FEBRUARY 1989, VOLUME 36, NUMBER 2 Notices of the American Mathematical Society 0444705449, 0444705139, 0444871292, 0444871195, 0444705384

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OTICES OF THE

AMERICAN MATHEMATICAL SOCIETY

Mathematics for a New Century, Lynn Arthur Steen

FEBRUARY 1989, VOLUME 36, NUMBER 2 Providence, Rhode Island, USA ISSN 0002-9920

page 133

Calendar of AMS Meetings and Conferences Thla calendar lists all meetings which have been approved prior to the date this issue of Notices was sent to the press. The summer

and annual meetings are joint meetings of the Mathematical Association of America and the American Mathematical Society. The meeting dates which fall rather far in the future are subject to change; this is particularly true of meetings to which no numbers have been assigned. Programs of the meetings will appear in the issues indicated below. First and supplementary announcements of the meetings will have appeared in earlier issues. Abatracta of papera presented at a meeting of the Society are published in the journal Abstracts of papers presented to the American

Mathematical Society in the issue corresponding to that of the Notices which contains the program of the meeting. Abstracts should be submitted on special forms which are available in many departments of mathematics and from the headquarters office of the Society. Abstracts of papers to be presented at the meeting must be received at the headquarters of the Society in Providence, Rhode Island, on or before the deadline given below for the meeting. Note that the deadline for abstracts for consideration for presentation at special sessions is usually three weeks earlier than that specified below. For additional information, consult the meeting announcements and the list of organizers of special sessions.

Meetings Meeting#

Date

Place

• April 15-16, 1989 Worcester, Massachusetts • May 19-20, 1989 Chicago, Illinois • August 7-10, 1989 850 Boulder, Coloradot (92nd Sul'lmer Meeting) October 21-22, 1989 851 Hoboken, New Jersey 852 • October 27-28, 1989 Muncie, Indiana November 18-19, 1989 Los Angeles, California 853 854 January 17-20, 1990 Louisville, Kentucky (96th Annual Meeting) March 16-17, 1990 Manhattan, Kansas August 8-11, 1990 Columbus, Ohio (93rd Summer Meeting) November 2-3, 1990 Denton, Texas January 16-19, 1991 San Francisco, California (97th Annual Meeting) August 8-11, 1991 Orono, Maine (94th Summer Meeting) Baltimore, Maryland January 8-11 , 1992 (98th Annual Meeting) Cambridge, England June 29-July 1, 1992 (Joint Meeting with the London Mathematical Society) San Antonio, Texas January 13-16, 1993 (99th Annual Meeting) January 5-8, 1994 Cincinnati, Ohio (100th Annual Meeting) • Please refer to page 165 for listing of special sessions. t Preregistration/Housing deadline is June 1 848

849

Abstract Deadline

Program Issue

Expired March 1 May 16

March April JulyI August

August 16 August 16 August 16 October 11

October October October December

Conferences May 26-May 30, 1989: AMS Pure Mathematics Symposium on Complex Geometry and Lie Theory, Sundance Resort, Sundance, Utah May 29-June 9, 1989: AMS-SIAM Summer Seminar on the Mathematics of Random Media, Virginia Polytechnic Institute and State University, Blacksburg, Virginia June 3-August 5, 1989: Joint Summer Research Conferences in the Mathematical Sciences, Humboldt State University, Arcata, California

July 10-30, 1989: AMS Summer Research Institute on Several Complex Variables and Complex Geometry, University of California, Santa Cruz, California August 7, 1989: AMS-SIAM-SMB Symposium on Some Mathematical Questions in Biology, Sex Allocations and Sex Change: Experiments and Models, University of Toronto.

Deadlines Classified Ads* News Items Meeting Announcements••

March Issue

April Issue

May/June Issue

July/August Issue

Feb 6, 1989 Feb 10, 1989 Feb 3, 1989

Mar 10, 1989 Mar 16 1989 Mar 9, 1989

April 21, 1989 April 27, 1989 April 20, 1989

June 12, 1989 June 12, 1989 June 5, 1989

• Please contact AMS Advertising Department for an Advertising Rate Card for display advertising deadlines. •• For material to appear in the Mathematical Sciences Meetings and Conferences section.

OTICES OF THE

AMERICAN MATHEMATICAL SOCIETY

DEPARTMENTS

ARTICLES

131 Letters to the Editor 133 Mathematics for a New Century Lynn Arthur Steen In his Survey Lecture for Action Group A5 (Post-Secondary Mathematics Education), presented at last summer's International Congress on Mathematics Education in Budapest, Professor Steen examines the impact that computers, new applications, research in learning, research in mathematics, and socio-economic trends will have on university mathematics programs. 139 Gordon Loftis Walker (1912-1988) William J. LeVeque Dr. LeVeque pays tribute to Gordon Loftis Walker who, during his tenure as Executive Director of the AMS (1959-1977), helped bring the Society to its present state of preeminence.

150 News and Announcements 154 Funding Information for the Mathematical Sciences 157 Meetings and Conferences of the AMS (Listing) 174 Mathematical Sciences Meetings and Conferences 184 New AMS Publications 186 AMS Reports and Communications Recent Appointments, 186 186 Miscellaneous Personal Items, 186 Deaths, 186

FEATURE COLUMNS 141 Computers and Mathematics Jon Barwise This month's column features Gian-Carlo Rota's article in memory of the mathematician Stanislaw Ulam, Larry Lambe's review of Scratchpad II, a programming language and computer algebra system, and Andrew Matchett's review of the program Graphical Aids for Stochastic Processes.

