sendhil Mullainathan, Dissecting racial bias in an algorithm used to manage the health of populations, Science 366 (2019), no. 6464, 447–453. [PB19] Samir Passi and Solon Barocas, Problem formulation and fairness, Proceedings of the conference on fairness, accountability, and transparency, 2019, pp. 39–48. [SCG+ 19] Maarten Sap, Dallas Card, Saadia Gabriel, Yejin Choi, and Noah A Smith, The risk of racial bias in hate speech detection, Proceedings of the 57th Annual Meeting of the Association for Computational Linguistics, 2019, pp. 1668–1678. [Swe13] Latanya Sweeney, Discrimination in online ad delivery, Communications of the ACM 56 (2013), no. 5, 44–54.

Yunyi Li

Maria De-Arteaga

Maytal Saar-Tsechansky Credits

Figures 1 and 2 are courtesy of the authors. Photo of Yunyi Li is courtesy of Jian Teng. Photo of Maria De-Arteaga is courtesy of the University of Texas at Austin. Photo of Maytal Saar-Tsechansky is courtesy of Lauren Gerson, McCombs School of Business, The University of Texas at Austin.

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Double-Anonymous Peer Review in Mathematics: Implementation for American Mathematical Society Journals Dan Abramovich, Henry Cohn, David Futer, and Robert Harington In March 2022, the American Mathematical Society (AMS) launched double-anonymous peer review across its journal program, beginning with Proceedings of the American Mathematical Society and Representation Theory. In February 2024 implementation expanded to the Transactions and Memoirs of the American Mathematical Society. Further rollout across AMS journals is ongoing. The goal of the double-anonymous peer review policy is to reduce implicit bias in peer review, including bias along gender, racial, and geographical lines, along with seniority bias. Peer review in mathematics has traditionally centered on single-anonymous peer review. In this model, reviewers are aware of the identities of an article’s author(s), but the reviewer(s) remain anonymous to authors. In double-anonymous peer review (formerly known as Dan Abramovich serves as managing editor of the Transactions and Memoirs of the AMS. He is L. Herbert Ballou University Professor of Mathematics at Brown University. His email address is [email protected]. Henry Cohn is a principal researcher at Microsoft, and an adjunct professor of mathematics at MIT. His email address is [email protected]. David Futer serves as managing editor of the Proceedings of the AMS. He is a professor of mathematics at Temple University. His email address is david [email protected]. Robert Harington is the chief publishing officer at the American Mathematical Society. His email address is [email protected]. For permission to reprint this article, please contact: [email protected]. DOI: https://doi.org/10.1090/noti3013

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double-blind), both reviewers and authors are anonymous to each other. Peer review is central to scholarship and is deployed across all academic disciplines. As Melinda Baldwin discusses in her excellent article [2] (see similar discussion in [1]): The most widely accepted story about peer review’s origin credits Henry Oldenburg with inventing it for the seventeenth-century Philosophical Transactions of the Royal Society, creating the impression that refereeing has been an unchanging part of science for over three hundred years. However, new historical work is beginning to shed more light on peer review’s development—and the real story is far more complicated than the neat tale of Oldenburg inventing refereeing out of whole cloth during the Scientific Revolution. Most existing histories of peer review have focused on the emergence of the scientific referee during the nineteenth century or on the inner workings of referee systems at particular journals. Those studies have shown that refereeing was not initially thought of as a process that bestowed scientific credibility and that many high-profile journals and grant organizations had unsystematic (or nonexistent) refereeing processes well into the twentieth century.

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Across much of the social sciences and humanities, double-anonymous peer review is the standard system of peer review. It is also becoming increasingly common in the natural sciences. In physics, the Institute of Physics Publishing (IOPP) launched double-anonymous peer review across all of its sixty-one journals in 2021 [3]. In astronomy, NASA practices double-anonymous peer review in its grant applications. In certain subdisciplines of computer science, including cryptography, double-anonymous peer review is quite common. Why did the AMS move to implement doubleanonymous peer review across its journals? The motivation for the AMS Council to adopt this change is so the referee’s first impressions of a paper are not dominated by its list of authors and their affiliations. Instead, double-anonymous refereeing aims to focus attention on the mathematics. It may still be possible to infer who wrote a submission, but it is hoped that double-anonymity lowers the likelihood of implicit bias and therefore supports inclusivity and diversity across mathematics. The AMS Council approved the transition to doubleanonymous peer review as a policy in January 2021. The AMS President, Ruth Charney, subsequently formed the Double-Anonymous Refereeing Committee, on which we served, charged with discussing how an implementation could work. During implementation discussions, it became clear that in mathematics, this form of peer review needed to be implemented with a light touch. While authors are required to submit manuscripts without author names or affiliations, they are not required to anonymize references, acknowledgments, or funding information, or to make other edits. Journal editors continue to have access to author and reviewer identities in double-anonymous peer review. In particular, editors can use this information in order to avoid conflicts of interest. The new policy is guided by the AMS’s strong belief in open dissemination of mathematics. There are no new restrictions on how authors choose to disseminate and publicize their work—for example, by giving talks, posting preprints that include author names, and discussing with colleagues. Referees are asked not to go out of their way to try to identify authors, but the AMS policy accepts that referees will sometimes already know who wrote the submission. For instance, a referee may already know the paper if they have seen it posted to the arXiv preprint server. Even in those cases, double-anonymous refereeing is a statement of principle about how submissions should be evaluated. Authors submitting articles to AMS journals employing double-anonymous peer review are tasked with the following tasks that represent a light touch to doubleanonymous peer review.

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At article submission, authors are asked to submit a version which does not include their names or affiliations in the preamble, headers, or footnotes of the paper. Authors may also choose to reword other instances in the paper that would tend to identify them, but this is not required. For example, a phrase like “we showed” or “the second author showed” could be replaced by “Smith showed,” referring to the author in the third person. If a paper is accepted, such wording can be adjusted prior to publication. At initial submission, AMS staff screens for author names in the paper itself, in the running heads, and in the affiliations list. A typical communication used in requests and reminders typically reads: “The PDF must not contain any information that can identify the author(s). In particular, there must be no authors’ names nor authors’ affiliations/email addresses listed in the paper, nor any links to their identities in any way.” In addition, identifiable versions are never released to referees. Implementation of this light-touch approach to doubleanonymous peer review is underway. As the AMS rolls this peer review model out across the journal portfolio, it is important to gather data and assess the effects of doubleanonymous peer review on authors, reviewers, and editors. The AMS is in the early stages of developing an organization-wide approach to demographic data collection for both our membership and publishing/research communities. In order to monitor the existence of many types of bias in our research community this data will need to be collected both at submission of a manuscript and post decision. In the meantime, we can choose to look at geographical data and institutional data. The data to be collected will help the AMS see to what extent the goals of the policy are achieved. But, first and foremost, we need to know that no serious harm is done. Concerns raised by journal editors included: (i) Will people continue to agree to provide expert opinions and write detailed referee reports? (ii) Will authors continue to submit papers appropriate to the journal’s portfolio? (iii) Will colleagues continue to agree to serve as editors? A previous introduction of a double-anonymous policy in 1975 was abandoned in 1980 on these counts [4]. After double-anonymous peer review was rolled out to Proceedings and Representation Theory, we were able to examine the first two of these concerns. Both Proceedings and Representation Theory transitioned to double-anonymous peer review in 2022. As illustrated in Figure 1, double-anonymous peer review does not seem to have had any impact that sets them apart from our other journals.

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[4] Everett Pitcher, A history of the second fifty years, American Mathematical Society, 1939–1988, Vol. I, American Mathematical Society, Providence, RI, 1988, DOI 10.1007/bf01017168. MR1002190

Figure 1.

Figure 2.

Dan Abramovich

Henry Cohn

David Futer

Robert Harington

Credits

Figure 2 is a preliminary look at reviewer denial data for the two AMS journals utilizing double-anonymous peer review, versus all other AMS journals. This data indicates that reviewers are not turning down an opportunity to review for the AMS due to journals embracing double-anonymous peer review. For the third concern (editors’ willingness to serve), the data sample is too small to conduct statistics. Suffice it to say that the AMS Editorial Boards Committee did not face difficulties in filling editorial vacancies in the two journals in which the policy was implemented.

Figures 1 and 2 are courtesy of the AMS. Photo of Dan Abramovich is courtesy of Deidre Confar. Photo of Henry Cohn is courtesy of MFO. CC BY-Sa 2.0 DE. Photo of David Futer is courtesy of the AMS. Photo of Robert Harington is courtesy of the AMS.

References

[1] Melinda Baldwin, In referees we trust?, Physics Today 70 (2017), no. 2, https://doi.org/10.1063/PT.3 .3463. [2] Melinda Baldwin, Scientific Autonomy, Public Accountability, and the Rise of “Peer Review” in the Cold War United States, Isis 109 (2018), no. 3, https://www.journals .uchicago.edu/doi/full/10.1086/700070. [3] Rachael Harper, IOP Publishing commits to adopting double-anonymous peer review for all journals (2020), https://ioppublishing.org/news/iop -publishing-commits-to-adopting-double-blind -peer-review-for-all-journals/.

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WASHINGTON UPDATE

The AMS Marks Twenty Years of Sending Mathematicians to Congress Karen Saxe In the fall of 2005, mathematician David Weinreich joined a few dozen other scientists—each sponsored by a scientific society—bringing their expertise to Congress. David was the first Congressional Fellow sponsored by the AMS, though the program began in 1973, with a class of seven Fellows. AMS Congressional Fellows are part of a larger program bringing scientific expertise to the US government; the 30– 35 Congressional Fellows are joined by about 250–275 who work in executive branch agencies. There is also one Fellow placed in the judicial branch. Fellows bring scientific expertise to government decision-makers. The question of where Congress gets their science and tech information is a great one and has a complex answer. The first answer is that members of Congress have science policy advisors, and some have strong scientific backgrounds. From 1974–1995, the Office of Technology Assessment (OTA) served as a primary support, providing Congress with objective analyses of science and technology issues.1 This Karen Saxe is senior vice president, Government Relations, at the AMS. Her email address is [email protected]. For permission to reprint this article, please contact: [email protected].

office influenced legislation and fostered relationships between Congress and the scientific and technological community. Congressional Fellows help fill the gap left by the OTA’s disbandment. After the fellowship year, AMS Congressional Fellows follow different career paths. Almost all AMS Fellows have come from academia. A few returned to academia, but most have not. The AMS Congressional Fellowship has proved transformational for the careers of individual mathematicians for twenty years. 2005–2006 Fellow David Weinreich focused on a wide range of issues during his year working for Representative Robert Andrews from southwestern New Jersey, issues from agriculture to water resources. One success during his fellowship was a provision of law that prevented logging in Alaska’s Tongass National Forest. His fellowship was followed by full-time employment in the US House of Representatives. For four years he was the Legislative Director for Representative Bob Etheridge of North Carolina, followed by helping establish the office for first-year Representative Hansen Clarke of Detroit as Policy Director. In 2011, he left Congress and founded a consulting firm, the Weinreich Strategic Group, and he is very active as the Director of Policy and Government Relations of

DOI: https://doi.org/10.1090/noti3003 1

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Washington Update STM.2 STM is the leading association of scholarly publishers and its members publish, roughly, two-thirds of all published papers in science, technology, medicine, social sciences, and humanities. “The fellowship not only transformed my career trajectory, but enabled me to bring my analytical skills and knowledge of the research community to inform public policy,” David said. “It also led to many productive conversations with fellow mathematicians about funding policy and how to effect change in the government.” 2019–2020 Fellow Lucia Simonelli is a senior climate researcher at Giving Green. Following her year working for Rhode Island’s Senator Sheldon Whitehouse on climate and energy policy, she transitioned to full-time work in climate. “The AMS Congressional Fellowship enabled me to witness the importance of integrating scientists into policymaking and reflect on how the skills derived from mathematical training can transfer to a broader context. The experience helped restore my respect for, and faith in the government, and it taught me the power and importance of creating a strong network—not for self-gain, but more as a collective that can work together in various ways to support the advancement of common goals and causes.” 2021–2022 Fellow AJ Stewart, who spent his fellowship year working on economic policy for Georgia’s Senator Raphael Warnock, is now a policy advisor at the US Department of the Treasury. There, he investigates national security issues stemming from foreign investment in the United States. “I was always good at math and even though it took me a while to find my way towards becoming a mathematician, once I did I was hooked. However, I assumed that there was only one way to be a mathematician, by performing research at a university. This fellowship opened my eyes to how mathematics is applied across government in amazingly unique ways. Being able to tangibly apply mathematics every day and see the effects of that work has changed my whole view of what it means to perform mathematics.” This short article highlights the post-fellowship paths of only three of the amazing AMS Congressional Fellow alumni group. Read more about all alumni fellows, and find out how you can join this wonderful group (applications due February 1), at: https://www.ams.org/government /government/ams-congressional-fellowship.

Karen Saxe Credits

Photo of Karen Saxe is courtesy of Macalester College/David Turner.

2

The International Association of Scientific, Technical and Medical Publishers, known as STM, has more than 140 members across the globe including all the major commercial publishers, professional society publishers, and university presses. The AMS is a member.

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Joint Prizes JPBM Communications Award This award is given each year to reward and encourage communicators who, on a sustained basis, bring mathematical ideas and information to nonmathematical audiences. About this award. This award was established by the Joint Policy Board for Mathematics (JPBM) in 1988. JPBM is a collaborative effort of the American Mathematical Society, the Mathematical Association of America, the Society for Industrial and Applied Mathematics, and the American Statistical Association. Up to two awards of US$2,000 are made annually. Both mathematicians and nonmathematicians are eligible. Next prize. January 2025 Nomination period. Open Nomination procedure. Nominations should be submitted on MathPrograms.org. Note: Nominations collected before September 15 in year N will be considered for an award in year N+2. Information on how to nominate can be found here: https://www.ams.org/jpbm-comm-award.

Fellowships and Programs Joan and Joseph Birman Fellowship for Women Scholars The Joan and Joseph Birman Fellowship for Women Scholars is a midcareer research fellowship specially designed to fit the unique needs of women. This program is made possible by a generous gift from Joan and Joseph Birman. One award will be made for the 2024–2025 academic year in the amount of US$50,000. AMS membership will also be offered to the recipient for the duration of the fellowship. About this fellowship. The fellowship seeks to address the paucity of women at the highest levels of research in

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mathematics by giving exceptionally talented women extra research support during their midcareer years. The most likely awardee will be a midcareer woman whose achievements demonstrate significant potential for further contributions to mathematics. Applications will be accepted from mathematicians currently holding a tenured, tenuretrack, postdoctoral, or comparable (at the discretion of the selection committee) position at a US institution. The fellowship will be directed toward those for whom the award will make a real difference in the development of their research career. Candidates must have a statement regarding the applicant’s overall program of research, past and planned, that is meaningful to mathematicians who are not specialists. The statement should be no more than three pages, including bibliographical references. Special circumstances (such as time taken off for care of children or other family members) may be taken into consideration in making the award. Awardees may use the fellowship in any way that most effectively enables their research—for instance, for release time, participation in special research programs, travel support, childcare, etc. The award is issued through the recipient’s institution, and no part of it may be utilized for indirect costs. Application period. Applications will be collected via MathPrograms.org July 15, 2024–September 30, 2024 (11:59 p.m. ET). Find more information at https://www .ams.org/birman-fellow. For questions, contact the Programs Department at [email protected].

Centennial Research Fellowship The AMS Centennial Fellowship Program makes an award annually to an outstanding mathematician to help further their career in research. One award will be made for the 2024–2025 academic year in the amount of US$50,000. Acceptance of the fellowship cannot be postponed. AMS membership will also be offered to the recipient for the duration of the fellowship.

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

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About this fellowship. Eligibility: The eligibility rules are as follows: The primary selection criterion for the Centennial Fellowship is the excellence of the candidate’s research. • Preference will be given to candidates who have not had extensive fellowship support in the past. • Recipients may not hold the Centennial Fellowship concurrently with another research fellowship such as a Sloan, NSF Postdoctoral fellowship, or CAREER award. • Under normal circumstances, the fellowship cannot be deferred. • A recipient of the fellowship shall have held his or her doctoral degree for at least three years and not more than twelve years at the inception of the award (that is, received between September 1, 2013, and September 1, 2022). • Applications will be accepted from mathematicians currently holding a tenured, tenure-track, postdoctoral, or comparable (at the discretion of the selection committee) position at a US institution. Applications should include a detailed research plan for the fellowship period that is contextualized by the research statement. The plan should include a description of how the fellowship will support the applicant’s success. It should be no more than one page. The selection committee will consider the plan in addition to the quality of the candidate’s research and will try to award the fellowship to those for whom the award would make a real difference in the development of their research careers. Work in all areas of mathematics, including interdisciplinary work, is eligible. Application period. Applications will be collected via MathPrograms.org July 15, 2024–September 30, 2024 (11:59 p.m. ET). Find more information at https://www .ams.org/centfellow. For questions, contact the Programs Department at [email protected].

Claytor-Gilmer Fellowship

may be utilized for indirect costs. Given the aims of the fellowship, the most likely awardee will be a midcareer Black mathematician whose achievements demonstrate significant potential for further contributions to mathematics. Applications will be accepted from mathematicians currently holding a tenured, tenure-track, postdoctoral, or comparable (at the discretion of the selection committee) position at a US institution. Application period. Applications will be collected via MathPrograms.org July 15, 2024–September 30, 2024 (11:59 p.m. ET). Find more information at https://www .ams.org/claytor-gilmer. For questions, contact the Programs Department at [email protected].

Stefan Bergman Fellowship The Stefan Bergman Fellowship was established in 2023 with the proceeds of the Stefan Bergman Trust to support the advancement of the research portfolio of a mathematician who specializes in the areas of real analysis, complex analysis, or partial differential equations. One award will be made for the 2024–2025 academic year in the amount of US$25,000. AMS membership will also be offered to the recipient for the duration of the fellowship. About this fellowship. Applications will be accepted from mathematicians at a US institution who have not received tenure or comparable (at the discretion of the selection committee) and have not held significant fellowship support. Awardees may use the fellowship in any way that most effectively enables their research—for instance, for release time, participation in special research programs, travel support, childcare, etc. The award is issued through the recipient’s institution, and no part of it may be utilized for indirect costs. Application period. Applications will be collected via MathPrograms.org July 15, 2024–September 30, 2024 (11:59 p.m. ET). Find more information at https://www .ams.org/bergman-fellow. For questions, contact the Programs Department at [email protected].

The AMS established the Claytor-Gilmer Fellowship to further excellence in mathematics research and to help generate wider and sustained participation by Black mathematicians. One award will be made for the 2024–2025 academic year in the amount of US$50,000. AMS membership will also be offered to the recipient for the duration of the fellowship. About this fellowship. Awardees may use the fellowship in any way that most effectively enables their research—for instance, for release time, participation in special research programs, travel support, childcare, etc. The award is issued through the recipient’s institution, and no part of it SEPTEMBER 2024

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Executive Director Report

Lucy Maddock, Interim Executive Director

DOI: https://doi.org/10.1090/noti2992

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Both transition and growth marked 2023 for the American Mathematical Society (AMS). After seven years as AMS executive director (ED), Dr. Catherine Roberts, resigned in March, and the next several months were dedicated to transition planning. In May, the AMS Board of Trustees (BT) appointed me to serve as interim ED, in addition to my ongoing position as chief financial officer (CFO). Meanwhile, an Executive Director Search Committee began a nine-month, nationwide search. In January 2024, the BT announced the appointment of Dr. John Meier, provost and David M. and Linda Roth Professor of Mathematics of Lafayette College, as the new ED. He will begin a five-year term on July 1, 2024, and we are extremely excited to welcome him aboard. Like elsewhere, change has been a consistent theme at the AMS for the past four years. As the world navigated challenges of the COVID-19 pandemic, many companies experienced increased employee turnover in a period known as the “Great Resignation.” The AMS was no exception; however, employee numbers have since stabilized. Particularly noteworthy is that 2023 marked the year with the lowest AMS employee turnover since 2018, and we expect this stabilization to continue in 2024. In August, the BT approved a new organizational structure for the Society upon the recommendation of RW Jones, a strategic consulting firm focused on education. As a result, seven AMS divisions were transformed into three, allowing for greater collaboration and a streamlined reporting hierarchy. Additionally, Ashley Northington, MPA, the senior vice president and managing director of RW Jones, was brought on as the AMS’s interim chief external relations officer. Despite these ongoing transitions, staff have successfully adapted and remain steady in supporting the AMS mission of advancing research and connecting the mathematical community.

