OTICES OF THE

AMERICAN MATHEMATICAL SOCIETY

Rademacher on J( 1), Poincare Series of Nonpositive Weights and the Eichler Cohomology Marvin I. Knopp page 385

Columbus Meeting (August 8-11) page 447 First Announcem~t

The National Science Foundation Budget Request for Fiscal 1991 page 394

Mathematics Outside of Mathematics Departments page 408

APRIL 1990, VOLUME 37, NUMBER 4 Providence, Rhode Island, USA ISSN 0002-9920 Calendar of AMS Meetings and Conferences

This calendar lists all meetings which have been approved prior to Mathematical Society in the issue corresponding to that of the Notices the date this issue of Notices was sent to the press. The summer which contains the program of the meeting, insofar as is possible. and annual meetings are joint meetings of the Mathematical Associ­ Abstracts should be submitted on special forms which are available in ation of America and the American Mathematical Society. The meet­ many departments of mathematics and from the headquarters office ing dates which fall rather far in the future are subject to change; this of the Society. Abstracts of papers to be presented at the meeting is particularly true of meetings to which no numbers have been as­ must be received at the headquarters of the Society in Providence, signed. Programs of the meetings will appear in the issues indicated Rhode Island, on or before the deadline given below for the meet­ below. First and supplementary announcements of the meetings will ing. Note that the deadline for abstracts for consideration for pre­ have appeared in earlier issues. sentation at special sessions is usually three weeks earlier than that Abstracts of papers presented at a meeting of the Society are pub­ specified below. For additional information, consult the meeting an­ lished in the journal Abstracts of papers presented to the American nouncements and the list of organizers of special sessions. Meetings

Abstract Program Meeting# Date Place Deadline Issue 858 • April 19-21, 1990 Albuquerque, New Mexico Expired March 859 • August 8-11 • 1990 Columbus, Ohio May 18 JulyI August t (93rd Summer Meeting) 860 * October 20-21,1990 Amherst, Massachusetts August 6 October 861 * November 2-3, 1990 Denton, Texas August 6 October 862 * November 10-11,1990 Irvine, California August 6 October 863 January 16-19, 1991 San Francisco, California October 10 December (97th Annual Meeting) March 16-17, 1991 South Bend, Indiana March 22-23,1991 Tampa, Florida August 8-11, 1991 Orono, Maine (94th Summer Meeting) October 12-13, 1991 Philadelphia, Pennsylvania October 25-26, 1991 Fargo, North Dakota January 8-11, 1992 Baltimore, Maryland (98th Annual Meeting) March 27-28, 1992 Springfield, Missouri June 29-July 1, 1992 Cambridge, England (Joint Meeting with the London Mathematical Society) January 13-16, 1993 San Antonio, Texas (99th Annual Meeting) August 15-19, 1993 Vancouver, British Columbia (96th Summer Meeting) (Joint Meeting with the Canadian Mathematical Society) January 5-8, 1994 Cincinnati, Ohio (100th Annual Meeting) January 1 0-13, 1996 Orlando, Florida (1 02nd Annual Meeting) *Please refer to page 485 for listing of special sessions. t Preregistration/Housing deadline is June 6

Conferences

June 7-July 4, 1990: Joint Summer Research Conferences in August 2-3, 1990: AMS-SIAM-SMB Symposium on Some the Mathematical Sciences, University of Massachusetts at Mathematical Questions in Biology, Neural Networks, Amherst, Massachusetts. Vancouver, Canada. June 18-29, 1990: AMS-SIAM Summer Seminar on Vortex August 6-7, 1990: AMS Short Course, Combinatorial Games, 'Dynamics and Vortex Methods, University of Washington, Columbus, Ohio. Seattle, Washington. July 8-28, 1990: AMS Summer Research Institute on Differential Geometry, University of California, Los Angeles, California. Deadlines

May-June Issue July-August Issue September Issue October Issue Classified Ads* April 23, 1990 June 14, 1990 July 30, 1990 August 27, 1990 News Items April 25, 1990 June 18, 1990 August 3, 1990 August 30, 1990 Meeting Announcements** April 16, 1990 May 29, 1990 July 19, 1990 August14, 1990 * 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

ARTICLES DEPARTMENTS

385 Rademacher on J(r), Poincare Series of Nonpositive Weights and the 383 Letters to the Editor Eichler Cohomology Marvin I. Knopp 412 Forum NEW DEPARTMENT Starting with the Fourier series expansion of J ( r) and specific formulas In this issue, the Notices has instituted a new department called Forum. This department, for its coefficients, Rademacher was able to prove that J ( r) is a modular created in response to a recommendation of invariant. Applying similar reasoning to automorphic forms of negative the AMS Science Policy Committee, will pub­ weight, one is naturally led to cocycle conditions and Eichler cohomology. lish short articles on issues which are of Marvin Knopp describes the fascinating connections. interest to the mathematical community. We inaugurate the Forum with two articles con­ 394 The NSF Budget Request for Fiscal Year 1991 cerning mathematics education and the pro­ fessional mathematician. Fiscal year 1990 didn't bring the budget increases that NSF was hoping for, but the picture will be rosier in 1991 if the President's requested 440 News and Announcements increase of 14% for fiscal 1991 actually materializes. Part of the increase 444 Funding Information for the will go toward starting a new batch of Science and Technology Centers Mathematical Sciences and strengthening the NSF's education activities. In this annual report, Allyn Jackson examines the major components of the budget request. 445 AMS Elections 447 Meetings and Conferences of the 408 Mathematics Outside of Mathematics Departments Solomon A. AMS Garfunkel and Gail S. Young Columbus, OH In the last twenty years, there has been a dramatic rise in the number AugustB-11,447 of bachelor's degrees awarded in science and engineering. Because AMS Short Course, 482 these areas have become increasingly mathematical, one would expect a Invited Speakers, 485 concomitant increase in enrollments in advanced mathematics courses. Symposium on Some But, despite large increases in calculus enrollments, enrollments in Mathematical Questions in advanced mathematics have remained steady. In their study, the authors Biology, 488 found that more students are enrolled in advanced mathematics courses Call for Topics, 489 outside of mathematics departments than inside them. They discuss their 491 Mathematical Sciences Meetings findings and the implications for the mathematical community. and Conferences FEATURE COLUMNS 503 New AMS Publications 506 AMS Reports and 417 Computers and Mathematics Jon Barwise Communications The issue of what goes to make up good courseware for undergraduate Recent Appointments, 506 mathematics is addressed in an article by Keith Devlin. In addition, there Reports of Past Meetings, 506 are reviews of several pieces of mathematical software: FFTLIB, Phaser, Officers, 507 and three programs for the NeXT: Groups, Rubik Algebra, and Orbit. 508 Miscellaneous 435 Inside the AMS Personal Items, 508 Deaths, 508 H. Hope Daly, the AMS Director of Meetings, gives a behind-the-scenes Visiting Mathematicians look at what it takes to put on a January Meeting. (Supplement), 508 438 Washington Outlook 509 New Members of the AMS Hans J. Oser discusses testimony of Lauro Cavazos, Secretary of 513 Classified Advertising Education and Erich Bloch, Director of the National Science Foundation, 523 Forms given before the House Committee on Science, Space, and Technology.

APRIL 1990, VOLUME 37, NUMBER 4 381 From the Executive Director . .. Books: Acquisition and Distribution The Society is a publisher of books. The books published by the Society are principally for the communication of research mathematics. They are pub­ AMERICAN MATHEMATICAl. SOCIETY lished in series (Colloquium, Mathematical Surveys and Monographs ... ) and out-of-series (for example, I Have a Photographic Memory by Halmos). The first book published by the Society was the Proceedings of the Inter­ national Mathematical Congress of 1893. That Congress also spawned the

EDITORIAL COMMITIEE Colloquium Series which the Society began in 1896. The Colloquium Series Robert J. Blattner (Forum Editor) was the only book series for the first fifty years. Today, there are nearly thirty Michael G. Crandall such series, and this year the Society has plans to publish over seventy books. Robert M. Fossum (Chairman) When one thinks of publishing books, one thinks of the processes of editing, Lucy J. Garnett, D. J. Lewis Nancy K. Stanton composing/keyboarding, typesetting and printing. These are extremely im­ Robert E. L. Turner (Letters Editor) portant aspects of publishing and ones which the Society performs very well. MANAGING EDITOR However, there are two other important aspects of publishing that determine Donovan H. Van Osdol its content and its success in communication: acquisition of manuscripts and ASSOCIATE EDITORS distribution of published materials. These are the aspects of the Society's Ronald L. Graham, Special Articles publishing program that involve the mathematical community and depend Jeffrey C. Lagarias, Special Articles on that community for its quality and success. SUBSCRIPTION INFORMATION The acquisition of all Society books comes under review of editorial commit­ Subscription prices for Volume 37 (1990) are tees made up of volunteer research mathematicians appointed by the AMS $113 list; $90 institutional member; $68 individual member. (The subscription price for members is Council. These committees respond to submitted materials by evaluating the included in the annual dues.) A late charge of appropriateness to the series (if it is for a book in series) and the quality 10% of the subscription price will be imposed of the mathematical content. However, the committees, in general, do not upon orders received from nonmembers after solicit manuscripts nor have they ever been charged with such an acquisition January 1 of the subscription year. Add for post­ age: Surface delivery outside the United States activity. Individuals have served as volunteer acquisition editors, but this is and lndia-$11; to lndia-$22; expedited deliv­ not the rule. At present the Society's acquisition activity is small. There has ery to destinations in North America-$24; else­ been debate in the governing bodies of the Society regarding the purpose where-$49. Subscriptions and orders for AMS of the Society's publication program, how it should complement publishing publications should be addressed to the Amer­ programs of commercial publishers and whether the Society should become ican Mathematical Society, P.O. Box 1571, An­ nex Station, Providence, Rl 02901-1571. All or­ more of an initiator of proposals for publication. This is not simply a mat­ ders must be prepaid. ter of publishing more books, but rather of identifying, encouraging and ADVERTISING realizing materials that will enrich the Society's efforts to communicate re­ Notices publishes situations wanted and classi­ search mathematics. A successful acquisition program requires planning and fied advertising, and display advertising for pub­ thought about the purpose of the Society's book publication program. lishers and academic or scientific organizations. Copyright@ 1990 by the American Mathemat­ The distribution of Society books has been low key. It depends to a large ex­ ical Society. All rights reserved. Printed in the tent on standing orders, regular orders from libraries that purchase all books United States of America. in given series. Yet it is responsive to the needs of the individual mathemati­ The paper used in this journal is acid-free and cian and provides substantial member discounts. In distribution of its books, falls within the guidelines established to ensure as in acquisition, the Society has tended to respond rather than initiate. For permanence and durability. €9 Most of this publi­ example, one is not likely to find AMS books regularly displayed on shelves in cation was typeset using the TEX typesetting sys­ tem. campus bookstores. However, this is changing. The governing bodies of the [Notices of the American Mathematical Society is Society have authorized an effort for the active distribution of AMS books to published ten times a year (January, February, bookstores. This plan calls for the support of mathematicians, who will act March, April, MayjJune, July/August, Septem­ as local representatives of the AMS on their campuses and as liaisons with ber, October, November, December) by the Amer­ their campus-area bookstores, and ranges from this individual activity to the ican Mathematical Society at 201 Charles Street, establishment of a broad distribution activity. Along with this distribution Providence, Rl 02904-2213. Second class post­ age paid at Providence, Rl and additional mailing effort, the Society plans to introduce a new book series that provides books offices. POSTMASTER: Send address change no­ usable as texts in graduate studies in mathematics. tices to Notices of the American Mathematical So­ The real news in the Society's book publishing program is happening where it ciety, Customer Service Department, American Mathematical Society, P. 0. Box 6248, Provi­ has most contact with the mathematicians. By expanding our acquisition and dence, Rl 02940-6248.] Publication here of the distribution activities, we are enlarging the forum in which mathematicians Society's street address, and the other informa­ can publish their work, and at the same time increasing the availability of tion in brackets above, is a technical requirement published works to the community. of the U. S. Postal Service. All correspondence should be mailed to the Post Office Box, NOT the William Jaco street address.

382 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY I certainly never intended to lend minds and careers in a stimulating Letters support to that viewpoint. contemporary challenge" should help Third, when I wrote I was think­ it with his own efforts. to the Editor ing of different times, when repres­ Now, Rumanian mathematicians sion in the USSR was as much of an need: issue as it is in China today. Dur­ 1. the latest (good) books in the ing that period I was one of many field of mathematics and spe­ Confidentiality individuals who visited the USSR, cialized reviews (lithographed The Supreme Court, in a decision combining scientific trips with visits or published), on a case against the University of to the Moscow Refusnik Scientific 2. rapid copying apparatus (xe­ Pennsylvania by the Equal Employ­ Seminar. I like to think that our rox), ment Opportunity Commission, has visits, support and friendship were 3. computers and specialized teach­ ruled that confidential scholarly as­ among the factors which ultimately ing programs, sessments of the work of a candidate lead to the release of many (but 4. grants of 1-6 months for those for tenure at a University must be not all) members of that oppressed professors and researchers who disclosed to Federal Investigators. group. However, China in 1990 is want to learn what is new in This step is a clear violation of not the USSR circa 1971. education and in mathematics, confidentiality, basic to the mainte­ On the balance, I find that I agree 5. grants for doctorates. nance of quality at Universities. In much more with Professors Fuchs Over 100 mathematicians work in consequence, I will therefore decline and Wu than with many of those the Polytechnic Institute of Bucharest to write my letters to Universities who are now lending support to the and all are open-minded and willing about the work of candidates for present administration in China, and to make progress in mathematics. tenure. I recommend such action as withdraw my earlier opposition to the only proper response to this at­ boycott. Policy on Letters to the Editor tack on academic freedom. Joan S. Birman Letters submitted for publication in Notices Saunders Mac Lane Columbia University are reviewed by the Editorial Committee, (Received February 12, 1990) whose task is to determine which ones are University of Chicago suitable for publication. The publication (Received January 18, 1990) schedule normally requires from two to Rumanian Mathematicians four months between receipt of the letter in Providence and publication of the earliest Boycotting Scientific Visits to Ask for Help and Cooperation issue of Notices in which it could appear. China The revolution in Romania, an his­ Publication decisions are ultimately Editor's Note: The letter by Profes­ torical act of the whole people, makes made by majority vote of the Editorial sors Fuchs and Wu referred to in possible the direct contact between Committee, with ample provision for prior the following letter is the letter titled the mathematicians in this country discussion by committee members, by mail or at meetings. Because of this discussion Boycotting Scientific Visits to China and those in foreign countries. These period, some letters may require as much in the March 1990 issue of Notices contacts could be as follows: as seven months before a final decision is page 263. That letter, signed by 52 1. exchanges of didactic and sci­ made. Letters which have been, or may be, mathematicians, was submitted for entific literature, published elsewhere will be considered, but publication by Professors Fuchs and 2. exchanges of professors, re­ the Managing Editor of Notices should be informed of this fact when the letter is sub­ Wu. searchers and students, mitted. The January issue of Notices con­ 3. long-term contracts on common The committee reserves the right to tained a letter which I wrote about scientific research. edit letters. boycotting scientific visits to China. I Because of the lack of free con­ Notices does not ordinarily publish would like the opportunity to modify vertible currency, the financial basis complaints about reviews of books or arti­ cles, although rebuttals and correspondence my position and correct misimpres­ of all these contacts should be barter­ concerning reviews in Bulletin of the Amer­ sions evidently created by it. type relations. ican Mathematical Society will be consid­ First, my letter was written in re­ The aim of this letter is to inform ered for publication. All published letters sponse to a letter by Professors Fuchs the Mathematical Community in the must include the name of the author. and Wu that was, unfortunately, not World about these new possibilities Letters should be typed and in legible form or they will be returned to the sender, published in the same issue. and to involve it in these forms possibly resulting in a delay of publication. Second, I see that there are many of help and cooperation. Everybody Letters should be mailed to the Editor people within our community who who is interested in the development of Notices, American Mathematical Soci­ would like to return to "business as of mathematics "as a vibrant, liv­ ety, P.O. Box 6248, Providence, RI 02940, usual" with regard to China, and ing discipline ready to engage their and will be acknowledged on receipt.

APRIL 1990, VOLUME 37, NUMBER 4 383 Letters to the Editor

The fields of our didactic and scien­ procity agreements between these as­ compete with experienced people. (A tific interests are: mathematical edu­ sociations. variant of this scheme would be to cation, mathematical logic, algebraic Constantin N. Udri~te support all new Ph.D.s for, say 5 geometry, linear and multilinear alge­ Polytechnic Institute years, perhaps at some lower level, bra, real functions, measure and in­ Bucharest, Romania and divide the remaining funds as tegration, functions of complex vari­ (Received January 22, 1990) above.) Although this scheme is not ables, special functions, ordinary dif­ completely fair (does there exist a ferential equations, partial differen­ A Simple Proposal for Federal completely fair scheme?) I suggest tial equations, Fourier analysis, op­ Support of Mathematicians that it is fairer than the current dis­ erational calculus, integral equations, On page 135 of the February 1990 tribution system by several orders of functional analysis, operator theory, issue of Notices the federal support magnitude. For one thing it utilizes optimization, differential geometry, of Mathematics, NSF and Non NSF, the refereeing system for research pa­ analysis on manifolds, probability in FY 1990 is given as $147,100,000. pers which, with all its faults, works theory, statistics, numerical analysis, Let's round it off to 150 million. This much, much better than the current mechanics, and control systems. Un­ would support 10,000 research math­ grant distribution system. fortunately, with very few exceptions, ematicians at $15,000 each. Who is a It is worth noting one feature of we were unable to participate in "research mathematician"? One who this simple proposal: research funds international meetings, conferences, publishes mathematics articles in re­ go to research people. In contrast, the congresses, and summer seminars on search journals. If there are more current system funds a small army mathematics held abroad. than 10,000 such people just support of beaurocrats, including Secretaries Finally, as a member of the Amer­ the "top" 10,000 (count papers or and Deans, most of whom do no ican Mathematical Society and of the pages). research at all. Society of Mathematical Sciences of Is this scheme fair? No. A Godel Anton Zettl Romania, I suggest establishing reci- might be missed. New Ph.D.s would Northern Illinois University (Received February 8, 1990)

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The purpose of this book is to explain systemati­ Aimed at research mathematicians in the field of cally and clearly many of the most important techniques nonlinear PDEs, this book should prove an important re­ set forth in recent years for using weak convergence source for understanding the techniques being used at the methods to study nonlinear partial differential equations. forefront of this vital area of research. This work represents an expanded version of a series of ten talks presented by the author at Loyola University of Chi­ cago in the summer of 1988. The author surveys a wide collection of tech­ 1980 Mathematics Subject Classifi­ niques for showing the existence of solutions to various cations: 35,46 nonlinear partial differential equations, especially when ISBN 0-8218-0724-2, LC 89-27844, strong analytic estimates are unavailable. The overall ISSN 0160-7642 guiding viewpoint is that when a sequence of approxi­ 88 pages (softcover), January 1990 mate solutions converges only weakly, one must exploit All Individuals $12, List price $19 the nonlinear structure of the PDE to justify passing to limits. To order, please specify CBMS/7 4NA The author concentrates on several areas that are rapidly developing and points to some underlying viewpoints common to them all. Among the several themes in the book are the primary role of measure theory and real All prices subject to change. Shipment will be made by surface. analysis (as opposed to functional analysis) and the con­ For air delivery add, 1st book $5, each additional book $3, tinual use in diverse settings of low amplitude, high fre­ maximum $100. Prepayment required. Order from American quency periodic test functions to extract useful informa­ Mathematical Society, P.O. Box 1571, Annex Station, Providence, tion. The author uses the simplest problems possible to Rl 02901-1571, or call toll free 800-321-4AMS (321-4267) in the U.S. illustrate various key techniques. and Canada to charge with VISA or MasterCard.