187 New Members of the AMS 190 Classified Advertising 207 Forms

149 Inside the AMS: Progress in Mathematics The innovative new lecture series, which will be inaugurated at this summer's Joint Mathematics Meetings in Boulder, is presented.

FEBRUARY 1989, VOLUME 36, NUMBER 2

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Summer Meetings AMERICAN MATHEMATICAL SOCIETY

EDITORIAL COMMITTEE Robert J. Blattner, Ralph P. Boas Lucy J. Garnett, Mary Ellen Rudin Nancy K. Stanton, Steven H. Weintraub Everett Pitcher (Chairman) MANAGING EDITOR James A. Voytuk ASSOCIATE EDITORS Ronald L. Graham, Special Articles Jeffrey C. Lagarias, Special Articles SUBSCRIPTION INFORMATION Subscription prices for Volume 36 (1989) are $1 08 list; $86 institutional member; $65 individual member. (The subscription price for members is included in the annual dues.) A late charge of 10% of the subscription price will be imposed upon orders received from nonmembers after January 1 of the subscription year. Add for postage: Surface delivery outside the United States and lndia-$10; to lndia-$20; expedited delivery to destinations in North America-$15; elsewhere-$38. Subscriptions and orders for AMS publications should be addressed to the American Mathematical Society, P.O. Box 1571, Annex Station, Providence, Rl 02901-9930. All orders must be prepaid. ADVERTISING Notices publishes situations wanted and classified advertising, and display advertising for publishers and academic or scientific organizations. Copyright @ 1989 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. § [Notices of the American Mathematical Society is published ten times a year (January, February, March, April, MayfJune, July/August, September, October, November, December) by the American Mathematical Society at 201 Charles Street, Providence, Rl 02904. Second class postage paid at Providence, Rl and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, Membership and Sales Department, American Mathematical Society, P. 0. Box 6248, Providence, Rl 02940.] Publication here of the Society's street address, and the other information in brackets above, is a technical requirement of the U. S. Postal Service. All correspondence should be mailed to the Post Office Box, NOT the street address.

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Over the past few years registration at the AMS-MAA Summer Meetings has been on the decline. Except for the Centennial Celebration last year which drew almost 1500 participants, the summer meetings have been averaging about 700 registrants. This is a substantial decrease from the meetings in the 1970s where it was common to have 1000 mathematicians register for a meeting. The decline is ever more stiking when you realize that the membership for both the AMS and the MAA has increased by about 50% since the mid 70s. There is ever more irony to the situation since the past two regular summer meetings in Laramie, Wyoming and Salt Lake City, ,Utah boasted some of the best scientific programs presented at any meeting, summer or annual. One can guess at some of the reasons for the decline. First, specialized meetings that concentrate on a single topic and foster greater interaction between participants have been on the increase. The AMS in part is responsible for this change with the introduction of the Summer Research Conferences in 1982. Second, travel funds from research grants are not as readily available as they were some years ago -a:ncnriai.Vfdua:lsuare rrnich more selective when it comes to making their meeting plans. One other reason may stem from the advances in communication technology which allows a researcher to exchange information with colleagues via electronic mail as opposed to direct contact at a meeting. Another interesting fact, observed by analyzing the attendance at a summer meeting, is the decline in the number spouses and children accompanying the registered participants. In the past, the summer meeting may have been combined with the family vacation but times have changed and the trip across country by car to a mathematics meeting may be a relic of the past. Whatever the reasons are, the AMS-MAA Summer Meeting is an important part of the mathematical infrastructure in this country and ways should be found to renew interest in this meeting. One step in this direction is the introduction of a new lecture series, Progress in Mathematics, that will begin at this year's meeting in Boulder, Colorado. The "Inside the AMS" section of this issue of Notices contains a description of the lecture series and this year's program. It is hoped that this series will be as popular as the current AMSMAA Invited Lecture Series that draw overflow crowds at the annual and summer meetings. The Society, along with MAA, will continue to examine the situation and also invites comments and reactions from the mathematical community on the summer meetings. You may address your remarks to Hope Daly, American Mathematical Society, P.O. Box 6248, Providence, RI 02940. However for the present, the meeting this summer in Boulder will again have an outstanding scientific program and together with the Progress in Mathematics lecture series will prove to be an excellent meeting. Hope that you will be in Boulder, August 7-10.

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

Letters to the Editor

Service to the Profession

In the last few years, while sitting on the AMS Science Policy Committee and the NSF Advisory Committee, I have frequently heard from governmental agency directors of the difficulties they have in attracting highly qualified mathematicians to serve in temporary positions as program directors, rotators, or coordinators of national initiatives. Setting aside the nontrivial demands of a temporary move to the DC area, a very prohibitive factor for talented mathematicians considering such assignments is that many of their peers and senior colleagues regard taking on these duties as a decision to leave research, veer from the blessed way, and a sure indication of a desire for bourgeois comfyness. But as a consequence our profession is the loser. The attitudes described above are markedly different from that of other disciplines, for instance engineering. Service to the profession, through work with professional societies or duty assignments with federal agencies, is regarded as a mark of distinction and a factor to be considered in evaluating the worth of an individual, including his or her evaluation for promotion or merit raises. Such service really does count in the balance against research and teaching, as contrasted to the usual campus or departmental service to which we pay lip service, but usually not dollars. As we press forward to meet the funding challenges of the "David Report," the influence of computing in mathematics research, and the concomitant reshaping of our curriculum, we will want our best talent speaking and working for us at the