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In January, the first in-person reimagined Joint Mathematics Meetings (JMM) (and first in-person annual meeting since January 2020) took place in Boston, with sixteen partner societies and 5,140 participants. The Meetings Department worked closely with partners to arrange their programs and exhibits, including soliciting their input for the grand opening reception. Meeting in-person again was an enormous success, and seventeen partners were secured for 2024. The AMS also held seven sectional meetings in Atlanta; Cincinnati; Fresno, California; Buffalo, New York; Omaha, Nebraska; and Mobile, Alabama. The spring Eastern Sectional was held virtually. Growth and retention in membership were notable achievements in 2023. There were 6,769 new member enrollments, a record number and 1,400 more than in 2022. This growth includes 2,250 new dues-paying members. Increased efforts in membership retention paid off with 952 more members renewing in 2023 than in 2022. This will continue to be an area of focus. Meanwhile, the AMS continued to expand its services and resources for the mathematical community. Two new programs were launched: the Stefan Bergman Fellowship and the AMS-Simons Research Enhancement Grants for PUI Faculty, and five new prizes were established: the Ivo and Renata Babuˇska Thesis Prize (awarded in 2024), the Elias M. Stein Prize for New Perspectives in Analysis (awarded in 2024), the Elias M. Stein Prize for Transformative Exposition (to be awarded in 2025), the Elias M. Stein Mentoring Award (to be awarded in 2026), and the I. Martin Isaacs Prize for Excellence in Mathematical Writing (to be awarded in 2025). Additionally, the Office of Equity, Diversity, and Inclusion and the Division of Meetings and Professional Services (now both part of External Affairs) secured NSF supplemental funding for a new AMS partnership with the Inclusive Graduate Education Network (IGEN), an alliance of disciplinary societies, research centers, and other organizations dedicated to advancing equity in STEM graduate education. The proposed IGEN Mathematics Initiative (IGEN-Math) is a one-year national capacity-building project. It’s team will work closely with internal and external stakeholders, including IGEN Alliance partner organizations and the IGEN-Math Advisory Group, to engage the mathematics community in developing the framework for a centralized hub of bridge programs in mathematics intended to improve equity and inclusion in mathematics graduate education. As part of the AMS’s ongoing accessibility initiative, MathViewer (accessible HTML) was expanded to include all primary journals and added to ePub production workflows, beginning with a retrospective conversion of the Graduate Studies in Mathematics series. The MathViewer

SEPTEMBER 2024

journal article output increased from 275 articles to 939. The AMS is now in full compliance with existing open access (OA) journal mandates through zero-embargo Green OA, Diamond OA, and Gold OA (on the B journals). In addition, the new user interface of MathSciNet was released in June, the first major revision since 2006. The updated interface incorporates a modern look, greater use of the database to help users refine their searches, and improved accessibility, especially for users with vision impairments or fine motor control limitations. Each year, speakers bring science directly to Capitol Hill via congressional briefings organized by the Office of Government Relations (OGR). These speakers offer stories of how federal investment in basic research in math and science pays off for American taxpayers and helps the nation remain a world leader in innovation. Beginning in 2023, each briefing highlighted work connected to one of the National Science Foundation (NSF)-funded Mathematical Sciences Institutes; last year’s was in partnership with the Institute for Pure and Applied Mathematics (IPAM). The AMS helped organize and host the following briefings with various coalitions: “Investing to Win: The Essential Role of Federally Funded Research,” “Federally Funded Research and the Advent of Artificial Intelligence: A TFAI Deconstructing Event,” “The National Imperative to Develop STEM Talent: Why the Investment in Education Matters,” and “STEM 101.” The Office of Government Relations also made more than 100 visits to Congressional and Executive Branch offices in 2023, both with AMS leadership and in partnership with various coalitions. In addition, it supported the work of the Advisory Group on Artificial Intelligence (AI) and the Mathematical Community, which is charged with focusing on issues at the forefront of developments in AI, including: the role of mathematics in the development and deployment of artificial intelligence, the use of AI in publications, education, and research, and its impact on research in mathematics and our community. As OGR built relationships in Washington, DC, our Communications and Marketing Department focused on building community and conveying the AMS brand through the organization’s messaging. Together with Creative Services, they implemented a marketing and promotion campaign planning process to help departments meet their messaging objectives and to advance AMS strategic priorities. This process created a cohesive visual and messaging brand for all the pieces of a campaign, including social media, advertising, meeting materials, brochures, website graphics, and email. Additionally, by planning, creating, and implementing content for each type of platform, there was a 234-percent increase in users of AMS social

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platforms and a 221-percent upsurge in impressions, the number of times AMS content was seen. Led by the Secretary’s Office, the AMS is conducting a comprehensive analysis of the bylaws to ensure that they align with the current legal landscape, that they serve the organization as well as possible, and that they reflect the AMS’s values. As part of this analysis, legal counsel was asked to assist in bringing the bylaws into compliance with ambient law and to make additional suggestions for improvement. A Bylaws Review Taskforce was created, and its recommendations will be brought to Council in January 2025. The Secretary’s Office also worked with Information Services at the AMS to create a search portal for members of the Committee on Committees and Nominating Committee to better illuminate past membership in AMS committees. This tool will also help in recruiting election candidates and committee members, and the hope is that it can eventually be expanded. In the area of planned giving, the AMS saw several estate gifts of more than one million dollars, following a decade of preparation and stewardship. With guidance and encouragement from the CFO and the Development Committee, the Development Department revised the gift acceptance policy and the AMS Book Fund to provide both more flexible and more enduring support for the AMS. Also, the hiring of a development communications officer has increased the AMS’s capacity to apply for grants from foundations and government agencies. In other highlights, beginning in 2022 and continuing into 2023, Human Resources designed and implemented a formal internship program for graduate and undergraduate students. The AMS will look to partner with local colleges to provide recurring opportunities for both workstudy students as well as traditional internship opportunities. This internship program allows students to become involved with the organization and connect with other members of the mathematical community. The AMS applied for the Employee Tax Retention Credit (ERTC) under the Coronavirus Aid, Relief, and Economic Security Act (CARES Act). This is a payroll tax credit aimed at employers impacted by the COVID-19 pandemic. As a result, the AMS will receive $2,727,926 in future credits. Additionally, some 1,100 statements and 800 payments went out to authors in mid-April for 2023 sales and royalties. More than $350,000 in royalty payments were processed. Finance Division staff continue to attempt to contact some 500 authors who are missing tax or other documentation needed to process payment. The Computer Sciences Division participated in two critical projects, one to become compliant with version four of the Payment Card Industry’s Data Security Standards (PCI-DSS) and a second to select and implement a

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software package for collection and remittance of sales tax in all states in which the AMS has nexus. In addition, more applications were integrated into Single Sign-On with multifactor authentication, and the network firewalls in Providence were upgraded to next-generation models with enhanced threat detection. Although 2023 was marked by both significant transitions and notable accomplishments, the AMS is wellpositioned to build upon its successes. Employee turnover stabilized, and the launch of a new internal organizational structure created a more collaborative environment for staff. The Society organized and held its first in-person JMM since January 2020 and saw achievements in accessibility, outreach, development, campaign planning, membership, programs offered to the math community, and more. Despite the challenges of 2023, the AMS thrived, and I look forward to what is ahead for the organization under Dr. Meier’s leadership. Lucy Maddock Interim Executive Director May 2024

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 71, NUMBER 8

MP: a-ballot-opener

FROM THE AMS SECRETARY

Election 2019Election 2024

Update in master page

Special Section Section Special 4 PICAS

Member at Large Member at LargeCS: ofBthe Council of the Council (five to be elected) (five to be elected) Alejandra Dan FreedAlvarado Benjamin Fernando Antieau Q. Gouvêa Dawei Chen D. Hacon Christopher Emily DanielClader Krashen Carla SusanCotwright Loepp Lenhard Ng Dan Isaksen Kasso A.Lai Okoudjou Yvonne Maria Cristina Pereyra Christopher J. Leininger Hal Schenck Adriana Salerno Melanie Matchett Pham Huu Tiep Wood

1 PICA

Ballots

Nominating Committee Nominating Committee (three to be elected) (three to be elected) Sami H. Assaf David Fisher Ricardo Cortez Aimee S.Garcia A. Johnson Rebecca Lily Signe Khadjavi Yongbin Ruan Kasso Savitt A. Okoudjou David Gigliola Staffilani Deane Yang — SINGLE LINE SPACE — Jared Wunsch Editorial Boards Committee (two to be elected) Editorial Boards Committee Ian (twoAgol to be elected) David Marker Ivan Corwin James McKernan Irene Fonseca Terence Tao Hacon Christopher Michael J. Larsen

Online Ballots ing online by August 20, or a paper ballot by September 20.

guishable from the original, will be sent by first class or airmail. However, the deadline for receipt of ballots will not be extended. Biographies of Candidates

If you do not receive this information by that date,for please AMS members will receive email with instructions votcontact the AMS (preferably before October 1) to request ing online by August 12. If you do not receive this inforcall the AMS a ballot. emailplease to [email protected] mation bySend that date, contact the AMSor (preferably beat 800-321-4267 (within the US or Canada) or 401-455fore October 1). Send email to [email protected] . The 4000 (worldwide). The deadline foris receipt of ballots is deadline for receipt of online ballots November 1, 2024. November 2, 2018. Starting in 2024, the AMS Election will be conducted fully online without the ability to receive a paper ballot.

The next several pages contain biographical information Biographies of Candidates PS: AHEAD about all candidates. All candidates were given the opporThe next several apages contain biographical tunity to provide statement of not more thaninformation 200 words about all candidates. All candidates were to given the oppor (400 words for presidential candidates) appear at thetunity to provide a statement of not more than 200 end of their biographical information. Photos werewords supto appear the end of their biographical information. plied by theatcandidates. Photos were supplied by the candidates.

AMS members will receive email with instructions for vot-

Write-in Votes It is suggested that names for write-in votes be accompaWrite-in Votes by the institution or web the individual for Itnied is suggested that names for address write-inofvotes be accompawhom the vote is cast. nied by the institution or web address of the individual for whom the vote is cast.

Description of Offices Description of Offices The vice president and the members at large of the Coun-

cil serve three years onthe the members Council. That bodyof deterThe viceforpresident and at large the mines all serve scientific policyyears of the createsThat and overCouncil for three onSociety, the Council. body sees numerous committees, appoints the treasurers determines all scientific policy of the Society, creates and and Replacement Ballots members of the Secretariat, makesappoints nominations of candioversees numerous committees, the treasurers A member who has not received a ballot by September 20, dates for future elections, and determines chief editors and members of the Secretariat, makes the nominations of 2018, or who has received a ballot but has accidentally of several key editorial boards. Typically, each of these new candidates for future elections, and determines the chief spoiled it, may write to [email protected] or Secretary of members the Council will alsoboards. serve on one of the Socieditors ofofseveral key editorial Typically, each of the AMS, 201 Charles Street, Providence, RI 02904-2213, ety’s six policy committees. Current and past members of these new members of the Council will also serve on one USA, asking for a second ballot. The request should inof the Society’s five policy committees. Current and past clude the individual’s member code and the address to of the Council may be found here: www.ams which the replacement ballot should be sent. Immediately SEPTEMBER 2024 NOTICES OF THE AMERICAN Mmembers ATHEMATICAL SOCIETY 1089 .org/comm-all.html#COUNCIL. upon receipt of the request a second ballot, indistin105

Notices of the American Mathematical Society

Update in

Volume 66, Number 1

PS: TEXT NO INDENT

Vice President Vice (onePresident to be elected) CS: I (one be elected) SaratoBilley Minerva Abigail Cordero Thompson Malabika — Pramanik SINGLE LINE SPACE — Board of Trustees Board (one toofbeTrustees elected) (one to be elected) Matthew Ando Rick Miranda Henry Cohn Brooke E. Shipley

PS: NEWS SUBHEAD

PS: TEXT NO INDENT

List of Candidates–2019 Election

List of Candidates

Election Special Section

the Council may be found here: https://www.ams.org

A Note from AMS Secretary Boris Hasselblatt

/comm-all.html#COUNCIL.

The choices you make in these elections impact the direction the Society takes in its publications, conferences, programs, and policies. On behalf of the other officers and Council members, I urge you to take a few minutes to review the candidates’ biographies, fill out your ballot, and submit it. The Society belongs to its members and by voting, you will influence its policies and priorities. Also, I invite you to consider other ways of participating in Society activities. The Nominating Committee, the Editorial Boards Committee, and the Committee on Committees are always interested in learning of members who are willing to serve the Society in various capacities. Names are always welcome, particularly when accompanied by a few words detailing the person’s background and interests. Self-nominations are probably the most useful. Recommendations can be transmitted through an online form (https://www.ams.org/committee-nominate) or sent directly to the secretary: [email protected] or Office of the Secretary, American Mathematical Society, 201 Charles Street, Providence, RI 02904-2213 USA.

The Board of Trustees, of whom you will be electing one member for a five-year term, has complete fiduciary responsibility for the Society. Among other activities, the trustees determine the annual budget of the Society, prices of journals, salaries of employees, dues (in cooperation with the Council), registration fees for meetings, and investment policy for the Society’s reserves. The person you elect will likely serve as chair of the Board of Trustees during the fourth year of the term. Current and past members of the Board of Trustees, as well as the full charge for a trustee, may be found here: https://www.ams.org/comm -all.html#BT. The candidates for vice president, members at large, and trustee were suggested to the Council either by the Nominating Committee or by petition from members. While the Council has the final nominating responsibility, the groundwork is laid by the Nominating Committee. The candidates for election to the Nominating Committee were nominated by the current president, Bryna Kra. The three elected will serve three-year terms. The main work of the Nominating Committee takes place during the annual meeting of the Society, during which it meets over three days. The Committee then reports its suggestions to the spring Council, which makes the final nominations. Current and past members of the Nominating Committee, as well as the full charge, may be found here: https://www .ams.org/comm-all.html#NOMCOM. The Editorial Boards Committee is responsible for the staffing of the editorial boards of the Society. Members are elected for three-year terms from a list of candidates named by the president. The Editorial Boards Committee makes recommendations for almost all editorial boards of the Society. Managing editors of Communications of the AMS, Journal of the AMS, Mathematics of Computation, Proceedings of the AMS, and Transactions of the AMS; and chairs of the Colloquium, Mathematical Surveys and Monographs, and Mathematical Reviews editorial committees are officially appointed by the Council upon recommendation by the Editorial Boards Committee. In virtually all other cases, the editors are appointed by the president, again upon recommendation by the Editorial Boards Committee. Current and past members of the Editorial Boards Committee, as well as the full charge, may be found here: https://www.ams.org/comm-all.html#EBC. Elections to the Nominating Committee and the Editorial Boards Committee are conducted by the method of approval voting. In the approval voting method, you can vote for as many or as few of the candidates as you wish. The candidates with the greatest number of the votes win the election.

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NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 71, NUMBER 8

MP: a-ballot-opener

FROM THE AMS SECRETARY

Election 2019Election 2024

Update in master page

SpecialBiographies Section Candidate 4 PICAS

ology, UT Southwestern, Dallas, TX, March 2023; SemiVice PS: NEWS SUBHEAD ListPresident of Candidates–2019 Election fields and other finite algebraic structures in coding algorithms, UT Arlington MAA Student Chapter 20th AnniverMember at Large Nominating Committee sary Celebration, March 2023; Advocating For Diversity in (one to be elected) CS: I Professor of Mathematics of the Council (three to be elected) B and CS: Mathematics from a Personal and Professional Perspective, Sara Billey to be elected) Sami H. Assaf Vice Provost for (five Faculty Success, EDGE25, Celebrating 25 years of EDGE, Bryn Mawr ColAbigail Thompson Freedat ArRicardo Cortez The University Dan of Texas lege, October 2023. Rebecca Garcia Fernando Q. Gouvêa — SINGLE LINE SPACE — lington Synergistic activities: M. Ruan Cordero, T. Jorgensen, and B. Board of Trustees Christopher D. Hacon Yongbin PhD: University of Iowa, Shipman, DesigningDavid contracts and honors thesis projects (one to be elected) Daniel Krashen Savitt 1989. in mathematics, Chapter in The other culture: Science Matthew Ando Susan Loepp Deane 13 Yang Selected publications or and mathematics education in honors,LINE Ed. E. —B. Buckner — SINGLE SPACE Rick Miranda Lenhard Ng other forms of scholarship: Collegiate Council, LinKasso A.´Okoudjou and K. Garbutt, National Editorial BoardsHonors Committee 1. with J. A. Mendoza Alvarez,Pereyra Maria Cristina be elected) Sciences Research coln, 2012; Chair of(two thetoMathematical Considerations Hal for Schenck increasing Ian Agol Institute (MSRI) Human Resources Advisory Committee, participation of minoritized Melanie Matchett Wood David Marker M. Cordero, J. EpperBerkeley, California, 2013–2015; ethnic and racial groups in James McKernan son, and T. Jorgensen, Linking mathematics research to mathematics, Notices Amer. Math. Soc. 68 (2021), no. Terence TaoThe GK-12 MAVS program, secondary school classrooms: 2, 235–239, MR4202342; 2. with M. Mast, Valuing and Proceedings EDULEARN14: 6th International Conference on 1 PICA supporting work in mathematics education: An adminisEducation and New Technologies, Spain, guishable from theLearning original, will be sentBarcelona, by first class or Ballots trative perspective, Chapter 3 in Mathematics education: A 2014, pp. 4715–4723; Member of Harvard’s Pipelines into airmail. However, the deadline for receipt of ballots will AMS members will receive email with instructions for vot spectrum of work in mathematical sciences departments, AWM Biostatistics Advisory Board, 2015–present; P. Harris et al., not be extended. ing online by August a paper ballotFractional by September 20. Series, Springer, 2016;20, 3. or with L. Chen, dimenEditors, Testimonios: Stories of Latinx and Hispanic MatheIf you semifield do not receive thisNote information by that date, please sional planes, Mat. 32 (2012), no. 2, 57– PS: AHEAD Mathematical maticians, A co-publication of the American Biographies of Candidates contact the AMS4. (preferably before October 1) to request 61, MR3071793; with V. Jha, Fractional dimensions in Society and the Mathematical Association of America, SepThe next several pages contain biographical information or call AMS a ballot. Send email to [email protected] semifields of odd order, Des. Codes Cryptogr. 61the (2011), tember 2021, 59–70. at 800-321-4267 (within the 5. USSemifield or Canada) or 401-455about all candidates. All candidates were given the opporno. 2, 197–221, MR2826957; planes of order Candidate statement: I amofhonored tothan be nominated 4000 (worldwide). 4 2 The deadline for receipt of ballots is tunity to provide a statement not more 200 words p and kernel GF(p ), J. Geom. 83 (2005), no. 1-2, 5–9, and considered for the role of Vice President of the AmeriNovember 2, 2018. to appear at the end of their biographical information. MR2193222. can Mathematical Society (AMS). As an advocate for mathPhotos were supplied by the candidates. Selected addresses or public presentations: Nonematics and mathematicians, I have long admired the AMS Write-in Votes associative algebraic structures in cryptography, National and its profound commitment to advancing the field of It is suggested that names for write-in votes be accompaDescription of Offices Conference of the Society for Advancing Chicanos and mathematics on both national and international fronts. nied by the institution or web address of the individual for The vice president and the members at large of the Native Americans in Science, SACNAS, Long Beach, CA, Since its founding in 1888 the AMS has been a beacon whom the vote is cast. Council serve for three years on the Council. That body October 2016; Discovering mathematics in everything of excellence in promoting mathematical inquiry, commudetermines all scientific policy of the Society, creates and we do, Smithsonian Museum, Futures That Inspire Hall, Replacement Ballots nication, and application. Through its diverse array of puboversees numerous committees, appoints the treasurers March 2022; Advocating For Diversity in STEM, Susan G. lications, meetings, advocacy efforts, and programs, the A member who has not received a ballot by September 20, Komen’s Big Data For Breast Cancer initiative, Breast Canand tirelessly membersstrives of thetoSecretariat, makes nominations of 2018, or who has received a ballot but has accidentally AMS fulfill its mission. The mission of cer Hackathon Challenge, Women in Computational Bicandidates for future elections, and determines the chief spoiled it, may write to [email protected] or Secretary of the AMS resonates deeply with my own values and aspiraeditors several key editorial boards. Typically, each of the AMS, 201 Charles Street, Providence, RI 02904-2213, tions as of a mathematician and educator. I am committed these new members of the Council will also serve on one USA, asking for a second ballot. The request should inof the Society’s five policy committees. Current and past clude the individual’s member code and the address to of the Council may be found here: www.ams which the replacement ballot should be sent. Immediately SEPTEMBER 2024 NOTICES OF THE AMERICAN Mmembers ATHEMATICAL SOCIETY 1091 .org/comm-all.html#COUNCIL. upon receipt of the request a second ballot, indistin105

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Minerva Cordero Vice President

Election Candidate Biographies

to upholding and furthering these ideals, advocating for the continued advancement of mathematics and the empowerment of mathematicians from diverse backgrounds. If entrusted with the role of Vice President, I pledge to work tirelessly in service to the AMS community, collaborating with fellow members to chart a course that promotes inclusivity, excellence, and innovation in the mathematical sciences. Together, we will continue to build upon the rich legacy of the AMS, ensuring that mathematics remains a vibrant and indispensable force for understanding the world around us.

Malabika Pramanik Professor, University of British Columbia, Vancouver PhD: University of California, Berkeley, 2001. AMS offices and committees: Transactions and Memoirs Editorial Committee, 2011– 2019; Western Section Program Committee, 2022–2024; Chair, Fellows Program Selection Committee, 2022–2025. Selected publications or other forms of scholarship: 1. with I. Łaba, Maximal operators and differentiation theorems for sparse sets, Duke Math. J. 158 (2011), no. 3, 347–411, MR2805064; 2. with T. Collins and A. Greenleaf, A multi-dimensional resolution of singularities with applications to analysis, Amer. J. Math. 135 (2013), no. 5, 1179–1252, MR3117305; 3. with J. Kim, L2 bounds for a maximal directional Hilbert transform, Anal. PDE 15 (2022), no. 3, 753–794, MR4442840; 4. with Y. Liang, Fourier dimension and avoidance of linear patterns, Adv. Math. 399 (2022), Paper No. 108252, 50 pp., MR4384610; 5. On some properties of sparse sets: a survey, ICM—International Congress of Mathematicians. Vol. IV. Sections 5–8, 3224–3248, 2023, MR4680359. Selected addresses or public presentations: Plenary lecture, AMS Spring Sectional Meeting, USA, April 2021; Fourier Analysis @ 200, International Centre for Mathematical Sciences (ICMS), June 2022; Invited address, 44th Summer Symposium in Real Analysis, Paris, June 2022; International Congress of Mathematicians (virtual), Analysis session, July 2022; Barnett Public Lecture, University of Cincinnati, April 2023. Synergistic activities: Lead organizer, ”Diversity in Mathematics,” an undergraduate summer school for women and gender minorities, funded by Pacific Institute for Mathematical Sciences (PIMS), Fields Institute, Centre de Recherches Math´ematiques (CRM), 2016–2020; Lead organizer, ”Diversity in Mathematics,” a math camp

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for high school students, funded by Natural Science and Engineering Research Council of Canada (NSERC) PromoScience, 2016–2020; Scientific co-director, Canadian Mathematical Society National Meeting, Vancouver, December 2018; Co-organizer, ”Fourier Restriction Online,” virtual program during COVID, December 2020–March 2021; Speaker, participant, and member, Indian Women in Mathematics, 2020–present; Sadosky Award Committee, Association for Women in Mathematics (AWM), 2022– 2024; Co-chair, Travel Grants Sub-Committee, International Congress of Mathematicians, International Mathematical Union-Centre for Developing Countries (IMUCDC), 2026. Additional experience/qualifications you bring to the position: Vice President for the Pacific Region, Executive Committee of the Canadian Mathematical Society, 2017–2019; Canadian Journal of Mathematics and Canadian Mathematics Bulletin, 2019–present; Inaugural Fellow of the Canadian Mathematical Society, 2019; Scientific director, Banff International Research Station (BIRS), 2020– 2025; Fellow of the American Mathematical Society, 2021; Editor-in-Chief, Research in Mathematical Sciences, Springer, 2022–present; Review panelist for Natural Science and Engineering Research Council of Canada (NSERC), 2022– 2024, and National Science Foundation (NSF), 2008– 2011, 2022–2023. Candidate statement: The AMS is a vibrant community alive with ideas, initiatives, and activities. Through its meetings, publications, travel grants, MathSciNet, Mathematics Research Communities, and more, it has something to offer to every mathematician, regardless of their background, interests, or career stage. Its impact spans research, teaching, pedagogy, art, industry, and government policy. It connects mathematics and the wider society where its members live and work. In a world increasingly fraught with differences, it celebrates the wealth of expertise, perspectives, and lived experiences within the mathematical community. The AMS has been a backdrop of my academic journey for the last three decades. My first paper as a graduate student was in an AMS journal. Over the years, I have been deeply inspired by the talent that AMS nurtures, and fortunate to serve on its committees and editorial boards, while experiencing firsthand its impact around the world. I am honored by this opportunity to give back to the community that has been a constant source of professional support for me. If elected, I will continue to build an inclusive environment in mathematics and raise awareness of the new opportunities and challenges for mathematicians in a rapidly changing world.

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Board of Trustees Henry Cohn Senior Principal Researcher and Adjunct Professor of Mathematics, Microsoft Research New England and MIT PhD: Harvard University, 2000. AMS offices and committees: Journal of the AMS, Associate Editor, 2012–2016; Robbins Prize Committee, 2012; Fellows Selection Committee, 2015–2017; AMS Council, 2016–2021; Committee on Publications, 2016–2018; Committee on Committees, 2017–2020; Executive Committee, 2018–2021; Long Range Planning Committee, 2019–2020; Committee on Equity, Diversity, and Inclusion, 2020–2021; Doubly Anonymous Refereeing Committee, 2021; Conant Prize Committee, 2022–2024; Liaison Committee with the American Association for the Advancement of Science, 2022–2024; Committee on Education, 2024–2027. Selected publications or other forms of scholarship: 1. with J. Blasiak, T. Church, J. A. Grochow, E. Naslund, W. F. Sawin, and C. Umans, On cap sets and the grouptheoretic approach to matrix multiplication, Discrete Anal. (2017), Paper No. 3, 27 pp., MR3631613; 2. with A. Kumar, S. D. Miller, D. Radchenko, and M. Viazovska, The sphere packing problem in dimension 24, Ann. of Math. (2) 185 (2017), no. 3, 1017–1033, MR3664817; 3. with F. Gon¸calves, An optimal uncertainty principle in twelve dimensions via modular forms, Invent. Math. 217 (2019), no. 3, 799–831, MR3989254; 4. with C. Borgs, J. T. Chayes, and S. Ganguly, Consistent nonparametric estimation for heavy-tailed sparse graphs, Ann. Statist. 49 (2021), no. 4, 1904–1930, MR4319235; 5. with N. Afkhami-Jeddi, T. Hartman, and A. Tajdini, Free partition functions and an averaged holographic duality, J. High Energy Phys. (2021), no. 1, Paper No. 130, 42 pp., MR4257711. Selected addresses or public presentations: Lecture in Combinatorics Section, International Congress of Mathematicians, August 2010; AMS Arnold Ross Lecture, Salt Lake City, November 2014; Math Encounters talk, National Museum of Mathematics, September 2018; Conant Lecture, Worcester Polytechnic Institute, November 2018; Invited Address, joint AMS/VMS meeting, Quy Nhon, Vietnam, June 2019. Synergistic activities: OurCS Conference for Undergraduate Women in Computer Science, research project

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mentor, October 2007, March 2011, and October 2013; PROMYS Foundation, Trustee, 2011–present; Advisory Council for National Museum of Mathematics, 2011– present; Advisory Board for Building Computational Thinkers project, Boston Museum of Science, 2013–2016; Program in Mathematics for Young Scientists (PROMYS), teaching a daily class to high school students for six weeks each summer, 2015–present. Additional experience/qualifications you bring to the position: My combination of academic and industrial experience gives me a perspective I hope would be useful for the Board of Trustees. Candidate statement: I am grateful to be a candidate for the Board of Trustees. If elected, I will do my best to ensure that the business affairs of the AMS are conducted efficiently and transparently on behalf of the members, and that the Society remains on a solid financial footing. It’s important for the health of the mathematical community that the AMS both continue its important work in supporting fundamental research in mathematics and broaden the scope of its activities to include those who have been overlooked or excluded in the past. These goals reinforce each other, and neither could be achieved in isolation. Making progress requires thoughtful decisions by the Board of Trustees, and I would be honored to assist in this process.