384 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Rademacher on J( r), Poincare Series of Nonpositive Weights and the Eichler Cohomology Marvin I. Knopp Dedicated to Paul T. Bateman

all r in 1i and V E f(1). (See [1, Chapter 1], [32, Chapter This article is the twenty-second in the series of Special 3] and §1.2). Furthermore, among modular invariants Articles published in the Notices. Marvin I. Knopp received J ( r) has the distinction that it generates the whole field the Ph.D. degree in 1958 from the University of Illinois of modular functions over the complex field C [ 17, (Urbana). His dissertation, written under the direction of Theorem 1E, p. 345]. It has the Fourier series expansion

Paul T. Bateman, dealt with modular forms of nonpositive 00 weights. He has continued to study modular forms, focus­ (3) 123 J(r) = e-27riT + 744 + L ene 21rinr, L E 'H, ing upon their Fourier coefficients, the Eichler cohomology n=l theory and the theory of modular integrals and their Mellin transforms. He has held professorships at the University of where the en are rational integers [33, p. 56]. Wisconsin, the University of Illinois at Chicago Circle, Bryn In 1938 Rademacher, unaware that Petersson [25] Mawr College and has been a Professor of Mathematics at had already done so, published an exact formula for Cn Temple University since 1976. [30]. (For specifics see (12).) Starting with that formula, in [31] he adopts an entirely fresh viewpoint concerning J ( r), taking it to be defined (anew) by ( 3) and the exact I. Introduction. formula (12). He poses the problem: to show from this 1. Rademacher's perspective upon J(r). new definition that J ( r) is a modular invariant. Since In 1939 Hans Rademacher presented a new construc­ the two transformations Sr = r + 1, Tr = -1/r generate tion of the modular invariant J(r) in a seminal-yet r(1) [12, p. 7] and since J(r + 1) = J(r) follows directly little known-paper. This note examines Rademacher's from (3), this problem reduces (an odd word, considering construction from the perspective of a half-century of the difficulty involved) to that of deriving, from (3) and further advances in the theory of modular and auto­ (12) alone, the equation J( -1/r) = J(r), now far from morphic forms. Especially important is the connection obvious. Rademacher solves this problem by carrying with the Eichler cohomology theory [3], developed some out a profound transformation of the function defined twenty years later. by (3) and (12), representing J(r) as what we may now The modular group r( 1), is the group of linear term a "modified Poincare series." fractional transformations Vr = (ar + b)f(er +d), r complex, with a, b, e, d E Z (Z is the set of rational 2. Poincare's construction of automorphic functions. integers) and ad-be = 1. The absolute modular invariant Poincare series appear for the first time in Poincare's cel­ J ( r) is defined, for r in the upper half-plane 'H., by ebrated 1882 memoir on Fuchsian functions [28]. They provide a perspective indespensible for understanding Rademacher's construction and the later work [8, 9, 10, 11, 21, 3 5] based directly upon it. with Poincare deals with groups of linear fractional trans­ formations acting on the unit disc, in particular, with (2) Gk(r)= I:' (mr+n)-k,kEZ,k?:.3, the construction of their invariant functions. For consis­ m,nEZ tency with our introductory remarks we consider instead groups r of linear fractional transformations acting on the Eisenstein series of weight k. (The notation E' the upper half-plane 'H.; that is, we assume that r is indicates omission of the term form= n = 0.) The well­ a discrete group of mappings Vr = (ar + b)f(er +d), known behavior of Gk(r) under r(1) implies directly that with a, b, e, d real and ad - be > 0. If r is finite, it J(r) is invariant with respect to f(1): J(Vr) = J(r), for is an easy matter to construct a meromorphic function

APRIL 1990, VOLUME 37, NUMBER 4 385 Rademacher, Poincare Series and Eichler Cohomology

F( r), automorphic (that is, invariant) with respect to r, series (5) to the rigid motions of hyperbolic geometry. by forming the finite sum This revelation took place immediately following a two week period during which he thought intensely about the (4) F(r) = L f(Vr), question, but with inconclusive partial results. Poincare VEr ultimately obtained a complete solution, but only after several further similar occurrences, equally unexpected with f chosen meromorphic in H. That F does not and sudden. Of particular interest is Poincare's immedi­ reduce to a constant can be guaranteed by a suitable ate recognition, in each instance, that the new idea would choice of f. be fruitful, before working through any of the details. When r is infinite, in contrast, this simple construc­ For Poincare's own account see [29, pp. 52-55]. There is tion does not suffice, because the series in (4) may a discussion of this episode as well in [5, pp. 12-15]. fail to converge. To overcome this difficulty, Poincare introduced the series which bear his name: 3. Elliptic functions and Eisenstein series. A problem different in technical detail, but virtually f(Vr) (5) Fk(r;f) = L ( d)k, Vr = (ar + b)f(cr +d); identical in spirit, is one resolved before Poincare's VEr cr + birth: that of constructing elliptic functions, that is to say, functions meromorphic in the complex plane C in ( 5) f is a rational function and k a positive integer and automorphic with respect to a group of translations chosen large enough to guarantee absolute-uniform con­ in two independent directions. Suppose w1 and w2 vergence of the series (5) in compact subsets of H. (The are complex numbers with lm(w1jw2) =1= 0. Then the existence of such k follows from the discreteness of r.) "lattice" L = {mw1 + nw2lm, n E Z} is discrete in C, The function Fk ( r; f) so formed, while merom orphic in and when k ~ 3 the "Mittag-Leffler sum", H, fails to have the desired simple automorphic property 1 (8) Ek(z)=Ek(z;w1,w2)= )k' L (z+mw1 +nw2 (6) F(Mr) = F(r), all MEr, m,nE Z

characteristic of functions F defined by (4). However, represents a function (in fact, the simplest one) with a the absolute convergence of ( 5) implies readily that for pole of order k at each of the lattice points. When k ~ 3 the series (8) obviously converges to a function mero­ all M = (; ~) E r, morphic inC, but much more is true. The Mittag-Leffler construction, designed only to produce meromorphic functions with poles at a prescribed discrete set of points (with prescribed principal parts, as well), actually yields since (yr + f5)k(cMr + d)k = (c'r + d')k, where c'r + elliptic functions invariant with respect to the group d' is the lower row of V M. A function satisfying G = G(wi> w2) = {z-+ z + wlw E L}. This obtains since the transformation formulae (7) and certain regularity (8) can be rewritten, in analogy with (4) and (5), as conditions is called an automorphic form of weight k on r. (We note that the Eisenstein series Gk(r), defined by (9) Ek(z) = L fk(Vz),fk(z) = z-k. (2), is virtually the same as the Poincare series (5), with VEG f = 1 and r = r(l). In any event, Gk(r) is a modular Here, once again, the Eisenstein series Gk(r) come to form-that is an automorphic form on rc 1)-of weight mind, for defining k, since Gk(r) satisfies (7) for M E r(l).) A function satisfying ( 6) is called an automorphic function on r. To reach his original goal, construction of nontrivial meromorphic functions possessing the (absolute) invari­ we have E;(O; r; 1) = Gk(r). ance property (6) with respect to r, Poincare forms For k = 2 the nice convergence (absolute-uniform, the quotient Fk(r;fi)/Fk(r;/2), with rational jj and h on compact subsets of C which do not contain any chosen to have distinct singularities in H. The latter of the lattice points mw1 + nw2) of (9) fails, but-as condition ensures that this quotient does not reduce to a is well understood-this difficulty is inessential, easily constant. overcome by subtraction of "convergence terms" from This work of Poincare on the problem of constructing the summands: automorphic (he called them "Fuchsian") functions pro­ 1 I { 1 vides the context for the well-known story of his sudden E z =-+ 2( ) z2 L (z + mw1 + nw2)2 revelation while stepping on a bus to go on an excursion, ( 10) m,n,EZ and in the midst of an unrelated conversation. The un­ expected insight was the relationship of his (Poincare) (mw1 ~ nw2) 2 } •

386 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Rademacher, Poincare Series and Eichler Cohomology

(Here, E' has the same meaning as in (2).) "Nice" (r, D) as above, such that Sis conformally equivalent to convergence of (10) follows readily, and £ 2(z) clearly r\D? The affirmative answer given by the uniformization has a pole of order 2 at each lattice point. However, in theorem means that the theory of Fuchsian groups is co­ re-establishing convergence we apparently have sacrificed extensive with the theory of Riemann surfaces. However, invariance with respect to G(w1,w2). It appears, in there does not seem to be a simple, natural interpreta­ fact, that invariance has given way to the characteristic tion of automorphic forms of arbitrary real weights on a functional equation of an elliptic integral: Fuchsian group in terms of the corresponding Riemann surface. For further details about the history of automorphic forms and their role in contemporary mathematics, I where Cv is a constant dependent upon V. As it turns refer the reader to the Historical Development chapter out, the sacrifice is only apparent, since E2(z) is an even (chapter 1) of [ 17] and to the references supplied there. function (£2( -z) = E 2(z)) and this fact implies directly that Cv = 0 for all V E G. Thus, E2(z) is a genuine II. Rademacher's work on J(r). elliptic function, not an elliptic integral. Of course, £ 2(z) 1. J ( 1:) as a parabolic Poincare series. is the well-known Weierstrass function p(z) [7,§9]. The exact formula of Petersson and Rademacher for the coefficients Cn in the expansion (3) of J(r) is 4. Remarks on automorphic forms. The modular invariant J (1:) was first studied by Dedekind 2n ~ 1 (4ny'n) and, independently, by Klein in 1877, about fifty years (12) Cn=y'n.f...JrAt(n)/1 -£- ,n;:::l. after the development of elliptic functions. Shortly there­ f=l after, Picard used J(r) and the monodromy theorem to Here, Ae ( n) is a Kloosterman sum defined by prove his famous "little" theorem: an entire function ( 13) omitting more than one complex value from its range is constant. Nowadays Picard's little theorem is most often Ae(n) = L exp [ -~ni(nh+h')] ,hh' =-1 (mod£), h(mod 1 J proved using instead of J(r) the related function A.(r), (h.l)=l invariant with respect to the principal congruence sub­ group of level 2, a normal subgroup of index 6 in r( 1). while / 1 is the modified Bessel function of the first kind, (A.(r) has a simple expression in terms of the Weierstrass given by the power series elliptic functions p(z) and p'(z).) About the same time H. A. Schwarz and Poincare used the theory of automor­ oo (x 12 )2j+ 1 ( 14) /l(x) = L .,(. 1)' phic functions in studying ordinary second-order linear j=O ). J + . differential equations. This was the context for Poincare's fundamental Rademacher obtains this formula in [30] by a refinement work [28], which effectively initiated a systematic theory of his own variant [32] of the Hardy-Ramanujan "circle of automorphic forms and automorphic functions with method" (in contrast to Petersson's entirely different respect to Fuchsian groups, discrete groups of linear approach involving modular forms of weight 2). In [31] fractional transformations acting on a half-plane or disc. he proves the following result, which brings to light a In the background of Poincare's work (and dominant in completely new way of viewing the fundamental modular the contemporaneous work of Klein) is the idea of a invariant J ( 1:). Riemann surface. In light of the later "uniformization Theorem 1. [31, (4.1)]. J(r) has the representation theorem," proved completely in 1912 after thirty years ( 15) of effort by a number of mathematicians, the theory of 123 J(r) =e-2nir + e2nifr + 731 Riemann surfaces can be viewed as contained in the theory of Fuchsian groups. The relationship between the two theories is close at hand: given a Fuchsian group r acting on a disc or half-plane D, one can introduce a (f,m)=l natural topology on the set of orbits S = r\D in such a way that the structure is analytic, the topological space - exp( -2nim' /f)}, S a Riemann surface. Then the meromorphic functions on S correspond to the automorphic functions with where m' is any integral solution of the congruence respect to r, the (first-order) differentials on S to the mm' = -1(mod£) and£' is the rational integer defined automorphic forms of weight 2 with respect to r. by-£'= (mm' + 1)/£. The problem of uniformization deals with the con­ In the historical introduction to his influential work, verse: given an arbitrary Riemann surface is there a pair Discontinuous groups and automorphic functions, J. Lehner

APRIL 1990, VOLUME 37, NUMBER 4 387 Rademacher, Poincare Series and Eichler Cohomology has expressed the opinion that this formula for J ( r) "is where a and b are so chosen that Vc,d = ( ~ ~) E r( 1). as striking and elegant as the classic identities of Euler We next take advantage of the fact that the summands in and Jacobi". [17, p. 41] Lehner further refers to the for­ ( 17) are unchanged under replacement of the pair (c, d) mula ( 15) as "an expansion of J into partial fractions," by the pair ( -c, -d), to rewrite the sum ( 17) as and he compares it with the definition (10) of the Weier­ strass function p, noting in particular, "the subtracted 21 "'LJ "'LJ {exp( -2niVc,dr)- exp( -2nialc)}. 'convergence summand' in each case" [17, pp. 40-41]. l~lci~K 1 ~.~; 11f( To appreciate more fully the insight which informs this remark, we rewrite ( 15) to resemble closely a modified But, the condition (c, d) = 1 implies that c = 0 occurs form of the Poincare series (5), namely, the "parabolic only with d = ±1, and d = 0 only with c = ±1. In the Poincare series" introduced by Petersson [24]: former case we can choose the matrices to be ± ( b~), and in the latter, ± ( ~ [/). Thus, with the definition

( 16) e-27Ciajc { c=fO s(c,d)= 0 ' c = 0' Here, v is an arbitrary integer, k is an integer, }, > 0 the expression ( 15) becomes is the minimal width of a translation in r (e.g., A. = 1 for r = r(1)) and the notation I:* indicates that-in 123J(r) = 732 + 1~ ~ 2: 2: contrast to the summation over all V E r as in ( 5)-in ( 18) lci~K ldi$K (<.d)= I ( 16) the sum is confined to V E r with distinct lower row c, d. This restriction arises naturally as a necessary {exp( -2niVc,dr) - s(c, d)}. condition for convergence since the rational function Comparison of (18) with (16) now clearly justifies f of ( 5) has given way in ( 16) to the exponential our viewing the right-hand side of ( 18) as a parabolic function e 27Civrf).. For, the periodicity of e 27Civrf). implies Poincare series of weight k = 0 on the group r( 1), for v = that the numerator e 27CivVrf). in ( 16) is independent of -1, but with the order of summation prescribed explicitly the upper row of V, and from this it follows directly (lattice points in expanding squares) and modified by the that, regardless of the size of k, the full sum on all subtracted convergence summands s(c, d). We stress that V E r cannot converge, since each summand would then without this prescription and modification, there can be occur infinitely often. On the other hand, the assumption no hope of convergence for a Poincare series of weight k > 2 assures absolute uniform convergence of the series k = 0. (For r( 1) absolute convergence occurs only if in (16) on compact subsets of 1i [17, pp. 276-277]. k > 2.) Even with them, convergence remains far from As with the Poincare series (5), Petersson's modified obvious. The proof comes naturally out of Rademacher's Poincare series ( 16) are automorphic forms of weight k development. on r as long as k is an integer greater than two. The proof is the same in both cases. The parabolic Poincare 2. Invariance of J ( r). series have two clear advantages over the Poincare series As we remarked earlier in reference to the definition ( 5): (i) the analytic behavior of Gk (r; v) can be controlled (10) of p(z) (= E2(z)), modification by the convergence completely at the parabolic cusps of r; (ii) the Gk ( r; v) summands s(c, d) seems to disturb the invariance under behave well with respect to the Petersson inner product, r(1) of the right-hand side of (18), leading to the well enough, indeed, to make possible a direct inference introduction of additive periods in the transformation that they form a basis for all automorphic forms of formulae (as in ( 11)). At first glance, then, Rademacher's weight k on r which are holomorphic in 1i and at the formula (18) appears to imply only that J(r) is an finite cusps [17, pp. 284-289]. abelian integral with respect to r( 1), but not necessarily To compare the expression ( 15) with the parabolic a modular invariant (i.e. that the periods vanish). But, as Poincare series ( 16) we begin by recalling that the in the case of p(z), invariance does follow quite directly full modular group r( 1) is the group of invariance from the structure of the formula ( 18). for J(r). Furthermore, cr + d occurs as the lower row To see this, recall that invariance under all of r( 1) will of a transformation in r( 1) precisely when c and d result from the single transformation formula J ( -1 I r) = are relatively prime integers. With a simple change of J(r). We note from (18) that to prove this it will suffice notation the double sum in ( 15) becomes to demonstrate the invariance under r ---+ -1 I r of the ( 17) finite sum L L {exp (-2ni~; + ~)- exp(-2nialc)}, LK(r) = L L exp(-2niVc,dr). l

388 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Rademacher, Poincare Series and Eichler Cohomology

However, At this point Rademacher divides the multiple sum in (21) into the two parts

t lim L e-2~rim' /l { 2ni } N--+oo f(fr- m) f=l lm(:";N = L L exp( -2niVd,-cr), (m.f)=l lci~K ldi:";K (c.d)~l + ~ lim "" e-2~rim'/f ~ _!._ { 2ni }P' L... N--+oo L... L... p! f(fr- m) f=l lmi:";N p=2 (m.l)=l stnce· (ac b)d ( 01 -I)0 = (d b -a)-c = vr d,-c· Now we infer l:K(-1/r) = EK(r) by matching the a step justified on the grounds that the second is abso­ pair (c, d) with the pair (d, -c). lutely convergent as a triple sum and first is convergent by virtue of its appearance as the left-hand side in 3. Sketch of the proof of Theorem 1. Rademacher's Lemma. [31, p. 238, (2.1)]. Suppose We present a very brief account of Rademacher's deriva­ r E 'H. Then, tion of the expression ( 15) from the definition of J ( r) given by (3) and (12). At the heart of the method is a (22) justifying rearrangement of a difficult technical lemma t lim L { exp( -2nim' /f)} certain conditionally convergent double series. N--+oo f(fr- m) f=l lmi:";N Rademacher begins by inserting the expressions ( 12) (m.l)=l and ( 13) into ( 3), and then inverting the order of K summation in the double sum so obtained. The validity = K--+ooL...lim "" ""L... { }, of this step relies crucially upon the estimate of Weil i=l (m(:";K [36], (m.f)=l Ae(n) = O(f 112H),E 0, > with m' defined as in the statement of Theorem 1. uniformly inn. (Actually, any nontrivial estimate At(n) = (Convergence of the right-hand side of (22) implies O(f 1- 0 ),o > 0, would suffice for the purpose.) The inter­ directly the convergence of ( 18), the (modified) parabolic change of summations implies that 123J(r), as defined Poincare series of weight k = 0.) by (3) and (12), equals Applying the Lemma to the first sum and absolute ( 19) convergence to the second, he obtains e-2~rir+ 744 + 2n t ~ L e-2~rih' /f (23) f=l h(modf) K 3 J(r) =e-21rir 732 lim "" "" 12 + + K-+ooL... L... ~ . 1 (4ny'n) f=l lmi:";K xL...exp[2nm(r-h/f)]v'n/1 -f- . (m.f}=l n=l e-2~rim' /l ~ _!._ { 2n i }P the power series (14), performs L... p! f(fr- m) Next, he replaces / 1 by p=l the salient another interchange of summations and-in K feature of his calculation-makes use of the Lipschitz =e-21rir + 732 + lim L L sum formula [ 12, p. 65], K--+oo f=l lmi:";K (20) (m.f}=l 00 l:nPe2~rinr e-2~rim' f£ {exp ( 2ni ) - 1}. f(fr- m) n=l I I 1' '\"N ( · · )-1 -2 21r lmN--+oo Liq=-N -l'f lq , Finally, (15) results from (23) upon separation of the = { + + _E_ '\"00 ( . + . )-p-1 single term for m = 0 (with f = 1), and application of ' p E z+. (21r)P+I Wq=-00 -l'f lQ the simple identity This transforms ( 19) into '/ 1 -m'r -f' (21) -m f + f(fr-m) = fr-m e-21rir + 732 (m', f' defined as in the statement of Theorem 1) + ~ lim "" e-2~rim' /l ~ _!._ { 2n i }P L... N--+oo L... L... p! f(fr- m) and the invariance of the summand under the map f=l lm!:";N p=l (m.f)=l (f, m)-+ ( -f, -m).

APRIL 1990, VOLUME 37, NUMBER 4 389 Rademacher, Poincare Series and Eichler Cohomology

III. Connection with Eichler cohomology. This can be proved fork:::; 0 without invoking (25) ([12, 1. Generalization to modular forms of nonpositive weight. pp. 24-30] or [17, pp. 166-176]). A number of mathematicians have developed Rade­ 2. When k = 0, f.L = 1 and a_ 1 = 1, then (25) reduces macher's ideas further, extending them (i) to discrete to the series (12) for the coefficient of J(r). groups of real linear fractional transformations other 3. Both Lehner [19, 20] and Petersson [23] have than r(l ); (ii) to automorphic forms of weights k :::; 0. derived (25) in the broader setting in which a general (See §1.2, following (7), for the definition.) Here we H-group r [17, p. 266] replaces r(l). In this more general emphasize the generalization to forms of negative weight, case, the structure of the series for an remains unchanged as this leads directly to the Eichler cohomology theory. in its essentials. We confine our attention to the case of automorphic The Rademacher-Zuckerman formula (25) makes forms F on r( 1) (that is, modular forms) of weight available in the wider context of modular forms with k :::; 0, with k an even integer and with "multiplier weights k :::; 0 precisely the same viewpoint that Rade­ system" identically one. This means F satisfies (7), with macher adopts in [31] toward J(r). Specifically, one can k even and :::; 0, for all M E r( 1). (For a definition define the function F(r) by the series (24) and (25), and discussion of multiplier systems, see [12, pp. 12- and ask-as Rademacher did concerning J(r)-how (or, 13] or [17, pp. 267-268].) The definition of modular more tentatively, whether) it is possible to show from form requires, as well, that F be holomorphic in H and this definition that F(r) does in fact satisfy (7), the expressible there as an exponential series of the form characteristic functional equation for a modular form of weight k. This perspective, indeed, forms the basis for 00 much of the work I undertook in the early 1960's [8, 9, F(r) = L ane2ninr_ (24) 10, 11, 13, 14]. As it turns out, it is impossible to show n=-Ji that all functions F ( r) so defined are modular forms, not (Note that periodicity of F follows from (7), with because of a defect in the method, but rather because they are not all modular forms. We shall comment upon M= (6:).) this further in §111.2, below. Rademacher derived the exact formula (12) for the Rademacher's method yields: coefficients in the expansion of J ( r) by refining a method Theorem 2. [11, p. 28, (3.07)]. For r E H define the he and Zuckerman [32] had devised earlier to calculate function the coefficients an in the exponential series (24) for an 00 arbitrary modular form F of negative weight k ("positive Fv(r) = e-2nivr + (-l)kf2(2n) Lan(v)e2ninr, dimension"-k in the terminology of [32]). The formula n=l of Rademacher and Zuckerman for the an, in the special case when k is even (and the multiplier system is with v a positive integer and an(v) the infinite sum on identically one), is f occurring in (25). Put r = -k, a positive even integer. Then, Fv ( r) is holomorphic in 1i and it there has the !i 00 representation an= (-l)kf2(2n) I:a-v L (25) v=l f.=l Fv(r) = e-2nivr + O:v + r' { exp(2niv jr) r 1Ae,v(n)(v jn)(-k+l)/2Lk+t { ~n v'nl/}, n ~ 1, r 1 } - L 1 (2niv jr)1 where /=0 t. (26) K (28) + lim '"" '"" (lr- m)' Ae,v(n)= L exp[-~ni(nh+vh')],hh'=-l(modl), K -+oo L....i L....i h(modl} f=l l~imi~K (h.l)~l m'r+l') { exp ( -2niv fr _ m and

1 oo (x j2)2j-k+l . , / 1 ( 2niv ) } - exp ( -2nlm v f)~~ tT l(h _ m) , (27) Lk+t(X) = '1(.- k 1)1 . Lj=O ]. } + . where o:v is a constant depending only upon v and r. (See Remarks. 1. The expression (25) implies that a mod­ [11, p. 27], where o:v is denoted Cv.) ular form of weight k < 0 which is bounded at ioo (that The proof of (28) requires that a general­ is, a_ 1 = a_2 = · · · = a_!i = 0 in (24)) is identically zero. ization of the Rademacher Lemma in which

390 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Rademacher, Poincare Series and Eichler Cohomology