national level. Those talented persons who want to speak and work for mathematics should be encouraged by their chairpersons and colleagues, and not made to feel like the next step after taking on a rotator position in Washington is applying for Social Security benefits. How about instead, a semester leave upon return to get back in the research groove? In a recent federal research report there appeared the quip that the only difference between chemists and physicists was that when the chemists formed a wagon circle in response to an attack, the guns pointed inward. We may be guilty in part of the same strategy if we do not make a greater effort to encourage and reward those who enter the fray, instead of shooting them out of the saddle. David A. Sanchez Lehigh University (Received November 17, 1988)

The Society's Position on SDI

Even people who do not share any political views with Mr. Constantine (Notices, September 1988, page 964) would probably feel some sympathy with his irritation at what he sees as the politicisation of the AMS (for politicisation of the Society would be in no-one's interest)-were it not for his gratuitously offensive language. However he is simply wrong about the Society being politicised. As you have been kind enough to allow me space to say here earlier [Notices, March 1988, page 380], the Society has many overseas members of diverse political colours and if it is to continue to be a world body of the kind it has so successfully been so far, it must not only be seen to be dissociated from any politically motivated excesses perpetrated by any U.S. administration, but must even be seen not to be associated with anything believed to be a politically motivated excess perpetrated by a U.S. administration-or for that matter anyone else. Rightly or wrongly, SDI,

FEBRUARY 1989, VOLUME 36, NUMBER 2

is felt by many millions of people to be such an excess, and it is entirely proper for the Society to be seen to have nothing to do with it. If Mr. Constantine believes what he says about mathematics being apolitical then he should applaud the Society's stance and resume paying his subscription. I hope he will. Thomas E. Forster University of Cambridge, England (Received November 14, 1988)

Soviet-American Cooperation

Last events in the USSR, namely "perestroika" and democratization of the political life at last makes possible the direct contacts between the educational and scientific institutions in this country and foreign Policy on Letters to the Editor Letters submitted for publication in Notices are reviewed by the Editorial Committee, whose task is to determine which ones are suitable for publication. The publication schedule requires from two to four months between receipt of the letter in Providence and publication of the earliest issue of Notices in which it could appear. Publication decisions are ultimately made by majority vote of the Editorial Committee, with ample provision for prior discussion by committee members, by mail or at meetings. Because of this discussion period, some letters may require as much as seven months before a final decision is made. Letters which have been, or may be, published elsewhere will be considered, but the Managing Editor of Notices should be informed of this fact when the letter is submitted. The committee reserves the right to edit letters. Notices does not ordinarily publish complaints about reviews of books or articles, although rebuttals and correspondence concerning reviews in Bulletin of the American Mathematical Society will be considered for publication. Letters should be typed and in legible form or they will be returned to the sender, possibly resulting in a delay of publication. Letters should be mailed to the Editor of Notices, American Mathematical Society, P.O. Box 6248, Providence, RI 02940, and will be acknowledged on receipt.

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Letters to the Editor

universities avoiding the usual bureaucratic bars. These contacts could have a form of a long-term contracts on common scientific research, exchanges of students, researchers and faculty. Because of the lack of free convertible currency the financial basis of all these contacts should be a barter-type relations. The aim of this letter is to inform the American Mathematical community about these new possibilities and to involve it in these forms of cooperation. Being a visiting professor at University of Texas in 1978-1979 school year I have seen the high level of the American technology and a high level of mistrust to Soviets in the academic circles of the USA. I have to say that if in those years this mistrust had sometimes a real basis, now the time has changed and everybody who is interested in the development of science, culture and peace in the world should help it with his own efforts. Me as a reviewer of the MR was doing my best as I it understand to the good of our beautiful science more than 15 years and now as a chairman of the Mathematical Department of the Kiev Polytechnical Institute (Chernigov branch) I have a right and a possibility to cooperate in the above mentioned sense with the Mathematical Department of any university in the USA, Canada, UK or any other country. Our institute is comparatively young, open-minded and looking ways to progress and prosperity.

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If anybody is interested, please write to the below mentioned address. The fields of scientific interests of our faculty are: differential equations (ordinary and partial, especially nonlinear), dynamical systems, computer simulation, software and databases development, computer science, numerical analysis, mathematical education and the computer usage in the education, mathematical education of the engineers. Yuri V. Kostarchuk Kiev Polytechnical Institute (Received November 28, 1988)

Environmental Values

Recent Letters to the Editor (see

Notices, September 1988) have expressed criticism of any AMS view that does not support or is wary of military funding for mathematics or mathematicians. This criticism has been for such reasons as the failure of such funding expressing a lack of sensitivity for the isolation of military research workers in mathematics, or else because the failure of such funding represents a politicization of the society that implies leanings toward the left. Please let me address those specific issues from my own unique point of view. It is hard for a mathematician to be more isolated from their colleagues than am I. Because of my conviction that industrial and military technologies threaten the adaptive biodiversity, and even the