Brooke E. Shipley Professor, Department of Mathematics, Statistics, and Computer Science (MSCS), University of Illinois at Chicago PhD: Massachusetts Institute of Technology, 1995. AMS offices and committees: Academic Freedom, Tenure, and Employment Security, 2004–2007; AMS-IMSSIAM Committee on Summer Research Conferences in the Mathematical Sciences, 2004–2007; Proceedings Editorial Committee of the AMS, 2009–2013; AMS-Simons Travel Grants Committee, 2013–2016; Committee on Publications, 2018–2021; AMS Council, Member at Large, 2018– 2021; Search Committee for the Notices Chief Editor, 2020–2021; Doubly Anonymous Refereeing Committee, 2021–2022; Committee on the Profession, 2024–2027. Selected publications or other forms of scholarship: 1. with M. Hovey and J. Smith, Symmetric spectra, J. Amer. Math. Soc. 13 (2000), no. 1, 149–208, MR1695653; 2. with D. Dugger, K-theory and derived equivalences, Duke Math. J. 124 (2004), no. 3, 587–617, MR2085176; 3. 𝐻ℤ-algebra spectra are differential graded algebras, Amer.

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J. Math. 129 (2007), no. 2, 351–379, MR2306038; 4. with K. Hess, M. Kedziorek, and E. Riehl, A necessary and sufficient condition for induced model structures, J. Topol. 10 (2017), no. 2, 324–369, MR3653314; 5. with J. P. C. Greenlees, An algebraic model for rational torusequivariant spectra, J. Topol. 11 (2018), no. 3, 666–719, MR3830880. Selected addresses or public presentations: Invited Address, AMS Sectional Meeting, Boulder, CO, 2003; Wolfson Lecture Series, Manchester, England, 2006; Lecture Series, Workshop on Algebraic Topology, MSRI, 2013; Plenary Speaker, Nebraska Conference for Undergraduate Women in Mathematics, Lincoln, NE, 2017; Panelist, Critical Issues in Mathematics Education, MSRI, Berkeley, CA, 2022. Synergistic activities: NSF ADVANCE Co-PI, UIC, Women in Science and Engineering System Transformation (WISEST), 2009–2012; Interim Director, WISEST, 2012–2013; AWM Committee on Committees, 2013– 2016; Executive Advisory Board, Department of Education HSI-STEM program, UIC Latinos Gaining Access to Networks for Advancement in Science (L@S GANAS), 2017– 2020; Co-Director, UIC Young Scholar’s Program (fourweek summer program open to all Chicago high school students), 2020–present; Graduate Research Opportunities Workshop (GROW) Steering Committee, 2020– present; AWM Research Networks Committee, 2021–2024. Additional experience/qualifications you bring to the position: NSF Postdoctoral Research Fellow, 1995; NSF Career Award, 2002; Sloan Research Fellow, 2002; AWM Noether Lecture Selection Committee, 2009–2012; ELATE Fellow, Drexel University, 2014–2015; Head, UIC Department of Mathematics, Statistics, and Computer Science (MSCS), 2014–2022; AMS Fellow, 2015; Co-Chair, UIC Faculty Equity Committee, 2017–2024; NSF Institute for Mathematical and Statistical Innovation (IMSI), Co-PI and Board of Advisors, 2020–present; Senior Berwick Prize, London Mathematical Society, joint with John Greenlees, 2022. Candidate statement: I am honored to be considered as a candidate for Trustee of the AMS. Of the many experiences and roles listed above, two main roles are especially relevant to financial stewardship and to furthering the multifaceted mission of the AMS. For eight years (2014 to 2022), I was Head of a department that encompasses pure and applied mathematics, statistics, mathematical computer science, and mathematical education at one of the most diverse research universities in the country. Since 2020, I have served as Co-PI and member of the Board of Advisors for the NSF Institute for Mathematical and Statistical Innovation (IMSI). Scientific activity at IMSI is focused on applications of the mathematical sciences, em-

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phasizing questions of importance to society at large. The two other pillars of IMSI’s mission are improving and facilitating communication and broadening access and participation. Throughout my career, I have been dedicated to serving the broader mathematical community, and I look forward to the opportunity to continue to do so as Trustee for the AMS.

Member at Large Alejandra Alvarado Professor of Mathematics, Eastern Illinois University PhD: Arizona State University, 2009. AMS offices and committees: Central Section Program Committee, 2022–2024; MAASIAM-AMS Hrabowski-GatesTapia-McBay Lecture Selection, 2023–2025. Selected publications or other forms of scholarship: 1. with J.-J. Delorme, On the Diophantine equation x4 +y4 +z4 +t4 =w2 , J. Integer Seq. 17 (2014), no. 11, Article 14.11.5, 14 pp., MR3291083; 2. with C. R. Price, Academic preparation for business, industry, and government positions, Assoc. Women Math. Ser., 18, Springer, 2019, 79– 87, MR4061882. Selected addresses or public presentations: Arithmetic Progressions on Curves, Conference on Strengthening Community in Research Mathematics, invited speaker, Pomona College, Claremont, CA, 2023; EDGE: The Early Years, invited panelist for Mobilizing the Power of Diversity: Celebrating 25 Years of EDGE, 2023. Synergistic activities: EDGE Co-director, summer 2016; CSU Channel Island faculty mentor, summer REU, 2017. Additional experience/qualifications you bring to the position: Government employee, 2018–2020; AWM Executive Committee Clerk, 2022–present. Candidate statement: Thank you for considering me for Member at Large. If elected, I hope that I can contribute towards making mathematics a welcoming community and bring awareness to its contribution not just in the sciences but to modern society. I am concerned that mathematics research is not being valued and appreciated as it could be, such as the proposed cuts of West Virginia University graduate programs,

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and the loss of our master’s programs in mathematics and mathematics education. Math is the backbone of many of the sciences. I appreciate that the AMS offers travel grants for PUI faculty to attend JMM. As a recipient of the AMS-Simons grant, this is especially important as I am at a PUI. My experience in government has allowed me to see firsthand the applications of mathematics and given me unique insight on faculty transition to (and from) nonacademic positions. I am committed to the increase and advancement of women and underserved students in the mathematical sciences. I hope that my actions and experience will serve as evidence of my candidacy.

Benjamin Antieau Professor, Northwestern University PhD: UIC, 2010. AMS offices and committees: Associate Editor, Journal of the AMS, 2023–2027. Selected publications or other forms of scholarship: 1. with A. Mathew, M. Morrow, and T. Nikolaus, On the Beilinson fiber square, Duke Math. J. 171 (2022), no. 18, 3707– 3806, MR4516307; 2. with B. Bhatt and A. Mathew, Counterexamples to Hochschild-Kostant-Rosenberg in characteristic p, Forum Math. Sigma 9 (2021), Paper No. e49, 26, MR4277271; 3. with T. Nikolaus, Cartier modules and cyclotomic spectra, J. Amer. Math. Soc. 34 (2021), no. 1, 1–78, MR4188814; 4. Periodic cyclic homology and derived de Rham cohomology, Ann. K-Theory 4 (2019), no. 3, 505–519, MR4043467; 5. with D. Gepner and J. Heller, K-theoretic obstructions to bounded t-structures, Invent. Math. 216 (2019), no. 1, 241–300, MR3935042. Selected addresses or public presentations: The nilpotency of v1 in the K-theory of ℤ/pn , Workshop on p-adic Hodge theory and applications, Clay Math Institute, 2022; The cyclotomic t-structure (two talks), Drinfeld seminar, online, 2022; Derived algebraic geometry (nine lectures), MSRI summer school, 2023; Integral models for spaces, Conference on generalized Lie algebras in derived algebraic geometry, Utrecht, 2023; Motivic filtrations (five lectures), Masterclass on THH and zeta values, Copenhagen, 2023. Synergistic activities: Co-founder (with David Dumas) of the Math Computing Laboratory at UIC, which introduced undergraduate students to visualization and exploration techniques in mathematics through term-long

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projects; Co-organizer of many schools and workshops, such as the electronic Algebraic K-Theory Seminar and Vitamin K1. Candidate statement: I am interested in research for its own sake as well as its broad use in many other human endeavors. Increasingly, I am also interested in computer algorithms for understanding highly complex objects arising in algebraic and arithmetic geometry. Finally, I have also always been interested in pedagogy and the role of the university in the development of science and scientists. The AMS is an important companion to us in each of these realms: it is crucial in helping individual mathematicians flourish, inside and outside of the academy, in advising national leaders on science policy, and in helping to craft academic policy in research and teaching. I am excited about the prospect of serving the AMS, and the mathematical community at large, as a Member at Large of the Council of the AMS and being a part of this work. I hope that by doing so, I can help further participation in a subject beloved to me and also help to influence the organization’s response to various opportunities and challenges facing the (mathematical) world at present.

Dawei Chen Professor of Mathematics, Boston College PhD: Harvard University, 2008. Selected publications or other forms of scholarship: 1. with I. Coskun, Extremal effective divisors ℳ 1,n , Math. Ann. 359 (2014), no. 3-4, 891–908, MR3231020; 2. with M. Möller and D. Zagier, Quasimodularity and large genus limits of Siegel-Veech constants, J. Amer. Math. Soc. 31 (2018), no. 4, 1059–1163, MR3836563; 3. with M. Bainbridge, Q. Gendron, S. Grushevsky, and M. Möller, Compactification of strata of Abelian differentials, Duke Math. J. 167 (2018), no. 12, 2347–2416, MR3848392; 4. with M. Möller, A. Sauvaget, and D. Zagier, Masur-Veech volumes and intersection theory on moduli spaces of Abelian differentials, Invent. Math. 222 (2020), no. 1, 283–373, MR4145791; 5. with M. Möller and A. Sauvaget, Masur-Veech volumes and intersection theory: the principal strata of quadratic differentials, Duke Math. J. 172 (2023), no. 9, 1735–1779, MR4608330. Selected addresses or public presentations: IAS-PCMI Summer School on Moduli Spaces of Riemann Surfaces, Park City, UT, Invited Talk, 2011; Algebraic Geometry Northeastern Series, Stony Brook, NY, Plenary Speaker,

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2011; The Abel Symposium, Lofoten, Norway, Plenary Speaker, 2017; Algebraic Geometry Fall School, Berlin, Germany, Minicourse, 2019; Texas Geometry and Topology Conference, Fort Worth, TX, Plenary Speaker, 2023. Synergistic activities: Co-organizer for over 20 conferences, such as Algebraic Geometry Northeastern Series (AGNES), workshops at BIRS, HMI, ICERM, and an online seminar series Moduli Across the Pandemic (MAP); Mentor for the 2015 Algebraic Geometry Summer Research Institute Bootcamp and the 2023 AGNES Summer School; Mentor for over 20 undergraduate, graduate, and postdoctoral research and independent study projects; Notices of the AMS Early Career Section article author (“Developing Relationships with Experts”); MathSciNet reviewer (over 110 reviews) and zbMath reviewer (over 150 reviews); NSF panelist (4 times); Proposal reviewer for the European Research Council, French National Research Agency, and Chile National Fund for Scientific and Technological Development; Letter writer and reference for over 60 people; Referee for over 40 journals; Boston College (BC) University Core Renewal Committee member, 2014–2017; BC Mathematics Graduate Program Director, 2016–2018, 2020–2021; BC Undergraduate Educational Policy Committee member, 2020–2023. Additional experience/qualifications you bring to the position: Clay Liftoff Fellow, 2008; NSF standard grants, 2011–2014, 2020–2023, 2023–2026; NSF CAREER Award, 2014–2020; AIM SQuaRE, 2017–2019; IAS von Neumann Fellow, 2019; Simons Fellow, 2024. Candidate statement: I am honored to be nominated to run for the position of Member at Large of the AMS Council. Today, the mathematical community faces many new challenges and diverse voices. Nevertheless, creating an environment of equality and inclusivity, promoting diversity in education and research, and nurturing the growth of young mathematicians are shared ideals and goals that I have always been dedicated to. For example, during the pandemic, to provide young scholars with opportunities to showcase their research, I organized a series of online conferences where all speakers were graduate students or postdoctoral scholars, with half of them being women or from other underrepresented groups. I also contributed an article to Notices of the AMS, introducing how early-career researchers can develop connections with experts in their fields. Furthermore, I volunteered for STEM activities at local schools, demonstrating surface geometry to elementary school students using donuts and pretzels. AMS is a big family, and I aspire to listen to and assist the needs of community members just like caring for family members, and to nurture the growth of the new generation of mathematicians as if they were my own children. I am committed to giving my best effort towards this endeavor.

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Emily Clader Associate Professor, San Francisco State University PhD: University of Michigan, 2014. Selected publications or other forms of scholarship: 1. Landau–Ginzburg/Calabi– Yau correspondence for the complete intersections 𝑋3,3 and 𝑋2,2,2,2 , Adv. Math. 307 (2017), 1–52, MR3590512; 2. Editor, with Y. Ruan, B-model Gromov– Witten theory, Birkhäuser, 2018, MR3967072; 3. with S. Grushevsky, F. Janda, and D. Zakharov, Powers of the theta divisor and relations in the tautological ring, Int. Math. Res. Not. (2018), no. 24, 7725–7754, MR3892277; 4. Why twelve tones? The mathematics of musical tuning, Math. Intelligencer 40 (2018), no. 3, 32–36, MR3851071; 5. with F. Janda and Y. Ruan, Higher-genus quasimap wallcrossing in the gauged linear sigma model, Duke Math. J. 170 (2021), no. 4, 697–773, MR4280089. Selected addresses or public presentations: ”Double ramification cycles and tautological relations,” Western Algebraic Geometry Symposium (WAGS), October 2016; ”Wall-crossing in Gromov–Witten theory,” Algebraic Geometry Northeastern Series (AGNES), October 2017; ”Wall-crossing in Gromov–Witten theory,” plenary lecture, Texas Algebraic Geometry Symposium, April 2018; ”The moduli space of curves and its tautological ring,” colloquium, UC Berkeley, February 2020; ”Permutohedral complexes and curves with cyclic action,” colloquium, Brown University, April 2022. Synergistic activities: Mentor of eleven master’s theses (six by women) and three undergraduate research groups (five of ten students women, seven of ten Black or Latinx), San Francisco State University, Fall 2016–present; Faculty advisor, Mathematistas (student group for gender equity in mathematics), San Francisco State University, 2017–present; Volunteer, San Quentin Math Circle (at San Quentin State Prison), Fall 2018; Invited speaker, San Francisco Nerd Nite (public lecture for an audience of about 200 people), Spring 2019; Author of numerous expository texts, including a forthcoming undergraduate-level algebraic geometry textbook Algebra and Geometry (with Dustin Ross) and a forthcoming invited article ”CurveCounting and Mirror Symmetry” in the Notices of the AMS; Faculty advisor and founder, Math Department PhD Application Group (to mentor students through the PhD application process), San Francisco State University, Fall 2022–present; Co-developer, Math Department course on Quantitative Reasoning for Civic Engagement, San

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Francisco State University, Fall 2023–present; Research project mentor at four graduate workshops (Women in Algebraic Geometry, ICERM, 2020; AGNES Summer School on Intersection Theory on Moduli Spaces, Brown University, 2023; Workshop on Combinatorics of Moduli of Curves, BIRS, 2024; Women in Algebraic Geometry 2, IAS, 2024). Additional experience/qualifications you bring to the position: NSF DMS grant, 2018–2021; Participant, UndocuAlly training (to better serve undocumented students), San Francisco State University, Spring 2019; Chair, San Francisco State University Math Department Curriculum Committee, Fall 2020–present; Co-organizer, San Francisco State University AlgebraGeometry-Combinatorics Seminar, 2020–present; NSF panelist, 2021; NSF CAREER grant, 2022–2027. Candidate statement: I am honored to be nominated as a candidate for the position of Member at Large of the Council of the AMS. I am committed to thinking deeply and concretely about how mathematics can be made accessible to the broadest possible community, by (among other things) creating texts and teaching materials that strive for effective communication, by mentoring students of many backgrounds through key transitions, and by choosing seminar and conference speakers from a broad range of demographic communities, subject areas, and career stages. If elected to serve, I would bring to the position my experience as a faculty member at a primarily undergraduate institution whose student body is diverse in myriad ways. San Francisco State University in general, and its Math Department in particular, has a long history of attempting to infuse its curriculum with equity, inclusivity, and social relevance. My firsthand experience with both the successes and the shortfalls of this mission equip me to bring creative ideas but also a critical eye to the crucial conversations that the AMS leads, and by which our community is shaped.

Selected publications or other forms of scholarship: 1. A Mathematician’s Journey to Public Service, in: S. D’Agostino, S. Bryant, A. Buchmann, M. Guinn, and L. Harris (eds.), A Celebration of the EDGE Program’s Impact on the Mathematics Community and Beyond, Association for Women in Mathematics Series, vol. 18, Springer, 2019, MR4061899. Selected addresses or public presentations: MAA Distinguished Lecturer Series, on the intersection of Math & Policy, October 2023. Synergistic activities: Throughout my career I have endeavored to give back to my personal community as well as the math community through invited talks and panels at various diversity-focused events online and in person. I consistently mentor budding mathematicians, helping them navigate the challenges of undergraduate and graduate school. Most recently I was asked to serve at the next Infinite Possibilities Conference. I have also influenced mathematics awareness through my service with the AMS CSP and COE. Additional experience/qualifications you bring to the position: Former AMS Congressional Fellow; Former Tenure-Track Assistant Professor. Candidate statement: I believe it’s important to have diversity throughout the ranks of the AMS leadership. This diversity of thought and experience serves to greatly enrich the international community of the AMS membership. The face of math continues to evolve along with opportunities to engage and communicate, building new and cultivating ongoing research collaborations. In addition, the opportunity is great to reintroduce and reinvigorate mathematical science foundations and bridge interdisciplinary sciences gaps for the expansion of emerging technologies including artificial intelligence (AI) and machine learning (ML) within the math community. I hope to have the opportunity to continue to work with AMS in this distinguished capacity. Thank you for your consideration.

Carla Cotwright

Dan Isaksen Supervisory Applied Research Mathematician, Department of Defense PhD: The University of Mississippi, 2006. AMS offices and committees: Committee on Science Policy, 2022–2025 (Chair, 2023–2025); Committee on Education, 2023–2025; Liaison Committee with the American Association for the Advancement of Science, 2024–2026.

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NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

Professor of Mathematics, Wayne State University PhD: University of Chicago, 1999. AMS offices and committees: Central Section Program Committee, 2024–2026. Selected publications or other forms of scholarship: 1. with D. Dugger, The Hopf condition for bilinear forms over arbitrary fields, Ann. of Math.

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(2) 165 (2007), no. 3, 943–964, MR2335798; 2. with G. Wang and Z. Xu, Stable homotopy groups of spheres, Proc. Natl. Acad. Sci. USA 117 (2020), no. 40, 24757– 24763, MR4250190; 3. with G. Wang and Z. Xu, Stable homotopy groups of spheres and motivic homotopy theory, Proceedings of the International Congress of Mathematicians 4 (2022), 2768–2790; 4. with G. Wang and Z. Xu, Stable homotopy groups of spheres: from dimension 0 to ´ 90, Publ. Math. Inst. Hautes Etudes Sci. 137 (2023), 107– 243, MR4588596. Selected addresses or public presentations: Introductory Workshop: Algebraic Topology, Mathematical Sciences Research Institute, Berkeley, 2014; Homotopy Theory in the Ecliptic, Reed College, 2017; Motivic, Equivariant, and Non-Commutative Homotopy Theory, mini´ course presenter, Institut des Hautes Etudes Scientifiques, France, 2020; Advances in Homotopy Theory IV, Beijing Institute of Mathematical Sciences and Applications, 2023; Homotopy theory in honor of Paul Goerss, Northwestern University, 2023. Synergistic activities: Founder and Organizer, Electronic Computational Homotopy Theory research community, 2017–current. Additional experience/qualifications you bring to the position: Editorial board member, Algebraic and Geometric Topology, 2010–current; Editorial board member, Homology, Homotopy and Applications, 2019–current; Editorial board member, Proceedings of the London Mathematical Society, 2023–current; Fellow of the American Mathematical Society, 2024. Candidate statement: Our society constantly changes, and mathematical research culture must constantly adapt. The modes by which people interact and communicate are currently changing rapidly. Traditionally, departments of mathematics (and analogous organizations in government and industry) were essential to create critical masses to foster healthy mathematical discussion and interaction. The rise of videoconferencing and other online communications tools makes physical proximity less relevant. For the long-term health of research mathematics, we should prepare for a future in which the Mathematical Conversation is carried out primarily online. Rising generations of mathematicians are and will continue to be fluent in this medium, and it is our responsibility to build infrastructure in which they will thrive. Online communication tools have the democratizing potential to include populations that were previously excluded from the Mathematical Conversation. We should be intentional about building an online mathematical culture that furthers mathematical opportunity for everyone who seeks it.

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Should the Society promulgate principles and best practices for online mathematical opportunities? Does the Society have a role to play in online conferences? How can the Society facilitate the increasingly online employment process? What can the Society do to encourage the interscholastic cooperation that is a key aspect of online activities?