,et+r(.er- m) (= .e 1-k(£r- m)) replaces the denomi­ parts condition" [23, Theorems 1 and 3] and his "gap nator £(£r- m) of (19). The analogy of (28) with (15) theorem" for automorphic forms [26, Theorem 1]. needs no clarification; when v = 1 and k = 0 (so r = 0), This state of affairs has made inevitable the frequent, (28) reduces to (15). Like the expression (15) for J(r), prominent appearance of functions having transforma­ (28) can be rewritten as a modified parabolic Poincare tion laws like (32) - that is, with additive polynomials - series of weight k < 0 on r (1 ), comparable to ( 18): observable in the recent (since (1957)) study of modular (29) and automorphic forms. These "Eichler integrals" - as Fv(r)- O!v = lim they are called - are functions F ( r) holomorphic in H K->oo L ""~ \c\~K ldi$K and satisfying {c.d)~l { (cr +d)' exp( -2n iv Vc,dr) - q( r; c, d)}, (33) (yr + J)-k F(Mr) = F(r) + PM(r), where q( r; c, d) is the polynomial of degree r = -k given for all M = (;;5) E r, a discrete group of real linear by fractional transformations. Here, k is an even integer (30) ::; 0, called the weight of the integral F(r), and PM(r) is e-2rriavfc I:' l ( 2rriv) I ( C! + dy-t =j= 0 a polynomial in r of degree at most -k. The PM(r) are 1=0 1! c ' c , q(r;c,d) = { of F(r). Combining (32) and the 0, c = 0. the period polynomials fact that Fv(r + 1) = Fv(r), we find that the functions integrals on r(1), since r--+ r + 1 and The "convergence terms" q( r; c, d) here replace the s(c, d) Fv(r) are Eichler r--+ -1/r generate the group. Naturally, if PM(r) = 0 for appearing in (18); when v = 1 and k = 0, q(r;c,d) MEr (33), then the Eichler integral F is really an reduces to s(c, d). all in automorphic form on r. importance in establishing a further non­ 2. The Eichler cohomology theorem. Of crucial trivial link between Eichler integrals and automorphic Since Fv(r + 1) = Fv(r) follows directly from the defini­ differentiation formula of G. Bol [2], tion of Fv (without regard to the nature of the an(v)), forms is the (34) showing that Fv(r) is a modular form of weight k on D(-k+tl{(yr + J)-k F(Mr)} = (yr + J)k-2F(-k+tl(Mr), r (1) reduces to proving the single transformation for­ mula where M = C:), with aJ- fly= 1. Clearly, (34) implies ( 31) that the ( -k + 1) 51 derivative of an Eichler integral of weight k on r is an automorphic form on r of weight However, the presence of the subtracted polynomials 2- k. (34) follows for differentiable F by induction on q( r; c, d) in (29) gives rise instead to a transformation -k and for analytic F by the Cauchy integral formula. formula of a more general kind, namely, An immediate consequence of (33) is the (cocycle) consistency condition

with Pv(r) a polynomial "period" of degree::; r = -k. In contrast to the case of J( r)(k = 0 and v = 1), when where for convenience we have introduced the slash k < 0 these polynomial periods do not always disappear. operator In fact, the identical vanishing of Pv(r) depends upon (¢1M)(r) = (yr + J)-~¢(Mr), the parameters k and v, and in the generic situation forM= ( ~;). When ¢ is a polynomial of degree ::; -k, Pv ( r) does not vanish. This, notwithstanding the fact that Pv(r):::: 0 for allv E z+ when k = 0, -2, -4, -6,-8 so is ¢1M. A collection of polynomials {PMIM E r} and -12. Conspicuously, k = 10 does not belong on satisfying (35) - thus, necessarily of degree ::; -k - forms this list. (Verification of these facts is left aside.) Thus, a (weight -k) cocycle on r. Given a fixed polynomial p while the exact formula (25) of Rademacher-Zuckerman of degree ::; -k it generates the cocycle {PMIM E r} by shows that every modular form of weight k < 0 is a means of PM= PIM-p. We call a cocycle of this special linear combination of the functions Fv(r), the converse form a coboundary and define the Eichler cohomology is not true (at least fork = -10 and even k ::; -14). Of group H~k(r) as the quotient vector space of weight -k course, with fixed k one can form a linear combination cocycles modulo weight -k coboundaries. of sufficiently many Fv(r) to force the disappearance of The identity ( 34) suggests a direct relationship be­ the additive polynomial and thus obtain a modular form. tween H~k(r) and automorphic forms of weight 2- k There are stringent conditions upon such linear combi­ on r, which we can establish as follows. Let G be such a nations, explained by Petersson through his "principal form and F a ( -k + 1)-fold anti-derivative of G. Then

APRIL 1990, VOLUME 37, NUMBER 4 391 Rademacher, Poincare Series and Eichler Cohomology by (34) F satisfies (33), with PM('r) a polynomial of IV. Concluding Remarks degree:::; -k for each MEr. This produces a mapping Although appearing eighteen years after [31 ], Eichler's into H~k(r), if we attach to G the cohomology class in work did not find its motivation in Rademacher's ap­ H~k(r) of the cocycle {PMIM E r}. Indeed, Eichler's proach to J(r). This is clear both from the internal classic paper [3], which initiated the study of H~k (r), evidence (Eichler's article itself) and from the fact that the necessary link is established not in Rademacher's identifies a distinguished subspace of H~k (r) with a di­ work, but in the extensions of it to negative weight in rect sum of two spaces of automorphic forms of weight [8, 10, 11 ], published between 1960 and 1962. That 2- k on r: one can consider Eichler cohomology an outgrowth of Eichler Cohomology Theorem. For k E Z, k :::; 0, Rademacher's work on J(r) is an instance of hindsight, H~k(r) is isomorphic to the direct sum C+(r, 2- k) EB an example illustrating the familiar, yet striking, fact C0 (r, 2- k), provided r is an H-group [17, p. 266]. that developments which seem unrelated at first can turn Remarks 1. c+(r, 2- k) is the (finite-dimensional) out with time to be aspects of the same mathematical space of entire automorphic forms on r of weight 2 - k, phenomenon. those forms for which at each parabolic cusp of r I have not described all of the applications now in the exponential expansion has no terms with negative the literature of Rademacher's method. These include: exponents. C0(r, 2- k) is the subspace of cusp forms in (i) the use of the method to construct Poincare series c+(r,2-k), those entire forms such that each expansion of weight 2 (in which case convergence problems arise contains only terms with positive exponents. in the definition ( 16)) [21, 35]; (ii) application to the 2. For simplicity I have stated only a restricted form construction of automorphic forms of real (not neces­ of the version of Eichler's theorem given in [6]. However, sarily integral) nonpositive weights [22]. In [21] Lehner this form of the theorem exhibits the essence of the full restricted his attention to Poincare series of weight 2 result. Other versions include Eichler's original result [3] on r( 1), while Smart [35] carried out a generalization and [34, 4, 16, 18, 15]. to certain subgroups of finite index in r(1). The work The proof of the Eichler theorem given in [6] depends of Niebur [22], while significant principally for its ex­ strongly upon Theorem 2, extended to general H -groups tension of Rademacher's method to nonintegral weights, r. This generalization applies directly to establish a provdies new insights even for negative integral weight: strong connection between the cocycle {PM} arising from when the weight is an integer the results of [22] do not Fv and the cocycle {PM-} arising from F_v (the result of reduce to those of [ 10], but strengthen them instead. replacing v by -v in the Fourier series definition of F11 ): According to Paul Bateman, Rademacher tried with­ out success to extend his method of [31] to nonintegral (36) PM-(r) =PM(!). weights, in particular to the function 1/IJ(T), which has weight -1/2. We may therefore safely assume that, could he have known of it, Rademacher would have been most (See [10, (4.8)].) Then, for a linear mapping suitably interested in Niebur's work. In the spring of 1963, well defined from automorphic forms of weight 2 - k into before that work was begun, I had the opportunity to tell H~k(r), the relation (36) yields a proof that the mapping Professor Rademacher about by own generalization to is one-to-one [6, pp. 570-571 ]. (This mapping necessarily negative integral weights. As we walked alone, near the keeps c+ and C0 disjoint, even though C0 c c+.) The university campus in Madison, Wisconsin, I broached proof that the range of this mapping consists of the entire the subject and he siezed upon it with apparent interest. space H~k(r) requires Petersson's generalized Riemann­ But, within moments, something distracted us; to my Roch Theorem [27, Theorem 9]. later regret, we never returned to the subject. The Eichler cohomology theorem may be regarded as stating that every polynomial cocycle arises as the system of period polynomials of some Eichler integral, and that this Eichler integral is uniquely determined by the References cohomology class of the given cocycle. Like the Riemann­ [1]. T. Apostol, Modular functions and Dirichlet series Roch theorem (more properly, Petersson's generalization in number theory, Spring-Verlag, New York, 1976. of it), the Eichler cohomology theorem establishes a [2]. G. Bol, Invarianten linearer differentialgleichung­ profound connection- only hinted at by (34) -between en, Abh. Math. Sem. Univ. Hamburg 16 (1949), 1-28. automorphic forms of weight 2- k(k E Z, k :::; 0) and [3]. M. Eichler, Eine Verallgemeinerung der Abelschen those of weight k. It shows that each entire automorphic Integrale, Math. Zeitschrift 67 ( 1957), 267-298. form of weight 2- k gives rise to an "obstruction" to the [4]. R. Gunning, The Eichler cohomology groups and existence of forms of weight k, and that each cusp form automorphic forms, Trans. Amer. Math. Soc. 100 (1961), in fact gives rise to two such obstructions. 44-62.

392 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Rademacher, Poincare Series and Eichler Cohomology

[5]. J. Hadamard, The psychology of invention in the S.-B. Heidelberger Akad. Wiss. Math. - Nat. Kl. 1950, mathematical field, Dover, New York, 1954. 417-494. [6]. S. Husseini and M. Knopp, Eichler cohomology [24]. H. Petersson, Theorie der automorphen Formen and automorphic forms, Illinois J. Math. 15 ( 1971 ), beliebigen reeller Dimension und ihre Darstellung durch 565-577. eine neue Art Poincarescher Reihen, Math. Annalen 103 [7]. K. Knopp, Theory offunctions, part two, tr. by F. ( 1930), 369-436. Bagemihl, Dover, New York, 1947. [25]. H. Petersson, Uber die Entwicklungskoeffizien­ [8]. M. Knopp, Automorphic forms of nonnegative ten der automorphen Formen, Acta Math. 58 ( 1932), dimension and exponential sums, Michigan Math. J. 7 169-215. (1960), 257-287. [26]. H. Petersson, Uber eine Metrisierung der au­ [9]. M. Knopp, Construction of a class of modular tomorphen Formen und die Theorie der Poincareschen functions and forms I, II, Pacific J. Math. 11 ( 1961 ), Reihen, Math. Annalen 117 (1940), 453-537. 275-293 and 661-678. [27]. Zur analytischen Theorie der Grenzkreisgrup­ [ 10]. M. Knopp, Construction of automorphic forms pen, II, Math. Annalen 115 ( 1937 /38), 175-204. on H -groups and supplementary Fourier series, Trans. [28]. H. Poincare, Memoire sur les fonctions Fuchsi­ Amer. Math. Soc. 103 (1962), 168-188. Correction, ennes, Acta Math. 1 ( 1882), 193-294. Trans. Amer. Math. Soc. 106 (1963), 341-345. [29]. H. Poincare, Science and Method, tr. by F. [ 11 ]. M. Knopp, Fourier series of automorphic forms Maitland, Thomas Nelson and Sons, London, 1914. of nonnegative dimension, Illinois J. Math. 5 ( 1961 ), [30]. H. Rademacher, The Fourier coefficients of 18-42. the modular invariant J(r), Amer. J. Math. 60 (1938), [12]. M. Knopp, Modular functions in analytic number 501-512. theory, Markham Pub. Co., Chicago, 1970. [31]. H. Rademacher, The Fourier series and the [13]. M. Knopp, On abelian integrals of the second functional equation of the absolute modular invariant kind and modular functions, American J. Math. 84 J(r), Amer. J. Math. 61 (1939), 237-248. (1962), 615-628. [31a]. H. Rademacher, Correction, Amer. J. Math. 64 [14]. M. Knopp, On generalized abelian integrals of ( 1942), 456. the second kind and modular forms of dimension zero, [32]. H. Rademacher and H. Zuckerman, On the Amer. J. Math. 86 ( 1964), 430-440. Fourier coefficients of certain modular forms of positive [ 15]. M. Knopp, Some new results on the Eichler dimension, Annals of Math. 39 ( 1938), 433-462. cohomology of automorphic forms, Bull. Amer. Math. [33]. B. Schoeneberg, Elliptic modular functions, Soc. 80 (1974), 607-632. Springer-Verlag, New York, 1974. [ 16]. I. Kra, On cohomology of Kleinian groups, Ann. [34]. G. Shimura, Sur les integrales attachees aux of Math. (2) 89 ( 1969), 533-556. formes automorphes, J. Math. Soc. Japan 11 ( 1959), [17]. J. Lehner, Discontinuous groups and automor­ 291-311. phic functions, Math. Surveys, no. 8, American Math. [35]. J. R. Smart, On modular forms of dimension- Soc., Providence, R.I., 1964. 2, Trans. Amer. Math. Soc. 116 (1965), 86-107. [18]. J. Lehner, The Eichler cohomology of a Kleinian [36]. A. Weil, On some exponential sums, Proc. Nat. group, Math. Ann. 192 (1971), 125-143. Acad. Sci. U.S.A. 34 ( 1948), 204-207. [ 19]. J. Lehner, The Fourier coefficients of automor­ phic forms belonging to a class of horocyclic groups, Michigan Math. J. 4 (1957), 265-279. The series of Special Articles was created to provide a place for arti­ The Fourier coefficients of automor­ cles on mathematical subjects of interest to the general membership [20]. J. Lehner, of the Society. The Editorial Committee of the Notices is especially phic forms on horocyclic groups, II, Michigan Math. J. interested in the quality of exposition and intends to maintain the 6 (1959), 173-193. highest standards in order to assure that the Special Articles will [21]. J. Lehner, On modular forms of negative di­ be accessible to mathematicians in all fields. The articles must be mension, Michigan Math. J. 6 (1959), 71-88. interesting and mathematically sound. They are first refereed for ac­ Construction of automorphic forms curacy and (if approved) accepted or rejected on the basis of the [22]. D. Niebur, breadth of their appeal to the general mathematical public. and integrals, Trans. Amer. Math. Soc. 191 (1974), Items for this series are solicited and, if accepted, will be paid 373-385. for at the rate of $250 per page up to a maximum of $750. Manu­ [23]. H. Petersson, Konstruktion der Modulformen scripts to be considered for this series should be sent to Ronald L. und der zu gewissen Grenzkreisgruppen gehorigen auto­ Graham or Jeffrey C. Lagarias, Associate Editors for Special Arti­ von positiver reeller Dimension und cles, Notices of the American Mathematical Society, Post Office Box morphen Formen 6248, Providence, Rhode Island 02940. die vollsHindige Bestimmung ihrer Fourierkoeffizienten,

APRIL 1990, VOLUME 37, NUMBER 4 393 The National Science Foundation Budget Request for Fiscal Year 1991

and Independent Agencies. This committee assignment This article is the 18th in an annual series of reports makes for some particularly difficult choices, for the outlining the President's request to Congress for the NSF must compete for increases with such programs as NSF budget. Last year's report appeared in the April Veterans Medical Care and Assisted Housing. 1989 issue of Notices, page 359. Little of the requested increase for FY 1990 actually materialized. Across-the-board reductions sawed off a It's that time of the year again when the Executive couple of favorite perches from which NSF advocates Branch presents its budgetary wish list to Congress. have crowed about doubling the NSF budget. For one Included is the budget request for the National Science thing, no new awards in the much-publicized Science and Foundation (NSF) for fiscal year 1991, which begins in Technology Centers (STC) program will be made in FY October, 1990. Well-respected for funding solid research, 1990, though the NSF intends to use FY 1991 money the NSF holds a unique place among science agencies. to fund some new centers out of the current batch of But in the last few years, that special position has meant proposals. The reductions also meant a scaling back of that the NSF has been asked to take responsibility for an the increases intended to offset inflation on continuing increasing number of tasks with only modest budgetary grants, a measure that frustrates NSF Director Erich increases. Mathematics and science education, academic Bloch's efforts to increase average grant size. research, developing stronger ties between universities Originally, the President's requested increase for the and industry, academic facilities modernization-can a NSF's FY 1990 budget was about 14%, with a sizeable relatively small government agency do all this? 15.5% increase for the Division of Mathematical Sci­ Congress and the Administration both enthusiasti­ ences (DMS). The House and Senate compromised on a cally support the NSF but seem to be at odds over 12% increase, but Gramm-Rudman-Hollings automatic its role. The typical pattern of recent years has the reductions, as well as an across-the-board reduction to President requesting double-digit increases for the Foun­ finance the "war on drugs," whittled it down to 8.5%. dation, with healthy hikes for research. Then Congress, With its 5% increase, the DMS did better than the 3% under severe strain to reduce the deficit, shaves per­ hike for the Mathematical and Physical Sciences (MPS) centages off the increase and asserts its own priorities directorate, the larger organizational structure containing by shifting funds to NSF's education directorate. And the DMS. Other MPS divisions fared even worse: for in FY 1990, over the opposition of the NSF and the example, the Physics Division received a 1% increase, a Administration, Congress instructed the Foundation to substantial loss after inflation.* fund modernization of academic research facilities. Congress also instructed the NSF to spend $20 million on the research facilities modernization program, despite NSF protests that the program is a luxury in a time of The FY 1990 Budget tight budgets. Everyone agrees that university labs and Formulation of the NSF budget request begins with nego­ research facilities need reconstruction, modernization, tiations between the NSF and the Office of Management and expansion; everyone also agrees that $20 million and Budget, which tries to translate the Administration's won't make a dent in solving the problem, which would priorities into budget numbers for the departments and run into billions of dollars to solve. agencies of the federal government. The resulting budget request is sent to both houses of Congress, each of which *These percentages were calculated by omitting $75 million ear­ reworks the budget to its liking. The NSF budget is marked for reconstruction of the collapsed Green Bank telescope and overseen by the House and Senate Committees for Vet­ therefore may not agree with analogous percentages calculated from eran's Administration, Housing and Urban Development, the accompanying tables.

394 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF Budget Request for 1991

Table I. National Science Foundation (Millions of Dollars) 1987 1988 1989 1990 1991 Actual Change Actual Change Actual Change Plan Change Request (1) Mathematical Sciences Research Support $59.9 6.5% $63.8 3.4% $66.0 4.7% $69.1 6.7% $73.7 (2) Other Research Support (Note A) 1390.3 3.2% 1434.7 6.8% 1532.5 9.9% 1684.3 12.5% 1895.2 (3) Education, Foreign Currency Program (Note B) 99.6 40.2% 139.6 22.6% 171.1 19.4% 204.3 22.9% 251.0 ( 4) Program Development and Management ("Overhead") (Note C) 77.8 8.6% 84.5 8.0% 91.3 8.4% 99.0 12.1% 111.0 (5) Science and Technology Centers 25.0 8.0% 27.0 93.0% 52.1 (6) Totals $1627.6 5.8% $1722.6 9.5% $1885.9 10.5% $2083.6 14.4% $2383.0 (7) (1) as% of (1) and (2) 4.13% 4.26% 4.13% 3.94% 3.74% (8) (1) as% of (6) 3.68% 3.70% 3.50% 3.32% 3.09% Note A. Scientific research facilities (excluding mathematics), including national and special research programs and certain national research centers. Note B. The programs provide support in every field, including mathematics. The foreign currency program was eliminated in FY 1989. Note C. Administrative expenses of operating the Foundation. These funds are not considered to directly support research. Also included is the cost of the Office of the Inspector General.

Table II. Directorate of Mathematical and Physical Sciences (Millions of Dollars) Division 1987 Actual 1988 Actual 1989 Actual 1990 Plan 1991 Request Mathematical Sciences $59.9 (12.9%) $63.8 (13.5%) $66.0 (13.1%) $69.1 (12.4%) $73.7 (11.9%) Astronomical Sciences 85.1 (18.3%) 85.8 (18.2%) 89.2 (17.7%) 91.7 (16.5%) 100.7 (16.2%) Physics 117.0 (25.2%) 117.9 (25.0%) 122.3 (24.3%) 123.8 (22.2%) 130.2 (21.0%) Chemistry 93.8 (20.2%) 94.0 (19.9%) 96.8 (19.2%) 98.0 (17.6%) 103.4 (16.6%) Materials Research 108.9 (23.4%) 110.6 (23.4%) 115.1 (22.9%) 116.9 (21.0%) 123.5 (19.9%) Science and Technology Centers 14.0 (2.8%) 14.5 (2.6%) 27.0 (4.3%) Major Research Equipment (See Note) 0.5 (0.0%) 43.0 (7.7%) 63.0 (10.1%) Totals $464.7 $472.0 $503.8 $557.0 $621.5 Note. This new budgetary heading includes a number of large-scale instrumentation projects, such as a gravitational wave observatory. A special Congressional appropriation of $75 million was made last year to reconstruct the Green Bank telescope, which collapsed in 1988. Half of the $75 million was carried over to FY 1990 from thr FY 1989 budget and is therefore excluded from these figures.

Table Ill. Compilation of the NSF Budget, 1986-1991 (Millions of Dollars) 1986 1987 1988 1989 1990 1991 Increase Increase Actual Actual Actual Actual Plan Request 1986-1989 1986-1991 (1) Mathematical Sciences Research Support $51.9 $59.9 $63.8 $66.0 $69.1 $73.7 27.2% 42.0% 1982 dollars* 51.8 58.3 59.7 58.8 13.5% (2) Other Research Support 1283.8 1390.3 1434.7 1532.5 1684.3 1895.2 19.4% 47.6% 1982 dollars 1281.2 1352.4 1342.2 1365.9 6.6% (3) Education, Foreign Currency Program 85.7 99.6 139.6 171.1 204.3 251.0 99.7% 192.9% 1982 dollars 85.5 97.0 130.6 152.5 78.4% (4) Program Development and Management ("Overhead") 71.8 77.8 84.5 91.3 99.0 111.0 27.2% 54.6% 1982 dollars 71.7 75.7 79.1 81.4 13.5% (5) Science and Technology Centers 25.0 27.0 52.1 1982 dollars 22.3 (6) Totals 1493.2 1627.6 1722.6 1885.9 2083.6 2383.0 26.3% 59.6% 1982 dollars 1490.2 1583.3 1611.5 1680.9 12.8% *Current dollars are converted to 1982 dollars using the wholesale/producer index.