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

survival of all life on the planet, I have always refused employment in any areas that might be adverse to these bioethical values, adversely impact the remaining pristine wild lands that are a source of primary global adaptability, and threaten the overall ecological basis for the survival of species and critical symbiosis. Such a viewpoint demands an entirely different attitude toward research politics than any of the usual political or economic goals suggest. Environmental extremism in science, like radical pacifism in ethics, intends to affirm and assist in sustaining the value of biocentric spirituality beyond mere political humanism, whether left, right, or even moderate. Given the potentially global catastrophe of nuclear ecocide, it is hard to believe that strong opposition to the continued militarization of mathematically related work must really have to do with either politics or economics except in the most general sense of universal interrelatedness. For these reasons I believe that the recent referendum of the AMS which criticizes military funding of research should not only be applauded, but should be a beginning to go much further in restoring the mathematical sciences to their lofty level of ethical and spiritual scholarship as a whole. Peter J. Bralver Earth First! Bio-Diversity Project Sherman Oaks, California (Received September 28, 1988)

MATHEMATICS FORA NEW CENTURY Lynn Arthur Steen

Implications for Mathematics Education Lynn Arthur Steen is Professor of Mathematics at St. Olaf College in Northfield, Minnesota. The following is his Survey Lecture for Action Group AS (Post-Secondary Mathematics Education) of the International Congress on Mathematical Education. The Congress was held in Budapest from July 27 to August 4, 1988.

Mathematics is an ancient discipline vested with modern authority. Mathematics empowers people with the capacity for control in their lives; it offers science a firm foundation for effective theories; and it promises society a vigorous economy. In all cultures, in all generations, children study mathematics to gain access to a better life. In this respect, today is no different than yesterday, the twentieth century no different than the eighteenth. But in other respects the world has changedsignificantly, profoundly, and irreversibly. Until recently, school mathematics (mostly arithmetic) sufficed as the language of commerce, while higher mathematics was viewed as the language of science. No more. Today international commerce is based on computer models of trading that employ such sophisticated tools as stochastic differential equations, while medical research relies nearly as much on mathematical models as on clinical evidence. Indeed, the language of advanced mathematics-from control theory to combinatorics, from differential geometry to statistics-has permeated business, medicine, and virtually every information system of modern society. Computer-based communication has created a world economy based as much on exchange of data and ideas as on shipment of goods. In industrialized countries, as much as half of the labor force is now engaged in information-based work, while in developing countries information-based industries are often the fastest growing segment of the economy. Mathematics is part of this new world order, an amplifier of the mind that bestows significant personal and economic advantage on those who possess it.

The new reality in which we all live, this high-tech prelude to the next century, augurs important changes for the context in which mathematics is taught in universities around the world. Different countries will experience these changes at different times and in different degrees. But none will escape the winds of change unleashed by the power of modern computers and innovative applications. One might naively expect that as mathematics increases in its applications, good students would flock to mathematics in increasing numbers. But what we observe is quite the opposite: high demand mixed with low quality threatens post-secondary mathematics in almost every country. Before trying to understand this paradox, let us examine the evidence: 1. Shortage of Secondary School Teachers.

As demand for mathematics rises, demand for instruction in mathematics also rises. But this same demand attracts graduates to jobs other than teaching, because pay and job conditions are often much better outside educational institutions. The combination of increased enrollments with decreased numbers of prospective teachers creates shortages which are already being felt in many countries: • From Western Europe: "Fairly soon, secondary school mathematics will be taught by people who are absolutely not qualified. It isn't yet as bad as the United States is right now, but it's pretty close." • From a Developing Country: "Many mathematics teachers in our secondary schools have not taken any post-secondary studies."

2. Weak Mathematics Preparation of Entering University Students.

Two factors account for this trend, which is widespread in all regions of the world. Wherever secondary school

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mathematics teachers are poorly prepared, their students are likely to be poorly prepared. Even in countries with well qualified mathematics teachers in secondary schools, the many attractive new career options in computer science, business, and medicine often distract good students away from serious mathematics even before they enter the university. When careers in mathematics become only a second or third choice for many able students, university mathematics departments are bound to notice both a decline in the average preparation of entering students and, perhaps more important, an increase in the variance: • From Western Europe: "The wide range of qualification levels at the end of secondary education prevents a uniform starting level in post-secondary education." • From Eastern Europe: "The growing difference in preparation presents some problems at the university." • From a Developing Country: "Since about 1970, the general level of mathematics preparation has been decreasing, and right now is very inadequate."

3. Declining Student Interest in Mathematics. The very professions that use mathematics extensively provide attractive and financially desirable alternatives for expression of mathematical talent. For most students who want to make a contribution to society, traditional pure mathematics appears to be of limited interest when competing with more visible careers such as engineering, medicine, computer science, and finance. Careers in these more "practical" fields now pose stimulating challenges to the mathematical interests of the young student, comparable in intellectual reward and attainment to traditional mathematics. • From Western Europe: "Our most serious problem is the attraction of engineering and computer science at the expense of mathematics." • From a Developing Country: "Intelligent students now choose areas such as health sciences and engineering; mostly second or third rate students choose to study mathematics."

The evidence is clear from around the world: As mathematics becomes more applicable, able students flock to the disciplines that use mathematics rather than to traditional pure mathematics. Student preferences pose a direct threat to the health of post-secondary education-where student choices are most noticeableand destabilize secondary education as well by reducing the number of well-prepared mathematics teachers. In

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extreme situations, declining student interest in mathematics can lead to chronic shortages of well-prepared teachers, thereby establishing a negative feedback loop which rapidly drives the whole system of mathematics education into a state of crisis. It is as if we wired our home thermostats to shut off whenever it got too cold: as fall turns to winter, we'd soon have no heat whatsoever.