Yvonne Lai Professor & Graduate Chair, Department of Mathematics, University of Nebraska-Lincoln PhD: University of California, Davis, 2008. AMS offices and committees: AMS Lecture on Education Selection Committee, 2024–2026; Notices Editorial Board Committee, 2025–2027. Selected publications or other forms of scholarship: 1. An effective compactness theorem for Coxeter groups, Geometriae Dedicata 145 (2010), no. 1, 195–217, MR2600954; 2. Teaching Undergraduate Mathematics, Critical Issues in Mathematics Education Series, vol. 4, Mathematical Sciences Research Institute, Berkeley, CA, 2012; 3. with M. A. Carlson and R. Heaton, Giving reason and giving purpose, in Mathematics Matters in Education: Essays in honor of Roger E. Howe, Springer, New York, 2017, 141– 179; 4. as member of writing team, Catalyzing Change in High School Mathematics: Initiating Critical Conversations, National Council of Teachers of Mathematics, Reston, VA, 2018, vi+107; 5. with G. Burrill, H. Cohn, D. Sinha, J. Y. Son, and K. E. Stevenson, Listening for common ground in high school and early collegiate mathematics, Notices Amer. Math. Soc. 70 (2023), no. 6, 798–805. Selected addresses or public presentations: Invited presentation at workshop on ”Mathematicians and School Mathematics Education” at the Banff International Research Station, Banff, 2014; Briefing of the US House Committee on Science, Space, & Technology on “STEM Education 101,” Washington, DC, 2023; Invited speaker for Forum for World Education on “Raising the Bar on Mathematics Education: Lessons from Far and Near,” New York City, 2024; JMM Invited Address on “Building Bridges in Mathematics Education” (Project NeXT Lecture on Teaching and Learning), San Francisco, 2024. Synergistic activities: Explore Math Program/Davis Math Circle: Co-Founder (2005–2008); Algebra Project: Invited Young Mathematicians (2007), Lesson Planning Team Member (2009–2010), co-PI (2021–2025); Mathematicians in Mathematics Education: Presenter (2011),

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Organizer (2012), Lead Organizer (2013–2014) with William McCallum, Deborah Ball, and Roger Howe; MAA Committee on the Mathematical Education of Teachers: Member (2016–2022), Chair (2022–2024), created NCTM liaison for committee, led review of NCTM policy document, conducted bi-monthly virtual listening tour of mathematics faculty across nation invested in the mathematical education of teachers; Metric Geometry and Gerrymandering Group’s Geometry of Redistricting Summer School: Instructor (2018); MAA Task Force on the Treatment of Teacher Education: Co-Chair (2021–2023), produced guiding structure for MAA CUPM Curriculum Guide Program Area Reports on Teacher Education; Board of Directors of the Mathematical Foundation of America (administers the Canada/USA Mathcamp): Member (2018– 2022), Vice Chair (2023–2025); AMS Committee on Education Forum: Evolving Curriculum in High School and Early Undergraduate Mathematical Sciences Education: Organizer and Moderator (2022); Nebraska State High School Mathematics Standards: Post-Secondary Advisor (2022); AMS Special Session on Mathematics Education, Standards, Policy, and Politics: Lead Co-Organizer (2023); Conference Board of the Mathematical Sciences (CBMS) Task Force on Modernizing Mathematics: Member (2023– 2024), produced consensus document proposed to membership of CBMS; CBMS Steering Committee for the Mathematical Education of Teachers III (MET III): Member (2024–2026). Additional experience/qualifications you bring to the position: Service to the profession. Association for Women in Mathematics (AWM) Educational Columnist; Bay Area Mathematics Olympiad for Teachers: “Chief Inspirer” to director Joshua Zucker, 2010; National Association of Math Circles: Advisory Board Member, 2012–2015; USAMO: Grader, 2015–2016; Associate Editor of AMS Blog on Teaching and Learning, 2018–2021; Associate Editor and Editorial Board for Problems, Resources, and Issues in Mathematics Undergraduate Studies, 2020–2024; Associate Editor of AMS Column on Teaching and Learning, 2022– 2025; Subcommittee of the AMS Committee on Education, 2023–2024. Memberships. American Educational Research Association (AERA), American Mathematical Society (AMS), Association for Women in Mathematics (AWM), Association of Mathematics Teacher Educators (AMTE), Canadian Mathematics Education Study Group/Groupe ´ Canadien d’Etudes en Didactique des Math´ematiques (CMESG/GCEDM), Consortium for Mathematics and its Applications (COMAP), Mathematical Association of America (MAA), National Association of Mathematicians (NAM), National Council for Teachers of Mathematics (NCTM), TODOS Mathematics for All (TODOS).

SEPTEMBER 2024

Grants. PI NSF award DUE-1726744, PI NSF award DUE-2408993, co-PI NSF award DRL-2101393, co-PI NSF award DUE-1747937, co-PI NSF award DUE-1439867, PI NSF award DGE-1445551, co-PI NSF award DUE-1035268, co-PI NSF award DMS-1135049. Candidate statement: I am deeply honored to be nominated to serve on the AMS Council. The AMS has a leading role in the health of the mathematical sciences community. If elected, I will work for the AMS to: (1) Advocate. The AMS depends on graduate education for its future vitality. I will advocate for graduate studies in the mathematical sciences, including identifying factors in success, persisting, and attrition. (2) Connect. The AMS is energized when it supports and sees value in a wide array of perspectives and experiences. I aim for the AMS to make intentional connections across industry, sciences, and education to strengthen the entire mathematics community. (3) Advance. The AMS should advance access to the mathematical sciences. Too many students and teachers have too few experiences of mathematics as beautiful, joyful, and powerful. I will work for the AMS to continue and expand efforts that foster full and equitable participation in the mathematical sciences. Throughout my career collaborating with researchers and teachers, I have been driven to connect communities and ideas across the mathematical sciences to improve education. If elected, I bring this commitment to the Council to influence policy and practice that sustains and nurtures the mathematical sciences community.

Christopher J. Leininger Professor, Rice University PhD: University of Texas, Austin, 2002. AMS offices and committees: AMS-Simons Travel Grants Committee, 2015–2018; Transactions and Memoirs Editorial Committee, 2024–2028. Selected publications or other forms of scholarship: 1. with M. Bestvina, K. Bromberg, and A. E. Kent, Undistorted purely pseudo-Anosov groups, J. Reine Angew. Math. 760 (2020), 213–227, MR4069890; 2. with S. Dowdall and I. Kapovich, Dynamics on free-by-cyclic groups, Geom. Topol. 19 (2015), no. 5, 2801–2899, MR3416115; 3. with M. Duchin and K. Rafi, Length spectra and degeneration of flat metrics, Invent. Math. 182 (2010), no. 2, 231–277, MR2729268; 4. with R. P. Kent, IV, Shadows of mapping class groups: capturing convex cocompactness, Geom.

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Funct. Anal. 18 (2008), no. 4, 1270–1325, MR2465691; 5. with E. Field, H. Kim, and M. Loving, End-periodic homeomorphisms and volumes of mapping tori, J. Topol. 16 (2023), no. 1, 57–105, MR4532490. Selected addresses or public presentations: Lawrence, KS, AMS Sectional Meeting 1081, Invited Address, 2012; Midway, Utah, Conference in Honor of Mark Feighn, Plenary Lecture, 2018; Luminy, France, Conference in Honor of Mladen Bestvina, Plenary Lecture, 2019; Columbia University, Conference in Honor of Walter Neumann, Plenary Lecture, 2022; Monte Verita, Switzerland, Conference in Honor of Ursula Hamenstädt, Plenary Lecture, 2023. Synergistic activities: Co-organizer of University of Warwick EPSRC Symposium, UK, 2017–2018; Associate editor for New York Journal of Mathematics (2020–2024), Advances in Mathematics (August 2022–present), Transactions and Memoirs of AMS (February 2024–present); Coorganizer for Conference: Geometry, Arithmetic, and Groups, University of Texas, Austin, 2022; Co-organizer for SLMath Semester program, Topological and Geometric Structures in Low Dimensions, Spring 2026; Supervised/mentored a diverse group of 20 PhD students (7 current) and 8 postdocs. Additional experience/qualifications you bring to the position: University of Illinois, Mathematics Executive Committee, 2012–2014; University of Illinois, Mathematics Promotion and Tenure Committee, 2017– 2019 (Chair, 2018–2019); University of Illinois, Campus Research Board (campus-wide research funding board), 2017–2020; Director of Graduate Studies, Rice University, 2021–present; Rice University Graduate Council, 2021– present. Candidate Statement: I am committed to advancing mathematical research, and promoting diversity and inclusiveness at all levels. I am happy to serve as a Member at Large, should I be elected.

Adriana Salerno Professor, Bates College PhD: University of Texas at Austin, 2009. AMS offices and committees: Task Force: AMS In Racial Discrimination, 2020–2021. Selected publications or other forms of scholarship: 1. with J. H. Silverman, Integrality properties of Bottcher coordinates for one-dimensional superattracting germs, Ergodic Theory and Dynamical Systems 40 (2020), no. 1, 248– 271, MR4038034; 2. with C. Doran, T. Kelly, S. Sper-

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ber, J. Voight, and U. Whitcher, Hypergeometric decomposition of symmetric K3 quartic pencils, Research in the Mathematical Sciences 7 (2020), no. 2, Paper No. 7, 81 pp., MR4078177; 3. with L. Schneps, Mould theory and the double shuffle Lie algebra structure, in Periods in Quantum Field Theory and Arithmetic: Proceedings of the ICMAT, Madrid, Spain, September 15–December 19, 2014, Springer Proceedings in Mathematics and Statistics, 339– 430, 2020, MR4100685; 4. with U. Whitcher, Hasse-Witt matrices and mirror toric pencils, Advances in Theoretical and Mathematical Physics 26 (2022), no. 9, 3345–3375; 5. with D. Banerjee and S. Chari, Higher dimensional origami constructions, Involve 16 (2023), no. 2, 297–312, MR4597247. Selected addresses or public presentations: Diagonal pencils and Hasse-Witt invariants, Newton Institute, Cambridge, UK, 2022; Teaching Math Is Hard (Haimo Award presentation), MAA MathFest, Tampa, FL, 2023; Arithmetic, Hypergeometric Functions, and Mirror Symmetry, Lathisms Cafe Con Leche lecture, online (https://youtu .be/Dc4tZwZ98eY?si=EC-7CxDo9RrD_2sf), 2023; The mathematics of secrets (plenary), MAA Golden Section meeting, University of California, Santa Cruz, CA, 2024; The mathematics of secrets (plenary), MAA North Central Section meeting, University of St. Thomas, Minneapolis, MN, 2024. Synergistic activities: Public awareness of mathematics: I have been the writer for several blogs for the AMS. Among them, PhD + epsilon: An early career mathematician blogs about her experiences and challenges (http:// blogs.ams.org/phdplus/), a biweekly blog I wrote from March 2011 to September 2015; Inclusion/Exclusion from January 2016 to January 2020, about issues of diversity and inclusion in the mathematical sciences; and the Joint Math Meetings blog from 2008 to 2019. I was also on the editorial board for MAA FOCUS and the American Mathematical Monthly, for which I wrote a few articles. I have also written book reviews and articles for a general audience for American Scientist, the Notices of the AMS, and the AWM Newsletter. Diversity, equity, and inclusion: I am a graduate of the AAAS-SACNAS-sponsored Linton-Poodry Summer Leadership Institute (summer 2016), and the HHMI-SACNAS Advanced Leadership Institute (summer 2018). I was in the core leadership for an HHMI grant to diversify STEM education at Bates. I have mentored women in mathematics as a co-leader of a project, with Ursula Whitcher, during Sage Days 50: Women in Sage, in July 2013, then as a coleader of a project, with Leila Schneps, during the Women in Numbers - Europe workshop, in October 2013. I coorganized the Roots of Unity workshop, a workshop for women of color in early graduate studies. I was one of

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the writers of the report “Towards a Fully Inclusive Mathematics Profession, Report of the Task Force on Understanding and Documenting the Historical Role of the AMS in Racial Discrimination,” with Tasha Inniss, Jim Lewis, Irina Mitrea, Kasso Okoudjou, Francis Su, and Dylan Thurston. Undergraduate research, applications, and computation: I have advised students in independent research both during the academic year and during the summer. I have advised several senior theses at Bates, two of which led to publications. Many of these theses have been in number theory and cryptography, and use a high level of computation and experimentation, including machine learning. I have also advised math education theses. Student-centered learning: I have developed several inquiry-based learning courses and team-based learning courses at Bates in a variety of topics, including complex and real analysis, number theory, calculus, and p-adic analysis. Additional experience/qualifications you bring to the position: I have been a rotating Program Officer at the NSF since September 2021. Candidate statement: Mathematicians are people who do math, so I believe that advancing mathematics research necessitates supporting and nurturing the mathematical community. Nobody should have to leave their identity at the door, mathematics is not neutral, and we are all better at mathematics if we create a community where everyone can flourish. I love doing math, and I love helping others do math. That’s a lesson I learned as an undergraduate at the Universidad Simon Bolivar in my home of Venezuela, just falling in love with the subject. After completing my PhD in number theory at UTAustin, I served as Professor of Mathematics and Chair at Bates College. As a liberal arts professor, my commitment to creating and developing a more inclusive and equitable math community has grown, through innovative pedagogy, equity work, and communication of mathematics. As an NSF Program Officer, I have expanded my perspective on the research community which continues to strive towards being a more inclusive and inviting place. I would be honored to join the AMS leadership to help advance mathematics and the people who love and do mathematics.

SEPTEMBER 2024

Pham Huu Tiep Joshua Barlaz Professor, Distinguished Professor of Mathematics, Department of Mathematics, Rutgers University PhD: Moscow State University, Moscow, Russia, 1989. AMS offices and committees: Proceedings Editorial Committee of the AMS, 2011– 2019; Joint AMS-Vietnamese Mathematical Society, Quy Nhon, 2017–2019; Mathematical Reviews Editorial Committee, 2017–2025. Selected publications or other forms of scholarship: 1. with G. Navarro, A reduction theorem for the Alperin weight conjecture, Invent. Math. 184 (2011), 529–565, MR2800694; 2. with G. Navarro, Characters of relative p′degree over normal subgroups, Annals of Math. 178 (2013), 1135–1171, MR3092477; 3. with R. Bezrukavnikov, M. Liebeck, and A. Shalev, Character bounds for finite groups of Lie type, Acta Math. 221 (2018), 1–57, MR3877017; 4. with M. Larsen and A. Shalev, Probabilistic Waring problems for finite simple groups, Annals of Math. 190 (2019), 561–608, MR3997129; 5. with R. M. Guralnick and M. Larsen, Character levels and character bounds for finite classical groups, Invent. Math. 235 (2023), 151–210. Selected addresses or public presentations: Invited Speaker, 2012 AMS Spring Western Section Meeting, Honolulu, HI, 2012; Plenary Speaker, Annual Meeting of the DFG Priority Programme on Representation Theory SPP 1388, Bad Boll, Germany, 2013; Invited Speaker, ICM 2018, Rio de Janeiro, Brazil, 2018; Plenary Speaker, 2019 Canadian Mathematical Society Summer Meeting, Regina, Canada, 2019; Principal Speaker, Groups St. Andrews, Newcastle, UK, 2022. Synergistic activities: Co-organizer, Global/Local Conjectures in Representation Theory of Finite Groups, BIRS, Banff, Canada, March 13–18, 2011; Co-organizer, An Introduction to Character Theory and the McKay Conjecture, Summer graduate school, MSRI, Berkeley, CA, July 11–22, 2016; Co-organizer, New Perspectives in Representation Theory of Finite Groups, BIRS, Banff, Canada, Oct. 15– 20, 2017; Lead organizer, Group Representation Theory and Applications, Semester program, MSRI, Berkeley, CA, Jan. 16–May 25, 2018; Co-organizer, Groups, Representations, and Applications: New Perspectives, Isaac Newton Institute for Mathematical Sciences, Cambridge, UK, Jan. 6–March 18, 2020, May 3–July 29, 2022; Co-organizer, Monodromy and Its Applications, Princeton, NJ, Dec. 7–9, 2023.

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Additional experience/qualifications you bring to the position: Fellow of the AMS, Inaugural Class, 2013; Simons Fellow in Mathematics, 2014–2015, 2022–2023; Clay Senior Scholar, Clay Mathematics Institute, 2016; ICM 2018, IMU Panel “Machine-Assisted Proofs,” Rio de Janeiro, Brazil, 2018; Chern Professor, MSRI, Berkeley, CA, Spring 2018; Kalman Prize for Best Paper, New Zealand Mathematical Society, 2021; Currently serves on the Editorial Boards of Annals of Mathematics (Associate Editor), Algebra and Number Theory, J. Pure Applied Algebra, and Springer Developments in Mathematics. Candidate statement: I am honored to be nominated for election as a Member at Large of the Council of the AMS. If elected, I would like to contribute my efforts to some of the challenges that we mathematicians are facing, which include (i) how we can advocate the role of mathematics to society and broaden the connections of mathematics to all areas of everyday life; (ii) how we can improve the quality of mathematics teaching at all levels, starting from K–12 schools to colleges to PhD programs; and (iii) how the AMS can broaden and strengthen its relationships with mathematicians in developing countries.

Nominating Committee David Fisher Milton B. Porter Professor of Mathematics, Rice University PhD: University of Chicago, 1999. AMS offices and committees: Joan and Joseph Birman Fellowship Selection Committee, 2023–2025. Selected publications or other forms of scholarship: 1. with G. Margulis, Local rigidity of affine actions of higher rank groups and lattices, Ann. of Math. (2) 170 (2009), no. 1, 67–122, MR2521112; 2. with A. Eskin and K. Whyte, Coarse differentiation of quasi-isometries I: Spaces not quasi-isometric to Cayley graphs, Ann. of Math. (2) 176 (2012), no. 1, 221–260, MR2925383; 3. with A. Eskin and K. Whyte, Coarse differentiation of quasi-isometries II: Rigidity for Sol and lamplighter groups, Ann. of Math. (2) 177 (2013), no. 3, 869–910, MR3034290; 4. with U. Bader, N. Miller, and M. Stover, Arithmeticity, superrigidity, and totally geodesic submanifolds, Ann. of Math. (2) 193 (2021), no. 3, 837–861, MR4250391; 5. with A. Brown and S. Hurtado, Zimmer’s conjecture: subexponen-

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tial growth, measure rigidity, and strong property (T), Ann. of Math. (2) 196 (2022), no. 3, 891–940, MR4502593. Selected addresses or public presentations: Harvard, Invited talk at Clay Mathematics Institute Annual Meeting, 2007; Notre Dame, Invited Address, AMS Sectional Meeting, 2010; Virtual, Vinberg Online Distinguished Lecture Series, 2022; Virtual, ICM invited address in 3 sections: Dynamics, Geometry, Topology, 2022. Synergistic activities: Edited three books, two with a special emphasis on high expository quality surveys; Editor at various points for a total of five different journals: Geometry and Topology, Journal of Topology and Analysis, Indiana University Math Journal, Journal of Lie Theory, Geometriae Dedicata; Organized math-related film series at Indiana University Cinema; Organized 42 conferences, summer schools, and workshops, including a special trimester at IHP in Spring 2024. Additional experience/qualifications you bring to the position: I work in an area of mathematics that crosses boundaries between several areas of analysis, dynamics, geometry, and topology. This leads to having a broad and deep network across many fields of mathematics, which will help with the task of finding mathematicians willing to do the important work of the AMS. Candidate statement: We live in a difficult time for mathematics, science, and higher education. Public trust in science and higher education are low. The AMS needs exceptional leadership to help mathematics research and education continue to thrive at all levels and for all people in the United States and abroad. I expect support for research and education to be challenged at both the state and federal levels and for diversity programs to continue to face particularly strong challenges. We need to be prepared to meet these problems adroitly and effectively. The Nominating Committee plays a key role in selecting the leadership that will be on the front lines; I would be very happy to help find people who are ready to fill those roles.

Aimee S. A. Johnson Professor, Swarthmore College PhD: University of Maryland, College Park, 1990. AMS offices and committees: AMS-Simons PUI Research Grants Committee, 2024–2026. Selected publications or other forms of scholarship: 1. with K. Madden, Putting the pieces together: understanding Robinson’s nonperiodic tilings, College Math. J. (1997), no. 3, 172–181, MR1444004;

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2. with A. Sahin, Isometric extensions of zero entropy ℤ𝑑 loosely Bernoulli transformations, Trans. Amer. Math. Soc. 352 (2000), no. 3, 1329–1343, MR1670842; 3. with K. Madden and A. Sahin, Discovering discrete dynamical systems, Classr. Res. Mater. Ser., MAA, MR3677179; 4. with D. McClendon, Topological speedups of ℤ𝑑 -actions, Dyn. Syste. 37 (2022), no. 2, 222–261, MR4430568; 5. with V. Cyr, B. Kra, and A. Sahin, The complexity threshold for the emergence of Kakutani inequivalence, Israel J. Math. (2022), no. 1, 271–300, MR4555896. Selected addresses or public presentations: University of Wroclaw Wandering Seminar, March 2017; Little School Dynamics, Feb. 2021; Expanding Dynamics virtual Conference Series, May 2021; Brigham Young University Colloquium, March 2022; Special session in Ergodic Theory, Symbolic Dynamics, and Related Topics, AMS Joint Meeting, San Francisco, Jan. 2024. Synergistic activities: Organizer of Special Sessions at AMS Meetings, Jan. 2002, Jan. 2014, Jan. 2017; Cofounder and Leadership Team Member, Philadelphia Area Math Teachers’ Circle, 2011–2020; Rubin Scholar Mentor for students from underrepresented groups in STEM, 2011– 2024; Lester R. Ford Awards Committee for MAA, 2012– 2016. Additional experience/qualifications you bring to the position: Chair of the Department of Mathematics and Statistics, 2016–2020, 2021–2024; member of AWM, MAA, and Sigma Xi. Candidate statement: I would be honored to serve on the AMS Nominating Committee. The AMS plays a crucial role in advocating, promoting, and supporting our discipline. Through its organization of meetings, it promotes the dissemination of knowledge and the creation of community. Through its publications, it advances our scholarship. And through its advocacy, it continually advances mathematical research and education. Since all of these activities are vitally important, it is crucial that the AMS officers include representation from a diverse group of people who can use their passion for mathematics to further the goals of the society. Although my research is in pure mathematics, the six years I have spent as Chair of a department of mathematics and statistics, which includes a strong applied math component, has given me a wide view of the pathways people take in their mathematical careers. I have also seen how important it is for the future of our discipline to have representation along all axes of our society. As a member of the Nominating Committee, I would work to find a diverse slate of candidates that can bring their energy and vision to the leadership of the AMS.

SEPTEMBER 2024

Lily Signe Khadjavi Professor of Mathematics, Loyola Marymount University PhD: U. C. Berkeley, 1999. AMS offices and committees: Committee on Equity, Diversity, and Inclusion, 2022– 2025; Council, 2022–2025; Human Rights of Mathematicians, 2023–2026. Selected publications or other forms of scholarship: 1. Driving while black in the City of Angels, Chance 19 (2006), no. 2, 43–46, MR2247023; 2. Edited with G. Karaali, Mathematics for social justice: Resources for the college classroom, Classroom Resource Materials, vol. 60, MAA Press, Providence, RI, 2019, vii+277 pp., MR3967051; 3. with R. Bryant, R. Buckmire, and D. Lind, The origins of Spectra, an organization for LGBT mathematicians, Notices Amer. Math. Soc. 66 (2019), no. 6, 875– 882, MR3929579; 4. with R. Malek-Madani and T. Moore, Navigating an Uncharted Path: The Life and Legacy of Dr. Gladys B. West, Notices Amer. Math. Soc. 68 (2021), no. 3, 357–364, MR4218169; 5. with T. Moore and K. Weems, The Infinite Possibilities Conference: Creating Moments of Belonging, Notices Amer. Math. Soc. 71 (2024), no. 3, 349–355, excerpted from Count Me In: Community and Belonging in Mathematics, edited by D. Haunsberger and D. Dumbaugh, Classroom Resource Materials, vol. 68, MAA Press, Providence, RI, 2022, 145–155, ISBN: 978-1-47046566-7. Selected addresses or public presentations: Invited speaker, AMS Special Session on Number Theory and Cryptography, JMM, Seattle, 2016; Plenary address on “Women and Mathematics: Inspiration, Obstacles, and Opportunities,” Celebrating the Mathematical Legacy of Professor Maryam Mirzakhani, UCLA, 2017; Plenary address on “Policing and the Issue of Racial Profiling in Los Angeles,” Latinx in the Mathematical Sciences Conference, IPAM, 2018; Plenary address, Math for All conference, virtual/New Orleans, February 2022; Invited speaker, “Empowering students through authentic engagement,” Educating at the Intersection of Data Science and Social Justice, ICERM, Providence, 2023. Synergistic activities: NSF award 1135426, 2011–2012; Co-chair of the Infinite Possibilities Conference supporting BIPOC women in mathematics, 2012, 2015, 2018, and Board Member of Building Diversity in Science; CoPI, NSF award 1464089, 2015–2016; Co-PI Board Member, Harvard Gender Sexuality Caucus, 2016–2018; Principal Investigator, NSF award 1642548, 2016–2021; Coorganizer, “The Mathematics and Mathematicians Behind

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Hidden Figures,” JMM, 2017; Principal Investigator, NSF award 2015440, for AWM Travel/Mentoring grants, 2020– 2024; Member, SLMath Broadening Participation Advisory Committee (formerly MSRI HRAC), 2020–2023; Board member, Spectra, 2022; Steering Committee Member, Infinite Possibilities Conference, 2025. Additional experience/qualifications you bring to the position: Appointee of the Attorney General of California to the Racial Identity and Profiling Advisory Board, Stop Data Analysis Subcommittee, State and Local Policies and Accountability Subcommittee, 2020–2024; Chair, Loyola Marymount University Mathematics Department, 2021–2024; Representative, AAAS Human Rights Coalition, 2022–2025; Mary and Alfie Gray Award for Social Justice, inaugural awardee, AWM, 2023. Candidate statement: I am deeply honored to be considered for the Nominating Committee. Serving on the AMS Council has given me a practical perspective into the many functions of the Society, from supporting the internal health of the mathematical community to serving as a public face of the mathematical sciences. I believe the AMS is strengthened when an array of perspectives and experiences is represented, as we promote mathematical research and address the needs of the community. My service on advisory boards, overseeing NSF grants with the AWM, and organizing conferences aimed at broadening participation in mathematics have all given me the wonderful opportunity to interact with researchers across a broad range of mathematical fields and at many different types of institutions. If serving on the Nominating Committee, I would be excited to collaborate with committee members to develop a diverse slate of candidates. In this way, AMS can strengthen its commitment to supporting meetings, publications, and other scientific programming; engaging in advocacy for mathematics; addressing systemic issues around inclusion; and responding to sudden challenges, such as the vulnerable position of students and professionals in the face of travel bans.