APRIL 1990, VOLUME 37, NUMBER 4 395 NSF Budget Request for 1991

R&D Increases The NSF received about 150 proposals for the second In the federal government research and development round of STCs, to be funded during FY 1990. The picture as a whole, the NSF is small-fry: its projected Foundation's decision to delay funding to FY 1991 came budget of $2.38 billion is only about 3.5% of the as a result of the double blow of a smaller increase than amount requested for R&D overall. "Big science" projects requested, followed by the across-the-board spending feature prominently in the federal R&D budget, and reductions. Realistically, though, the funding won't be some rival the NSF budget in size. For example, the much delayed. The interdisciplinary panel reviews and Administration has requested increases for the Strategic site visits originally set the award date in August 1990, Defense Initiative ($4.66 billion, up 22%) and the space and it's been pushed forward to October 1990, the station ($2.63 billion, up 36%). The basic research beginning of FY 1991. category of the federal budget is to rise by 8%, compared Ann Boyle, a program director for Algebra and to a requested 12% increase for NSF's research activities. Number Theory and DMS coordinator for the STC Among the Foundation's major areas of increase for program, says she's received inquiries from investigators FY 1991 is the Computer and Information Science and concerned about the future of the program. Uncertainty Engineering directorate, set to rise by $23.6 million over whether Director Bloch, a strong proponent of (13.8%). Most of the increase will go toward the NSF the STC program, will be reappointed in August has Supercomputing Centers and networking activities, both increased worries over the fate of the program. However, of which support research in all areas of science, math­ Boyle says the NSF would lose a lot of credibility if, ematics, and engineering. The Geosciences directorate after all the effort put into the current batch of proposals, will receive an increase of 18.1 %, mostly as part of the it made no new STC awards. In addition, because this government-wide global change initiative. is the second STC competition, she notes, there was a The FY 1991 budget for the DMS is set to increase greater proportion of strong proposals, since proposers by 6. 7% to $73.7 million, a significant scaling-back from now have a better idea of what the program is all about. the request for FY 1990, when the NSF asked for Boyle says there were about eight mathematics pro­ $76.1 million for mathematics. The lower request level posals in the current competition, but, given the inter­ is probably in response to tough budgetary times for the disciplinary nature of the program, other proposals may government as a whole. Indeed, other NSF divisions saw have mathematics components. Thirty site visits to the a similar belt-tightening in the FY 1991 request levels. top proposing institutions are currently under way. In A portion of the increase requested for the DMS this round, the NSF plans to add eight to ten new 'STCs. for 1991 will support a number of Foundation-wide activities, such as the new Faculty Awards for Women program, designed to provide multi-year support for outstanding senior women scientists. The Special Projects Office in the DMS will handle curriculum development in undergraduate mathematics, extending reform projects Educational Activities to courses other than calculus. The DMS will continue For FY 1991, the NSF itself has come out with a hefty efforts to expand interactions between mathematics and proposed increase of 23% for Science and Engineering other fields of science, especially biology, and expects to Education, a perennial favorite for Congressional bud­ continue the "regional geometry institutes." The DMS getary boosts. The largest increase, a whopping 4 7.1 %, also plans to set aside about $1 million, distributed would go to the division of Undergraduate Science, among the different disciplinary programs, to provide Engineering, and Mathematics Education (USEME). If grants to young investigators. the request becomes reality, the Course and Curriculum program, which oversees the calculus curriculum reform projects, would double its current budget of $4 million. Science and Technology Centers Part of the new funds are intended to allow USEME The STC program got off the ground in late 1988, when to participate financially in the curriculum development the Foundation funded eleven centers. Mathematics re­ projects to be developed in the research divisions. search is supported on a number of these grants (see Under the FY 1991 request, USEME's Instrumenta­ Notices, February 1989, page 154). The STCs are de­ tion and Laboratory Improvement (ILl) program would signed to incorporate research, education, and technology see an increase of nearly 50%. The increase would come transfer to address scientific and engineering problems mostly as a result of consolidating the program within requiring an interdisciplinary approach. Some in the USEME; right now, ILl grants to doctoral-granting insti­ scientific community have expressed apprehension that tutions are handled through the research divisions, rather the centers may eventually drain funds for individual than through USEME. In addition, plans are being de­ researchers. veloped to allow more comprehensive ILl proposals,

396 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF Budget Request for 1991 including personnel and materials development, rather Much of the current excitement in mathematics stems than only equipment. from the interactions between various subdisciplines and between mathematics and other areas of science and Allyn Jackson engineering: Staff Writer • The Geometry Supercomputer Project based at the University of Minnesota involves researchers from across the country and Europe. Its Minimal Surface The following text was prepared by the staff of the Division of Math­ Team has developed a "Surface Evolver" program that ematical Sciences of the NSF and was submitted to Congress as models geometrically the evolution of surfaces driven part of the Administration's Budget Request for the Fiscal Year by various forces such as surface tension, crystalline 1991. surface energy, gravity and pressure. The program will be an aid in developing and testing conjectures about surfaces. It is available to the public and will run on several different machines. Mathematical Sciences • How closely may equal-sized spheres be packed in Summary of Request space? This problem, and its analogues in more than The FY 1991 Request for the Mathematical Sciences three dimensions, is very important in communica­ Subactivity is $73.72 million, an increase of $4.6 million, tions theory where the packing solutions are used to or 6. 7 percent, over the FY 1990 Current Plan of $69.12 determine efficient error-correcting codes. Recently million. Dr. Noam Elkies of Harvard University used seem­ ingly unrelated concepts from number theory to find sphere packings, in higher dimensions, that are denser (Millions of Dollars} than any previously known. FY 1990 The above examples reflect the impact of mathe­ FY 1989 Current FY 1991 Program Element Actual Plan Request matics research on fields as diverse as materials science Classical Analysis $5.81 $5.97 $6.22 and information technology. They are typical of recent Modern Analysis 5.78 5.96 6.21 developments throughout mathematics and exhibit two Geometric Analysis 5.89 6.31 6.68 related phenomena: Topology & Foundations 7.04 7.24 7.55 • Many of the most fundamental advances are being Algebra & Number Theory 9.85 10.12 10.56 Applied Mathematics 8.13 8.30 8.70 made by people with deep understanding of a wide Statistics & Probability 7.26 7.44 7.81 range of mathematical and scientific topics. Computational Mathematics 4.74 4.91 5.22 • Application and theoretical development are operat­ 14.77 Special Projects 11.52 12.87 ing in mutually enriching ways, with the computer's Total, Subactivity $66.02 $69.12 $73.72 extension of the reach and power of mathematics a driving factor. These observations have profound implications for Scientific Overview the training of graduate students and for the continuing The Mathematical Sciences Subactivity fosters the cre­ development of established researchers. ation and development of mathematical ideas, methods, The Mathematical Sciences Subactivity also supports, and techniques and promotes their use in improving through its Office of Special Projects, various efforts our understanding of physical, biological, engineering, that cut across the mathematical sciences, including: and social phenomena. Support for significant research research institutes and centers; postdoctoral research and related infrastructure activities ensures the continu­ fellowships; research conferences, workshops and special ing vitality and long-range health of the discipline. All years; shared scientific computing research equipment; subdisciplines of the mathematical sciences, from those and undergraduate programs managed in collaboration with the sharpest intrinsic focus to those that reach out with the Science and Engineering Education Activity. to other areas of knowledge, are supported to encourage During the last few years, support for the mathe­ interaction and provide a healthy balance among them. matical sciences has emphasized improving the pool of The core of the Foundation's program in the math­ talent entering the field. Significant gains have been made ematical sciences is the support of standard research through added support for graduate students, postdoc­ projects. Collaborative research efforts, inclusion of grad­ toral researchers and undergraduate faculty and students. uate students and postdoctoral researchers, and develop­ Support for undergraduate curriculum development in ment and use of computational techniques are encour­ calculus was initiated in FY 1988 in cooperation with aged where appropriate. Science and Engineering Education.

APRIL 1990, VOLUME 37, NUMBER 4 397 NSF Budget Request for 1991

The chart below illustrates the crucial role played by in calculus. Emphasis is also placed on interactions NSF in the federal support of academic research in the with other disciplines, computational mathematics, ap­ mathematical sciences. In all areas of the mathemati­ propriate support for small research groups, and human cal sciences, Foundation-supported research involves a resources issues of importance to the field. broader range of basic research topics than the more project-oriented research sponsored by the mission agen­ cies. FY 1991 Budget Request NSF coordinates its support of research in the mathe­ matical sciences with its counterpart federal agencies {Millions of Dollars) FY 1990 through the Interagency Committee for Extramural Current FY 1991 Change Mathematics Programs (ICEMAP). This group meets Program Element Plan Request Percent regularly to share information on policies of support and Classical Analysis $5.97 $6.22 4.2% to discuss areas of emphasis and of unusual scientific Modern Analysis 5.96 6.21 4.2 opportunity. Geometric Analysis 6.31 6.68 5.9 Topology & Foundations 7.24 7.55 4.3 Algebra & Number Theory 10.12 10.56 4.3 Federal Academic Mathematics Funding Applied Mathematics 8.30 8.70 4.8 Statistics & Probability 7.44 7.81 5.0 Computational Mathematics 4.91 5.22 6.3 Special Projects 12.87 14.77 14.8 Total, Subactivity $69.12 $73.72 6.7%

The FY 1991 Budget Request of $73.72 million is $4.6 million (6. 7 percent) above the FY 1990 Current Plan. The main elements of this increase are expanded participation of women and minorities, enhanced un­ dergraduate programs, and strengthened support for standard research projects with emphasis on graduate students and equipment. Within Special Projects, the increase is directed to the following priorities: • Undergraduate course and curriculum development. Changes Between FY 1990 Request Efforts will continue in calculus and expand the focus and FY 1990 Current Plan to other introductory courses such as precalculus, discrete mathematics, geometry, and statistics. {Millions of Dollars) • Activities in conjunction with the Alliances for Mi­ FY 1990 FY 1990 Current Change nority Participation program. The focus will be on the Program Element Request Plan Percent critical transitions between high school and under­ Classical Analysis $6.47 $5.97 -7.7% graduate education and between undergraduate and Modern Analysis 6.46 5.96 -7.7 graduate education, with an emphasis on mentoring. Geometric Analysis 7.16 6.31 -11.9 • NSF Faculty Awards for Women. New awards will Topology & Foundations 7.87 7.24 -8.0 Algebra & Number Theory 11.15 10.12 -9.2 recognize and enhance the research of the increas­ Applied Mathematics 9.21 8.30 -9.9 ing numbers of mid-career women with outstanding Statistics & Probability 8.21 7.44 -9.4 research credentials in the mathematical sciences. Computational Mathematics 5.85 4.91 -16.1 Across other programs: Special Projects 13.67 12.87 -5.9 • Improving the accessibility of Foundation programs Total, Subactivity $76.05 $69.12 -9.1% to new investigators is the focus of a $1.0 million in­ crease. Approximately twenty more new investigators The FY 1990 Current Plan is $69.12 million, a will be supported than would otherwise be possible. decrease of $6.93 million (9.1 percent) from the FY • Increased support for graduate students and equip­ 1990 Budget Request due to Congressionally mandated ment is the emphasis of an increase of $1.4 million. reductions. This represents an increase of $3.10 million • Special attention will be given to strengthening re­ (4. 7 percent) over the FY 1989 level. search support in areas of opportunity such as interac­ Within the FY 1990 allocation, priority will be given tions with other disciplines, computational mathemat­ to modern geometric mathematics and to undergrad­ ics, and incorporating geometric ideas and methods uate course and curriculum development, particularly throughout the mathematical sciences.

398 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF Budget Request for 1991

application. Researchers at Harvard, Carnegie-Mellon The following text was prepared by the staff of the Division of Com­ and the University of California at Berkeley have in­ puter and Computation Research in the directorate of Computer and Information Science and Engineering at the NSF and was sub­ vented a method to create software that checks itself mitted to Congress as part of the Administration's Budget Request against bugs, worms, and viruses. Their techniques will for the Fiscal Year 1991. allow software designers to certify that their programs will be secure or correct. Both academic and industrial computer and com­ putation research are supported by several agencies of the Federal government. However, with the exception of Computer and Computation Research NSF, most Federal programs support mission-directed re­ Summary of Request search. As with industry, most of these Federal programs The FY 1991 Request for the Computer and Computa­ concentrate their support at a few academic institutions. tion Research (CCR) Subactivity is $24.28 million, an NSF provides support in all areas of computer and increase of $2.30 million or 10.5 percent over the FY computation research to a broad range of academic 1990 Current Plan of $21.98 million. research institutions. This strengthens the Nation's long­ term scientific and technology potential by directly in­ volving teaching faculty, graduate and undergraduate stu­ dent assistants in research, and by providing incentives interaction. About 50 percent of (Millions of Dollars) for university-industry FY 1990 academic research in software and 7 5 percent of research FY 1989 Current FY 1991 in computing theory is funded by this Subactivity. Close Program Element Actual Plan Request collaboration is maintained with other federal agencies Computer & Computation Theory $6.42 $6.42 $6.89 through the Office of Science and Technology Policy, Numeric & Symbolic Computation 2.83 3.33 3.99 formal interagency agreements and informal interaction Computer Systems Architecture 3.10 3.10 3.21 of program officers. Software Systems 6.08 5.87 6.13 Software Engineering 2.74 3.26 4.06 Total, Subactivity $21.17 $21.98 $24.28 Changes Between FY 1990 Request and FY 1990 Current Plan

(Millions of Dollars) Scientific Overview FY 1990 Computer and computation research discovers the laws FY 1990 Current Change which govern problem-solving procedures and develops Program Element Request Plan Percent computing systems which test and utilize the laws. Topics Computer & Computation Theory $6.71 $6.42 -4.3% for study include: Numeric & Symbolic Computation 2.95 3.33 12.9 Computer Systems Architecture 3.60 3.10 -13.9 • Strategies and algorithms for solving problems, meth­ Software Systems 6.87 5.87 -14.6 ods of representing and transforming information; Software Engineering 3.54 3.26 -8.2 • Programs and software systems for solving large Total, Subactivity $23.67 $21.98 -7.2% problems or controlling large systems; and • Machines for executing programs. Parallel computation is a basic theme for much of the research supported in the Subactivity. Promising new The FY 1990 Current Plan is $21.98 million, a de­ parallel computer architectures require new algorithms, crease of $1.70 million (7.2 percent) from the FY languages, tools and software systems to be effective. 1990 Request of $23.67 million, due to Congressionally­ To develop them, new research is required in theory, mandated reductions. The effects of these reductions are: problem solving, design and implementation. ( 1) fewer awards than planned in four programs, (2) Research on large software systems is of current na­ decreased emphasis on new computer architectures in tional importance. The Subactivity concentrates on fun­ order to emphasize software research for high perfor­ damental issues in this area, including formal methods mance computing, and (3) reduction in scope of new in software engineering, the systematic analysis of soft­ efforts in software engineering. The Numeric and Sym­ ware artifacts, and distributed tools and programming bolic Computation increase of $0.38 million results from environments for collaborative software development. the transfer of research on automatic theorem proving Fundamental research in formal methods and the the­ from Software Systems to facilitate coordination between ory of software engineering often has important practical closely related research topics.

APRIL 1990, VOLUME 37, NUMBER 4 399 NSF Budget Request for 1991

FY 1991 Budget Request (Millions of Dollars) FY 1990 (Millions of Dollars) FY 1989 Current FY 1991 FY 1990 Program Element Actual Plan Request Current FY 1991 Change Knowledge and Database Systems $7.68 $7.75 $9.12 Program Element Plan Request Percent Robotics and Machine Intelligence 6.04 6.24 6.55 7.3% Computer & Computation Theory $6.42 $6.89 Interactive Systems 2.66 2.66 3.05 3.33 19.8 Numeric & Symbolic Computation 3.99 Information Technology Computer Systems Architecture 3.10 3.21 3.5 and Organizations 2.76 3.33 4.16 Software Systems 5.87 6.13 4.4 Software Engineering 3.26 4.06 24.6 Total, Subactivity $19.14 $19.98 $22.88 Total, Subactivity $21.98 $24.28 10.5%

The FY 1991 Request increases emphasis on funda­ Overview mental software research throughout all program ele­ Scientific This Subactivity supports research to improve basic ments of the Subactivity. It focuses previous efforts in understanding and design of information systems that parallel and distributed systems and scientific comput­ ability to work and live in complex en­ ing on the problems of high performance computing by enhance our vironments. Research in this Subactivity focuses on increasing support in the Numeric and Symbolic Com­ how to provide the best computational structures and puting program element by $0.66 million. It adds support physical devices to facilitate the use of information. for basic research on methods of software creation by Research to improve understanding of economic and so­ increasing the Software Engineering program element by cietal consequences of advanced network technologies in $0.80 million. organizations and other distributed work environments In both public and private sectors, unreliable and is also supported. inefficient software continues to be a major source of risk Databases have become central elements of social to life, property and money. Expansion of fundamental and economic functioning, and key factors in scientific software research will provide methods for increasing research. Similarly, as ways were discovered to automate software reliability and performance, for testing and the transformation of knowledge about processes into validating critical software systems to reduce the risk physical work, robots and other automated machines of failure, and for improving computer, network, and have proliferated. Both phenomena have obvious ef­ information security and integrity. The potential payoff fects on industrial development and growth, and major from this research is exemplified by recent results which for national economic competitiveness. indicate that certain kinds of computer programs can consequences Past technology improvements provide only a hint be made practically immune to defects. Using a method usefulness of electronic information of constructing software called "k-testability," Richard of the potential processing in the future. Continued progress requires Lipton of Princeton University has demonstrated that improvements in both theories and techniques for ma­ the probability of software failure can be reduced greatly nipulating and using information as well as exploiting the by slowing slightly the speed or performance of the power of new computing machinery. Major opportunities software. Such tradeoffs may be desirable for high risk the utilization of automated systems computer applications. exist to improve and shared knowledge bases, and to exploit computers as Finally, new research will be started in software for use intelligent agents in performing complex tasks. of application-specific computing systems, jointly with Progress requires fundamental research and engi­ the Microelectronics Information Processing Systems representing knowledge in Subactivity. neering experimentation on machines, and automating complex processes of sens­ learning, reasoning The following text was prepared by the staff of the directorate of ing and understanding, inference, Science and Engineering Education at the NSF and was submitted and problem solving. Equal attention is needed to im­ to Congress as part of the Administration's Budget Request for the prove the interaction between people and computers, Fiscal Year 1991. by understanding the cognitive requirements of human information processing, and by improving the modes of Information, Robotics, and interaction, such as graphic display, multi-media input, Intelligent Systems and high-level mechanisms for human-machine dialogue. Summary of Request Fundamental research is also needed on problems re­ The FY 1991 Request for the Information, Robotics, and lated to advanced scientific communications network Intelligent Systems (IRIS) Subactivity is $22.88 million, applications, such as computer mediated communica­ an increase of $2.90 million, or 14.5 percent, over the tions, digital journals, collaborative work systems, and FY 1990 Current Plan of $19.98 million. retrieval of information from distributed databases.

400 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF Budget Request for 1991

This subactivity is a primary source of Federal sup­ FY 1991 Budget Request port for basic interdisciplinary research in computing, neuropsychology, and the behavioral and social sciences (Millions of Dollars) dealing with both theory and experimental applications FY 1990 Current FY 1991 Change of information processing technology. It provides about Program Element Plan Request Percent 50 percent of the total Federal support for these ar­ Knowledge and Database Systems $7.75 $9.12 17.7% eas. Other federal agencies, particularly Department of Robotics and Machine Intelligence 6.24 6.55 4.9 Defense (DOD), provide the balance, concentrating on Interactive Systems 2.66 3.05 14.6 sharply defined and mission-oriented applications. Information Technology Industrial laboratories play an important role as and Organizations 3.33 4.16 25.0 consumers of basic university research, but they tend to Total, Subactivity $19.98 $22.88 14.5% focus on short term goals leading to marketable products rather than on fundamental knowledge improvement. The FY 1991 Request will maintain base support in Significant ambitious national and international efforts all program elements to take advantage of computing in these areas are under way in Japan and in Europe. technologies such as new sensors, parallel computers and Although research in this field is relatively new, high bandwidth communication networks. These tech­ significant progress has been made. Of particular impor­ nologies, unavailable just a few years ago, are rapidly tance are advances in the areas of computer vision, and changing the way information from the physical envi­ sensor-based robotic systems. For example, a project at ronment is collected, stored, and used as knowledge for Columbia University has created a robot hand which personal and organizational activities. combines tactile and visual sensors, providing and inte­ Most of the increment ($2.60 million) will be allocated grating information needed for handling soft materials to enhance research in two priority areas located primar­ and accomplishing complex tasks. The experimental sys­ ily in two program elements: Knowledge and Database tem advances the state of the art for automating complex Systems, and Information Technology and Organiza­ and delicate tasks such as final product assembly and tions. computer assisted surgery. Both of these activities will support work at the intersection of computer science, artificial intelligence, and a variety of scientific and engineering disciplines. Changes Between FY 1990 Request Multi-disciplinary teams will be supported to encourage and FY 1990 Current Plan cross-fertilization and to facilitate knowledge transfer between academic and industrial research communities. (Millions of Dollars) • Scientific Databases ($1. 75 million): This increment FY 1990 FY 1990 Current Change in the Knowledge and Database Systems and Inter­ Program Element Request Plan Percent active Systems programs is to initiate research to Knowledge and Database Systems $8.21 $7.75 -5.8% advance the database technology essential for im­ Robotics and Machine Intelligence 6.77 6.24 -7.8 proving productivity and enabling new forms of data Interactive Systems 3.39 2.66 -21.5 analysis and knowledge processing in scientific and Information Technology engineering fields. The goal is to develop new method­ and Organizations 3.27 3.33 1.8 ologies and enterprises that will be sustained by user Total, Subactivity $21.64 $19.98 -7.7% disciplines. Application areas in the biological and geological sciences will be targeted so that research results will be useful to initiatives involving global The FY 1990 Current Plan is $19.98 million, a de­ change and the human genome. crease of $1.67 million (7.7 percent) from the FY • Collaboration Technology ($0.85 million): The goal 1990 Request of $21.64 million, due to Congressionally­ of this research, to be funded primarily within the mandated reductions. This represents a $0.84 million Information Technology and Organizations program, increase over FY 1989. Priorities have been maintained is to improve methods of using information tech­ in coordination theory and technology within the In­ nologies to improve coordination, communication formation Technology and Organization Program and and resource management of human collaborative machine vision and sensing within the Robotics and activity. Results of this research will contribute to Machine Intelligence Program. In line with protecting new knowledge and improved tools for humans to these priorities, Interactive Systems is level funded. function more efficiently in the distributed workplace.