Standards of Stability Comparisons with the past are of limited value when analyzing problems of today. It is too easy to fall prey to the merchants of nostalgia who wish for restoration of a dream from the past. What is needed instead are standards by which the post-secondary system can be judged, independent of particular transient effects. Data from many countries around the world reveal some patterns that appear to transcend the great differences in educational systems. For example: • In most industrialized countries, between two and three out of every million citizens earn a doctor's degree in mathematics. Only in developing countries and in the United States are doctoral productivity patterns significantly below this level. • About 3% of university degrees are in mathematics, in both industrialized and developing countries. Despite wide variation in the availability of university education, in virtually all countries this percentage varies only from 1.5% to 4%. • Most countries produce annually from 20 to 40 university mathematics graduates per million in the population, although a few industrialized countries are much more productive, with averages in the range of 60 to 80. For a hypothetical country with a stable population, universal secondary education, and university education for 20% of the population, one would expect for each million citizens to need about 25-30 university graduates each year just to maintain the supply of secondary school teachers, and about 3 doctoral graduates to replenish the supply of university faculty. Although neither these data nor estimates are based on very firm evidence, they do suggest that the norm in most industrialized countries at this time is that the educational systems produce just about enough mathematics graduates to replace themselves, with little left over to support an increasingly mathematicized society. It is not surprising, therefore, that in the face of increased demand for mathematically trained graduates, most nations are reporting shortages of teachers and weaknesses among mathematics students. Changes in the nature of mathematics and societyincreased usefulness of mathematics, increased mathematization of society-have shifted significantly the

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

Mathematics for a New Century

delicate balance between supply and demand of mathematically-educated university graduates. This shift suggests the need for structural changes in the way mathematics is taught in institutions of higher education, to insure that post-secondary mathematics programs continue to meet the increasingly diverse needs of society. The direction of change will be determined by the impact of several significant forces that influence university mathematics programs: computers, new applications, research in learning, research in mathematics, and socio-economic trends.

Impact of Computers The most obvious force for major change in university mathematics is the rapidly increasing impact of electronic computers on the way mathematics is practiced. Computers have influenced mathematics in two quite different ways: they have made mathematics more powerful than ever before, and they have altered the very nature of the mathematical sciences. Computers are now commonly used in mathematical research of all kinds, both pure and applied; they have altered the balance among subdisciplines, posed new problems for theory, and provided new tools for exploration and proof. Unfortunately, this new intellectual balance within the mathematical sciences is rarely reflected in school or university curricula. In most countries, computers have had only a very slight effect on post-secondary mathematics education, and then only in a few industrialized countries. Indeed, many countries report no significant post-secondary curricular change in the last 25 years-a period during which the entire computer revolution has occurred. University mathematics curricula are virtually immune to change; in mathematical terms, we might say that the mathematics curriculum is invariant under intellectual revolutions. It is surprising that computers have had so little impact on the university mathematics curriculum. Lack of hardware, a very real problem in many countries and many universities, is more a symptom than a cause. The real issue lies much deeper: a lack of will. Mathematicians are notoriously conservative in curricular matters, and resist changing their courses even if they regularly use newer techniques in their own professional lives. Apart from actual use of computers, the impact of algorithmic approaches to mathematics cries out for a prominent role in the undergraduate curriculum. Such subjects as computational complexity, dynamical systems, scientific computation, and visual data analysis have blossomed in the last quarter century, but are hardly noticeable in the current curriculum. Much could be done to introduce these topics early even without actual use of computers. But most faculties do not make

that effort, preferring instead to rely on the comfortable classical curriculum. Is it any wonder that good students no longer find mathematics an attractive subject?

Impact of New Applications A second factor that suggests the need for major revitalization in undergraduate mathematics is the ciramatic increase in breadth of application of the mathematical sciences. No longer are engineering and physics the prime users of undergraduate mathematics. Sophisticated techniques from yesterday's research are routinely used in today's applications. Consider the biological sciences: differential equations are used in physiology, as are combinatorial methods in genetic sequencing, knot theory in modelling DNA, graph theory in neurophysiology, mathematical modelling in protein engineering, statistical methods in clinical trials, and probability theory in epidemiology. The list of new applications could go on indefinitely: mathematical biology is one of the most exciting frontiers of applied mathematics today, a great showcase for the power and versatility of mathematics. These mathematical tools have helped push biological research to the frontier of science: understanding life and intelligence is the scientific challenge of our age, and mathematics plays a central role in this venture just as it did a century ago in the quest to understand the nature of matter. Similar mathematical methods are used increasingly in environmental science, in natural resource modelling, in economics and sociology, in psychology, and in cognitive science. In these newer disciplines, mathematics offers structures for understanding (and hence controlling) many of the factors that affect the quality of life on earth. Mathematical ideas are even finding their way into the fine arts, as computer tools are used by graphic artists, film makers, and musicians. New applications have unquestionably changed the character of the modern mathematical sciences. They have changed not only how mathematics is used, but also the problems on which mathematicians work. Science has always inspired much of the most interesting mathematics, and this decade is no exception. Especially because of the natural attraction of diverse applications to so many students, it is perhaps surprising that these applications have had so little impact on the post-secondary curriculum. The answer, once again, seems to lie in the natural curricular conservatism of university mathematicians. By maintaining a curriculum from the 1950s that is narrowly focused on traditional topics from mathematics with applications to engineering and physics, we give students the impression that mathematics is a discipline rooted more in the past than in the future. Nothing could

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be further from the truth, although one cannot blame students for not knowing this. We never tell them.