Kasso A. Okoudjou Professor, Tufts University PhD: Georgia Institute of Technology, 2003. AMS offices and committees: Member at Large, Council, 2019–2022; Committee on Science Policy, 2019–2022; CoChair, Task Force on Understanding and Documenting the Historical Role of the AMS in Racial Discrimination, 2020– 2021; Executive Committee,

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2020–2024; Student Mathematics Library Editorial Committee, 2024–2028. Selected publications or other forms of scholarship: 1. with R. S. Strichartz, Weak uncertainty principles on fractals, J. Fourier Anal. Appl. 11 (2005), no. 3, 315– 331, MR2167172; 2. with A. Benyi, K. Grochenig, and L. Rogers, Unimodular Fourier multipliers on modulation spaces, J. Funct. Anal. 246 (2007), no. 2, 366– 384, MR2321047; 3. with M. Ehler, Minimization of the probabilistic p-frame potential, J. Statist. Plann. Inference 142 (2012), no. 3, 645–659, MR2853573; 4. Extension and restriction principles for the HRT conjecture, J. Fourier Anal. Appl. 25 (2019), no. 4, 1874–1901, MR3977139; 5. with T. R. Inniss, W. J. Lewis, I. Mitrea, A. Salerno, F. Su, and D. Thurston, Towards a fully inclusive mathematics profession—one year later, Notices Amer. Math. Soc. 69 (2022), no. 7, 1214–1219, MR4454145. Selected addresses or public presentations: Keynote Speaker, International Conference on Technology, Engineering & Mathematics (TEM’18), Kenitra, Morocco, March 2018; Invited Speaker, 24th Conference for African American Researchers in the Mathematical Sciences (CAARMS 24), Institute for Advanced Study, Princeton, NJ, July 2018; Keynote Speaker, Undergraduate Research Conference, Georgia Institute of Technology, Atlanta, GA, April 2022; Lecture at the 2022 Workshop on Operator Theory with an Eye on Linear Systems and Hypercomplex Analysis, Chapman University, Orange, CA, May 2022; 8th International Conference on Computational Harmonic Analysis, Ingolstadt, Germany, September 2022; Spring 2023 Christie Lecturer, Bowdoin College, April 2023. Synergistic activities: IMU Volunteer Lecturer Program, IMSP, Benin, January 2014; Member of MIT’s Department of Mathematics’ Diversity and Community Building Committee, 2018–2020; Co-Chair, Task Force on Understanding and Documenting the Historical Role of the AMS in Racial Discrimination, 2020–2021; Co-Organizer of the (virtual) Gene Golub SIAM Summer School, AIMS South Africa, 2021; Member of Tufts University’s AS&E Diversity Fund Committee, 2021–2022. Candidate statement: I am honored to run for election as a member of the Nominating Committee. Aligned with its mission statement, the AMS is dedicated to advancing the interests of mathematical research and scholarship, contributing to the national and international community through publications, meetings, advocacy, and various programs. In pursuit of this mission, the AMS is actively fostering inclusivity by promoting the involvement of all mathematicians in its initiatives. Moreover, achieving these goals necessitates integrating diverse perspectives into the AMS governance structure. Therefore, if elected, I

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commit to collaborating with the Nominating Committee to inspire greater engagement from mathematicians and enhance their participation in the AMS and its governance.

Gigliola Staffilani Abby Rockefeller Mauze Professor, MIT PhD: University of Chicago. AMS offices and committees: Invited Address Committee For National Meetings, 2009–2012 (Chair, 2011–2012); Graduate Studies in Mathematics, 2010–2024 (Chair, 2020– 2024); AMS-MAA Joint Lecture Committee, 2018–2019; Bulletin Chief Editor Search Committee, 2018–2020; Committee on the Profession, 2018– 2021; Council, 2018–2023 (Member at Large, 2018–2021, Executive Committee Representative, 2019–2023); Chair, Bôcher Memorial Prize Selection Committee, 2019–2020; Executive Committee of the Council, 2019–2023; Long Range Planning, 2020–2022; Editor, Communications of the American Mathematical Society, 2020–2025; Nominating Committee of the ECBT, 2021–2022; Journal of the AMS Associate Editor, 2022–2026; Committee on Committees, 2023–2025; Invited Address Committee For National Meeting, 2023–2026 (Chair, 2023–2024). Selected publications or other forms of scholarship: 1. with J. Colliander, M. Keel, H. Takaoka, and T. Tao, Global well-posedness and scattering for the energycritical nonlinear Schrödinger equation in ℝ3 , Ann. of Math. (2) 167 (2008), no. 3, 767–865, MR2415387; 2. with J. Colliander, M. Keel, H. Takaoka, and T. Tao, Transfer of energy to high frequencies in the cubic defocusing nonlinear Schrödinger equation, Invent. Math. 181 (2010), no. 1, 39–113, MR2651381; 3. with A. R. Nahmod, T. Oh, and L. Rey-Bellet, Invariant weighted Wiener measures and almost sure global well-posedness for the periodic derivative NLS, J. Eur. Math. Soc. (JEMS) 14 (2012), no. 4, 1275–1330, MR2928851; 4. with D. Mendelson, A. R. Nahmod, N. Pavlovi´c, and M. Rosenzweig, Poisson commuting energies for a system of infinitely many bosons, Adv. Math. 406 (2022), Paper No. 108525, 148 pp., MR4441152; 5. with M. A. Garrido, R. Grande, and K. M. Kurianski, Large deviations principle for the cubic NLS equation, Comm. Pure Appl. Math. 76 (2023), no. 12, 4087–4136, MR4655361. Selected addresses or public presentations: Hendrik Lecture Series, MAA, Colorado, 2018; Simons Foundation Public Lecture, NYC, 2021; Floer Lectures, University of Bochum, Germany, 2023; Inaugural Noether Lecture Se-

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ries, IAS, Princeton, 2023; Alice Roth Lecture, ETH, Zurich, 2023. Synergistic activities: Co-designer of the online single variable calculus class offered on MITx, 2015–2016; Organizes the Women in Math group at MIT (co-designed the web page https://math.mit.edu/wim/); Chairs the committee on Diversity and Community Building in the math department at MIT (co-designed the web page https://math.mit.edu/diversity/). Candidate statement: I am honored to have been nominated to run as a member of the Nominating Committee of the American Mathematical Society. Over the years I have served in several committees of the AMS, and I have appreciated the thoughtfulness, dedication, and willingness to serve the community that the AMS officers have demonstrated. The AMS is a wonderful society that is helping the mathematical community in a variety of manners, and the officers are some of the most important parts of its organization. If elected I will be proud to continue to give back to the society by recruiting the most capable and dedicated future officers. I am sure members of the AMS have realized that leading the society has proved to be a challenging experience. It is important to have officers who will maintain the highest standards of our discipline, who will value diversity of backgrounds, and who will work not just to benefit our mathematical community, but also to make the value of analytic thinking and evidence-based arguments more popular.

Jared Wunsch Professor of Mathematics, Northwestern University PhD: Harvard, 1998. AMS offices and committees: Central Section Program Committee, 2011–2013; Prize Oversight Committee, 2019– 2026 (Chair, 2019–2024); Mathematical Surveys and Monographs Editorial Committee, 2023–2027; Invited Address Committee for National Meetings, 2024–2027; Collected Works Editorial Committee, 2024–2028. Selected publications or other forms of scholarship: 1. with R. Melrose, Propagation of singularities for the wave equation on conic manifolds, Invent. Math. 156 (2004), no. 2, 235–299, MR2052609; 2. with A. Hassell, The Schrödinger propagator for scattering metrics, Ann. of Math. (2) 162 (2005), no. 1, 487–523, MR2178967; 3. with M. Zworski, Resolvent estimates for normally hyperbolic trapped sets, Ann. Henri Poincar´e 12 (2011),

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no. 7, 1349–1385, MR2846671; 4. with D. Baskin and A. Vasy, Asymptotics of radiation fields in asymptotically Minkowski space, Amer. J. Math. 137 (2015), no. 5, 1293– 1364, MR3405869; 5. with D. Lafontaine and E. A. Spence, For most frequencies, strong trapping has a weak effect in frequency-domain scattering, Comm. Pure Appl. Math. 74 (2021), no. 10, 2025–2063, MR4303013. Selected addresses or public presentations: MSRI Evans Lecture, MSRI, 2008; CMI Summer School on Evolution Equations, 2008; AMS Fall Central Sectional meeting, 2010; Journ´ees EDP, Biarritz, France, 2012; S´eminaire Laurent Schwartz, IHES France, 2016. Synergistic activities: Organized summer schools in Zürich (2008) and at Northwestern (2019, 2024); Mentor for Causeway Postbaccalaureate program, 2021–2022. Candidate statement: I have been an AMS member for my whole professional life, and have been grateful for the Society’s efforts to further our work as mathematicians and to share that work with the larger world. I have had some direct involvement in these efforts through service on AMS committees, most notably as Chair of the Prize Oversight Committee from 2019–2024. My committee service and my time as Department Chair at Northwestern (2012–2015) have given me experience in coping with distinct—and sometimes competing—demands on an organization with finite resources. My own research is mainly in pure analysis and PDE, with a flavor of mathematical physics. An additional recent research direction has been in the more applied direction of numerical analysis. I consequently have a broad view of what is interesting in mathematics and where it can be found. In 2024 I served on the Invited Address Committee, and especially enjoyed the process of selecting some dishes from the rich banquet table of current mathematical research. The Nominating Committee has a related mission, and I would be excited for the chance to cast a wide net in seeking excellent candidates for AMS leadership positions. The Nominating Committee, like the AMS as a whole, faces challenges of balancing the Society’s role as the advocate for mathematics research and its missions in education, outreach, and the broadening of participation in the profession. These missions work in synergy, and I aim to advance them all.

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Editorial Boards Committee Ivan Corwin Professor, Columbia University PhD: New York University, 2011. AMS offices and committees: Centennial Fellowship Selection Committee, 2017– 2019 (Chair, 2018–2019); Associate Editor for Bulletin Articles, 2018–2028. Selected publications or other forms of scholarship: 1. with G. Amir and J. Quastel, Probability distribution of the free energy of the continuum directed random polymer in 1+1 dimensions, Comm. Pure Appl. Math. 64 (2011), no. 4, 466–537, MR2796514; 2. The Kardar-Parisi-Zhang equation and universality class, Random Matrices Theory Appl. 1 (2012), no. 1, 1130001, MR2930377; 3. with A. Borodin, Macdonald processes, Probab. Theory Related Fields 158 (2014), no. 1-2, 225–400, MR3152785; 4. with A. Hammond, Brownian Gibbs property for Airy line ensembles, Invent. Math. 195 (2014), no. 2, 441–508, MR3152753; 5. with N. O’Connell, T. Seppäläinen, and N. Zygouras, Tropical combinatorics and Whittaker functions, Duke Math. J. 163 (2014), no. 3, 513– 563, MR3165422. Selected addresses or public presentations: ICM Seoul, Invited Lecture, 2014; Mahler, Lipschitz, ChernSimon, Pinsky Lectures. Synergistic activities: Scientific board member for ICERM (2020–2023) and SLMath (2021–present); Lead organizer of GROW 2024 and 2025 conferences. Additional experience/qualifications you bring to the position: Editorial board member at 10 journals in general mathematics as well as probability and mathematical physics. Candidate statement: I am happy to contribute to the ongoing mission of ensuring excellent editorial oversight of AMS journals and book series.

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Election Candidate Biographies

Irene Fonseca Kavˇci´c-Moura University Professor of Mathematics, Carnegie Mellon University PhD: University of Minnesota, 1985. AMS offices and committees: Representative to AMSIMS-SIAM Evaluation Panel for the NSF Mathematical Sciences Postdoctoral Research Fellowships, 2006–2009; AMS Nominating Committee, 2009–2011; AMS Short Course Subcommittee, 2016–2019; Mathematics Research Communities Advisory Board, AMS, 2017– 2020; AMS Fellows Selection Committee, 2017–2020; AMS Prize Oversight Committee, 2019–2025; CAMS, Communications of the AMS, Senior Editor, 2020–2025; AMS Bôcher Memorial Prize Selection Committee, 2021–2024; AMS Editorial Boards Committee, 2022–2025; AMS-SIAM Committee to Select the Winner of the 2024 Birkhoff Prize, Chair, 2023–2026; AMS Vice President, 2024–2027. Selected publications or other forms of scholarship: 1. with G. Bouchitt´e and L. Mascarenhas, A global method for relaxation, Arch. Rational Mech. Anal. 145 (1998), 51–98, MR1656477; 2. with S. Müller, 𝒜-quasiconvexity, lower semicontinuity, and Young measures, SIAM J. Math. Anal. 30 (1999), 1355–1390, MR1718306; 3. with G. Dal Maso and G. Leoni, Asymptotic analysis of second order nonlocal Cahn-Hilliard-type functionals, Trans. Amer. Math. Soc. 370 (2018), 2785–2823, MR3748585; 4. with N. Fusco, M. Morini, and G. Leoni, A model for dislocations in epitaxially strained elastic films, J. Math. Pures Appl. 111 (2018), 126–160, MR3760751; 5. with R. Choksi, J. Lin, and R. Venkatraman, Anisotropic surface tensions for phase transitions in periodic media, Calc. Var. Partial Differential Equations 61 (2022), no. 107, 41 pp., MR4404852; 6. with N. Fusco, G. Leoni, and M. Morini, Global and local energy minimizers for a nanowire growth model, Ann. Inst. H. Poincar´e Anal. Non Lin´eaire 40 (2023), 919–957. Selected addresses or public presentations: AWMSIAM Sonia Kovalevsky Lecturer, SIAM Annual Meeting, Boston, 2006; Invited Lecture, ICM 2022, July 6–14, 2022. Synergistic activities: Mentoring of junior faculty (pre-tenure) at the Department of Mathematical Sciences, CMU, 2016–2023; AWM (Association for Women in Mathematics) Scientific Advisory Committee (Chair, 2021). Additional experience/qualifications you bring to the position: Director of the Center for Nonlinear Analysis; Past President of SIAM, 2013–2014; SIAM SIAG APDE Chair, 2019–2020; SIAM SIAG MS Chair, 2023–2024; PI SEPTEMBER 2024

of large network grants; Co-PI in international mobility programs, including with SISSA (Trieste); Serves on 20 Editorial Boards, including Advances in Calculus of Variations, Archive for Rational Mechanics and Analysis, Communications of the AMS (CAMS), ESAIM: COCV (SMAI), Journal of Nonlinear Science, Mathematical Models and Methods in Applied Sciences (M3AS), and SIAM Journal on Mathematical Analysis; Member of the ICM 2026 Local Organizing Committee, and Chair of several advisory and scientific boards of research centers and institutes, international prize committees, and review and evaluation panels of multiple universities in the US and abroad. Candidate statement: I am honored and grateful to accept the nomination to join the EBC. It is a responsibility that I take seriously, and I welcome this opportunity to continue upholding the standards of excellence set forth by the AMS and to contribute to the advancement of our field through editorial guidance.

Christopher Hacon McMinn Presidential Endowed Chair, Distinguished Professor, University of Utah PhD: UCLA, 1998. AMS offices and committees: Fellows Program Selection Committee, 2013–2016; Cole Prize Selection Committee, 2015; Western Section Program Committee, 2015–2016 (Chair, 2016). Selected publications or other forms of scholarship: 1. with C. Birkar, P. Cascini, and J. McKernan, Existence of minimal models for varieties of log general type, J. Amer. Math. Soc. 23 (2010), no. 2, 405–468, MR2601039; 2. with J. McKernan, Existence of minimal models for varieties of log general type II, J. Amer. Math. Soc. 23 (2010), no. 2, 469–490, MR2601040; 3. with J. McKernan and C. Xu, ACC for log canonical thresholds, Ann. of Math. (2) 180 (2014), no. 2, 523–571, MR3224718; 4. with C. Xu, On the three-dimensional minimal model program in positive characteristic, J. Amer. Math. Soc. 28 (2015), no. 3, 711–744, MR3327534; 5. with J. McKernan and C. Xu, On the birational automorphisms of varieties of general type, Ann. of Math. (2) 177 (2013), no. 3, 1077–1111, MR3034294. Selected addresses or public presentations: AMS Fall Sectional Meeting, University of California, Riverside, Plenary Speaker, 2009; Algebraic and Complex Geometry Session, ICM, Invited Lecture, 2010; British Mathematical Colloquium, Edinburgh, Plenary Speaker, 2010; European

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Election Candidate Biographies

Congress of Mathematics, Krakow, Plenary Speaker, 2012; JMM, Baltimore, Plenary Speaker, 2014. Additional experience/qualifications you bring to the position: Clay Research Award, 2007; Frank Nelson Cole Prize in Algebra, 2009; Editor, Journal of Algebraic Geometry, since 2009; Associate Editor, Journal of the American Mathematical Society, 2009–2017; Antonio Feltrinelli Prize in Mathematics, Mechanics, and Applications, 2011; MSRI Science Advisory Committee, 2012–2018 (Co-chair, 2015– 2018); Fellow of the AMS, 2013; Associate Editor, Annals of Mathematics, 2013–2020; Associate Editor, Bollettino dell’Unione Matematica Italiana, since 2013; ICM committee for the selection of sectional speakers in Algebraic and Complex Geometry, 2014 and 2022; Selection Committee, Alfred P. Sloan Research Fellowships in Mathematics, 2015–2021; Associate Editor, Cambridge Journal of Mathematics, since 2016; Associate Editor, Journal of Pure and Applied Algebra, since 2016; E. H. Moore Research Article Prize, 2016; Member of the American Academy of Arts and Sciences, 2017; Breakthrough Prize, 2018; Member of the National Academy of Sciences, 2018; Selection Committee, Breakthrough Prize in Mathematics, 2019–present (Chair, since 2021); Fellow of the Royal Society, 2019. Candidate statement: I am honored to be nominated for election to the AMS Editorial Boards Committee. The books and journals published by the AMS are an important resource of the highest quality for mathematicians worldwide. I believe that if elected to the AMS Editorial Boards Committee, my experience as an author and editor will allow me to contribute to the ongoing success of these publications.

Michael J. Larsen Distinguished Professor, Mathematics, Indiana University, Bloomington PhD: Princeton University, 1988. AMS offices and committees: Transactions and Memoirs Editorial Committee, 2000– 2001; Committee on Publications, 2014–2017; AMS Council, Member at Large, 2014– 2017; Transactions and Memoirs Editorial Committee, 2015–2019; Frank Nelson Cole Prize Selection Committee, 2016–2017; Journal of the AMS, Associate Editor, 2016–2021; Journal of the AMS Editorial Committee, 2021–2025. Selected publications or other forms of scholarship: 1. with M. H. Freedman, A. Kitaev, and Z. Wang, Topological quantum computation. Mathematical challenges of

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the 21st century (Los Angeles, CA, 2000), Bull. Amer. Math. Soc. (N.S.) 40 (2003), no. 1, 31–38, MR194313; 2. with R. Pink, Finite subgroups of algebraic groups, J. Amer. Math. Soc. 24 (2011), no. 4, 1105–1158, MR2813339; 3. with A. Shalev and P. H. Tiep, Probabilistic Waring problems for finite simple groups, Ann. of Math. (2) 190 (2019), no. 2, 561–608, MR3997129; 4. with V. A. Lunts, Irrationality of motivic zeta functions, Duke Math. J. 169 (2020), no. 1, 1–30, MR4047547; 5. with L. Hesselholt and A. Lindenstrauss, On the K-theory of division algebras over local fields, Invent. Math. 219 (2020), no. 1, 281–329, MR4050106. Selected addresses or public presentations: Algebraic Geometry: Seattle (plenary lectures), 2005; Texas A & M University Frontier Lecture Series, 2008; Binghamton University Dean’s Speaker Series in Geometry/Topology, 2011; Cornell University Chelluri Lecture, 2022; International Congress of Mathematicians (online), 2022. Synergistic activities: Founded the Bloomington Math Circle; Wrote Putnam Exam problems, originally as a member of the Putnam Committee (MAA), and recently as an ”additional contributor.” Candidate statement: AMS publications have meant a lot to me over my professional life. My shelves are crammed with them, from the beaten-up old volumes of Automorphic Forms, Representations, and L-functions, to recent acquisitions, like the handsome collected works of Tate. My hope is that in the face of rapid change, the AMS can remain a relevant and viable alternative to high-priced commercial publishers. I have only been involved with AMS books as a reader, but on the journal side, I have served as an AMS editor since 2015 (first for the Transactions and the Memoirs and now for JAMS). In that capacity, I have come to appreciate the enormous amount of labor that referees put into making the journal system work. If I am elected to the Editorial Boards Committee, I will try to ensure that our editors are worthy of that effort: fair, broadly knowledgeable, mathematically open minded, and as transparent as possible given the constraints of the anonymous refereeing process.

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 71, NUMBER 8

MP: a-ballot-opener

FROM THE AMS SECRETARY

Election 2019Election 2025

Update in master page

Section CallSpecial for Suggestions 4 PICAS

PS: NEWS SUBHEAD

Vice President Member at Large Nominating Committee the Nominating Committee, for the following contested seats in the 2025 AMS elections: (one to be elected) CS: I of the Council CS: B (three to be elected) Sara Billey to be elected) Sami H. Assaf vice president, trustee, and five(five members at large of the Council. Abigail Thompson Dan Freed Ricardo Cortez Deadline for SPACE suggestions: November 1, 2024Q. Gouvêa Fernando Rebecca Garcia — SINGLE LINE — Board of Trustees Christopher D. Hacon Yongbin Ruan (one to be elected) Daniel Krashen David Savitt president, for the following contested Matthew the Ando Susan Loepp seats in the 2025 AMS elections: Deane Yang — SINGLE LINE SPACE — Rick Miranda Lenhard Ng three members of the Nominating Committee and two members of the Editorial Boards Kasso A. Okoudjou Editorial Boards Committee Committee. Maria Cristina Pereyra (two to be elected) Ian Agol Deadline for suggestions: JanuaryHal 31,Schenck 2025 Melanie Matchett Wood David Marker James McKernan boards the Editorial Boards Committee, for appointments to various editorial Terence Tao of AMS publications.

Deadline for suggestions: Can be submitted any time guishable from the original, will be sent by first class or airmail. However, the deadline for receipt of ballots will AMS members will receive email with instructions for votnot be extended. ing online by August 20, or a paper ballot by September 20. Send your suggestions for any of the above to: If you do not receive this information by that date, please Boris Hasselblatt, Secretary Biographies of Candidates PS: AHEAD contact the AMS (preferably before October 1) to request American Mathematical Society The next several pages contain biographical information a ballot. Send email to [email protected] or call the AMS 201 Charles Street at 800-321-4267 (within the US or Canada) or 401-455about all candidates. All candidates were given the opporProvidence, RI 02904-2213, USA 4000 (worldwide). The deadline for receipt of ballots is tunity to provide a statement of not more than 200 words [email protected] November 2, 2018. to appear at the end of their biographical information. or submit them online at Photos were supplied by the candidates. www.ams.org/committee-nominate

Write-in Votes

It is suggested that names for write-in votes be accompanied by the institution or web address of the individual for whom the vote is cast.