APRIL 1990, VOLUME 37, NUMBER 4 401 NSF Budget Request for 1991

The following text was prepared by the staff of the Division of Ad­ processes at levels of complexity and detail that inform vanced Scientific Computing in the directorate of Computer and our most advanced notions of reality. Information Science and Engineering at the NSF and was submit­ The range and diversity of research addressed is vast: ted to Congress as part of the Administration's Budget Request for the Fiscal Year 1991. from modelling the surfaces of viruses to simulating the dynamics of thunderstorms; from performing cal­ culations to reveal new features of nucleon masses; to building models of international economic activity. As a result, the centers serve scientists and engineers from all Advanced Scientific Computing the Foundation activities. Each year, more than 5,000 Summary of Request university faculty, postdoctoral, graduate and undergrad­ The FY 1991 Request for the Advanced Scientific Com­ uate students, and industrial researchers from across the puting (ASC) subactivity is $62.58 million, a decrease of nation use the Centers. These researchers come from all $0.25 million, or 0.4 percent, from the FY 1990 Current types of institutions and from every field. Plan of $62.83 million. The NSF Supercomputing program provides re­ searcher William Krakauer of the College of William (Millions of Dollars} and Mary ready access to the advanced high perfor­ FY 1990 mance computing technology required for his research FY 1989 Current FY 1991 on tungsten. In physical processes such as catalysis and Program Element Actual Plan Request corrosion, the surface of a material are where all the Centers $53.79 $60.21 $59.59 action takes place. Using the Cornell National Super­ New Technologies 1.79 2.62 2.99 computer Facility and proceeding from equations repre­ Total, Subactivity $55.58 $62.83 $62.58 senting atomic interactions, Dr. Krakauer has simulated the behavior of a small number (30) of tungsten atoms in an attempt to predict surface properties at a variety Scientific Overview of temperatures. Faster supercomputers in the future The Advanced Scientific Computing Subactivity provides will allow the simulation of larger numbers of atoms for the science and engineering basic research community more realistic sample sizes. access to advanced computational facilities through sup­ To meet the growing demand for access and speed, port of four nationally accessible Supercomputer Re­ the computing systems at the Centers were upgraded in search Centers: 1989. Major equipment was added for data storage and • The San Diego Supercomputer Center at the Univer­ scientific visualization, and new supercomputers were sity of California at San Diego; acquired. These upgrades resulted in a doubling of raw • The National Center for Supercomputing Applica­ processing capacity, and a very substantial improve­ tions at the University of Illinois; ment on the capability available to the U.S. research • The Pittsburgh Supercomputing Center; and community. • The Cornell National Supercomputer Facility at Cor­ In addition to providing access to supercomputers, nell University. the Centers provide a broad array of critical user services: The Subactivity also supports basic research on meth­ specialized graphics and visualization systems, applica­ ods and technologies for high performance scientific tions software libraries and consulting services, training computing. workshops, and related activities. Educational activities The Centers serve the U.S. academic research com­ range from special programs for high-school students and munity, support software development to maximize pro­ undergraduates, through established disciplinary research ductivity, and train new users. In addition, the Centers, programs at the postdoctoral level. working jointly with industry, are leaders in research The Centers also receive support from industrial on supercomputing systems including state-of-the-art su­ affiliates, and work with them on new ways to use percomputers and software, peripherals for data storage, high performance computing for increased productivity. retrieval and display, network connections, and special For example, ALCOA Laboratories' partnership with the purpose processors. Pittsburgh Supercomputing Center has resulted in the State-of-the-art high performance computers are es­ ability to test a new can design more quickly and at a sential to leadership in contemporary scientific and en­ cost of about $2,000 per test, rather than $100,000 per gineering research, as they permit researchers to explore test required for traditional physical prototyping. otherwise unmanageable problems. Operating at speeds NSF activities in this area are coordinated with of from ten to one hundred times faster than mainframe the more mission-oriented activities of other Federal computers, or up to billions of operations per second, agencies, most notably the Department of Energy, De­ supercomputers are used to analyze data and to simulate partment of Defense, the National Oceanic and Atmo-

402 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF Budget Request for 1991 spheric Administration, and the National Aeronautics element, with special attention to topics where the and Space Administration. This cooperation encourages research domains of computer science and disciplinary the exchange of skills, applications, software operations computing intersect. and management experience, and optimizes facility uti­ The Subactivity will support incorporation of new lization. highly-parallel supercomputers into operations at the na­ tional centers. Research related to mapping promising applications onto these new systems, and increasing sup­ Changes Between FY 1990 Request port for parallel computing on conventional production and FY 1990 Current Plan supercomputer systems will be accelerated.

(Millions of Dollars) Overall system performance and research productivity FY 1990 can be significantly increased by using more efficient FY 1990 Current Change software and algorithms. Along with expansion of appli­ Program Element Request Plan Percent cation libraries, upgrading dataset management systems, Centers $69.03 $60.21 -12.8% and improvement of visualization and graphics capabil­ New Technologies 2.66 2.62 -1.7 ities, the Subactivity will expand its support of research Total, Subactivity $71.69 $62.83 -12.4% on new computational algorithms and software develop­ ment. The Subactivity is funded at $62.83 million in FY Research will be supported to develop automated tools 1990, $8.86 million ( 12.4 percent) below the Request for improvement of user software and to provide users level. The reduction is taken in the Supercomputer with a more robust, friendly computing environment. Centers program element and reflects the lower costs due The interface to systems is central in this regard. Software to the phasing out of support for the John von Neumann is required to enable researchers to move seamlessly from Center.This nets out to a $6.42 million real increase over personal workstations, through local computing systems, FY 1989, for systems upgrades at the four remaining to the national centers, accessing applications software centers. The New Technologies research program element and tools in each environment. Implementation of a is maintained near the FY 1990 Request level, which standard operating environment in all centers systems is is an increase of $0.83 million, or 46.4 percent, over central to this goal. FY 1989, in order to bolster software research for high In order to enhance the scientific environment of the performance computing applications. centers, additional support will be offered for research associate programs, visitors programs, research workshop series, summer research institutes and extended programs FY 1991 Budget Request for select disciplinary topics. (Millions of Dollars) Cooperative efforts between the national centers in FY 1990 such areas as development of on-line tools and doc­ Current FY 1991 Change umentation, management of large databases, scientific Program Element Plan Request Percent visualization and software development will be initiated. Centers $60.21 $59.59 -1.0% Emphasis will be placed on development of generic tools New Technologies 2.62 2.99 14.3 for use at the national centers with distribution to state Total, Subactivity $62.83 $62.58 -0.4% and local supercomputer centers. The centers have developed extensive education and The Request supports existing upgrade programs at the training programs, resources, and expertise and have four national centers at the cooperative agreement levels, put into place mechanisms for propagating these efforts. and enhances research activities in technologies support­ Education and training programs will span the entire user ing high performance computing. Although the decrease base, with special emphasis on introducing the methods for the Centers program element is $0.62 million, this and experience of scientific computing to undergraduate represents a net increase of $7.38 million over the FY students. The centers will continue to integrate use of the 1990 Current Plan for the four remaining centers, which supercomputer systems into undergraduate curricula of will support renewal and upgrades. Computing services many disciplines at both local and non-local institutions. and resources and educational activities will be extended Opportunities for training scientists and students at to a broader national constituency. A wider spectrum of institutions lacking formal affiliation with the centers computational science and engineering research activities will be made available, along with special programs for will be addressed through the New Technologies program researchers at minority institutions.

APRIL 1990, VOLUME 37, NUMBER 4 403 NSF Budget Request for 1991

The following text was prepared by the staff of the Division of Net­ Collaborative development of national and interna­ working and Communications Research and Infrastructure in the tional networking with other federal agencies results in directorate of Computer and Information Science and Engineering effective interaction accross the Nation. The NSFNET at the NSF and was submitted to Congress as part of the Adminis­ composed of tration's Budget Request for the Fiscal Year 1991. backbone is the centerpiece of the internet connected Federal agency networks, regional mid-level networks, and local wide area networks. Together, they form the prototype of a national research and education network. Networking and Communications Research and NSF is leading the Federal interagency implemen­ Infrastructure tation of the national research and education network. Summary of Request Cooperating closely with five other agencies to intercon­ The FY 1991 Request for the Networking and Communi­ nect their cross-country research networks, the NSFNET cations Research and Infrastructure (NCRI) Subactivity backbone has been upgraded and reconfigured to accom­ is $31.29 million, an increase of $9.10 million, or 41.0 modate substantial growth in usage. percent, over the FY 1990 Current Plan of $22.19 Although operational and pilot networks are the most million. visible parts of the Subactivity, progress depends on an aggressive program of research in communications sciences and engineering. Projects are supported at the (Millions of Dollars) FY 1990 forefront of technology and application. For example, re­ FY 1989 Current FY 1991 search performed by principal investigators Ezio Biglieri Program Element Actual Plan Request and Kung Yao at the University of California, Los An­ NSFNET $13.59 $15.96 $22.04 geles, has resulted in new architectures to implement Networking and advanced signal processing algorithms used in high­ Communications Research 4.02 6.23 9.25 speed data communication systems. The new machines Total, Subactivity $17.61 $22.19 $31.29 are fault tolerant, and a major advance for insuring reliable, secure computer networks. Research often benefits from substantial industrial Scientific Overview participation and is positioned to transfer directly The Networking and Communications Research and In­ into practice, thereby enhancing the Nation's indus­ frastructure Subactivity supports the development and trial strength. This is demonstrated in a joint initiative operation of the NSFNET computer network for re­ of NSF, DARPA and U.S. industry, which began in search and education, and fundamental scientific and 1989. It involves the creation and operation of national engineering research on communications theory and data "testbed" networks for advanced research on problems networks. The research and network service functions are associated with super-high speed data transmission. Each integrated, creating the mechanism for fast translation of testbed uses an advanced scientific research application research into practice and driving research toward new to stress and improve the underlying technology. A signif­ problems based on experience. icant feature of the project is very substantial industrial This provides a vehicle for testing research results participation and cost sharing. Industry partners include and a natural opportunity for university/ industry collab­ the nation's major communications common carriers oration. In addition to stimulating academic research in and computer vendors and their research laboratories. a highly leveraged way, this approach affords the use of They are working with several National Laboratories, new technology developed and deployed by the telecom­ NSF Supercomputer Centers, and university researchers munications industry, such as high bandwidth fiber optic in a truly national program. communications lines, at low marginal cost. NSFNET, a network of computer networks, or "inter­ Changes Between FY 1990 Request net", was created in 1986 to provide access to the NSF and FY 1990 Current Plan Supercomputer Centers for the nation's computational science community. It has grown into a transcontinental (Millions of Dollars) backbone network interconnecting more than a dozen FY 1990 autonomous regional networks, which in turn connect FY 1990 Current Change nearly 300 academic, industrial and governmental re­ Program Element Request Plan Percent search institutions. Through this network, researchers NSFNET $19.63 $15.96 -18.7% can link to each other for research collaboration, and Networking and 7.50 6.23 -16.9 to unique resources such as advanced supercomputers, Communications Research radio telescope arrays and biotechnology databases. Total, Subactivity $27.13 $22.19 -18.2%

404 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF Budget Request for 1991

The FY 1990 Current Plan is $22.19 million, a Science and Engineering Education decrease of $4.94 million ( 18.2 percent) from the request Summary of Request level of$27.13 million, due to Congressionally-mandated The FY 1991 Request for the Science and Engineering reductions. The decrease is spread over all programs with Education (SEE) Activity is $251.00 million, an increase priority given to assistance to regional networks of the of $46.73 million, or 22.9 percent, over the FY 1990 NSFNET and to primarily undergraduate institutions. Current Plan of $204.27 million. The FY 1990 Current Plan also represents an increase of $4.58 million over the FY 1989 level, which will sustain (Millions of Dollars) the planned NSFNET activity. FY 1990 FY 1989 Current FY 1991 Subactivity Actual Plan Request Teacher Preparation FY 1991 Budget Request and Enhancement $63.66 $81.00 $89.60 Materials Development, Research, (Millions of Dollars) and Informal Science Education 43.99 48.00 61.50 FY 1990 Undergraduate Science, Engineering, Current FY 1991 Change and Mathematics Education 28.00 34.00 50.00 Program Element Plan Request Percent Research Career Development 30.98 36.90 44.90 NSFNET $15.96 $22.04 38.1% Studies and Program Assessment 4.50 4.37 5.00 Networking and Total, Activity $171.13 $204.27 $251.00 Communications Research 6.23 9.25 48.5 Total, Subactivity $22.19 $31.29 41.0% NSF Role The responsibility of the Science and Engineering Ed­ The FY 1991 Request continues the capacity expan­ ucation (SEE) Activity is to define and fund programs sion of the NSFNET backbone network, primarily for and projects that support the educational aspects of the increasing the transmission speed from 1.5 million bits Foundation's mission. SEE also plays a major role in per second (Tl) to 45 million bits per second (T3). developing human resources for science and engineering. Additional funding will be provided to regional networks The magnitude of the educational effort in the United to assist their capacity expansion in cases where scientific States and the long lead times needed for new programs, need is demonstrated. User services will be expanded materials, and methods require a continuing and signif­ and more closely coordinated with those offered by other icant involvement in this area. Sustaining this level of agencies supporting research networks. This will initiate commitment, visibility, and continuity is responsive to service integration, and create a single point of contact the national concern with science, mathematics, and en­ and assistance for Internet users. gineering education and will draw the best, most creative The requested increase for the Networking and Com­ people into the process. munications research program element will be used for SEE fulfills this responsibility by conducting leader­ three purposes: ship activities that inform and stimulate other sectors, • The number of awards for disciplinary fundamental and by supporting original work and other merit-based, research will be increased; high-leverage activities that serve as prototypes and mod­ • Funding is increased for the network technology els of excellence for the Nation. A major objective is component of program in coordination theory and to encourage appropriate cooperation among academic collaboration technology funded jointly with two scientists, engineers, educators and the private sector for other Subactivities in CISE; and intellectual partnerships as well as for leveraging funds. • The coordinated interagency program of research and The National Science Foundation is active at all levels technology demonstration in networks operating at of education-precollege, undergraduate, graduate, and speeds in excess of a billion bits per second will postgraduate. SEE serves as the focal point for education continue into its third year. at NSF. In this role SEE manages the programs for which it has sole budgetary responsibility, and serves in a coordinating and facilitating capacity for other administrative units throughout the Foundation. The following text was prepared by the staff of the directorate of The role of SEE is to help insure that: Science and Engineering Education at the NSF and was submitted • a high-quality precollege education in science and to Congress as part of the Administration's Budget Request for the Fiscal Year 1991. mathematics is available to enable those who are interested and talented to pursue technical careers, especially in science and engineering, as well as to

APRIL 1990, VOLUME 37, NUMBER 4 405 NSF Budget Request for 1991

provide a base for understanding by all citizens; Changes Between FY 1990 Request • those who select scientific and engineering careers and FY 1990 Current Plan have available the best possible professional educa­ tion in their disciplines. While other NSF programs (Millions of Dollars) provide apprenticeship training for the next gener­ FY 1990 FY 1990 Current Change ation of scientists and engineers through research Subactivity Request Plan Percent support, SEE's efforts (in addition to graduate and Teacher Preparation postdoctoral fellowships) concentrate on the educa­ and Enhancement $68.50 $81.00 18.2% tional structure leading to this stage; that is, on Materials Development, Research, the teachers, students, laboratory, and classroom re­ and Informal Science Education 49.00 48.00 -2.0 Undergraduate Science, Engineering, sources that remain important through undergraduate and Mathematics Education 30.00 34.00 13.3 and graduate studies; Research Career Development 38.00 36.90 -2.9 • opportunities are available at the college level for Studies and Program Assessment 4.50 4.37 -2.9 interested nonspecialists to broaden their science, Total, Activity $190.00 $204.27 7.5% mathematics, and technical perspective and under­ standing to familiarize them with the principles, The FY 1990 Current Plan is $14.27 million more than practices, techniques, and limits of science; the FY 1990 Request of $190.00 million, and $33.14 • informal science education programs are available to million more than FY 1989, a 19.4 percent increase. The maintain public awareness of and interest in scientific distribution of the increase, as specified by Congress, is and technological developments affecting their lives; to the pre-college Teacher Preparation and Enhancement and Subactivity and to the Career Access Program of the • the educational pipelines leading to careers in science, Undergraduate Science, Engineering, and Mathematics engineering and mathematics yield sufficient numbers Education Subactivity. All Subactivities are then reduced of well-educated individuals to meet the needs of the by the Gramm-Rudman-Hollings sequestration and other U.S. technical workforce. Increasing the participation mandated across-the-board reductions to arrive at the in such careers of minorities, women, and the disabled Current Plan. receives special attention at all educational levels. Across the Activity in FY 1990: Several SEE programs have specific requirements • precollege programs total $140.37 million ($21.22 for cost sharing and other forms of funds leveraging. million more than FY 1989, a 17.8 percent increase); Data for FY 1989 demonstrate that precollege teacher • undergraduate programs total $34.00 million ($6.00 enhancement and teacher preparation projects generate million more than FY 1989, a 21.4 percent increase); non-NSF funds equal to one-half of NSF funds invested, • graduate programs total $29.9 million ($5.92 million while projects in instructional materials development and more than FY 1989, a 24.7 percent increase). informal science education generate external funding equal to three times the amount of the NSF awards. At the undergraduate level, there is a 1: 1 matching requirement for equipment purchases. FY 1991 Budget Request SEE coordinates its activities closely with other rele­ (Millions of Dollars} vant areas of NSF and with other agencies, e.g. NASA, FY 1990 the Department of Energy and the Department of Educa­ Current FY 1991 Change tion (DoEd). Such coordination is intended to promote Subactivity Plan Request Percent maximum use of expertise in the agencies, to minimize Teacher Preparation program overlap, and to optimize the use of limited and Enhancement $81.00 $89.60 10.6% Materials Development, Research, Federal resources. Within the Foundation, the under­ and Informal Science Education 48.00 61.50 28.1 graduate instrumentation support is an example of an Undergraduate Science, Engineering, integrated management effort led by the SEE Activity and Mathematics Education 34.00 50.00 47.1 and involving all of the Research Directorates. Exter­ Reseach Career Development 36.90 44.90 21.7 nally, NSF and DoEd have collaborated in supporting Studies and Program Assessment 4.37 5.00 14.4 a number of projects, including the International Study Total, Activity $204.27 $251.00 22.9% of Mathematics, the International Science Study, Inter­ national Comparison of 13-year-olds, and a study of The Teacher Preparation and Enhancement Subactiv­ state-based science and math indicators at the precollege ity will continue to focus on improvement of precollege level. NSF, NASA, and the Department of Energy have science and mathematics education through effective jointly co-funded the National Academy of Science's preservice and inservice teacher training. In FY 1991, "Mathematical Sciences Education Board". emphasis will be placed on the Statewide Initiatives

406 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY NSF Budget Request for 1991 to forge strong Federal-State partnerships to restructure completing the planned doubling of the program from its pre-college education throughout the Nation. This effort level of 560 new fellowships in FY 198 7. This number will assist states in meeting the national education goals includes the continuation of a specially-focused "Women that are being developed as a result of the September in Engineering" offering to attract more women into 1989 Education Summit. the Nation's engineering faculty. The funding will also The Materials Development, Research, and Informal permit an expansion of the Young Scholars Program Science Education Subactivity will continue to focus on for talented high school and middle school students the need for a consistent pattern of elementary and who may be candidates for careers in mathematics, secondary school science and mathematics instruction. engineering, and the sciences. The amount requested A major effort will be continued in FY 1991 to de­ for this Subactivity in FY 1991 includes an increase velop improved instructional materials for science in in the fellowship stipend to $13,500 per year, and the secondary schools. Informal science activities and reflects the third step in an administrative adjustment in research about the teaching and learning process will also the funding schedule for the Graduate Fellowships and be emphasized. Programs to develop and demonstrate Minority Graduate Fellowships Programs. the use of modern technology in addressing educational The Studies and Program Assessment Subactivity will problems will continue. continue its studies of national and international trends Programs in the Undergraduate Science, Engineering, relevant to science education, the collection and analysis and Mathematics Education Subactivity will increase of data on science and engineering education issues, and substantially in FY 1991. In addition to the continued the support of related policy studies. Efforts to assess the SEE emphasis on strengthening college instrumenta­ results of NSF education programming will be expanded. tion and laboratories (College Science Instrumentation Program), the effort will be expanded by assuming re­ Summary by Educational Level sponsibility for all types of institutions. Comprehensive The allocation to programming at the various educational regional centers for minorities and other model colle­ levels of FY 1990 appropriations and the funds requested giate programs to stimulate the participation of women, for FY 1991 are shown in the following table. minority, and disabled students in curricula leading to science careers will be substantially increased to support (Millions of Dollars) approximately 15 centers throughout the Nation by FY FY 1990 1991. Efforts to enhance the currency of undergraduate Current FY 1991 Change faculty and to improve the undergraduate curricula will Education Level Plan Request Percent also increase significantly. Pre-college $140.37 $165.10 17.6% Undergraduate 34.00 50.00 47.1 The Research Career Development Subactivity will Graduate 29.90 35.90 20.1 continue its current efforts. The number of new three­ year graduate fellowships will be increased to 1100, Total, Activity $204.27 $251.00 22.9%

GROUP ACTIONS AND INVARIANT THEORY A. Bialynicki-Birula, J. Carrell, P. Russell, and D. Snow, Editors (Conference Proceedings, Canadian Mathematical Society, Volume 10)

This volume contains the proceedings of a conference, 1980 Mathematics Subject Classifications: 14-02, 14L30, 1406, 2006; sponsored by the Canadian Mathematical Society, on 20G05, 14025 Group Actions and Invariant Theory, held in August, 1988 in ISBN 0-8218-6015-1, LC 89-17605 ISSN 0731-1036 Montreal. The conference was the third in a series bringing 240 pages (softcover), September 1989 together researchers from North America and Europe Individual member $20, List price $33, (particularly Poland). The papers collected here will provide Institutional member $26 an overview of the state of the art of research in this area. The To order, please specify CMSAMS/10NA conference was primarily concerned with the geometric side of invariant theory, including explorations of the linearization All prices subject to change. Shipment will be made by surface. problem for reductive group actions on affine spaces (with For air delivery add, 1st book $5, each additional book $3, max­ a counterexample given recently by J. Schwarz), spherical imum $100. Prepayment required. Order from American Math­ and complete symmetric varieties, reductive quotients, ematical Society, P.O. Box 1571, Annex Station, Providence, RI automorphisms of affine varieties, and homogeneous vector 02901-1571, or call toll free 800-321-4AMS (321-4267) in the bundles. U.S. and Canada to charge with VISA or MasterCard.