Impact of Research on Learning Those who study the way students learn have accumulated convincing evidence for something that observant teachers have known for generations-that students retain more of things they learn by themselves than whatever they memorize for a test. Effective teaching requires that students be engaged by what they learn, not only that they pass tests based on effort of preparation. The conclusion of this research should not surprise any experienced teacher. What is new, however, is evidence suggesting a mechanism that helps explain why learning takes place this way. Teachers normally act as if each student's mind is a blank slate-or an empty computer disk-on which effective teachers can record whatever information they like. Research in cognitive science suggests otherwise: each student's mind is more like a computer program than a computer disk. Each student brings to the mathematics classroom a rich set of prior mathematical experiences that provide a unique mental framework in which the student creates new patterns derived from new experiences. Learning occurs not in the act of remembering, but in the gradual development of mental frameworks unique to each individual. In other words, students learn by modifying their mind's program, not by storing new data in their mind's memory. Some of each student's experience is based on school lessons, but much of it is learned from the ambient environment in which the child lives. Those at this Congress who are discussing ethno-mathematics can document example after example of student-generated mathematical algorithms derived from street experience rather than from school. Scientists who supervise students in laboratory work know how much more is learned with the hands than with the mind alone. Students who work with computers learn incredible detail simply by interaction with the machine-detail that they have never been formally taught and that would be very difficult to teach by any other means. Learning requires engagement, not just activity. I hardly need point out that most post-secondary mathematics-indeed, most mathematics at any levelis taught by lecture, with homework exercises for practice and examinations for enforcement. Lecturing and examining may be the easiest way to teach mathematics (or perhaps just the cheapest way), but they are by no means the most effective. Few students can learn mathematics well from lectures and homework alone. In the past, those few may have been all the students in our classes, sufficient in number to meet the needs of society. Today these few are the teachers and professors.

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But now, with increased need for mathematics by students of many different interests, the deficiencies of this style of pedagogy are increasingly apparent. Effective teaching for today's students requires a more diverse repertoire of approaches, including in addition to lectures, homework, and examinations, new opportunities for group work, for extensive writing, for oral practice, for exploration and experimenting, for modelling projects, and for computer activities. Students need to experience mathematics as they learn it, and not simply study it in preparation for exams.

Impact of Research in Mathematics Despite its ancient roots, mathematics continues to grow and change. What we call classical mathematics-in theory and applications-is largely the mathematics of the nineteenth century. In the first two thirds of the twentieth century the emphasis of mathematical research shifted to abstraction and axiomatic foundations, even as applications of mathematics to the physical sciences reached the point of what Eugene Wigner called "unreasonable effectiveness." Virtually all mathematics taught in university courses today is this classical mathematics of the nineteenth and early twentieth centuries. However, in the last thirty years, mathematics itself has undergone a renaissance as creative and breath-taking as the parallel but better known revolutions in biology and computer science. These discoveries have affected both theory and applications, changing significantly the very character of mathematics: • Number Theory. The digital character of computation has thrust number theory once again into center stage of mathematics, with new approaches to classical problems (for example, elliptic curves) producing unexpected dividends in the theory of computation, in mathematical logic, and in applications to the security of data transmission. • Statistical Science. As data deluges science from every source-from economics, from telemetry, from laboratories-statistical methods form a continually changing interface between information from the world and analysis by mathematical methods. • Optimization. Beginning with the simplex algorithm for linear programming and moving through recent interior methods devised by Karmarkar, mathematical optimization has joined the frontiers of mathematics both in challenging problems (to find optimal solutions) and in effective applications (to improve efficiency and reduce waste). • Visualization. Converging ideas from research in perception, in computer graphics, and in geometry are now producing innovative theories of perception that draw on such diverse specialities as differential

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

Mathematics for a New Century

geometry, combinatorial algorithms, data structures, and engineering. • Dynamical Systems. Recent research on fractals has shown how iterated order often begets chaos. Models of chaotic systems have permeated science and spawned whole new areas of research in mathematics; applications range from data compression in storage of photographs to theoretical models of instability (in stock markets as well as in the solar system). • Decision Theory. Whereas continuous models from analysis provide apt models for the physical sciences, the discrete models of game theory, social choice functions, and expert systems provide more appropriate tools for the human sciences which depend on decisions, votes, and choices rather than on continuous change.

During the last thirty years, mathematics has been transformed into a rich collection of mathematical sciences, tightly coupled to each other through interlocked theory and linked to the world of science and business by an increasing network of applications. Unfortunately, few university students see mathematics in this way-as a vibrant, living discipline ready to engage their minds and careers in a stimulating contemporary challenge. More often than not, students see mathematics as part of history, as a collection of tools developed in some previous era whose scientific problems are now passe.

Impact of Socio-Economic Factors It is no longer true in any country, if it ever was, that uni-

versities can function as islands of intellectual innocence amid an ocean of turbulent social change. All over the world, governments and students are increasing the pressure on universities to better meet the needs of society. Often these pressures are conflicting, overwhelming, and contrary to the traditional mission of universities. Like all subjects, mathematics is weakened when the strains on the universities become too great. Four issues seem widespread in reports from around the world: • Overcrowding. Public pressure for increased enrollment is common in virtually every country. Sometimes this pressure is caused simply by increases in the number of university age students; other times it is part of a conscious government decision to increase access to higher education as a means of furthering the educational expertise of the country. Moreover, since jobs are often hard to find while the status of "student" is frequently a comfortable refuge from adult responsibility, students often pursue very leisurely routes to their degrees. It is not uncommon in some countries for the majority of students to never finish.