Description of Offices

The vice president and the members at large of the Council serve for three years on the Council. That body determines all scientific policy of the Society, creates and Replacement Ballots oversees numerous committees, appoints the treasurers A member who has not received a ballot by September 20, and members of the Secretariat, makes nominations of 2018, or who has received a ballot but has accidentally candidates for future elections, and determines the chief spoiled it, may write to [email protected] or Secretary of editors of several key editorial boards. Typically, each of the AMS, 201 Charles Street, Providence, RI 02904-2213, these new members of the Council will also serve on one USA, asking for a second ballot. The request should inof the Society’s five policy committees. Current and past clude the individual’s member code and the address to of the Council may be found here: www.ams which the replacement ballot should be sent. Immediately SEPTEMBER 2024 NOTICES OF THE AMERICAN Mmembers ATHEMATICAL SOCIETY 1109 .org/comm-all.html#COUNCIL. upon receipt of the request a second ballot, indistinNotices of the American Mathematical Society

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Ballots

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List of Candidates–2019 Election

YOUR SUGGESTIONS ARE WANTED BY:

MP: a-ballot-opener

FROM THE AMS SECRETARY

Election 2019Election 2025

Update in master page

Special Section Nominations by Petition 4 PICAS

The president will name at least six candidates for these List of Candidates–2019 Election three places, among whom may be candidates nominated One position of vice president and member of the Council

by petition in the manner described in the rules and proex officio a term of three years is to be filled in the elecVicefor President Member at Large Nominating Committee cedures. tion of 2025. Council to nominate least two (one to beThe elected) of at the Council (three to be elected) CS: intends I CS: B The candidate’s assent and petitions bearing at least 100 candidates, among whom may be candidates nominated Sara Billey (five to be elected) Sami H. Assaf valid signatures are required a name to be placed on by petition described in the rules and procedures below. AbigailasThompson Dan Freed Ricardo for Cortez the ballot. In addition, several other rules and procedures, Five positions of member at large of the Council for a Fernando Q. Gouvêa Rebecca Garcia — SINGLE LINE SPACE — described below, should be followed. term Board of three are to be filled in the same election.D. Hacon of years Trustees Christopher Yongbin Ruan (one to beintends elected)to nominate at least ten Daniel Krashen David Savitt The Council candidates, Rules and Procedures Matthew Andobe candidates nominated Susan Loepp Deane Yang among whom may by petition in Use separate copies of the form—for eachLINE candidate SINGLE SPACE — for vice Rick Miranda Lenhard Ng the manner described in the rules and procedures below. president, member at large, or member of the Nominating Kasso A. Okoudjou Editorial Boards Committee Petitions are presented to the Council, which, according or Editorial Boards Committees. Maria (two to be elected) to Section 2 of Article VII of the bylaws, makes theCristina nomi- Pereyra Hal Schenck Agol must be addressed to Sec1. To be considered,Ian petitions nations. Melanie Matchett Wood David Marker Society, 201 Charles retary, American Mathematical Prior to presentation to the Council, petitions in supJames McKernan Street, Providence, RI 02904-2213, USA, and must arport of a candidate for the position of vice president or Terence rive by 24 February 2025.Tao of member at large of the Council must have at least fifty 2. The1name of the candidate must be given as it appears valid signatures and must conform to several rules and proPICA in the American membership cedures, which are described below. Petitioners can facilguishable from theMathematical original, will Society’s be sent by first class or Ballots records and must be accompanied by memberwill itate themembers procedurewill by receive accompanying theinstructions petitions with a airmail. However, the deadline for receiptthe of ballots AMS email with for vot code. If the member code is not known by the cansigned statement from the giving by consent. not be extended. ing online by August 20, candidate or a paper ballot September 20. didate, it may be obtained by the candidate contactIf you do not receive this information by that date, please Editorial Boards Committee PS: AHEAD ing the AMS headquarters in Providence ( amsmem@ams Biographies of Candidates contact the AMS (preferably before October 1) to request .org ). several pages contain biographical information Two places Send on the Editorial Boards Committee The next or callwill the be AMS a ballot. email to [email protected] 3. The petition for a single candidatewere may given consist ofoppor sev- filled by election. There will the be four continuing at 800-321-4267 (within US or Canada) members or 401-455about all candidates. All candidates the eral sheets each bearing the statement of the petition, of the Editorial Boards The Committee. 4000 (worldwide). deadline for receipt of ballots is tunity to provide a statement of not more than 200 words of their the position, and information. signatures. The president will name at least four candidates for November 2, 2018. toincluding appear atthe thename end of biographical The name of the candidate must be exactly the same these two places, among whom may be candidates nomiPhotos were supplied by the candidates. Write-in Votes on all sheets. nated by petition in the manner described in the rules and On the next of page is a sample form for petitions. Petiprocedures. It is suggested that names for write-in votes be accompa- 4.Description Offices tioners may make and use photocopies or reasonable The assent and petitions bearing at least 100for niedcandidate’s by the institution or web address of the individual The vice president and the members at large of the facsimiles. valid signatures areis required for a name to be placed on whom the vote cast. Council serve for three years on the Council. That body 5. A signature is valid when it is clearly that of the memthe ballot. In addition, several other rules and procedures, determines all scientific policy of the Society, creates and Replacement Ballots ber whose name and address is given in the left-hand described below, should be followed. oversees numerous committees, appoints the treasurers A member who has not received a ballot by September 20, column. members of the Secretariat, makes nominations of Nominating Committee 2018, or who has received a ballot but has accidentally 6.and When a petition meeting these various requirements candidates for future elections, and determines the chief or Secretary spoiled it,on may to [email protected] Three places thewrite Nominating Committee will be filled of appears, the secretary will ask the candidate to indicate editors of several editorial Typically, each of AMS, 201 Charles Providence, RI 02904-2213, by the election. There will beStreet, six continuing members of the willingness to bekey included on boards. the ballot. these new members of the Council will also serve on one USA, asking for a second ballot. The request should inNominating Committee. of the Society’s five policy committees. Current and past clude the individual’s member code and the address to www.ams members of the Council may be found which the replacement ballot should be sent.OF Immediately 1110 NOTICES THE AMERICAN MATHEMATICAL SOCIETY VOLUMEhere: 71, NUMBER 8 .org/comm-all.html#COUNCIL. upon receipt of the request a second ballot, indistin105

Notices of the American Mathematical Society

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PS: NEWS SUBHEAD

PS: TEXT NO INDENT

Vice President or Member at Large

Election Nominations by Petition

Nominations by Petition ____________________________________________________________________________ The undersigned members of the American Mathematical Society propose the name of ______________________________________________________________ as a candidate for the position of (check one):

□ Vice President (term beginning 02/01/2026) □ Member at Large of the Council (term beginning 02/01/2026) □ Member of the Nominating Committee (term beginning 01/01/2026) □ Member of the Editorial Boards Committee (term beginning 02/01/2026)

of the American Mathematical Society. Return petitions by February 24, 2025 to: Secretary, AMS, 201 Charles Street, Providence, RI 02904-2213, USA

Name, address, and AMS member code, if available (printed or typed)

Signature

Signature

Signature

Signature

Signature

Signature

SEPTEMBER 2024

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

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NEWS

AMS Updates Early-Bird Registration Open for December Joint International Meeting Register now for the joint meeting of the New Zealand, Australian, and American mathematical societies, taking place December 9–13, 2024, in Auckland, New Zealand. Early-bird registration is available until October 25, 2024. Plenary speakers at the joint meeting of the NZMS, AustMS, and AMS include Persi Diaconis (Stanford University), Rachael Ka’ai-Mahuta (Auckland University of Technology), Svitlana Mayboroda (University of Minnesota, ETH Zurich), Michael Miller (Victoria University of Wellington), and James Saunderson (Monash University). General plenary speakers at the joint meeting include Lara Alcock (Loughborough University), Richard Kenyon (Yale University), Eamonn O’Brien (University of Auckland), Priya Subramanian (University of Auckland), Katharine Turner (Australian National University), and Geordie Williamson (University of Sydney). Learn more and register at https://ms-meet-2024 .blogs.auckland.ac.nz/registration/. —AMS Communications

Last Call for AMS Fellowship Applications September 30, 2024, is the deadline to apply for the following AMS fellowships for 2025–2026: • Stefan Bergman Fellowship ($25,000), an earlycareer research fellowship for mathematicians who specialize in the areas of real analysis, complex analysis, or partial differential equations; • Joan and Joseph Birman Fellowship for Women Scholars ($50,000), a mid-career research fellowDOI: https://doi.org/10.1090/noti3014

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ship for women mathematicians, specially designed to fit the unique needs of women; • AMS Centennial Research Fellowship ($50,000), a research fellowship for mathematicians currently holding a tenured, tenure-track, postdoctoral, or comparable (at the discretion of the selection committee) position at an institution in North America. Recipients of the fellowship shall have held their doctoral degree for at least three years and not more than twelve years at the inception of the award; • AMS Claytor-Gilmer Fellowship ($50,000), a mid-career research fellowship to further excellence in mathematics research and to help generate wider and sustained participation by Black mathematicians. More information is available at https://www.ams .org/programs/ams-fellowships/ams-fellowships. —AMS Communications

Teens Win Menger Awards at ISEF The American Mathematical Society (AMS) presented the Karl Menger Awards at the 2024 Regeneron International Science and Engineering Fair (Regeneron ISEF), held in Los Angeles on May 17, 2024. The winners were high-school students who earned the right to compete at the Regeneron ISEF by winning a top prize at a local, regional, state, or national science fair. All winners received a one-year AMS membership and a booklet on Karl Menger. Quang Tran of Patrick F. Taylor Science and Technology Academy, Harvey, LA, received the first-place prize of $2,000 for Divisors. Second awards ($1,000): Anna Oliva, Carnegie Vanguard High School, Houston, TX, Symmetry, Fixed Points and Quantum Billiards and Emma Rueter, LeibnizGymnasium Berlin, Berlin, Germany, Integration of Sequences

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 71, NUMBER 8

Third awards ($500): Arda Ozcelebi, Izmir Ozel Ege Lisesi, Izmir, Turkey, p-Euler-phi Partitions and Their Properties; Yoonsang Lee, Korea Science Academy of KAIST, Seoul, South Korea, A Study on Arc Index of Theta Curves; Anay Aggarwal and Manu Isaacs, Jesuit High School, Portland, OR, Fast Modular Exponentiation with Factored Modulus; Helena Welch, Los Alamos High School, Los Alamos, NM, Modeling an Ancient Musical Instrument Certificates of Honorable Mention: Yunjia Quan, Charlotte Country Day School, Charlotte, NC, Enhancing Ethereum’s Security With LUMEN; Austin Luo, Morgantown High School, Morgantown, WV, Injective Chromatic Index of Packet Radio Networks; Ayush Jain, Shri Ram School - Aravali Campus, Gurgaon, Haryana, India, Detecting Causality Using Symplectic Quandles; Joseph Vulakh, Paul Laurence Dunbar High School, Lexington, KY, Twisted Homogeneous Racks; Arav Chand, Half Hollow Hills High School West, Dix Hills, NY, Proofs of Fibonacci Analogues of Two Theorems; Songtianze Huang, Hangzhou Foreign Languages School, Hangzhou, Zhejiang, China, Group of seventh chord transformations; Sarah Lu, Centro Residencial de Oportunidades Educativas de Mayaguez, Mayaguez, Puerto Rico, Enhancing Federated Learning Using Math and Coding AMS participation in the Regeneron ISEF is supported in part by funds from the Karl Menger Fund, which was established by the family of the late Karl Menger. Awards at the fair are given to pre-college students in mathematics as well as to mathematically oriented projects in computer science, physics, and engineering. —AMS Communications

Deaths of AMS Members Jacob P. Murre, of the Netherlands, died on April 9, 2023. Born on September 18, 1929, he was a member of the Society for 68 years. William W. Adams, of Silver Spring, Maryland, died on February 15, 2024. Born on July 23, 1937, he was a member of the Society for 63 years. George L. Csordas, of Honolulu, Hawaii, died on April 23, 2024. Born on September 20, 1941, he was a member of the Society for 57 years.

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NEWS

Mathematics People Sarnak Wins 2024 Shaw Prize Peter Sarnak of the Institute for Advanced Study (IAS) and Princeton University was awarded the 2024 Shaw Prize in Mathematical Sciences for his development of the arithmetic theory of thin groups and the affine sieve by bringing together number theory, analysis, combinatorics, dynamics, geometry, and spectral theory. Sarnak is the Gopal Prasad Professor of Mathematics at IAS and Eugene Higgins Professor of Mathematics at Princeton, where he also has served as department chair. Born in Johannesburg, South Africa, Sarnak received his PhD in Mathematics from Stanford University in 1980. He has taught at Stanford and the Courant Institute of Mathematical Sciences, New York University. Sarnak is a member of the US National Academy of Sciences and a Fellow of both the American Mathematical Society (AMS) and the Royal Society of London. The Shaw Prize Foundation awards three annual prizes, in astronomy, life science and medicine, and mathematical sciences, each bearing an award of US$1.2 million. The presentation ceremony is scheduled for November 12, 2024, in Hong Kong.

“Together with his PhD advisor, Ennio De Giorgi, Ambrosio founded the theory of free discontinuity problems, a class of problems in the calculus of variations that involves the combination of volume and surface energies. In this class, it is possible to frame problems coming from image segmentation and fracture mechanics,” according to a press release. “In the second part of his career, Ambrosio moved to the theory of currents in geometric measure theory, introducing a far-reaching extension of the Federer-Fleming theory to metric spaces and to the theory of flows associated to non-smooth vector fields. His present research interests include optimal transport and analysis in metric measure spaces.” Ambrosio was a plenary speaker at the 2018 International Congress of Mathematics (ICM) and previously gave an invited section lecture at ICM 2002. He was awarded the Caccioppoli Prize (1999), Fermat Prize (2003), Balzan Prize (2019), and Riemann Prize (2022). Northwestern’s biennial Nemmers Prizes recognize top scholars in earth sciences, economics, and mathematics for their lasting contributions to new knowledge, outstanding achievements, and the development of significant new modes of analysis.

—AMS Communications

Ambrosio Receives 2024 Nemmers Prize Luigi Ambrosio, Scuola Normale Superiore, Pisa, Italy, received the Frederic Esser Nemmers Prize in Mathematics. Ambrosio was honored for his “deep and numerous contributions to calculus of variations and geometric measure theory, and broad and far-reaching influence on these fields,” as announced by Northwestern University. Ambrosio, a professor of mathematical analysis, will receive US$300,000 and will present lectures, participate in seminars, and engage with faculty and students in other scholarly activities. DOI: https://doi.org/10.1090/noti3015

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—AMS Communications

Daubechies Named to Royal Society Ingrid C. Daubechies of Duke University has been elected a Foreign Member of the Royal Society, the United Kingdom’s national academy of sciences. In 2024, more than 90 researchers from around the world were elected to the Fellowship of the Royal Society as fellows and foreign members. “This new cohort has already made significant contributions to our understanding of the world around us and continues to push the boundaries of possibility in academic research and industry,” said Sir Adrian Smith, president of the Royal Society, in a press release.

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

VOLUME 71, NUMBER 8

Mathematics People

NEWS At Duke, Daubechies is James B. Duke Professor, Department of Mathematics and Department of Electrical and Computer Engineering. She received her PhD in 1980 from Vrije Universiteit Brussel.

2020. Kupers’ work is “motivated by the modern version of a classical problem: the classification of smooth manifolds,” according to a press release. —Centre de recherches math´ematiques

—AMS Communications

Nelson, Newton, Thorne Named Clay Research Awardees for 2024 The Clay Mathematics Institute announced the 2024 Clay Research Awards, which will be presented at the Clay Research Conference in Oxford, UK, on October 2, 2024. A joint award was made to James Newton (University of Oxford) and Jack Thorne (University of Cambridge) “in recognition of their remarkable proof of the existence of the symmetric power functorial lift for Hilbert modular forms.. . . The proof marks a milestone in work on the Langlands program.” Paul Nelson (Aarhus University) received a Clay Research Award in recognition of his contributions to the analytic theory of automorphic forms. “His work has resulted in the first convexity breaking bounds for a large class of L-functions on the critical line (including all the standard ones of GL(n)),” according to a press release. “This marks a significant advance in a field initiated one hundred years ago by Hermann Weyl in the context of the Riemann Zeta function.” —Clay Mathematics Institute

Kupers Awarded 2024 Aisenstadt Prize The 2024 André Aisenstadt Prize in Mathematics was awarded to Alexander Kupers, University of Toronto Scarborough (UTSC). Created in 1991 by the Centre de recherches mathématiques (CRM), the Aisenstadt Prize recognizes outstanding research results in pure or applied mathematics by a young Canadian mathematician. It includes a scholarship and a medal. Born in the Netherlands, Kupers received his PhD in 2016 from Stanford University. Following a postdoctoral position at the University of Copenhagen and a Benjamin Peirce Fellowship at Harvard University, he joined UTSC’s department of computer and mathematical sciences in

SEPTEMBER 2024

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

1115

AMS Graduate Student

Apply for support for doctoral student travel to the Joint Mathematics Meetings

• Connect with fellow researchers • Advance your career • Present your research • Explore new mathematical areas • Expand your knowledge of professional and educational matters

Submission deadlines and eligibility info:

www.ams.org/student-travel

Classified Advertising Employment Opportunities NEW JERSEY Program in Applied and Computational Mathematics Princeton University Postdoctoral Research Associate The Program in Applied and Computational Mathematics invites applications for Postdoctoral Research Associate or more senior positions, to join in research efforts of interest to its faculty. Domains of interest include nonlinear partial differential equations, computational fluid dynamics, material science, dynamical systems, numerical analysis, stochastic analysis, graph theory and applications, mathematical biology, financial mathematics, mathematical approaches to signal analysis, information theory, structural biology and image processing. The term of appointment is based on rank. Positions at the postdoctoral rank are for one year with the possibility of renewal pending satisfactory performance and continued funding; those hired at more senior ranks may have multi-year appointments. For details on specific faculty members and their research interests, please go to

Applicants must submit a cover letter, CV, bibliography/publications list, statement of research and three letters of recommendation online at https://www .mathjobs.org/jobs. PhD is required. This position is subject to the University background check policy. The work location for this position is in-person on campus at Princeton University. Princeton University is an Equal Opportunity/ Affirmative Action Employer: https://rrr.princeton .edu/2023/equal-opportunity-policy/equal -opportunity-policy and all qualified applicants will re-

ceive consideration for employment without regard to age, race, color, religion, sex, sexual orientation, gender identity or expression, national origin, disability status, protected veteran status, or any other characteristic protected by law. 4

https://www.pacm.princeton.edu/sites/default /files/2023-10/Faculty%20Interests%2023-24 .pdf.

The Notices Classified Advertising section is devoted to listings of current employment opportunities. The publisher reserves the right to reject any listing not in keeping with the Society’s standards. Acceptance shall not be construed as approval of the accuracy or the legality of any information therein. Advertisers are neither screened nor recommended by the publisher. The publisher is not responsible for agreements or transactions executed in part or in full based on classified advertisements. The 2024 rate is $3.65 per word. Advertisements will be set with a minimum one-line headline, consisting of the institution name above body copy, unless additional headline copy is specified by the advertiser. Headlines will be centered in boldface at no extra charge. Ads will appear in the language in which they are submitted. There are no member discounts for classified ads. Dictation over the telephone will not be accepted for classified ads. Upcoming deadlines for classified advertising are as follows: November 2024—August 23, 2024; December 2024—September 27, 2024. US laws prohibit discrimination in employment on the basis of color, age, sex, race, religion, or national origin. Advertisements from institutions outside the US cannot be published unless they are accompanied by a statement that the institution does not discriminate on these grounds whether or not it is subject to US laws. .Submission: Send email to [email protected]. DOI: https://doi.org/10.1090/noti3017

SEPTEMBER 2024

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

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NEW BOOKS

New Books Offered by the AMS Discrete Mathematics and Combinatorics

General Interest The Mathematical Playground

Exploring Discrete Geometry

People and Problems from 31 Years of Math Horizons

Thomas Q. Sibley, St. John’s University, Collegeville, MN Together with its clear mathematical exposition, the problems in this book take the reader from an introduction to discrete geometry all the way to its frontiers. Investigations start with easily drawn figures, such as dividing a polygon into triangles or finding the minimum number of “guards” for a polygon (“art gallery” problem). These early explorations build intuition and set the stage. Variations on the initial problems stretch this intuition in new directions. These variations on problems together with growing intuition and understanding illustrate the theme of this book: “When you have answered the question, it is time to question the answer.” Numerous drawings, informal explanations, and careful reasoning build on high school algebra and geometry. This item will also be of interest to those working in geometry and topology. Anneli Lax New Mathematical Library, Volume 56 August 2024, 156 pages, Softcover, ISBN: 978-1-47047807-0, LC 2024015227, 2020 Mathematics Subject Classification: 52–01, List US$69, AMS Individual member US$51.75, AMS Institutional member US$55.20, MAA members US$51.75, Order code NML/56 bookstore.ams.org/nml-56

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Alissa S. Crans, Loyola Marymount University, Los Angeles, CA, and Glen T. Whitney, Prison Math Project, Phoenix, AZ, Editors Welcome to The Mathematical Playground, a book celebrating more than thirty years of the problems column in the MAA undergraduate magazine, Math Horizons. Anecdotes, interviews, and historical sketches accompany the puzzles, conveying the vibrancy of the “Playground” community. The lively prose and humor used throughout the book reveal the enthusiasm and playfulness that have become the column’s hallmark. Each chapter features a theme that helps illustrate community: from the Opening Acts—chronicling how interesting questions snowball into original research—to the Posers and Solvers themselves. These stories add an engaging dimension beyond the ample mathematical challenge. A particular highlight is a chapter introducing the seven editors who have produced “The Playground”, revealing the perspectives of the individuals behind the column. The Mathematical Playground has plenty to offer both novice and experienced solvers. The lighthearted, conversational style, together with copious hints, a problem-solving primer, and a detailed glossary, welcomes newcomers, regardless of their background, to the puzzle-solving world. The more seasoned solver will find over twenty new problems plus open-ended challenges and suggestions for further investigation. Whether you’re a long-time Math Horizons reader, or encountering “The Playground” for the first time, you are invited into this celebration of the rich culture of recreational mathematics. Just remember the most important rule … Have fun!