APRIL 1990, VOLUME 37, NUMBER 4 407 Mathematics Outside of Mathematics Departments

Solomon A. Garfunkel Gail S. Young

in advanced mathematics would have risen in that The following article is based on a survey commissioned by period. Where are the students of advanced mathematics? the Exxon Education Foundation. Solomon A. Garfunkel is This was the question motivating this study, undertaken Executive Director of the Consortium for Mathematics and in 1988. What we found is that a rise in advanced Its Applications (COMAP) in Arlington, Massachusetts. Gail courses seems to be occurring, but it's not occurring in S. Young is professor of mathematics at Columbia Teacher's mathematics departments. College. The authors wish to express their deep appreciation to Dr. Richard F. Link, who prepared the sampling plan, oversaw data analysis, and kept two mathematicians statisti­ The Study cally on track. Anecdotal information and even a cursory look at col­ lege and university catalogs indicated that the content of many courses in departments (other than mathemat­ One could view the present state of American math­ ics!) consists entirely or mainly of advanced mathemat­ ematics as one of brilliant success or extreme crisis. ics. We found such courses not only in the expected Mathematics research is more fruitful than ever before places-departments of physics, engineering, computer in the history of the discipline. Mathematical methods science-but also in such departments as economics, po­ are making real contributions in more and more dis­ litical science, biology, and management. Therefore, the ciplines, and many new fields are coming into being. fact that enrollments in advanced mathematics courses Undergraduate mathematics majors who have learned have not risen may reflect an increase in enrollment some computer science can get jobs with starting salaries in mathematics courses being taught outside mathemat­ comparable to new mathematics Ph.D.s. New mathemat­ ics departments. This study was prepared to test this ics doctorates are in great demand and, in academia, can hypothesis and to provide a baseline for future studies. get starting salaries comparable to associate professors in We sampled 425 schools according to a statistically­ the humanities. based sampling plan (details are provided in the ap­ On the other hand, the number of mathematics pendix). Course catalogs were used to select the non­ Ph.D.s awarded each year has dropped steadily for mathematics department courses which appeared to con­ more than a decade, and more than half the new tain significant mathematical content and which used doctorates are going to foreign nationals. The number mathematics at or above the level of calculus. Statistics of bachelor's degrees in mathematics, after more than departments were specifically excluded from the study. doubling between 1960 and 1970, dropped during the In cases where it was difficult to assess the mathematical 1980s to the 1960 level. Calculus-level enrollments more content of a particular course, we telephoned department than trebled in the fifteen years following 1960. In chairs to get their appraisal. Once the courses were se­ 1985 calculus accounted for 39% of the mathematics lected, we obtained enrollment data from the institutions enrollments, whereas remedial and precalculus courses (in addition to other information about the courses). accounted for 52% of the enrollments; advanced courses Our study determined that, each year, there are over comprised just 9% (approximately 147,000). 170,000 enrollments in advanced mathematics courses This last fact is especially curious. Consider that, being taught outside mathematics departments. These between 1960 and 1983, engineering bachelor's degrees data are presented in the tables at the end of this article, rose from 38,000 to 72,000; biology degrees from 18,000 which also sort the numbers according to school size, to 44,000; social science degrees from 16,000 to 25,000. type (private or public, university or college), and region. Mathematics has become increasingly important in all Engineering departments recorded the largest num­ of these fields, so one would assume that enrollments ber of enrollments (83,854) with significant mathematics

408 NOTICES OF THE AMERICAN MATHEMATICAL SOCIE!TY Mathematics Outside of Mathematics Departments content. Of the courses represented, 86% developed the of the comments are vitriolic, many appreciate some mathematical content before it was applied; this was of the problems of mathematics departments, and some especially true of electrical and chemical engineering regard the present state as inevitable and even desirable. courses. Among the mathematical prerequisites men­ But overall the comments are critical, and very few of tioned for engineering courses were: calculus, linear the respondents are really happy with the mathematics algebra, discrete mathematics, and modeling. The term departments at their institutions. mathematical maturity was used in several responses. The comments can be classified roughly into five (Interestingly, engineering courses also had the largest categories: average class size: 113 students.) 1) The mathematics faculty does not know or appreciate Next in order of enrollments were business courses. applications. Calculus, linear algebra, and statistics were common pre­ (This perception is held even in the case of depart­ requisites, with "the mathematics involved in operations ments which we believe are strongly involved in applied research" mentioned quite often. The survey found 61 mathematics.) Some typical quotations: courses in "business forecasting", which seems to em­ • "[There is an] inability of mathematicians to come to ploy large amounts of probability and modeling; these grips with a difference between 'pure' and 'applied' courses are probably required, because they had large mathematics. Mathematicians of the 'pure' strain look enrollments. down their noses at the other strains. Thus science The sciences use a great deal of mathematics, and and engineering departments feel that their students science instructors seem to be expending a great deal of will not be adequately serviced by math departments. time teaching mathematics before they actually use it in The math departments are unconcerned because they their courses. Courses, such as Mathematical Methods view this as 'bogus' math which they have no interest of Physics or Numerical Methods in Biology, specifically in offering anyway ... " train students in the mathematical applications of a • "Modern technical fields, with their complex appli­ scientific discipline. Many of these courses seem to cations of math/technology to real-world problems, be required. Another interesting development is the have requirements that can't easily be met by isolated, prevalence of probability being taught in genetics classes, largely theoretical, overly generalized presentations despite the fact that it is a prerequisite for almost every that are not (usually) presented as being useful or in­ one of these courses. teresting to a practical person. The math department Agriculture and the social sciences also recorded courses 'turn off' most of our kids and it's up to us courses with advanced mathematics. Many courses stress back in the professional departments to turn their the need for mathematical maturity, with calculus being enthusiasm back [on] again." the formal prerequisite. Discrete mathematics and prob­ • "Mathematics departments have become so abstractly ability are also common prerequisites in these areas. Our oriented that their courses are not given any applied deletion of statistics courses from the survey contributes content... Most math Ph.D.'s never take an applied to the small size of the listed enrollments for social sci­ math course, so why should they teach it? Our engi­ ence courses. For example, a course entitled "Psychology neering school tried to convince our math department Statistics" is present in over 50% of the departments to teach some applied courses but eventually gave up." surveyed. 2) Mathematics faculty teach mathematics as an art with Department Responses full abstraction, not as a tool. We sent a preliminary draft of the first sections of this • "The content of most math courses focuses on the­ report to the chairpersons of responding departments, oretical development. This is not 'bad' per se but partly as a courtesy, but also in the hope that many leaves most students wondering about the 'what, would respond to the request in our cover letter that they when, where, and why' of applications ... seldom ad­ tell us why they think these courses exist and whether dressed in traditional math courses." their campuses know that so much mathematics is being • "Applied departments use math as a tool. An in­ taught outside the mathematics departments. We sent dividual topic is analogous to a hammer perhaps. out 714 drafts, and have received 292 replies (a response They wish to 'hammer' with it. On the other hand, rate of 41%). This is quite good considering the lack of math departments often become more interested in pressure in our letter, the fact that we are clearly not in its description and generalization of the 'hammer' their fields, and the lack of follow-up. itself.'' In fact, many of the replies are quite long and • "Engineers find math to be a need, not a love. detailed, showing considerable thought and concern. The Mathematicians ... are out of touch with the real nature of the comments does not depend on the size world and are more like mathematical artists than or status of the university or college. Although some real world scientists."

APRIL 1990, VOLUME 37, NUMBER 4 409 Mathematics Outside of Mathematics Departments

3) Topics span too many mathematics courses. Our follow-up questionnaire to non-mathematics de­ • "There is not room ... for every student to take the partments indicates that these departments believe that separate courses in differential equations (ordinary the substance of the survey is well-known and under­ and partial), vector and tensor analysis, complex stood on their campuses. In fact, it is clear from their variables, Fourier series, probability, ... all these are responses that the departments believe that mathematics covered in our one-year course by omitting the de­ faculty are responsible for and content with this state of tailed proofs and generalization ... " affairs. • "Other reasons [for such courses] include the claim In a preliminary form of this report we concluded that the mathematics needed cannot usually be found with, "Finally, we should like to stress that this report in one single mathematics course. Bits and pieces of makes no value judgements on the present state of several courses are wound together." affairs. It may or may not be healthy for undergraduate instruction in mathematics to be distributed among many departments." 4) The mathematics departments have not kept up with However, on a personal note we find both the survey new applied mathematics. results and the attitudes of non-mathematics departments • "Mathematics departments generally do a lousy job deeply disturbing. There is an indicated residue of ill­ with mathematics their faculty has no training for. feeling toward mathematics departments. Worse yet, Specifically, Shannon's information theory; automata; mathematics faculty and curricula are often seen as at transformation geometry; graph theory, particularly best irrelevant and at worst counterproductive. Moreover, as employed in practice; algebraic coding theory; respondent after respondent expresses the belief that polynomial rings and finite fields; computer ability; mathematics departments are unconcerned about issues and, one suspects, probability and statistics." of course offerings. Perhaps they are correct. We have shared the results of this survey with a number of mathematicians and 5) Mathematics courses do not give students the knowledge mathematics educators and seen no evidence of shock, or the mathematical maturity for further work. dismay, or surprise. We are anxious to get reaction from • "An attempt by mathematicians to do 'something the broader mathematics community. for everyone' in basic calculus courses rather than This survey was commissioned by the Exxon Educa­ concentrating on a generic mix of basic skills and tion Foundation. The purpose of this and several other concepts. Engineering has further suffered by the programs funded by the Foundation was to assess the loss of more preparatory subjects such as analytic state and health of undergraduate mathematics educa­ geometry." tion. Moreover these projects were designed to point the • "I cannot take it for granted that [students from way toward needed reform. Given these results, what, if calculus] are able to use their mathematical skills anything, should be done? in problem solving. What appears to be ... lacking is We would like to ask that question of the AMS the ability to formulate a problem quantitatively and membership. Are these results surprising? Are they trou­ then to solve it using the tools they learned in their blesome? What, if anything, should they spur us to do? At calculus course." the very least, we hope to begin a public debate. Please We are reporting these opinions as ones widely held write to us at: COMAP, 60 Lowell Street, Arlington, in other fields, and we are not endorsing or contradicting MA 02174. Your ideas and suggestions are important. them. They are, however, very widely held by our They can have a direct effect on programs designed to respondents, and therefore must be considered. improve the quality of undergraduate instruction and on the health of our profession. Conclusions As this is to our knowledge a baseline survey, it is Appendix: The Procedure difficult to measure any trends. However, anecdotal The four-year colleges and universities can be conveniently cat­ evidence indicates that there have been substantive egorized as public universities, private universities, public four­ private four-year colleges. To account for increases in enrollments in the courses surveyed. It is year colleges, and regional differences, we further stratified the four-year there are more possible certainly clear from the survey that institutions according to geographical region: East, South, Mid­ enrollments in advanced work in mathematics outside west, and West. The final variable considered was size of stu­ of mathematics departments than within (approximately dent population, for which we chose small (less than 3,000), 173,000 as compared to 147,000). It is unclear whether medium (3,000-10,000), and large (more than 10,000). Because this situation is fully understood at individual schools we wanted to cover as many enrollments as possible, as op­ (and in national assessments). posed to as many institutions as possible, we chose a sampling

410 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Mathematics Outside of Mathematics Departments plan which treated the institution type and geographic location • Amount of mathematical content with equal weights but gave a larger weight to the size variable. • Course function (development or the application of The effect of this weighted sampling technique is indicated by mathematics or both) the following numbers of students attending the institution type • Course prerequisites versus numbers of students covered by the sample population • Course text for each type: public universities (1,724,957 /1,241,497), pri­ • Accuracy of the catalog description. vate universities (586,781/453,330), public four-year colleges While much of the survey questionnaire contained strictly (3,484,586/1,453,747), and private four-year colleges objective information, such as course enrollment and frequency, (1,918,685/434,951). The 425 selected institutions (approxi­ the mathematical content of each course had to be determined mately 25% of all such institutions) were assigned control num­ by analyzing the prerequisites and the department chair's im­ bers to insure a "blind" study. Of these, 343 returned catalogs pression of the extent of mathematical content. In some rare within the time-frame necessary to be included in the survey. instances the coding required some degree of subjectivity; in The catalogs were used to select the non-mathematics courses these cases we used the catalog and a telephone survey follow­ which appeared to contain significant mathematical content. up to increase the likelihood of accurate results. Each course description was reviewed to determine whether it In order to organize the data more coherently, departments fell within the scope of our survey. Specifically, we were seeking were categorized into seven fields: agriculture, business, educa­ courses which required and used mathematics at, or above, the tion, engineering, humanities, science, and social science. This level of calculus. It should be noted that statistics departments categorization addresses the difficulty of comparing departments were explicitly excluded from the survey, along with any course with different names as well as the problem of accounting for cross-listed with the mathematics department. schools without certain departments. A survey questionnaire was designed to address the follow­ From the results of the survey, it was estimated that there ing issues: are 173,237 enrollments in courses that teach advanced math­ • Course frequency ematics outside of mathematics departments. The 95% confi­ • Course enrollment dence interval for the true number of enrollments is 173,237 ± 28,530. TABLES Distribution of enrollments by school size, school type, and region. SIZE LARGE MEDIUM SMALL TOTALS FIELD (+10,000) (3-10,000) (-3,000) AGRICULTURE 858 0 0 858 BUSINESS 32,419 15,908 341 48,668 ENGINEERING 70,142 10,696 3,016 83,854 SCIENCE 30,292 7,885 1,577 39,754 SOCIAL SCIENCE 59 44 0 103 TOTALS 133,770 34,533 4,935 173,237

TYPE Public Public Private Private FIELD Universities Colleges Universities Colleges TOTALS AGRICULTURE 165 627 66 0 858 BUSINESS 14,857 12,360 9,711 11,741 48,668 ENGINEERING 27,620 35,277 13,651 7,306 83,854 SCIENCE 12,670 16,601 3,821 6,662 39,754 SOCIAL SCIENCE 0 0 103 0 103 TOTALS 55,312 64,865 27,353 25,709 173,237

REGION FIELD EAST SOUTH MIDWEST WEST TOTALS AGRICULTURE 66 165 33 594 858 BUSINESS 18,196 7,742 10,564 12,166 48,668 ENGINEERING 16,960 30,967 16,300 19,627 83,854 SCIENCE 10,569 9,671 14,005 5,509 39,754 SOCIAL SCIENCE 22 0 37 44 103 TOTALS 45,813 48,545 40,939 37,940 173,237

APRIL 1990, VOLUME 37, NUMBER 4 411 Forum

is a role for some of those who, up to this point, In response to a recommendation of the Science Policy have been uninvolved, a role which might have the Committee and action by the Council, Notices is institut­ secondary effect of giving some of the future leaders ing with this issue a regular feature called Forum, which of the mathematical community some insight into the will publish short articles on issues which are of interest needs of school mathematics: to the mathematical community. There is much talk about the need for reform of math­ Readers are invited to submit articles for possible in- ematics education in the U.S., and there are currently clusion in Forum to: many initiatives for change. However, most mathemat­ Notices Forum Editor ical researchers have pretty much stayed out of the American Mathematical Society reform efforts so far. Many pay lip service, although in Post Office Box 6248 their heart of hearts they probably believe that there is Providence, Rl 02940 not much that they as researchers can do. Maybe they hope that their self-serving "prejudices" could actually Articles should be between 1000 and 2500 words long. be true, namely that the talented ones, those born to do We begin Forum this month with two articles concern- mathematics, will survive most any educational system ing mathematics education and the professional mathe­ equally well or poorly, and that those are the ones with matician by Herb Clemens and Harvey Keynes. whom the future of the subject lies. Also, we went through this discussion once before in our lifetimes, in the early sixties, when enlightened and mathematically correct reforms, based on curriculum led by mathematicians, were adopted. But Research Mathematicians and the change and there were serious political mistakes in that effort, and Education of Children - A Proposal many would say that things didn't get better. Herbert Clemens In any case, these days working research mathemati­ University of Utah cians with little time to spare are, and probably should This article is based on a 20-minute talk by the author in be, relatively marginal to the efforts to reform teacher the Special Session on Mathematicians and Educational training, school curricula, etc. Those problems are too Reform at the AMS-MAA annual meeting in Louisville, big, too absorbing, and too political, for those whose fun­ January 18, 1990. Although other mathematicians may damental activity is research in the subject itself. These be interested in the program proposed in the article, the researchers do not have the time or energy to give them­ article itself is addressed to a very restricted audience, selves to programs of educational reform. They often feel namely to those mathematicians, typically in their 30s that the mathematical community should do something and 40s, whose energies are directed primarily toward about education, but they are simply not willing to stop their own mathematical research. The article attempts doing research in order to take action themselves. So, for to find a limited yet very necessary role in elementary those not capable of making the "ultimate sacrifice" of and secondary educational reform for some of these stopping their research, what's to be done? mathematicians. In spite of the general tenor of remarks How can they continue their research with the nec­ in the January 31 issue of the Chronicle of Higher essary intensity, and yet do something useful? I feel Education, many mathematicians are deeply involved in that there is a way to resolve this dilemma, a way educational reform at all levels, and there is certainly not contradictory even to those who feel that their first no intention here of diminishing in any way their responsibility is to preserve, create and enhance good very commendable and successful activities. But there mathematics, and to communicate it to others. In fact,

412 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Forum

I feel that it is with an eye to protecting and cultivating more generous in sharing the beautiful mathematics that our subject for future generations that we should look for we do have. the researcher's principal role in mathematics education. For many, especially in our culture, mathematics is In this regard, I'd say that there is something essential more appealing when it is not an isolated, impersonal to the future of our discipline that is perhaps best done by exercise, but rather an act of quantitative communica­ those currently active in research. I feel that at least some tion which energizes giver and receiver. If many of us researchers can, and must, give a limited amount of time are demoralized by the character of our undergraduate and energy to cultivate the talented young people who teaching in this respect, maybe we are dealing with the can become successful and productive mathematicians. wrong group. Maybe by then it is too late in a young This is a role reserved to mathematicians-as a rule of person's life to strike the spark. Maybe the practical con­ thumb, you have to do some good mathematics before cerns of adulthood, and a certain amount of unconscious you can realize what's good and who has talent. And intellectual cynicism have already set in. among mathematicians, researchers, especially those who I think that the deeper quantitative contact with will occupy future roles of leadership in the mathematical students is possible, but the natural audience for the community, must play a part, if we are going to attract deeper kind of mathematical contact I'm talking about a new generation to our subject, and thereby insure its is much younger. Our younger children, especially the future in our society. quantitatively talented ones, are open to mathematical To most talented North American young people reasoning and are excited by it. Left unaddressed, this today, mathematics is simply not beautiful enough, or interest and excitement soon yields to other compelling human enough, to attract them away from other pursuits. influences. It is to this elite group of children of ages Is that primarily our fault? I don't think so-it is as much roughly 8-12 that we should direct some energy. Now our culture's fault as it is ours as mathematicians. If I'm not advocating that mathematical researchers should we have failed, it is an intellectual failure; we have become part-time grade school teachers. God forbid! not created or discovered quantitative knowledge that is Most would be terrible at that! Besides, most kids are beautiful or insightful enough. And it is hard to blame not made to be future mathematicians. In the matter ourselves for our own lack of intelligence-we can only of making true intellectual contact with those capable work with the smarts God gave us! of becoming future mathematicians, and inspiring them, But whether or not we are at fault for the current it's only a very small group we should be concerned with. situation is not the point. The future of our subject in My point is that that small group is not available for this country is at stake. We cannot build that future most of us at the University level, but that it is available on the basis of our ability to buy human talent from to us if we seek it out. For the future of our profession abroad, or solely on those whose parents' transplanted in this country, some of us must go out there and make values sustain them for one more generation. Not only contact with it. In this matter, there is just no substitute is such a policy not wise in historical terms, it may for personal contact. become practically impossible in the coming decades, So the primary duty of the researcher is to the if other regions of the world move ahead of the U.S. subject of mathematics. The role of such a small and economically. politically impotent group in the broader question of the So, if doing nothing is not an acceptable alternative, quantitative education of society in general is necessarily then let's begin to examine what the research community quite small. But our influence on a future mathematical might do. elite is essential. Therefore I feel that that is where In the making of the future mathematicians in this the research group must concentrate the very limited country, I don't believe that "enlightened programs" energies it has for social concerns. To do that, we must alone change young people very much. Of course, ex­ be ruthless. We must be willing to be called elitist, and posure to beautiful mathematics attracted us and it will correctly so! Our discipline is an elitist discipline-that is continue to attract some young people. But another es­ in the nature of things, nothing we do or say will change sential source of influence, I think, is contact with our that fundamental fact. So let's recognize it, accept it, and collective intelligence and our aesthetics. And here we put our social energies where they will do the most good fail! We simply do not put ourselves in situations in for everybody, namely by cultivating those who can and which we can share those things with a receptive audi­ will preserve and advance our subject in the future. ence. People change people. Right now, the intellectual So let me propose a program in math ed that the beauty of our subject and our own intellectual generosity AMS and some working research mathematicians should are simply not compelling enough to attract the neces­ get involved in. This program comes out of my own sary numbers of those who should come after us. If we experiences and those of other mathematicians who have haven't the capacity to make more beautiful mathematics spent some time with school kids in recent years. I for young people, we do have it within our power to be would like to propose that the AMS begin a program to