• Underfinancing. Some countries are struggling with run-away inflation; others are insisting on stricter accountability from higher education for producing graduates useful to society. In either case, universities everywhere are under considerable financial pressure. Severe cutbacks are not at all uncommon. Because of the increased world-wide demand for mathematics, these problems often divert mathematically trained personnel from one country to another, sometimes causing "brain-drain" that reduces the intellectual resources of those countries that can least afford the loss. • Links to Jobs. Many countries now experience the paradox of unemployed university graduates at the same time as positions requiring university degrees remain unfilled. Most often this is due to the difficulty of properly matching university work with job requirements; neither centrally planned economies nor free market systems seem able to solve this problem efficiently. Problems arising from socio-economic circumstances are significantly aggravated by a university mathematics curriculum that typically trails by fifty years the mathematical methods actually used in contemporary business and science. • Diminished Quality. As numbers prevail over standards, universities everywhere find that students are either unable or unwilling to undertake the type of rigorous curricula that have traditionally been the hallmark of university education. Elitism clashes with populism more directly on a university campus than anywhere else in society. Higher education is the crucible for this debate, and mathematics, to the extent that it provides the key to opportunity, is the principal fuel that heats this crucible.

Revitalizing Post-Secondary Mathematics Signs from around the world make clear the urg~nt need for revitalization of post-secondary mathematics education: • To attract good students to mathematics; • To encourage potential mathematics teachers for secondary schools; • To provide mathematical expertise for industry and government; • To support advances in science and engineering; • To help improve secondary education in mathematics. Many strong forces influence university mathematics education. We have touched on only a few: computers, applications, research in learning, research in mathematics and socio-economic factors. Because these .forces ' are so strong, change in post-secondary mathematics can only be accomplished by channeling these forces to constructive ends. Yesterday Geoffrey Howson advised us to build a surfboard-instead of continuing vain attempts

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to stem a tide generated by uncontrollable forces. I prefer a slightly different metaphor: a sailboat tacking against the wind. Like a captain of a sailboat whose course is set against the prevailing wind, mathematics educators must learn to use the winds of change to our own advantage. The most important task for post-secondary mathematics is to make it attractive as a subject of study for students with a wide variety of interests. University mathematics is good preparation for many different careers. Students these days see and expect personal and scientific challenges in many other fields, so they expect it also of mathematics. Those of us who teach in universities must: • Teach modern mathematics as well as classical topics, not just to advanced students but also to beginners; • Display the power of mathematics not only through its logic but also through the enormous variety of its applications;

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• Show students that computers are a natural ally in mathematical practice-as a source of ideas, a device for visualization, and a tool for calculation; • Engage students in the mathematics they study by developing new instructional strategies that involve students as active apprentices in the craft of mathematics; • Make mathematics a pump rather than a filter in the educational pipeline, to insure that students of many different interests benefit from university study of mathematics. Such changes will have significant beneficial effects on university mathematics programs, and by extension on school mathematics, as well as on science and society. Curriculum revitalization must involve new approaches, new priorities, and new examples that reflect not only mathematics of past centuries, which we all studied, but also mathematics for a new century, which our students will need.

NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY

GORDON LOFTIS WALKER 1912-1988

Gordon Walker, who retired as executive director of the American Mathematical Society in 1977, died in Providence on 18 December 1988. He had been ill for several months during the spring and summer, and then was convalescing well when he died following emergency surgery. Dr. Walker joined the staff in 1959. Between that time and his retirement, the number of AMS members increased from 6000 to 16,000, the staff expanded from 45 to 175 employees, and the annual budget increased tenfold to $5 million. In addition, the publication program, which consisted of five journals and seven book series at the start of his tenure, grew to 14 journals and 17 book series by his retirement. His term as executive director spanned a period of remarkable growth and development in mathematics and related areas.

Dr. Walker's efforts transformed the Society and helped to bring it to its present state of preeminence. Among the developments for which he deserves principal credit were the creation of the office's first computerized financial reporting system and the pioneering work of the Society in the area of

computer typesetting of mathematical texts. During his tenure the Soviet Union signed an international copyright agreement, and he immediately arranged for contracts for the translation of several of the principal Soviet journals. He attempted to computerize the operations of Mathematical Reviews, but that was not really feasible in the days of batch processing and Hollerith cards. According to Ellen Swanson, former head of the editorial department, he was the one who persuaded her to write her Mathematics Into Type, which has served as the standard reference for the preparation of mathematical manuscripts for many years. Dr. Walker was instrumental in the creation in 1967 of the Committee to Monitor Problems in Communication, and, as a member of it, he made the Committee aware of modern trends in handling scientific information. One result was the Mathematical Offprint Service, a pioneering effort in the selective dissemination of information. In the late 1960s, he helped to create a Commission on a National Information System in the Mathematical Sciences in order to devise a single system for dissemination of mathematical information. The Commission was an important step toward increasing cooperation in the mathematical community. Dr. Walker's ability to recognize and develop the potential of his employees was of great benefit to the Society. Most of the present department heads in the Providence office, and many of the other senior employees, flourished under his tutelage. The stability of the Society as a business operation depends on this cadre of senior staff, as they provide the support and know-how that any new director, coming from academia, must always find indispensable. Gordon Walker was born on 29 October 1912, in Salt Lake City, Utah. He received a B.S. in 1937 and an M.A. in 1938 from Louisiana State University, and his Ph.D. from Cornell University in 1942. Between 1942 and 1954 he served on the faculties of the University of Delaware, Temple University, Purdue University, and Cornell University. His research was in geometry and algebra; his publications include several papers and the book Military and Naval Maps and Grids (Dryden Press, 1942; with William W. Flexner). From 1954 to 1959 he was Head of the Mathematics Section at the Research Center of the American Optical Company. He served as a member of the Board of Directors of the National Federation of Science Abstracting and Indexing Services from 1966 to 1970, and as its president from 1968 to 1970. William J. LeVeque