Notices of the American Mathematical Society

Volume 71, Number 8

NEW BOOKS Problem Books, Volume 38 August 2024, 458 pages, Softcover, ISBN: 978-1-47047752-3, LC 2024014160, 2020 Mathematics Subject Classification: 00A07, 00A08, 00A27, 97D50, 01A07, 01A80, List US$39, AMS Individual member US$29.25, AMS Institutional member US$31.20, MAA members US$29.25, Order code PRB/38 bookstore.ams.org/prb-38

Teaching Mathematics Through Cross-Curricular Projects Elizabeth A. Donovan, Murray State University, KY, Lucas A. Hoots, Morehead State University, KY, and Lesley W. Wiglesworth, Centre College, Danville, KY, Editors This book offers engaging cross-curricular modules to supplement a variety of pure mathematics courses. Developed and tested by college instructors, each activity or project can be integrated into an instructor’s existing class to illuminate the relationship between pure mathematics and other subjects. Every chapter was carefully designed to promote active learning strategies. The editors have diligently curated a volume of twenty-six independent modules that cover topics from fields as diverse as cultural studies, the arts, civic engagement, STEM topics, and sports and games. An easy-to-use reference table makes it straightforward to find the right project for your class. Each module contains a detailed description of a cross-curricular activity, as well as a list of the recommended prerequisites for the participating students. The reader will also find suggestions for extensions to the provided activities, as well as advice and reflections from instructors who field-tested the modules. Teaching Mathematics Through Cross-Curricular Projects is aimed at anyone wishing to demonstrate the utility of pure mathematics across a wide selection of real-world scenarios and academic disciplines. Even the most experienced instructor will find something new and surprising to enhance their pure mathematics courses. This item will also be of interest to those working in math education. Classroom Resource Materials, Volume 72 August 2024, 351 pages, Softcover, ISBN: 978-1-4704-74669, LC 2024006104, 2020 Mathematics Subject Classification: 00–XX, 97–XX, List US$65, AMS Individual member US$48.75, AMS Institutional member US$52, MAA members US$48.75, Order code CLRM/72

Geometry and Topology Trees of Hyperbolic Spaces

Mathematical Surveys and Monographs Volume 282

Trees of Hyperbolic Spaces

Michael Kapovich, University of California, Davis, CA, and Pranab Sardar, Indian Institute of Science Education and Research, Mohali, India

Michael Kapovich Pranab Sardar

This book offers an alternative proof of the Bestvina–Feighn combination theorem for trees of hyperbolic spaces and de scribes uniform quasigeodesics in such spaces. As one of the applications of their description of uniform quasigeodesics, the authors prove the existence of Cannon–Thurston maps for inclusion maps of total spaces of subtrees of hyperbolic spaces and of relatively hyperbolic spaces. They also analyze the structure of Cannon–Thurston laminations in this setting. Furthermore, some group-theoretic applications of these results are discussed. This book also contains background material on coarse geometry and geometric group theory. Mathematical Surveys and Monographs, Volume 282 August 2024, 278 pages, Softcover, ISBN: 978-1-4704-74256, LC 2024009568, 2020 Mathematics Subject Classification: 20F67, 51F30, List US$135, AMS members US$108, MAA members US$121.50, Order code SURV/282 bookstore.ams.org/surv-282 Proceedings of Symposia in

PURE MATHEMATICS Volume 109

Frontiers in Geometry and Topology Paul M. N. Feehan Lenhard L. Ng Peter S. Ozsváth Editors

Frontiers in Geometry and Topology Paul M. N. Feehan, Rutgers, The State University of New Jersey, Piscataway, NJ, Lenhard L. Ng, Duke University, Durham, NC, and Peter S. Ozsváth, Princeton University, NJ, Editors

This volume contains the proceedings of the summer school and research conference “Frontiers in Geometry and Topology”, celebrating the sixtieth birthday of Tomasz Mrowka, which was held from August 1–12, 2022, at the Abdus Salam International Centre for Theoretical Physics (ICTP). The summer school featured ten lecturers and the research conference featured twenty-three speakers covering a range of topics. A common thread, reflecting Mrowka’s own

bookstore.ams.org/clrm-72

September 2024

Notices of the American Mathematical Society

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NEW BOOKS work, was the rich interplay among the fields of analysis, geometry, and topology. Articles in this volume cover topics including knot theory; the topology of three and four-dimensional manifolds; instanton, monopole, and Heegaard Floer homologies; Khovanov homology; and pseudoholomorphic curve theory. Proceedings of Symposia in Pure Mathematics, Volume 109 August 2024, 284 pages, Softcover, ISBN: 978-1-47047087-6, LC 2024004662, 2020 Mathematics Subject Classification: 53D45, 57K10, 57K18, 57K33, 57K41, 57R58, List US$139, AMS members US$111.20, MAA members US$125.10, Order code PSPUM/109 bookstore.ams.org/pspum-109

Mathematical Physics 32 FRANCK GABRIEL CLÉMENT HONGLER FRANCESCO SPADARO

Lattice Models and Conformal Field Theory

Lattice Models and Conformal FieldTheory Franck Gabriel, Université Lyon 1, France, Clément Hongler, EPFL, Lausanne, Switzerland, and Francesco Spadaro, Zürich, Switzerland

This book introduces the mathematical ideas connecting Statistical Mechanics and Conformal Field Theory (CFT). Building advanced structures on top of more elementary ones, the authors map out a well-posed road from simple lattice models to CFTs. Structured in two parts, the book begins by exploring several two-dimensional lattice models, their phase transitions, and their conjectural connection with CFT. Through these lattice models and their local fields, the fundamental ideas and results of two-dimensional CFTs emerge, with a special emphasis on the Unitary Minimal Models of CFT. Delving into the delicate ideas that lead to the classification of these CFTs, the authors discuss the assumptions on the lattice models whose scaling limits are described by CFTs. This produces a probabilistic rather than an axiomatic or algebraic definition of CFTs. Suitable for graduate students and researchers in mathematics and physics, Lattice Models and Conformal Field Theory introduces the ideas at the core of Statistical Field Theory. Assuming only undergraduate probability and complex analysis, the authors carefully motivate every argument and assumption made. Concrete examples and exercises allow readers to check their progress throughout. 1120

This item will also be of interest to those working in probability and statistics. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

Courant Lecture Notes, Volume 32 September 2024, approximately 205 pages, Softcover, ISBN: 978-1-4704-5618-4, LC 2024013482, 2020 Mathematics Subject Classification: 60–XX, 81–XX, 82–XX, List US$55, AMS Individual member US$40.88, AMS Institutional member US$43.60, MAA members US$49.05, Order code CLN/32 bookstore.ams.org/cln-32

New in Contemporary Mathematics Algebra and Algebraic Geometry CONTEMPORARY ATHEMATICS M

804

A Glimpse into Geometric Representation Theory

A Glimpse into Geometric RepresentationTheory Mahir Bilen Can, Tulane University, New Orleans, LA, and Jörg Feldvoss, University of South Alabama, Mobile, AL, Editors

This volume contains the proceedings of the AMS Special Session on Combinatorial and Geometric Representation Theory, held virtually on November 20–21, 2021. The articles offer an engaging look into recent advancements in geometric representation theory. Despite diverse subject matters, a common thread uniting the articles of this volume is the power of geometric methods. The authors explore the following five contemporary topics in geometric representation theory: equivariant motivic Chern classes; equivariant Hirzebruch classes and equivariant Chern-Schwartz-MacPherson classes of Schubert cells; locally semialgebraic spaces, Nash manifolds, and their superspace counterparts; support varieties of Lie superalgebras; wreath Macdonald polynomials; and equivariant extensions and solutions of the Deligne-Simpson problem. Each article provides a well-structured overview of its topic, highlighting the emerging theories developed by the authors and their colleagues. Mahir Bilen Can Jörg Feldvoss Editors

Notices of the American Mathematical Society

Volume 71, Number 8

NEW BOOKS Contemporary Mathematics, Volume 804 September 2024, 203 pages, Softcover, ISBN: 978-1-47047090-6, 2020 Mathematics Subject Classification: 14C17, 14M15, 14P20, 58A50, 17B56, 20G10, 05E05, 33D52, 14D24, 20G25, List US$135, AMS members US$108, MAA members US$121.50, Order code CONM/804

Memoirs of the American Mathematical Society, Volume 298, Number 1493 August 2024, 94 pages, Softcover, ISBN: 978-1-4704-70531, 2020 Mathematics Subject Classification: 13C14, 13H10, 14E16, 32S25, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/298/1493

bookstore.ams.org/conm-804

bookstore.ams.org/memo-298-1493

New in Memoirs of the AMS

Homotopy in Exact Categories Jack Kelly, Lincoln College, Oxford University, United Kingdom

Algebra and Algebraic Geometry

Memoirs of the American Mathematical Society, Volume 298, Number 1490 August 2024, 160 pages, Softcover, ISBN: 978-1-47047041-8, 2020 Mathematics Subject Classification: 18G35, 18N40, 18N70; 12J05, 18G80, 18M05, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/298/1490

Reconstructing Orbit Closures from their Boundaries

bookstore.ams.org/memo-298-1490

Paul Apisa, University of Wisconsin–Madison, Wisconsin, and Alex Wright, University of Michigan, Ann Arbor, Michigan

Analysis

Memoirs of the American Mathematical Society, Volume 298, Number 1487 August 2024, 141 pages, Softcover, ISBN: 978-1-4704-69115, 2020 Mathematics Subject Classification: 32G15, 37D40, 14H15, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/298/1487 bookstore.ams.org/memo-298-1487

Modular Representation Theory and Commutative Banach Algebras David J. Benson, University of Aberdeen, Scotland, United Kingdom This item will also be of interest to those working in analysis.

Multi-scale Sparse Domination David Beltran, Universitat de Valencia, Burjassot, Spain, Joris Roos, University of Massachusetts Lowell, Massachusetts, and Andreas Seeger, University of Wisconsin–Madison, Wisconsin Memoirs of the American Mathematical Society, Volume 298, Number 1491 August 2024, 104 pages, Softcover, ISBN: 978-1-4704-70425, 2020 Mathematics Subject Classification: 42B15, 42B20, 42B25, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/298/1491 bookstore.ams.org/memo-298-1491

Memoirs of the American Mathematical Society, Volume 298, Number 1488 August 2024, 118 pages, Softcover, ISBN: 978-1-4704-70296, 2020 Mathematics Subject Classification: 20C20; 46J99, 16T05, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/298/1488

Differential Equations

bookstore.ams.org/memo-298-1488

Dongxiao Yu, University of California, Berkeley, CA

Reflexive Modules on Normal Gorenstein Stein Surfaces,Their Deformations and Moduli

Memoirs of the American Mathematical Society, Volume 298, Number 1492 August 2024, 136 pages, Softcover, ISBN: 978-1-4704-70487, 2020 Mathematics Subject Classification: 35L70, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/298/1492

Javier Fernández de Bobadilla, Basque Foundation for Science, Bilbao, Basque Country, Spain, and Basque Center for Applied Mathematics, Bilbao, Basque Country, Spain, and Agustín Romano-Velázquez, Universidad Nacional Autónoma de México, Cuernavaca, Morelos, México

September 2024

Asymptotic Completeness for a Scalar Quasilinear Wave Equation Satisfying the Weak Null Condition

bookstore.ams.org/memo-298-1492

Notices of the American Mathematical Society

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NEW BOOKS

Number Theory

also reflect on the gender publication gap in mathematics and focus on one of the central pillars of zbMATH Open: the community of reviewers.

On p​-Adic L-Functions for Hilbert Modular Forms John Bergdall, Bryn Mawr College, PA, and David Hansen, Max Planck Institute for Mathematics, Bonn, Germany Memoirs of the American Mathematical Society, Volume 298, Number 1489 August 2024, 125 pages, Softcover, ISBN: 978-1-47047031-9, 2020 Mathematics Subject Classification: 11F67, 11F85; 11F41, 11F03, 11F80, 11F33, List US$85, AMS members US$68, MAA members US$76.50, Order code MEMO/298/1489

A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society.

June 2024, 110 pages, Softcover, ISBN: 978-3-98547-073-0, 2020 Mathematics Subject Classification: 01A74; 01–06, 01A60, 01A61, 01A65, List US$35, AMS members US$28, Order code EMSZBMATH bookstore.ams.org/emszbmath

Geometry and Topology

bookstore.ams.org/memo-298-1489

Configuration Spaces of Manifolds with Boundary

New AMS-Distributed Publications

Ricardo Campos, Institut de Mathématiques de Toulouse, Universiteé de Toulouse, CNRS, France, Najib Idrissi, Université de Paris, France, Pascal Lam brechts, Université catholique de Louvain, Louvain-la- Neuve, Belgium, and Thomas Willwacher, ETH Zúrich, Ramistrasse, Switzerland

General Interest 90 Years of zbMATH Klaus Hulek, Leibniz Universität Hannover, Germany, Octavio Paniagua Taboada, FIZ Karl sruhe, Germany, and Olaf Te schke, FIZ Karlsruhe, Germany, Editors zbMATH Open, the world’s most comprehensive and longest-running abstracting and reviewing service in pure and applied mathematics, was founded by Otto Neugebauer in 1931. It celebrated its 90th anniversary by becoming an open access database. In December 2019, the Joint Science Conference (Gemeinsame Wissenschaftskonferenz) agreed that the Federal and State Governments of Germany would support FIZ Karlsruhe in transforming zbMATH into an open platform. In the future, zbMATH Open will link mathematical services and platforms so as to provide considerably more content for further research and collaborative work in mathematics and related fields. This book explains how zbMATH Open has reacted to a rapidly changing digital era. Topics covered include: the linkage of zbMATH Open with different community platforms and digital maths libraries, the use of zbMATH Open as a bibliographical tool, API solutions, current advancements in author profiles, the indexing of mathematical software packages (swMATH), and issues concerning mathematical formula search in zbMATH Open. The authors 1122

The authors study ordered configuration spaces of compact manifolds with boundary. They show that for a large class of such manifolds, the real homotopy type of the configuration spaces only depends on the real homotopy type of the pair consisting of the manifold and its boundary. Moreover, they describe explicit real models of these configuration spaces using three different approaches. They do this by adapting previous constructions for configuration spaces of closed manifolds which relied on Kontsevich’s proof of the formality of the little disks operads. The authors also prove that our models are compatible with the richer structure of configuration spaces, respectively a module over the Swiss-Cheese operad, a module over the associative algebra of configurations in a collar around the boundary of the manifold, and a module over the little disks operad. This item will also be of interest to those working in algebra and algebraic geometry. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Notices of the American Mathematical Society

Volume 71, Number 8

NEW BOOKS Astérisque, Number 449 June 2024, 482 pages, Softcover, ISBN: 978-2-85629990-6, 2020 Mathematics Subject Classification: 55R80, 18M75, 18M70, 55P62, 55P48, List US$63, AMS members US$50.40, Order code AST/449 bookstore.ams.org/ast-449

Number Theory Les Suites Spectrales de Hodge-Tate Ahmed Abbes, Laboratoire Alexander Grothendieck, CNRS, IHES, Université Paris-Saclay, Bures-sur-Yevette, France, and Michel Gros, Université de Rennes, CNRS, France This book presents two important results in p​-adic Hodge theory following the approach initiated by Faltings, namely (i) his main p​-adic comparison theorem, and (ii) the Hodge-Tate

spectral sequence. The authors establish for each of these results two versions: an absolute one and a relative one. While the absolute statements can reasonably be considered as well as understood, particularly after their extension to rigid varieties by Scholze, Faltings’ initial approach for the relative variants has remained much less studied. Although the authors follow the same strategy as that used by Faltings to establish his main p​-adic comparison theorem, part of their proofs is based on new results. The relative Hodge-Tate spectral sequence is new in this approach. This item will also be of interest to those working in algebra and algebraic geometry. A publication of the Société Mathématique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list.

Astérisque, Number 448 May 2024, 482 pages, Softcover, ISBN: 978-2-85629-988-3, 2020 Mathematics Subject Classification: 11G25, 11F80, 14F05, 14F20, 14F30, 14F35, 14G20, List US$116, AMS members US$92.80, Order code AST/448 bookstore.ams.org/ast-448

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Meetings & Conferences of the AMS September Table of Contents The Meetings and Conferences section of the Notices gives information on all AMS meetings and conferences approved by press time for this issue. Please refer to the page numbers cited on this page for more detailed information on each event. Paid meeting registration is required to submit an abstract to a sectional meeting. Invited Speakers and Special Sessions are listed as soon as they are approved by the cognizant program committee; the codes listed are needed for electronic abstract submission. For some meetings the list may be incomplete. Information in this issue may be dated. The most up-to-date meeting and conference information can be found online at www.ams.org/meetings. Important Information About AMS Meetings: Potential organizers, speakers, and hosts should refer to https:// www.ams.org/meetings/meetings-general for general information regarding participation in AMS meetings and conferences. Abstracts: Speakers should submit abstracts on the easy-to-use interactive Web form. No knowledge of LaTeX is necessary to submit an electronic form, although those who use LaTeX may submit abstracts with such coding, and all math displays and similarly coded material (such as accent marks in text) must be typeset in LaTeX. Visit www.ams .org/cgi-bin/abstracts/abstract.pl . Questions about abstracts may be sent to [email protected]. Close attention should be paid to specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated.

Meetings in this Issue 2024 September 14–15 October 5–6 October 19–20 October 26–27 December 9–13

San Antonio, Texas Savannah, Georgia Albany, New York Riverside, California Auckland, New Zealand

p. 1127 p. 1129 p. 1131 p. 1133 p. 1134

2025 January 8–11 March 8–9 March 29–30 April 5–6 May 3–4 October 18–19 December 6–7

Seattle, Washington (JMM 2025) Clemson, South Carolina Lawrence, Kansas Hartford, Connecticut San Luis Obispo, CA St. Louis, Missouri Denver, Colorado

p. 1134 p. 1135 p. 1135 p. 1135 p. 1135 p. 1136 p. 1136

2026 January 4–7 March 7–8 April 18–19

Washington, DC (JMM 2026) Boise, Idaho Fargo, North Dakota

p. 1136 p. 1136 p. 1136

Associate Secretaries of the AMS Central Section: Betsy Stovall, University of Wisconsin– Madison, 480 Lincoln Drive, Madison, WI 53706; email: [email protected]; telephone: (608) 262-2933. Eastern Section: Steven H. Weintraub, Department of Mathematics, Lehigh University, Bethlehem, PA 180153174; email: [email protected]; telephone: (610) 758-3717. Southeastern Section: Brian D. Boe, Department of Mathematics, University of Georgia, 220 D W Brooks Drive, Athens, GA 30602-7403; email: [email protected]; telephone: (706) 542-2547. Western Section: Michelle Manes, University of Hawaii, Department of Mathematics, 2565 McCarthy Mall, Keller 401A, Honolulu, HI 96822; email: [email protected]; telephone: (808) 956-4679. September 2024

The AMS strives to ensure that participants in its activities enjoy a welcoming environment. Please see our full Policy on a Welcoming Environment at https://www.ams .org/welcoming-environment-policy.

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MEETINGS & CONFERENCES

Meetings & Conferences of the AMS IMPORTANT information regarding meetings programs: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See https://www.ams.org/meetings. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL. New: Sectional Meetings Require Registration to Submit Abstracts. In an effort to spread the cost of the sectional meetings more equitably among all who attend and hence help keep registration fees low, starting with the 2020 fall sectional meetings, you must be registered for a sectional meeting in order to submit an abstract for that meeting. You will be prompted to register on the Abstracts Submission Page. In the event that your abstract is not accepted or you have to cancel your participation in the program due to unforeseen circumstances, your registration fee will be reimbursed.

San Antonio, Texas University of Texas, San Antonio September 14–15, 2024 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 45, Issue 4

Meeting #1198

Deadlines

Central Section Associate Secretary for the AMS: Betsy Stovall

For organizers: Expired For abstracts: Expired

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html.

Invited Addresses James A M Alvarez, The University of Texas at Arlington, Leveraging Research on Mathematics Teaching and Learning to Reimagine Pathways to Mathematics. Jason R Schweinsberg, University of California San Diego, Using coalescent theory to analyze genetic data from growing tumors. Anne Shiu, Texas A&M University, Dynamics of Biochemical Reaction Networks.

Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl. Additive Number Theory and Modular Forms I, Debanjana Kundu, University of Texas - Rio Grande Valley, and Brandt Kronholm, University of Texas Rio Grande Valley.

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MEETINGS & CONFERENCES Advances in Coding Theory and Cryptography I, Henry Chimal-Dzul, University of Notre Dame, and Jingbo Liu, Texas A&M University-San Antonio. Advances in Differential Equations: Theory, Methods, and Applications I, Faranak Rabiei, Texas A & M University Kingsville, Aden Omar Ahmed, Texas A&M University-Kingsville, and Dongwook Kim, Texas A & M University Kingsville. Advances in Mathematical and Numerical Analysis of Partial Differential Equations for Application-Oriented Computations I, Bruce A Wade, University of Louisiana at Lafayette, Qin Sheng, Baylor University, Abdul Q.M. Khaliq, Middle Tennessee State University, JaEun Ku, Oklahoma State University, and Xiang-Sheng Wang and Yangwen Zhang, University of Louisiana at Lafayette. Applications of Algebraic Geometry I, Frank Sottile, Texas A&M University, Alperen Ergur, University of Texas at San Antonio, and Anne Shiu, Texas A&M University. Applications of model theory in analysis, topology and set theory I, Eduardo Dueñez and Jose N Iovino, The University of Texas at San Antonio. Applications of Probability in Biology I, Jason R Schweinsberg, University of California San Diego. A Showcase of Algebraic Geometry at Undergraduate Institutions I, David Swinarski, Fordham University, Julie Rana, Lawrence University, and Han-Bom Moon, Fordham University. Commutative algebra and connections to combinatorics I, Michael Robert DiPasquale, Louiza Fouli, and Arvind Kumar, New Mexico State University. Differential Geometry, Alvaro Pampano, Texas Tech University, Bogdan D. Suceava, California State University Fullerton, and Magdalena Daniela Toda, Texas Tech University and NSF. Dynamical systems: Statistical properties, spectral theory, and fractal geometry I, Mrinal Kanti Roychowdhury, School of Mathematical and Statistical Sciences, University of Texas Rio Grande Valley, and William R Ott, University of Houston. Enumerative Combinatorics I, Brian K. Miceli, Trinity University, and Lara Pudwell, Valparaiso University. Geometric Group Theory and Low-Dimensional Topology I, George Domat and Khanh Le, Rice University, Jing Tao, University of Oklahoma, and Christopher Jay Leininger, Rice University. Graph Theory I, Youngho Yoo and Chun-Hung Liu, Texas A&M University. Harmonic Analysis, Geometric Measure Theory and PDE I, Dorina I. Mitrea and Marius Mitrea, Baylor University. Homological and combinatorial methods in noncommutative algebra I, Amrei Oswald and Be”eri Greenfeld, University of Washington. Homological Commutative Algebra I, Luigi Ferraro, University of Texas Rio Grande Valley, and Alexis Hardesty, Texas Woman’s University. Inquiry Oriented Learning in the Mathematics Classroom I, Carolyn Luna, University of Texas At San Antonio, and Jennifer Austin, University of Texas at Austin. L-functions and Automorphic Forms I, Lea Beneish, University of North Texas, and Melissa Emory, Oklahoma State University. Link invariants and surfaces in 4-manifolds I, Michael Willis and Sherry Gong, Texas A&M University. Machine Learning, Data Science and Related Fields I, Hansapani Rodrigo, The University of Texas Rio Grande Valley, and Lakshmi Roychowdhury and Mrinal Kanti Roychowdhury, University of Texas Rio Grande Valley. Mathematical Modeling at the Interface of Ecology, Epidemiology, and Human Behavior I, Tamer Oraby, University of Texas - Rio Grande Valley, Lale Asik, University of the Incarnate Word, Ummugul Bulut, [email protected], and Md Rafiul Islam, University of the Incarnate Word. Mathematical Physics and Numerical Methods I, Vu Hoang and Jose Morales, University of Texas at San Antonio. Mathematics of Infectious Disease Emergence, Spread, and Control I, Zhuolin Qu, University of Texas at San Antonio, and Michael Andrew Robert, Virginia Tech. Mathematics: The gateway to Social Justice I, Juan B. Gutiérrez, University of Texas at San Antonio, James Broda, Washington and Lee University, Funda Gultepe, University of Toledo, Ron Buckmire, Occidental College, Matthew Salomone, Bridgewater State University, Joseph Edward Hibdon, Northeastern Illinois University, and Terrance Pendleton, Drake University. Methods & Applications of Data-driven Manufacturing I, Kristen Lee Hallas, The University of Texas Rio Grande Valley, and Benjamin Peters and Jianzhi Li, University of Texas Rio Grande Valley. Modeling and analysis in biological and epidemiological systems I, Michael Lindstrom, The University of Texas Rio Grande Valley, and Erwin Suazo and Zhaosheng Feng, University of Texas Rio Grande Valley. Non-Archimedean, Algebraic, Tropical Geometry and applications I, Jackson S. Morrow, University of North Texas, and Farbod Shokrieh, University of Washington.