APRIL 1990, VOLUME 37, NUMBER 4 413 Forum encourage some of its members to spend a little time in Finally, let's talk politics. If we as mathematicians a grade school "on their way to work." Such a program spend a little time with some bright young kids, we might, perhaps, take the following form: may make real intellectual contact with a few of them. 1. Each mathematician-volunteer visits a school 3 But there is another reason for spending a little time, times per week, about 1/2 hour per visit, during the even if we don't succeed in making that contact. Just course of one academic year, and works with groups being around some schoolkids every so often makes an of 10 or more children. The goal for the initial two­ important political statement to a far wider audience. year period is the participation of 100 mathematicians It is the statement that who those kids are and what throughout the U.S. those kids do is important to mathematicians, and to the 2. Each mathematician-volunteer gets one-quarter future of our subject. The statement is true, so let's say course teaching reduction from his or her home Uni­ it. versity. (Estimated replacement cost for lost teaching is $3,000-4,000, of which $2,500 would be covered by the Can Mathematicians Be Involved Program.) in Education and 3. AMS submits a proposal to NSF that they give a Still Survive $350,000 grant for the first two years of the program: a) in the Profession? $50,000 for half-time administrator, b) $25,000 operat­ Harvey B. Keynes ing budget, c) $25,000 equipment budget (to set up an University of Minnesota "all email" correspondence system and assemble useful I. The Issues materials), d) $250,000 to pay replacement costs to par­ Mathematics education is undergoing intensive examina­ ticipating Universities for a total of 100 quarter-course tion at all levels, and major reforms in curriculum and replacements. direction are now in the process of being implemented. 4. AMS asks NCTM to locate potential participating The need for mathematicians to be extensively involved school districts. Universities and individual mathemati­ in these endeavors has been widely recognized. A few cians wishing to participate go through the NCTM or have organized major innovative projects in teacher local contacts to line up a host school. training, networking for curricular reform, and innova­ 5. AMS asks NCTM to locate in each participating tive programs for underrepresented groups and gifted school a host teacher who serves as the facilitator for the students. Yet for the most part, mathematicians are ig­ interaction between the mathematician and the school noring these educational opportunities, and avoiding any kids. The host teacher also screens the mathematician real level of involvement. Despite excellent educational to insure his or her suitability for participation in funding opportunities at a time when research support is the Program, and chooses the children who should declining, and constant pleas for increased involvement participate, and familiarizes the mathematician with from university, government, and business leaders, aca­ NCTM guidelines for math learning and with the math demic mathematicians are clearly questioning whether program at the host teacher's school. their participation can be part of their professional ac­ 6. To the question "What do I do when I'm sitting tivities. We need to__ exaJlline and question some of the around for half an hour with a dozen fourth-graders?", reasons for thei(hesitlu1cy)tnd change many of the per­ the first quick answer is "Whatever the regular fourth ceptions if we hope tQ __ raise the educational involvement grade teacher suggests." The mathematical conversation of the mathematics community. usually begins with what the kids are doing in their At the heart of the issue is the belief that educational regular math class. If this collective conversation lags, or issues, concerns about teaching, and involvement in local resources are insufficient, an email "hotline" to a school curriculum are really not matters that are profes­ library with a resource person and some good materials sionally attractive to most university and college faculty. should help. One program officer at NSF described the typical reac­ 7. The short-term goal is to make early contact tion from chairs and senior faculty at all levels of state with mathematical potential; the long term goal is to universities to be that while education and teaching are locate and encourage future mathematicians in order to important for faculty at schools which should emphasize maintain the health of our profession. undergraduate education, their own school is primar­ For many researchers, the need to do something ily interested in research. This attitude is even seeping is compelling, but the limitations on their time and into some of the best small colleges, where emphasis energies are compelling too. The above is one suggestion on providing undergraduate research opportunities and of a way that some can involve themselves meaningfully, competition for faculty have led to increased interest in expending little more time than it takes to read the faculty research, especially for younger faculty. So vir­ morning paper. For some of us who have tried it, the tually all of the leading universities and colleges dismiss experience has been surprisingly satisfying. involvement in K-12 mathematics education in the name

414 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Forum of various research activities. The deeply held conviction all levels in our community, and professional rewards for that research interests and educational involvement are high quality, successful and significant activities become incompatible continually permeates discussions on these embedded in the system. An important aspect is not to matters. In fact, most mathematicians generally believe trivialize a good educational project as any non-routine that involvement in educational activities at any level is activity outside of research. The efforts required to orga­ a sign of disinterest or incapability in research activities. nize and implement a multi-year educational project can All too often, a faculty member who has lost interest be enormous and the chances for no real success can be in research and scholarship activities is described as au­ quite high. Education is unlike traditional mathematical tomatically being interested in teaching and education. research, and more like many experimental sciences. Even the few mathematicians with significant educa­ It is frequently not the conception of the project, but tional interests and research activities frequently feel the the enormous organizational and administrative efforts, stigma of being viewed as less dedicated and valuable together with sensitivity to classroom cultures in the than colleagues with only research activities. We must context of mathematics content, that makes for success­ examine ways to change this attitude of our colleagues ful educational programs. While these efforts may be and peers if we expect to see more mathematicians intellectually less challenging than standard research, the considering educational involvement. emotional, physical and organizational strain are usually Closely related to this view of education is the far more intense. We need to respect these levels of perception (and in most cases the reality) of the reward involvement, and reward those faculty that meet high system in our profession. In discussions with younger standards similar to those of good research. full professors who may be more willing to consider An important aspect of providing high-quality pro­ educational involvement, their opinion of the reward fessional involvement in mathematics education is to system in mathematics is a major deterrent to their encourage participation when personal circumstances are participation. They believe that even significant and right. Many active researchers might find a particular highly successful efforts in education will not provide educational program appealing, and wish to be involved merit salary increases and other professional respect and for a short period or even a few years. Because of fresh opportunities. Despite lip service to the contrary, this ideas and higher levels of enthusiasm, the involvement is the reality in many departments. In a recent essay of research mathematicians is critical to both new and on mathematics education, Bill Thurston states "What established programs. Many times, a reasonable research needs to change urgently is not so much the system program is maintained during this initial involvement, of advice, of teacher training or of mentoring, as the and in some cases, a new professional balance between system of professional rewards." In a few departments research and education is reached. We should encourage where such activities are rewarded, chairmen need to these elective shifts of emphasis. In the context of main­ keep these considerations virtually secret for fear of taining an appropriate research and scholarly program disapproval among the senior staff. Some highly respected for their department, younger faculty could be encour­ mathematicians would find it unacceptable if anything aged to have some exposure to educational activities. other than a theorem or an outside offer was to play a There are sufficient numbers of senior mathematicians role in a merit increase. Various presidents, provosts and who are re-examining their professional directions so deans publicly lament how faculty are not involved in that without really compromising tenure and promotion educational activities, yet seem to be unwilling to take standards, significant educational activities could still on the conventional wisdom and advocate changes in the take place. For certain senior mathematics faculty, pro­ reward system. fessional respect for educational issues might encourage So the impediments to more involvement in education them to seek new and productive involvements in edu­ are clear: professional respect among colleagues and cation when their research activities have peaked. With peers, and professional rewards. Since the forces to the current demographics of the community, this could change these issues reside primarily in our mathematics provide a growing source of mathematicians to meet the community, we need to address ways to influence our escalating needs in education. departments and professional societies. Which organizations are best equipped to influence these changes? Because the heart of the problem resides 2. What Can We Do? in the attitudes of research mathematicians, our research The key to any lasting reform will be changes in profes­ societies must play a leading role. The AMS is seriously sional attitudes of most mathematicians. This does not beginning to examine its role in education. Discussions mean that very many will change their professional goals at the 1990 meeting in Louisville could set the tone for or their personal value systems. What it does mean is a significant official level of recognition of educational that professional tolerance for colleagues who want to activities. SIAM has also maintained a serious interest become involved in education should be encouraged at in education, and would likely support any new AMS

APRIL 1990, VOLUME 37, NUMBER 4 415 Forum position. The MAA has always maintained its own edu­ on the other side. The balance is always shifting, but cational agenda, and their excellent ongoing efforts would in the post-Sputnik years moved heavily towards the re­ clearly support and supplement any changes within AMS search end. We currently have a culture that is generally and SIAM. Finally, the Mathematical Sciences Education comfortable with this role, and has built a professional Board (MSEB) and the NSF-funded Mathematicians and structure to support it. But recent changes in our society Education Reform Network (MER) can play appropriate now question the wisdom of the mathematical commu­ advocacy roles for shifts in educational interests within nity to continue to accept the current balance. Among the mathematics community. But the real change must these is the widespread recognition that the need to take place within the rank-and-file of the mathematics reform the K-12 mathematics curriculum necessitates profession, most specifically among the academic re­ the involvement of mathematicians at all levels to help search mathematicians. This will clearly be a long-term develop alternate approaches and curriculum. Under­ project, with many dissenting voices and calls for histori­ graduate and even graduate mathematics curriculum is cal standards. Such debate is healthy, and perhaps should also under careful review and revision. Because of the take place in our professional journals and at profes­ central role of mathematics in our changing society, sional meetings. Some courageous department heads and mathematics and mathematicians are now linked to two deans should take the initial steps in changing the reward central concerns of our society - economic competitive­ systems at their schools, and begin the process of pro­ ness and human resources. Future resource allocations fessionally integrating educational activities into the life to all sciences, including mathematics, will probably be of their mathematics departments. Most importantly, we heavily dependent on the country's perception of how need the research leaders in our community to actively effectively they contribute to the improvements of these support these new roles in mathematics education, and essential areas. To insure the vitality of our profession be willing to challenge other leaders with less tolerance. and to maintain attractive professional rewards, we as a community need to shift the professional balance at 3. Some Final Observations this time and re-affirm our interest in educational activi­ Especially in academic environments, mathematicians ties. We need to electively make this change within our have historically juggled two aspects of their professional own professional culture, and truly support our peers lives - their personal esthetics and research interests on and colleagues who engage in high quality and exciting one side, and their educational and service components educational programs.

UNFOLDINGS AND BIFURCATIONS OF QUASI-PERIODIC TORI H. W. Broer, B. Huitema, F. Takens, and B. L. J. Braaksma (Memoirs of the AMS, Number 421)

In the theory of dynamical systems, the occurrence of equilibria and periodic motions, as well as their general persistence and stability properties, are now fairly well understood. Researchers also have some systematic insight into the role of external parameters. This book aims to mimic this classical theory in the case of quasi-periodic motions. These motions are most familiar in the context of the conservative dynamics of classical mechanics, but they also occur with dissipative dynamics-for example, quasi-periodic attractors play a role in the onset of turbulence. In the first part of the book, the authors present a general treatment of the use of external parameters in various contexts, employing notions such as integrability and transversality. The second part, dealing only with dissipative cases, studies bifurcations when the hyperbolicity is mildly violated. Readers will appreciate the way the book systematically ties together a number of cases fo quasi-periodicity and the resulting improvement of accuracy. In addition, a number of new applications are presented.

1980 Mathematics Subject Classifications: 58, 34 All prices subject to change. Shipment will be made by ISBN 0-8218-2483-X, LC 89-18093 surface. For air delivery add, 1st book $5, each additional ISSN 0065-9266 book $3, maximum $100. Prepayment required. Order from 188 pages (softcover), January 1990 American Mathematical Society, P.O. Box 1571, Annex Individual member $13, List price $22, Station, Providence, RI 02901-1571, or call toll free 800-321- Institutional member $18 4AMS (321-4267) in the U.S. and Canada to charge with To order, please specify MEM0/421NA VISA or MasterCard.

416 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

Edited by Jon Barwise

Editorial notes Computers in research Are good research tools good teaching tools? This column is still anxious to get articles reporting on the use In last month's column, Phil Miles wrote an article about the of computers in mathematical research, as well as articles on the use of Derive in pre-calculus classes. He raised an important mathematics of computation. Submissions should be of about question: Does the use of such a program help or hinder our 3-5 pages in length, and informal in nature. Send them to: students' understanding of mathematical ideas? Jon Barwise Let's be clear what is at issue. Few doubt that students com­ CSLI ing out of mathematics courses need to have a basic under­ Ventura Hall standing of the computational tools that are available, and how Stanford, CA 94305 one would use them. Nor was Miles questioning whether the email: [email protected] computer can be an aid in teaching. His worry was whether After July 1, send it to me at my new affiliation: computer algebra systems help teach the basic ideas of mathe­ Jon Barwise matics. Department of Mathematics Several years ago I overheard some students, looking at a Indiana University returned calculus midterm, complaining about the rules of ex­ Bloomington, IN 47405 ponentiation. They thought the rules very unnatural. In partic­ email: [email protected] ular, one said "Mathematicians are really perverse to set up a rule like This month's column In addition to the article by Devlin, this month's column con­ rather than the more sensible rule ... " where you would multi­ tains short reviews of several pieces of mathematical software: ply exponents. His friend agreed. FFTLIB by PC Scientific, Phaser, and three programs for the What is disturbing about this conversation is not the error it­ NeXT: Groups, Rubik Algebra and Orbit. self, but the view of mathematics that the conversation demon­ strates. Would the use of a computer algebra system alleviate or exacerbate the problem? No doubt the answer depends on the system and how it is incorporated into the teaching. Miles ob­ servation is that there is a danger that these systems could make The Right Stuff things worse, not better, by making mathematics look even more like a formal game with arbitrary rules. Keith Devlin* The issue of what goes to make up good courseware for un­ dergraduate mathematics is addressed in an article by Keith De­ vlin in this month's column. Doug Child, one of the developers The right stuff: small or large? of one of the programs discussed, takes issue in a follow-up. It is now almost an axiom that the use of computers Further discussion is welcomed. in college-level teaching is A GOOD THING. After all, the Computer Algebra Systems Workshops students we produce will go out into a world dominated Speaking of the use of computer algebra systems in teaching, the by their use. But what about the use of computers in col­ National Science Foundation is sponsoring a number of CAS lege mathematics teaching? There too it seems a sensible Workshops. Here is a list of those coming up soon. move. Like the pocket calculator of yesterday and the • June 18-22, 1990: Mississippi State University, P.O. Drawer, slide-rule before that, the arrival of the readily available MS 39762. Contact Jimmy Solomon at MSU. electronic computer has changed for ever the manner in • July 15-20, 1990: Colby College, ME 04901. Contact Don Small at Colby College. *Dr. Keith Devlin is Carter Professor of Mathematics and Chair­ • July 22-27, 1990: Colby College, ME 04901. Contact Don man of the Mathematics Department at Colby College, Waterville, Small at Colby College. Maine 04901. His email address is [email protected].

APRIL 1990, VOLUME 37, NUMBER 4 417 Computers and Mathematics which large parts of mathematics are developed and used Available in the mathematics computer laboratory are in the world (setting aside research uses in university Mathematica, Maple, and various statistics packages, to­ mathematics departments). The mathematics we teach, gether with a host of other software. The use of TEX and the way we teach it, should certainly reflect those and J:b.TEX to prepare mathematical papers and student changes. exams is becoming increasingly common. But what makes a piece of mathematics software suitable for classroom use at the college level? It depends, I believe, on the nature of the course and on the level. The right hardware Features that would make a particular package an ideal First, some comments about the hardware. (For reasons tool to use in an upper-level class might make it quite that should become clear, I do not regard this as a inappropriate for use with freshmen. side-issue to this review.) In particular, a recent in-house workshop in computer The Macintosh computer, and especially the 5Mg use in mathematics teaching at my own college (Colby Macllcx with a color monitor that we have in our com­ College in Maine), led to the conclusion that all-purpose puter laboratory, is a very powerful teaching tool, both computer algebra systems (Maple, Mathematica, Cal­ for classroom demonstration using a screen projector (we culus T/L) have obvious uses for teaching upper-level have two of these for our teaching at Colby), and for courses to mathematics majors, but several major draw­ individual (or even better, student-pair) use. It is fast, backs as far as freshman-level instruction is concerned, robust, easy to use, and user-friendly to the point of a especially for classes made up of students whose interest love affair. (I have seen this level of attachment in well in mathematics is either peripheral or utilitarian. For over half the Macintosh users I know, and I know many!) lower-level courses, small, special-purpose packages seem The Mac came upon the world so rapidly that I suspect to be far more useful. a great many people still do not realize what a truly I should add that the above is my own conclusion, huge step foward it was. Quite simply, it revolutionized and likewise the views expressed in the remainder of this computing. Indeed, given what preceded it, it is highly article are my own. They do not constitute a consensus misleading to continue to call it a computer. From the view, carefully distilled from the reactions of each faculty user's point of view, it is something quite different. A member at Colby. It is not, however, a completely 'blind' "desktop friend and assistant" is the only phrase I can response. First of all, everything I say is in the light think of that conveys something of what it represents. To of what I learned and observed over the three days think of it in terms of a powerful von Neumann computer of our workshop. Secondly, my own research work in with a fancy graphics interface is to completely miss the mathematics over recent years has involved my looking point, analogous to the early description of the motor car very closely at issues of artificial intelligence, software as a 'horseless carriage.' (Though that particular example design, cognitive science, and linguistics. And for the should indicate just how pervasive can be the tendency past seven years I have frequently reviewed software in to fail to appreciate the scope of a truly revolutionary connection with a regular mathematics and computing development.) column I write for The Manchester Guardian. So what is it that singles out the Macintosh (and it One thing that should be made clear at the outset is should by now be clear that I am talking about the system that Colby is particularly well-equipped with computers. as encountered by the ordinary user, not the hardware Besides the three central Vax 'mainframe' facilities and a inside the case)? small number of Sun workstations, the campus is awash For one thing, the fact that once you have learned with Macintoshes. At the moment, roughly one-in-three to use one Macintosh application, you can easily use students purchase their own Mac at a discount from any other. Just as learning how to drive one kind of the campus bookstore, and the goal is for all students to car equips you to drive any other car, with only minor have their own computer. Plans exist for every dormitory adjustments. eventually to be wired into the campus Ethernet. The For another, learning how to use that first application standard equipment currently being provided for faculty (whatever it is) is extremely easy. By and large, once in the mathematics department is a 5Mgf80Mg Macllcx you know about mousing and pulldown menus, the with a color monitor. Outside my office is a student general rule is that the 'obvious' thing to do next almost computer laboratory containing 17 similar machines. invariably turns out to be the correct move. And if by There are three Laserwriter liNT printers, one for faculty, some chance it is not, the system is forgiving and asks the other two in the computer laboratory. All machines, you to try again before any serious damage is done (such office and laboratory, are connected to the campus as your accidently erasing the entire hard-disk). network and to two central Macintosh fileservers. And it is fast and responsive (at least when you are Software used by almost everyone on campus in­ into the realms of 5Mg llcx's). A bit like driving a BMW cludes Microsoft Excel, Microsoft Word, and MacDraw. but with the expansive comfort of a Buick.