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American Mathematical Society

Associate Executive Director Position Open The position of Associate Executive Director of the American Mathematical Society will become open in 1989. The Society is seeking applications and nominations of candidates for this position. The person filling this position will work in the Society's Providence office with the Executive Director and the continuing Associate Executive Director, Dr. James W. Maxwell. The Providence office employs about 155 people working within an administrative structure of four divisions and twelve departments. Each Associate Executive Director has the responsibility of administering the work of some part of this staff. The Society also maintains offices of Mathematical Reviews in Ann Arbor, which employs another 75 people. An important role of the administration in both of these offices is to facilitate communication between members of the staff and the Trustees, the Council, editorial and other AMS committees, governmental agencies, professional societies, and mathematicians throughout the world. The Society is looking for a candidate with sensitivity for the concerns of the mathematical research community and for the Society's commitment to service to that community. Such a candidate should • have earned a Ph.D. in one of the mathematical sciences • have knowledge and experience in computing, particularly as it may apply to office automation and information exchange/data processing • have a good command of the English language and be capable of writing well and easily • have an interest in administration and be able to work harmoniously with mathematicians and nonmathematicians alike. The initial appointment will be for three years and thereafter on an indefinitely renewable term or continuing basis. Applications and nominations should be sent to Dr. William H. Jaco Executive Director American Mathematical Society P. 0. Box 6248 Providence, RI 02940 Completed applications and appropriate letters of reference received by 15 March 1989 will be assured of full consideration. It is preferable (but not essential) that duties begin no later than 1 July 1989. The Society is an equal opportunity employer and has a generous fringe-benefit program including TIAA/CREF. Salary for the position will be commensurate with the background of the appointee.

Computers and Mathematics

Edited by Jon Barwise Editorial notes This month's column contains three pieces. The first article, written by Gian-Carlo Rota in memory of the mathematician Stanislaw Ulam, warms my heart, for it shows that Ulam, great mathematician that he was, embraced a view of mathematics as ever reaching out into new territory. He understood that real progress in computer science and artificial intelligence will require the development of a "mathematics of meaning." The real question, as I see it, is whether mathematicians will get involved in or otherwise support this development, and make sure it is serious mathematics, or whether they will watch from the sidelines, criticizing the efforts. The second article, written by Larry Lambe, is a review of Scratchpad II, a programming language and computer algebra system which currently runs on IBM mainframes and the IBM RT /PC. If you are interested in further information about the program and its availability, contact Sandy Wityak, Computer Algebra Group, IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598. The final article, written by Andrew Matchett, is a review of the program Graphical Aids for Stochastic Processes, a piece of courseware referred to briefly in last month's column. Help Wanted

This column is still in serious need of articles written by those with experience setting up computing facilities for mathematics departments of various sorts. What works, and what doesn't? If you could write such a piece, contact me at the address given below. Your colleagues need your expertise. Correspondence for this column should be sent to Professor Jon Barwise Center for the Study of Language and Information Stanford University Stanford, CA 94305 Email: [email protected]

The Barrier of Meaning In memorium: Stanislaw Ulam

Gian-Carlo Rota Massachusetts Institute of Technology

It took me several years to realize what Stan Ulam's real profession was. Many of us at the Laboratory who were associated with him knew how much he disliked being alone, how he would summon us at odd times to be rescued from the loneliness of some hotel room, or from the four walls of his office, after he had exhausted his daily rounds of long-distance calls. One day I mustered the courage to ask him why he constantly wanted company, and his answer gave him away. "When I am alone, he admitted, I am forced to think things out, and I see so much that I would rather not think about!" I then saw the man in his true light. The man who had the highest record of accurate guesses in mathematics, the man who could beat engineers at their game, who could size up characters and events in a flash, was a member of an all-but-extinct profession, the profession of a prophet. He shared with the men of the Ancient Testament and with the Oracle at Delphi the heavy burden of instant vision. And like all professional prophets, he suffered from a condition that Sigmund Freud would have labelled "a Proteus complex". It is unfortunate that Freud did not count any prophets among his patients. In bygone days, the Sybil's dark mumblings would be interpreted by trained specialists, Hermeneuts they were called, who were charged with the task of expressing her cryptic messages in lapidary Greek sentences.

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In Ulam's case, the Los Alamos Laboratory would instead hire consultants, charged with the task of expressing his cryptic messages in the dilapidated jargon of modern mathematics. Luckily, some of Stan Ulam's pronouncements were tape-recorded by Fran