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MEETINGS & CONFERENCES Noncommutative Geometry and Analysis I, Zhizhang Xie, Guoliang Yu, Bo Zhu, and Simone Cecchini, Texas A&M University. Operator algebras, quantum information and computation I, Jose A Morales Escalante, University of Texas at San Antonio, and Marius Junge, University of Illinois, Urbana and Champaign. Periodicity in Quantum Systems I, Long Li, Rice University, Wencai Liu, Texas A&M University, and Tal Malinovitch, Rice University. Quasi-periodic and Disordered Systems I, Alberto Takase, Rice University, Omar Hurtado, University of California, Irvine, and Matthew H Faust, Texas A&M University. Recent developments on local and nonlocal PDEs I, Fernando Charro, Wayne State University, and Thialita Nascimento, Iowa State University. Recent studies in topics related to ion channel problems I, Mingji Zhang, New Mexico Institute of Mining and Technology, and Saulo Orizaga, New Mexico Tech. Recent trends in differential equations applied to biological processes I, Rachidi B. Salako, University of Nevada, Las Vegas, and Markjoe O. Uba and Maria Amarakristi Onyido, Northern Illinois University. Research in Post-Secondary Teaching and Learning of Mathematics I, James A M Alvarez, The University of Texas at Arlington, and Paul Christian Dawkins, Texas State University. Spectral Theory of Schrödinger Operators and Related Topics I, Christoph Fischbacher, Fritz Gesztesy, and Jon Harrison, Baylor University. The many scales of mathematical analysis of fluid I, Xin Liu, Texas A&M University, Quyuan Lin, Clemson University, and Cheng Yu, University of Florida. Theoretical and Numerical Aspects of Nonlinear Dispersive Wave Equations I, Baofeng Feng, University of Texas Rio Grande Valley, and Geng Chen and Yannan Shen, University of Kansas. Topics in Convexity I, Zokhrab Mustafaev, University of Houston-Clear Lake.

Contributed Paper Sessions AMS Contributed Paper Session, Betsy Stovall, University of Wisconsin-Madison.

Savannah, Georgia Georgia Southern University October 5–6, 2024 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 45, Issue 4

Meeting #1199

Deadlines

Southeastern Section Associate Secretary for the AMS: Brian D. Boe

For organizers: Expired For abstracts: August 13, 2024

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html.

Invited Addresses Peter Bubenik, University of Florida, Topological Data Analysis: from geometry, algebra and combinatorics to analysis, learning and applications. Akos Magyar, University of Georgia, To Be Announced. Sarah Peluse, Princeton/IAS, Arithmetic Patterns in Dense Sets.

Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl. Advanced Topics in Graph Theory and Combinatorics. (Code: SS 4A), Songling Shan, Auburn University, and Zi-Xia Song, University of Central Florida. Advances in applied algebraic geometry (Code: SS 15A), Kisun Lee and Michael Byrd, Clemson University. September 2024

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MEETINGS & CONFERENCES Advances in the theory of integrable partial differential equations (Code: SS 27A), Barbara Prinari, University at Buffalo, and Zechuan Zhang, SUNY Buffalo. Algebraic, combinatorial and geometric aspects of representation theory. (Code: SS 19A), Cornelius Pillen, University of South Alabama, Aparna Upadhyay, University at Buffalo, SUNY, and Arik Wilbert, University of South Alabama. Applicable Analysis of Multi-physics Partial Differential Equations Systems (Code: SS 17A), George Avalos, University of Nebraska-Lincoln, and Justin Thomas Webster, University of Maryland, Baltimore County. Biological Systems Modeling and Analysis: recent progress and current challenges (Code: SS 16A), Dawit Denu, Georgia Southern University. Commutative Algebra (Code: SS 1A), Saeed Nasseh, Tricia Muldoon Brown, and Alina C. Iacob, Georgia Southern University. Control, PDEs and Inverse Problems. (Code: SS 29A), Tien Khai Nguyen, North Carolina State University, Loc Hoang Nguyen, UNC Charlotte, and Thuy T. Le, North Carolina State University. Convexity, Probability, and Asymptotic Geometric Analysis (Code: SS 12A), Galyna Livshyts, Georgia Institute of Technology, Steven Hoehner, Longwood University, and Stephanie Mui, Georgia Institute of Technology. Deterministic and Stochastic PDEs: Theoretical and Numerical Analyses (Code: SS 8A), Pelin Guven Geredeli, Clemson University, and Xiang Wan, Loyola University Chicago. Dynamical Systems and Control Systems with Applications (Code: SS 13A), Yan Wu, Georgia Southern University, and Liancheng Wang, Kennesaw State University. Ergodic theory and discrete analysis. (Code: SS 23A), Neil Lyall and Tomasz Szarek, University of Georgia. Exploring the Geometry for Teachers (GeT) Course (Code: SS 24A), Tuyin An, Georgia Southern University, and Erin Krupa, North Carolina State University. Extremal and structural graph theory. (Code: SS 9A), Ruth Luo, University of South Carolina, and Zhiyu Wang, Georgia Institute of Technology. Extremal Problems of Approximation Theory and Harmonic Analysis (Code: SS 35A), Yuliya Babenko, Kennesaw State University, and Scott Kersey, Georgia Southern University. Fluids, Waves, and Free Boundaries. (Code: SS 2A), David M. Ambrose, Drexel University, and Michael Siegel, New Jersey Institute of Technology. Game Theories in Network Security (Code: SS 32A), Zheni Utic, Georgia Southern University. Geometric Maximal Operators and Related Topics. (Code: SS 3A), Paul Hagelstein, Baylor University, and Alex Stokolos, Georgia Southern University. Harmonic analysis, fractals, and related topics in memory of Ka-Sing Lau and Robert Strichartz (Code: SS 14A), Sze-Man Ngai, Georgia Southern University, and Alexander Teplyaev, University of Connecticut. Interactions, Discrepancies, Approximations: From Energy Optimization to Dynamics (Code: SS 26A), Ryan W Matzke, Vanderbilt University, and Ihsan Topaloglu, Virginia Commonwealth University. Modules over Commutative Rings (Code: SS 10A), Laura Ghezzi, New York City College of Technology and The Graduate Center-Cuny, and Joseph P Brennan, University of Central Florida. Noncommutative Algebras, Quantum Groups, and Related Topics (Code: SS 31A), Garrett Johnson, North Carolina Central University, Xin Tang, Math & Computer Science, Fayetteville State University, and Xingting Wang, Louisiana State University. Nonlinear Dispersive Equations (Code: SS 34A), Iryna Petrenko, Florida International University, Justin Holmer, Brown University, and Svetlana Roudenko, Florida International University. Number theory and additive combinatorics (Code: SS 28A), Sarah Peluse, Princeton/IAS, and Giorgis Petridis, University of Georgia. Partitions and q-series (Code: SS 25A), Andrew V. Sills, Georgia Southern University, and Robert Schneider, University of Georgia. Poisson geometry, Diffeology and Singular Spaces. (Code: SS 21A), Yi Lin, Georgia Southern University, Jordan Watts, Central Michigan University, and Francois Ziegler, Georgia Southern University. Recent Advances in Contact and Symplectic Topology (Code: SS 30A), Nur Saglam, Georgia Tech, and Eduardo Fernández, University of Georgia. Recent advances in Molecular based Computational and Mathematical Bioscience (Code: SS 18A), Shan Zhao, University of Alabama, and Zhan Chen, Georgia Southern University. Recent Advances in Theory and Practice of Data Science (Code: SS 33A), Divine Wanduku and Ionut Iacob, Georgia Southern University.

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MEETINGS & CONFERENCES Recent Advances of PDEs in Modern Mathematical Physics: Theory and Applications (Code: SS 7A), Yuanzhen Shao, The University of Alabama, and Yi Hu and Shijun Zheng, Georgia Southern University. Recent developments in applications of complex analysis. (Code: SS 22A), Ashley Ran Zhang, Vanderbilt University, and Burak Hatinoglu, UC Santa Cruz. Recent Progress in Numerical Methods for PDEs (Code: SS 11A), Xuejian Li and Leo Rebholz, Clemson University. Topics in commutative algebra and algebraic geometry (Code: SS 6A), Prashanth Sridhar and Michael Brown, Auburn University. Topological Data Analysis, Theory and Applications (Code: SS 5A), Peter Bubenik and Kevin P. Knudson, University of Florida. Trees in many contexts. (Code: SS 20A), Hua Wang, Department of Mathematical Sciences, Georgia Southern University, and Heather Smith Blake, Davidson College.

Albany, New York University at Albany October 19–20, 2024 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 45, Issue 4

Meeting #1200

Deadlines

Eastern Section Associate Secretary for the AMS: Steven H. Weintraub

For organizers: Expired For abstracts: August 27, 2024

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html.

Invited Addresses Jennifer Balakrishnan, Boston University, Title to be announced. Jose Perea, Northeastern University, Title to be announced. Richard Rimanyi, UNC, Title to be Announced.

Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl. Agent-Based and Mean-Field Modeling for Complex Social Systems (Code: SS 1A), Daniel Brendan Cooney, University of Illinois Urbana-Champaign, Jeungeun Park, SUNY at New Paltz, and Rebecca Hardenbrook, Dartmouth College. Applied and Computational Topology (Code: SS 2A), Barbara Giunti and Håvard Bakke Bjerkevik, University at Albany, SUNY, and Justin Michael Curry, University at Albany SUNY. Data-Driven Modeling and Analysis of Complex Dynamical Systems (Code: SS 3A), Felix X.-F Ye, SUNY Albany, and Weiqi Chu, UMass Amherst. Ergodic Theory - In Memory of Nathaniel Friedman (1938 - 2020) (Code: SS 4A), Karin B. Reinhold, University at Albany, SUNY, Cesar E. Silva, Williams College, and Terry Adams, University at Albany. Explicit Methods in Arithmetic Geometry (Code: SS 5A), Manami Roy, Lafayette College, and Alexander J Barrios, University of St. Thomas. Generalized Schubert Calculus and Recent Progress (Code: SS 6A), Changlong Zhong, SUNY Albany, and Richard Rimanyi, UNC. Geometric Group Theory (Code: SS 7A), Matt Zaremsky, University at Albany, Emily Stark, Wesleyan University, and Daniel Studenmund, Binghamton University. Harmonic Analysis, Theory of Function Spaces and Their Applications (Code: SS 8A), Liding Yao, Chian Yeong Chuah, and Jan Lang, The Ohio State University. Holomorphic Function Spaces and Operators on Them (Code: SS 9A), Kehe Zhu, University at Albany, SUNY, and Zhijian Wu, University of Nevada, Las Vegas.

September 2024

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MEETINGS & CONFERENCES Homotopy Theory and Algebraic K-Theory (Code: SS 10A), Marco Varisco, University at Albany, State University of New York, and Brenda Johnson, Union College. Interactions Between Lie Theory and Combinatorics of Symmetric Functions (Code: SS 11A), Hadi Salmasian, University of Ottawa, and Siddhartha Sahi, Rutgers University, New Brunswick NJ. Invariants of Knots, Links, and Low-dimensional Manifolds (Code: SS 12A), Adam M. Lowrance, Vassar College, Patricia Cahn, Smith College, Moshe Cohen, State University of New York At New Paltz, and Caitlin Leverson, Bard College. Mathematics and the Arts in Memory of Nat Friedman (Code: SS 13A), David A Reimann, Albion College, Ergun Akleman, Texas A&M University, and Alex Feingold, Binghamton University State University of New York. Matroids, Quivers, 𝔽1-geometry, and Connections with Algebra (Code: SS 14A), Jaiung Jun, SUNY New Paltz, Chris Eppolito, The University of the South, and Alexander Sistko, Manhattan College. Multivariable Operator Theory (Code: SS 15A), Rongwei Yang, Hyun-Kyoung Kwon, Alea L Wittig, and Kate Howell, University at Albany. Nonlocal Analysis and Geometric Measure Theory (Code: SS 16A), Cornelia Mihaila, Saint Michael’s College, and Brian Seguin, Loyola University Chicago. Nonsmooth Analysis and Geometry (Code: SS 17A), Matthew Badger, University of Connecticut, Ryan Alvarado, Amherst College, and Lisa Naples, Fairfield University, Fairfield CT USA. Permutation Patterns (Code: SS 18A), Megan A. Martinez, Ithaca College, and Rebecca Nicole Smith, SUNY Brockport. Probabilistic and Analytic Aspects in Convexity (Code: SS 19A), Michael Roysdon, Case Western Reserve University, Sergii Myroshnychenko, University of the Fraser Valley, Kateryna Tatarko, University of Waterloo, Yiming Zhao, Syracuse University, and Elisabeth M Werner, Case Western Reserve University. Quantum Mathematics for Computation (Code: SS 20A), Hanmeng (Harmony) Zhan, Worcester Polytechnic Institute, and Christino Tamon, Clarkson University. Random Processes and Probability (Code: SS 21A), Martin V. Hildebrand, University at Albany, SUNY. Recent Advances in Harmonic Analysis (Code: SS 22A), Joshua Brough Isralowitz, University At Albany, SUNY, and David Cruz-Uribe, University of Alabama. Recent Advances in Vertex Operator Algebras (Code: SS 23A), Antun Milas, SUNY at Albany, and Shashank Kanade, University of Denver. Recent Developments in Automorphic Forms and Representation Theory (Code: SS 24A), Moshe Adrian, Queens College, City University of New York, and Anantharam Raghuram, Fordham University. Recent Developments in Graph Theory (Code: SS 25A), Nathan Kahl and John T. Saccoman, Seton Hall University, and Kerry E Ojakian, Bronx Community (CUNY). Recent Developments in Physics Informed Machine Learning for Inverse Problems (Code: SS 26A), Taufiquar Khan and Sudeb Majee, University of North Carolina at Charlotte. Regularity of Nonlinear Equations and Free Boundary Problems (Code: SS 27A), Maria Soria-Carro and Iñigo Urtiaga Erneta, Rutgers University, and Daniel Restrepo, Johns Hopkins University. Singularities in Commutative Algebra (Code: SS 28A), Josh Pollitz and Claudia Miller, Syracuse University, and Jason Howell, University at Albany - State University of New York. Symmetric Functions and Applications (Code: SS 29A), Olya Mandelshtam, University of Waterloo, and Rosa C. Orellana, Dartmouth College. Topics in Recreational Math and Finite Geometry (Code: SS 30A), Lauren L Rose, Bard College, Kelly Isham, Colgate University, and Elizabeth McMahon and Gary Gordon, Lafayette College.

Contributed Paper Sessions AMS Contributed Paper Session (Code: CP 1A), Steven H Weintraub, Lehigh University.

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MEETINGS & CONFERENCES

Riverside, California University of California, Riverside October 26–27, 2024 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 45, Issue 4

Meeting #1201

Deadlines

Western Section Associate Secretary for the AMS: Michelle Ann Manes

For organizers: Expired For abstracts: September 3, 2024

The scientific information listed below may be dated. For the latest information, see https://www.ams.org/amsmtgs /sectional.html.

Invited Addresses Matthew D. Blair, University of New Mexico, Title to be announced. Hannah K. Larson, UC Berkeley, Title to be announced. Tianyi Zheng, University of California San Diego, Title to be announced.

Special Sessions If you are volunteering to speak in a Special Session, you should send your abstract as early as possible via the abstract submission form found at https://www.ams.org/cgi-bin/abstracts/abstract.pl. Advances in Extremal Combinatorics (Code: SS 27A), Emily Heath and Shira Zerbib, Iowa State University. Advances in Understanding of Student Thinking in Lower Division Mathematics Courses (Code: SS 26A), Sara Lapan, University of California, Riverside, Jeffrey S Meyer, California State University, San Bernardino, and Rasha Issa, University of California, Riverside. Applied Partial Differential Equations and Inverse Problems (Code: SS 11A), Amir Moradifam, University of California at Riverside, and Yat Tin Chow, University of California, Riverside. Calculating Probabilities using Matrix Methods with Applications to Markovian, Gaussian or Queueing Models (Code: SS 22A), Alan Krinik, California State Polytechnic University, and Randall J. Swift, California State Polytechnic University, Pomona. Conformal Geometry, Einstein Metrics, and General Relativity (Code: SS 16A), Andrew K. Waldron and Jaroslaw Kopinski, University of California, Davis. Dynamical Systems (Code: SS 5A), Agnieszka Zelerowicz and Zhenghe Zhang, UC Riverside. Dynamics of Solutions to Wave Equations (Code: SS 21A), Michael McNulty and Willie Wong, Michigan State University, and Po-Ning Chen, UC Riverside. Finite groups, their representations, and related structures (Code: SS 6A), Nariel Monteiro, University of California Santa Cruz, Robert Boltje, University of California, Santa Cruz, and Mandi A. Schaeffer Fry, University of Denver. Gender Equity in the Mathematical Sciences (GEMS) of Combinatorics (Code: SS 2A), Aleyah Dawkins, George Mason University, Andrés Vindas Meléndez, University of Kentucky, and Katie Waddle, University of Michigan. Geometric and Categorical Representation Theory (Code: SS 14A), Carl Mautner, UC Riverside, and Tom Gannon, University of California - Los Angeles. Geometry and Topology of Contact and Symplectic Manifolds (Code: SS 9A), Bahar Acu, Pitzer College, Wenyuan Li, University of Southern California, and Hyunki Min, UCLA. Geometry, topology and dynamics of character varieties (Code: SS 24A), Filippo Mazzoli, University of Virginia, and Brian Collier, Unviersity of California, Riverside. Graphical Calculus in Representation Theory and Low-Dimensional Topology (Code: SS 19A), Emily McGovern, North Carolina State University, and Agustina Czenky, University of Oregon. Harmonic Analysis and Applications (Code: SS 7A), Rodolfo H. Torres, University of California, Riverside, and Arpad Benyi, Western Washington University. Harmonic Analysis, Partial Differential Equations, and Spectral Theory associated with Invited Address by Matthew Blair (Code: SS 4A), Matthew D. Blair, University of New Mexico, and Xiaoqi Huang, Louisiana State University. Logic in SoCal (Code: SS 10A), Meng-Che Ho, California State University, Northridge, Scott Cramer, California State University, San Bernardino, Sheila Miller Edwards, Arizona State University, and Name Trang, University of North Texas. September 2024

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MEETINGS & CONFERENCES Moduli associated with Invited Address by Hannah Kerner Larson (Code: SS 3A), Hannah K. Larson, UC Berkeley, Jesse Kass, USC, and Patricio Gallardo, UC Riverside. Non-commutative Algebras in Representation Theory and Topology (Code: SS 20A), Peter Samuelson and Pallav Goyal, University of California, Riverside, and Boris Tsvelikhovskiy, UC Riverside. Non-commutative birational geometry, cluster structures and canonical bases (Code: SS 8A), Jacob Greenstein, University of California Riverside, Vladimir Retakh, Rutgers University, and Arkady Berenstein, University of Oregon Eugene. Probability and Mathematical Physics (Code: SS 25A), David E Weisbart and Rahul D. Rajkumar, University of California Riverside. Random matrices, related structures, and applications (Code: SS 18A), John Peca-Medlin, University of Arizona, and Yizhe Zhu, University of California Irvine. Random walks on groups and dynamics of group actions associated with Invited Address by Tianyi Zheng (Code: SS 29A), Omer Tamuz, California Institute of Technology, Gil Goffer, University of California at San Diego, and Tianyi Zheng, University of California San Diego. Recent Advances in Modeling and Simulation of Complex Fluids (Code: SS 12A), Yiwei Wang and Weitao Chen, University of California, Riverside, and Siting Liu, University of California, Los Angeles. Several Complex Variables: New developments and trends (Code: SS 23A), Ziming Shi, University of California - Irvine, John N Treuer, Texas A&M University, and Bun Wong, University of California, Riverside. Structural Features in Mathematical Physics (Code: SS 17A), Adam M. Yassine, Pomona College, and Andrea Stine, University of California, Riverside. Surfaces, 3-manifolds and hyperbolic geometry (Code: SS 13A), Julien Paupert, Thi Hanh VO, and Puttipong Pongtanapaisan, Arizona State University. Topics in Algebraic Geometry (Code: SS 15A), Javier Gonzalez Anaya, Harvey Mudd College, Courtney George, University of California, Riverside, and Jose Gonzalez, University of California at Riverside. Topics on Geometric Analysis (Code: SS 1A), Xiaolong Li, Wichita State University, Lihan Wang, California State University, Long Beach, and Qi S Zhang, UC Riverside. Topological and Geometric Methods in Combinatorics (Code: SS 28A), Zoe Wellner, Carnegie Mellon University, and Zilin Jiang, Arizona State University.

Auckland, New Zealand December 9–13, 2024

Issue of Abstracts: To be announced

Monday – Friday Associate Secretary for the AMS: Steven H. Weintraub

Deadlines

Program first available on AMS website: To be announced

For organizers: Expired For abstracts: September 30, 2024

Seattle, Washington Seattle Convention Center and the Sheraton Grand Seattle January 8–11, 2025

Program first available on AMS website: To be announced Issue of Abstracts: Volume 46, Issue 1

Wednesday – Saturday

Deadlines

Meeting #1203 Associate Secretary for the AMS: Brian D. Boe

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MEETINGS & CONFERENCES

Clemson, South Carolina Clemson University March 8–9, 2025 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 46, Issue 2

Meeting #1204

Deadlines

Southeastern Section Associate Secretary for the AMS: Brian D. Boe, University of Georgia

For organizers: August 13, 2024 For abstracts: January 14, 2025

Lawrence, Kansas University of Kansas March 29–30, 2025 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: Volume 46, Issue 2

Meeting #1205

Deadlines

Central Section Associate Secretary for the AMS: Betsy Stovall, University of Wisconsin-Madison

For organizers: August 27, 2024 For abstracts: February 4, 2025

Hartford, Connecticut Hosted by University of Connecticut; taking place at the Connecticut Convention Center and Hartford Marriott Downtown April 5–6, 2025 Saturday – Sunday

Program first available on AMS website: To be announced Issue of Abstracts: To be announced

Meeting #1206

Deadlines

Eastern Section Associate Secretary for the AMS: Steven H. Weintraub

For organizers: September 17, 2024 For abstracts: February 11, 2025

San Luis Obispo, California California Polytechnic State University, San Luis Obispo May 3–4, 2025 Saturday – Sunday

Program first available on AMS website: Not applicable Issue of Abstracts: Volume 46, Issue 3

Meeting #1207

Deadlines

Western Section Associate Secretary for the AMS: Michelle Ann Manes

September 2024

For organizers: October 1, 2024 For abstracts: March 5, 2025

Notices of the American Mathematical Society

1135

MEETINGS & CONFERENCES

St. Louis, Missouri Saint Louis University October 18–19, 2025

Issue of Abstracts: To be announced

Saturday – Sunday Central Section Associate Secretary for the AMS: Betsy Stovall, University of Wisconsin-Madison Program first available on AMS website: To be announced

Deadlines For organizers: March 18, 2025 For abstracts: August 26, 2025

Denver, Colorado University of Denver December 6–7, 2025

Issue of Abstracts: Volume 46, Issue 4

Saturday – Sunday Western Section Associate Secretary for the AMS: Michelle Ann Manes Program first available on AMS website: Not applicable

Deadlines For organizers: May 6, 2025 For abstracts: October 14, 2025

Washington, District of Columbia Walter E. Washington Convention Center and Marriott Marquis Washington DC January 4–7, 2026

Issue of Abstracts: To be announced

Sunday – Wednesday Associate Secretary for the AMS: Betsy Stovall

Deadlines

Program first available on AMS website: To be announced

For organizers: To be announced For abstracts: To be announced

Boise, Idaho Boise State University March 7–8, 2026

Issue of Abstracts: To be announced

Saturday – Sunday Western Section Associate Secretary for the AMS: Michelle Ann Manes, AIM Program first available on AMS website: To be announced

Deadlines For organizers: To be announced For abstracts: To be announced

Fargo, North Dakota North Dakota State University April 18–19, 2026

Issue of Abstracts: To be announced

Saturday – Sunday Central Section Associate Secretary for the AMS: Betsy Stovall, University of Wisconsin-Madison

Deadlines For organizers: To be announced For abstracts: To be announced

Program first available on AMS website: To be announced 1136

Notices of the American Mathematical Society

Volume 71, Number 8

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IT’S HERE!

Your Daily Epsilon of Math Wall Calendar 2025

Keep your mind sharp all year long with Your Daily Epsilon of Math Wall Calendar 2025 featuring a new math problem every day! Let mathematicians Rebecca Rapoport and Dean Chung tickle the left side of your brain by providing you with a math challenge for every day of the year. The solution is always the date, but the fun lies in figuring out how to arrive at the answer, and possibly discovering more than one method of arriving there. 2024; 14 pages; ISBN: 978-1-4704-7800-1; List US$20; AMS members US$16; MAA members US$18; Order code MBK/151

Visit bookstore.ams.org/mbk-151