418 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

But most of all, it is fun. Given the right software, I out of it at all. Such experiences are not uncommon, have yet to see anyone who does not get 'hooked' by the as far as I can tell from talking to other colleges that beast. have tried computer-based calculus instruction, but as department chairman at a small (1600 students), private, highly-selective (expensive) liberal arts college, this was The Colby in-house workshop obviously a worry. Though we only graduate about 10 Now to the software. Finding out about this was not too or 12 mathematics majors each year, some 300 or so difficult. Colby has a national reputation for investigating students annually pass through our freshman calculus the use of computers in teaching. This stems largely teaching program. Only a small fraction (about 10%) of through the work of Don Small, the current director of this total was put into the computer-based courses, which the NSF "Computer Algebra System (CAS) Workshop" we still regard as under development. By and large these program, presently finishing a book with John Hosack were amongst our better student entry. on the use of CAS's in calculus teaching (to appear with McGraw-Hill later this year). In addition to Don's work, two of the faculty, Dale Skrien and Bruce Frech, have Computer algebra systems developed their own teaching software for the Macintosh. Let me start with what for me was the bad news, and (And prior to my arrival at Colby last year, I was in the what I think is one major factor behind the wide range of computer-rich environment of Stanford.) responses to our CAS-assisted calculus courses. Hitherto, But as is so often the case, practically all of this these have been structured around Maple and Calculus work has been done on an individual basis. At no stage T /L. Well, I do not think Maple is particularly well­ was a decision made that the department should "go for suited to freshman-level calculus teaching. And Calculus computers." It just happened, quite naturally. T /L strikes me as a disaster. When I arrived as chairman at Colby, I was excited I hasten to repeat that these are my own opinions. by all the activity going on around me, but unsure as They are offered in a constructive spirit, as part of the to just what was being done. So I asked Don Small to on-going development process of such systems. Maple is organize a small workshop for ourselves this last January. a very powerful research tool, and I can see it being a The basic format was: three days; hands on experience useful aid in higher-level mathematics courses involving interspersed with classroom instruction and discussion; a lot of student investigation. And it is only by developing pizza dinners together on two nights followed by a and examining the use of systems such as T /L (which discussion of what had been done during the day. In is designed as a calculus teaching tool-the mnemonic addition to us all becoming familiar with the two large "T /L" stands for "Teach and Learn") that we shall packages used in some Colby calculus courses, I wanted be able to make progress. So I wholeheartedly support each faculty member who made regular use of a certain anyone who sets out to develop any such system, and piece of software in their classes to demonstrate it to the would applaude their efforts. These are early days of trial rest. The workshop was small and intimate. Our faculty and error. consists of ten people. (A typical CAS workshop will The version of T /L we had available had so many have twenty or so participants.) faults, it is hard to know where to begin. (We had a Thus in both format and size, our workshop was pre-sale issue of Version 1.0, which Don told us is "very not quite the same as a typical NSF CAS workshop. close" to the version that Brooks/Cole will be bringing Likewise the intention and expected outcome were not. out later this year.) By their very nature, CAS workshops attract those who Calculus T /L sits on top of Maple, which means that either are, or else very much want to become, involved one is already far removed from the power and elegance in the use of CAS's in their teaching. The participants of the Macintosh graphical interface, and boy does this in our workshop ranged from the strongly committed show. Switching from an operation in one window to and highly experienced computer users to the skeptical one in another requires a clumsy two (separate) mouse computer novices who hardly ever use a computer either clicks. The menu layout within each window is not at for teaching or for anything else. Apart from Don Small, all easy to master. The main instruction method is the none of us were familiar with CAS's. familiar "programmed learning" routine that did not One problem that concerned me before we started work when it was presented in books and does not work was that the results that had been observed at Colby, very much better on the computer screen. The more so as far as computer-based freshman calculus instruction since this part of the program occupies a small, paged was concerned, were highly inconsistent. Some students window at the top of the screen, which means that one would emerge from such a course saying they had got is continually having to click forward and backwards a lot out of it, but there were others who came out of from one page to another. The result of all this to-ing it totally turned off mathematics, if indeed they came and fro-ing is that one very quickly gets lost, and with

APRIL 1990, VOLUME 37, NUMBER 4 419 Computers and Mathematics the loss of location comes the loss of the logic of the symbolic computations. But this package, remember, is argument, loss of the mathematics being presented, and supposed to explain the trapezoidal rule to beginning soon after that plain loss of interest. It took two of calculus students. us, both professors of mathematics with many years of Other complaints. Well, given the level of sophistica­ experience behind us, and both computer literate with tion of present day Macintosh graphics, there really is a great deal of Macintosh expertise, several increasingly no excuse for putting out a teaching program that uses frustrating attempts to follow a simple presentation of two disjointed minus signs and a greater-than sign for a the trapezoidal rule, and even when we had obtained function arrow, the word 'Pi' for the constant n, a freely an answer, it took a lot of effort to figure out what the hanging 1/2 to denote a square root, and the kind of output meant. misleadingly clumsy and just plain ugly boxed expression Figure 1 is a print-out of the screen display we shown in Figure 1 for the error function. (Incidentally, obtained. There was no additional explanation. What it was left to the user to guess what n denoted, and you see is what we got. As you will (might?) see, we likewise a and b, though I suppose even a beginning were trying to integrate sin x from 0 to n using the student should be able to guess this correctly.) trapezoidal rule with 8 subdivisions. Neither of the Following our disastrous experience with the trape­ two graphs we obtained was of the function we were zoidal rule, my colleague and I turned to summing integrating. The second derivative was evaluated and infinite series. Figure 2 shows what we got. Again there graphed, and we were presumably expected to use this were all the problems mentioned above concerning the to figure out the degree of error we were prepared small, paged instruction window at the top of the screen to tolerate, but despite being shown this part of the and the appalling screen display, with the word 'infinity' procedure by someone familiar with the program, we instead of the readily available (in a Macintosh font) were unable to reconstruct this step ourselves, and so oo symbol, and the wandering limits of summation that chose an error bound 'arbitrarily', thereby sidestepping only make sense to one already familiar with summation the most critical part of the entire process. in the first place. Also, there was considerable variation

It: x--> sin- sin

0.645964 harmonic series.

Pi (-l)n I S: n--> A(i) TError: n--> 1/12 ----- f. I.A: n-->-n- . i""' 1 2 !'=1·:·· ...... 1 ---===--~c.oo64596~ tE ....::: ...... ~) 20 • •n If I !=: ~~~~~~~~~~ ! Ia f (x) dx 1/16 Pi (4 sin (1/8 Pi) + 2 2 + 4 sin (3/8 Pi) + 2) - 1.91~231601 4 i -5-> i-.5833333333 i 15 i 1 6 i Figure I. Calculus TjL. Use of the trapezium rule to evaluate the 1=:m:n:m 1 integral of sin x from 0 to 1r. liJ • • • • • • • • • L------~-1 Being smart, Maple (which presumably performed 20 the actual calculation) realized that we were integrating a function that was symmetric over the range of integra­ tion, and collapsed the trapezoidal sum. Thus the final lim S(n) = undefined n->infinity expression contained only four terms instead of the ex­ pected eight, which would doubtless confuse the beginner Figure 2. Calculus T /L. Summation of the alternating harmonic trying to master the trapezoidal rule. And after a lot of series. thought we finally decided that the mysterious hanging First of all there was a mysterious "j" that kept 1/2 in the middle of the expression was an exponent on increasing by factors of 2 every time we asked for a the 2 beneath it, so that the middle term was in fact longer partial sum. We eventually figured out that it was 2v'2 (this being the same as 4/v'2). How did we figure the number of terms summed, but the way this appeared this out? Well, because we were very familiar with the obscured this from us for some time. Admittedly, we trapezoidal rule, familiar with the behavior of symmetric did not have a program manual at hand. After a brief functions, and had enough experience with computers explanation and demonstration of the program by Don (though not computer algebra systems themselves) to Small, we simply sat down and started to use the have a pretty good idea how they would perform their program. But as I argue below, that is exactly what

420 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics is required of good educational software aimed at the What makes for good software? beginning student. Moreover, this is exactly what the The Macintosh system offers such enormous potential Macintosh system is designed to provide, and not to take for introductory-level mathematics teaching, it seems to full advantage of it is to waste an extremely valuable me imperative that we take full advantage of its various resource. features and its capabilities. Good software should stay But far, far worse than the mysterious "j" was the close to the original Macintosh concept, making full use graph of the partial sums of our series. You will notice of the Macintosh system. The user should not have to from Figure 2 that these form a monotone increasing enter lines of syntactically exact code as she does with sequence. For the alternating harmonic series? Are you Maple. Good grief, there are even those pesky semi­ kidding? colons to worry about! (I am talking of the user of Of course, what has happened is that only every a freshman learning package now. For many advanced second partial sum has been plotted. Now you know uses, more standard programming paradigms are both that, and so do/. We also know that the partial sums of reasonable if not downright desirable.) The screen display this particular series oscillate! But what about the poor should be as close as possible in appearance to what freshman student struggling to grasp the notoriously appears in mathematical textbooks. The program should difficult concepts of sequences, series, convergence, and be easy to learn, without instruction, for anyone with a all the rest? To display a graph that hides the crucial modest familiarity with Macintosh operation. Attention factor about the alternating harmonic series that makes should be paid to the presentation of material within it converge, namely the oscillations of the partial sums the Macintosh framework. (The matrix package described about a limit-point, is inexcusable. And then to cap it all, below is a wonderful example of how to take advantage Maple (I presume it was Maple this time) told us that of Macintosh windows.) The program should be robust. the series did not converge, when the one thing the graph (Students rarely do the things expected of them. Neither did indicate was a pretty clear case of convergence! do their instructors for that matter!) The program should At this point, my colleague and I gave up in confusion be written with th ~ knowledge that it is to be used by a and dismay. human being, and that, of the human/computer pair, the But enough, surely, is enough. You get the picture. human is the one with the intelligence and the ability to So let me repeat that I am not at all opposed to reason and make decisions, the computer is the one with people striving to incorporate computer use into their the calculating and information processing speed and teaching. Since the graduates we produce will be seeking the reliability. And, most important of all, the program employment in a world dominated by computers, we should be transparent. The student should spend her would be negligent if we did not do this. But for heavens time thinking about the mathematics, not how to operate sake let us take full advantage of the technology available. the computer. There is simply no need to put on the market a package So what then did we see during our workshop that so that has so many faults as does Calculus T /L. impressed me (and all my colleagues)? As I understand it, Calculus T /L is largely the work of just one person, a mathematician. I applaud his effort and the motivation that lays behind it. But please, sir, Matrix Works don't let this get into the hands of beginning students. Well, the first of these was a program written by Dale Good educational software is very valuable. It is Skrien, a Colby faculty member. Called Matrix Works, also very costly. Except for the simplest of programs this package is designed for performing matrix arith­ (and maybe even then), it can only be produced by metic. The entire program is menu driven, and highly a team effort, involving mathematicians, mathematical visual. Rather than figuring out some fancy method for educationalists, educational psychologists, professional displaying arrays and brackets on the screen, Skrien did designers, and experienced software writers expert with the smart (but perhaps not obvious until you see it) the chosen machine. You would not entrust your child's thing, and used the Macintosh windowing system for his operation to a single doctor working in his living room. matrices. Each window is a matrix. A menu command You would expect a highly trained team having all allows you to specify, for each window you open up, what the facilities they required. Likewise with your child's size of matrix that window should represent. The name education. of the matrix appears in the window title bar. There are But enough about T /L. The good news is that once default matrix names supplied with each new window. you get away from those large, catch-all, CAS-type You can move matrices about on the screen using the systems, there are some really nice packages about. mouse in the usual way, so you can position them side by Before I tell you about the ones we looked at, let me tell side before you (say) add them together, thereby seeing you what, in general terms, I think makes them useful. at a glance if you are trying to add two incompatible

APRIL 1990, VOLUME 37, NUMBER 4 421 Computers and Mathematics matrices. All the usual matrix operations are available shown in Figure 5 (See next page). on the pulldown menus. The transpose of a matrix A Our introduction to this particular package illustrates is denoted by A l T, the inverse of A by A l -1. (A what good software can be like. Our instructor (who had commercially available version ought to replace these by used this program in his sophomore level differential AT and A -I , of course.) The solution vector to a system equations course) asked us to open the program in AX = B is named sol[AX = B] . And so on. In each the usual way, and began to explain how it was used. case, the user is free to rename a matrix at any stage, Within five minues at the most, no one in the room and the system makes all the necessary future naming was paying any attention to what he was saying, and changes this implies. he eventually sat down. Everyone was too engrossed in Figure 3 shows a screen dump of a typical session investigating various of the dozen or so menu-available, with Matrix Works. (Actually, we tidied it up a bit for sample differential equations supplied with the program clarity. But then, this is something that is easy to do with (all well-known examples from physics). Apparently our such a system.) instructor had had a similar experience when he used the r & rile Edit Modify Operations ~ program in class. His role rapidly changed from being u and answerer of the many questions that u• I teacher to helper 1.2 3.0 0.0 1.00 1.00 1.00 came up with. Interest-driven, explorative 6.99 0.00 0.00 0.27 0.38 0.02 the students 4.0 5.0 0.6 1.93 -1.65 -o. 75 0,00 -3.75 0.00 -0.03 0.03 0.06 -0.26 11.42 0.84 from the first moment. Great. 1.0 -2.0 -4.0 -1.04 learning 0.00 0.00 0.77 -0,41 -0.08

B ] detl8l B'T I so1(8H = 8( 1 27.2 1 1.00 2.00 3.00 Stella 0.39 3.0 I I ~·~ I 0.18 let me tell you about one further program. 8'15 Finally, 28 -0.74 (We did in fact look at several others. As I said at 1.25e12 1. 78e12 9. 72e10 2.40 6.00 0.00 r--- 2.41e12 3.44e12 1.88ell 8.00 10.00 1.20 the outset, Colby faculty have been particularly active -3.24e11 -4.64e1 I -2.57e10 2.00 -4.00 -8.00 in introducing computers in the classroom.) This was Stella, from Dartmouth College. This program shares all Figure 3. Matrixworks. A simple, menu-driven, window-based matrix arithmetic program. the features of ease-of-use and robustness enjoyed by the others we saw, though the only copy we had was virus­ Certainly one can think of ways to improve this infested and, being write-protected, we could not clean it program. A menu-available routine for truncating or up, so we had to run it (with care and much disinfectant rounding output matrix entries would be useful. So too afterwards) off the floppy disk, which probably slowed it would be a symbolic mode. (The present version of down a bit, but not enough to notice for the examples the program is purely arithmetic, though it does allow we gave it. for scientific notation.) But as it stands, it is a shining Stella is designed to help with mathematical mod­ example of good Macintosh software. The Macintosh­ elling of real-world phenomena in terms of differential familiar user needs only one instruction in order to equations. It forces the student to adopt a dynamical, become an expert user of this program: that each window logic-oriented approach to the problem, by requiring its represents a single matrix. The rest works just as you principal input in terms of a flow chart, as shown in would imagine. No need for a manual or an extensive Figure 6 (See page 424). This particular flowchart (which help facility. No need for an instructor standing at your the user enters using the mouse and a small palet of shoulder all the time. MacDraw-like symbols in the obvious way) models the growth of an infection in a closed community. There are DEGraph three groups of subjects, those susceptible to the infection The next program that was demonstrated for us was DE­ who do not yet have it, those infected, and those immune Graph, written by Henry Pinkham of Columbia Univer­ (this latter category could include those who have already sity. This is designed to provide graphical representation died as a result of the infection). Passage from one cat­ with routines for their of differential equations, together egory to another (which is particularly simple in this numerical solution. Figure 4 (See next page) shows one example, but could involve far more complex flow-lines) output from this program. What you see is the vector is indicated by a directed flow-pipe with a control tap (to field (shown using small arrows) for the solutions to the represent the rate of flow), in this case taps for 'infection' equation and 'recovery' (or 'death'). The presence of quantitative y' = sin(xy) dependencies are indicated by means of arrows. Thus, together with a family of particular solutions. The pro­ in the diagram, we see that the rate of infection depends gram also allows for the representation of time-dependent upon both the number of susceptibles and the number differential equations, such as the undamped pendulum: of infectives. The precise relationships are entered by .X = - sin y , y = x hand on a standard Macintosh-style questionnaire box

422 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

I I ' ' ' ' ' ' ' ' ' ( ' ' I ' ' ' ' ' ( ( ' ' I ' ' ' ' ' ' I I ( ' ' ' ' ' ' ' ( ' / ' '

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/ (

/ ( ' ' ' ' ' ' ' ' I / ' ' ' ' ' ' ( I ' ' ' ' I ' ' ' ' / ' ' ' ' ' ' / ' / ' Figure 4. DEGraph. Vector field and a family of particular solutions to the differential equation y' = sin(xy).

~ ~ / / / / / I / ' ' ' ' ' ' ' ' '

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/ / / / ' ' / / / I ' ' I I I / / " ' ' ' I I ' 'I ' ' I I / / / ' ' / I / ' ' I I / / ' ' / /

' ' / / / '' I I I ' ' ' ' / / " ' I ' I ' I I / I I ' ' / / / ''I I I ' ' ' ' / / "''"I I / I I ' ' / / / / I '\ ' I I I I I I I / / ' ' ' ' ' ' ' ' ' ' ' ' Figure 5. DEGraph. Vector field and some particular solutions to a time-dependent differential equation in two dependent variables.

APRIL 1990, VOLUME 37, NUMBER 4 423 Computers and Mathematics that the system displays at the appropriate time. In this (This particular system of equations is, in any case, not case, they are solvable by analytic means.) It transpired that there were S =-aS/ a number of theorems associated with this particular i =aS/- bl model, that provide a quantitative explanation (in terms of the initial values and the two parameters, a and b) R=bl of why some infections lead rapidly to an epidemic (where S denotes the number of susceptibles, I the while others simply do not get off the ground. I was not number of infectives, and R the number of 'removed' or familiar with these results, and doubtless they involve immunes, and where a and b are constants). some sophisticated mathematical analysis. But a few runs of the program using different parameters were all that was required for me to get a good sense of what these theorems said.

infection recovery Conclusion So what can we learn from all of this? Besides the Figure 6. Stella. Flowchart depicting the spread of an infectious specific issues of design and use mentioned above, just disease. The program interprets this mouse entered diagram as a this: The simpler the package the better. All-purpose differential equation. programs such as Maple and Mathematica are excellent Figure 7 shows the output graph for the particular for research and for use by more advanced, experienced, values a = 0.0003 and b = 0.095, starting from initial and highly computer-literate students. But they present values of S(O) = 999, /(0) = 1, and R(O) = 0, represent­ the beginning student (and the beginning mathematics ing the spread of an infection in a closed community of faculty member!) with such a huge obstacle that all but 1,000 souls starting with just one infected person. This the most dedicated (and probably some of them as well) graph clearly shows a pattern familiar to all those on fall by the wayside. Rather a toolbox of small, inde­ college campuses during the winter colds and flu season. pendent, user-friendly, well-designed, attractive, robust programs is what is required. 1 susceptible& 2 infectives 3 immunes Thus, if you are designing a computer teaching il 1000.00 aid, you should avoid the temptation to add every 1~ :s---- conceivable enhancement. Remember that your program 1 /~ il 750.00 is intended for use by a beginner. Your program should be transparent (to any user already familiar with the

500.00 Macintosh environment, or whatever). The student has il enough to cope with mastering the mathematics we throw on 250.00 \I at her. If a lengthy period of instruction is required il how to use the program, it has failed before it begins. an instruction manual. 0.0 • I I I I ,...... , Likewise if the program needs il 0.0 ~'=·-25.00 50.00 75.00 100.00 Though any good computer package should, of course, Time be accompanied by (well-written!) full instructions, just Figure 7. Stella. The output graph produced in response to the in case something does seem to go wrong, the average model shown in Figure 6. In this case, the infection reaches its peak just over thirty days after being introduced by a single carrier, and thereafter user does not want to bother taking this document out tails off, though the number of those that remain susceptible to the of the cellophane. Like many others, I have a tall stack disease does not fall to zero, providing mathematical confirmation to of Macintosh manuals that came with my machine. But the oft-observed fact that infections rarely die out altogether. I hardly ever open them, and some I have never opened! What happened when this particular program was (The one exception was when I had problems making demonstrated to us using an overhead screen-projection a hard-disk backup, and then, just as it should, the (I have already mentioned that we were forced to use manual explained what the problem was, namely that just one, disk-locked copy of this program), was that the backup program does not run reliably in conjunction a fascinating discussion rapidly developed concerning with Multifinder.) the significance of the two constants a and b, both in This is exactly the same as with all the other devices terms of the mathematical behavior of the model and I have in my home, my TV, my video-recorder, my hi-fi what these constants represented in real life. In other and my CD-player, my washer and dryer, my stove, and words, we immediately began to investigate the model, my car. I do not want to have to study a manual before without getting bogged down in details of the calculus. I can start to use them.

424 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

I list all of the above items deliberately, to emphasize for examples. I find Devlin's conclusion about the use that good educational software (indeed, good software in of CAS without foundation and, even worse, that his general) should allow the user to use his computer like methodology for investigating powerful, new software any other household item. Transparently. packages as teaching tools is seriously flawed. It seems to me that one of the problems we face at the In spite of the fact CAS' are far from perfect pro­ moment with regards to mathematics software is that, grams, they are having a dramatic impact on calculus not surprisingly, it is being developed by people who courses. Curricula are changing, new teaching methods like using computers, who love designing and writing are developing, and new methods for solving prob­ computer programs, and who have had many years of lems (which include step verification) are evolving. One experience both as mathematicians and computer users. wonders then, what was the context for Devlin's in­ What with all that excitement, the poor end-user can so vestigations? His conclusion is that for all entry level easily be forgotten. And then we are left bewildered when students, all kinds of courses, all teaching methods, and some of those end-users do not embrace our products all instructors, CAS should not be used. What an incred­ with the enthusiasm our creations evoke in ourselves. ible conclusion. So quickly obtained. In fact, there are Though I cannot claim to be a 100% convert to many interesting, 'successful' projects underway. Some Heideggerian philosophy, I lived in the Bay Area long of these will be covered in two MAA Notes Volumes enough for some of it to rub off, and I cannot see any currently being prepared. better way to end this article than by advising anyone who And now to Calculus T /L. The most bothersome has not already done so to read the book Understanding thing about Devlin's comments is that they are based Computers and Cognition: A New Foundation for Design, upon totally false ideas about what Calculus T /L is or by Terry Winograd and Fernando Flores (Addison­ how it should be used. I am an expert in such matters Wesley, 1986). The classroom computer should be like because I am its author. the Heideggerian hammer described in this book. You Calculus T /L began as a project to turn a command­ should be able to use it without being consciously aware line driven computer algebra system, Maple, into one of its presence. that utilizes the graphic interface of the Macintosh. It The computer, and in particular the Macintosh and implements a number of innovations including easily machines like it, has presented us with a tremendous reusable results, concrete mathematical objects that stu­ opportunity to elevate our mathematics teaching to a dents can "get their hands on", facilities for developing new level. But it has to be done properly, or it is best pattern matching skills, and an object-oriented drawing not done at all. Let's make sure we get it right. program environment for assembling solutions. A goal was to enhance the level of communication of symbolic information so that some basic cognitive problems of Almost No Stuff In, Wrong Stuff Out students might be investigated. Again, the issues run deep. Many fundamental questions remain open. User J. Douglas Child* interface design is an art that depends heavily upon trial and error. One of the reasons for having Calculus T /L I'm somewhat disturbed by the article written by Keith published is to make its new ideas (both good and bad) Devlin titled "The Right Stutr'. But then I've been available for serious examination by others. What's good using a CAS in the teaching of calculus for four years. will be used in future efforts by others; what's bad will be The issues are quite subtle and run more deeply than lost forever. I never dreamed that T /L would be ripped Devlin ever takes time to imagine. The assumption apart in print before it was even published. It's time to that only "appliance" software (i.e., simple and sharply­ consider the basis for Devlin's comments. focused software) is valuable for teaching beginning The main topics of the T /L portion of the previous students runs counter to all of the experiences I've had. article are the trapezoidal rule and a particular alter­ I believe appliance software is desirable, but certainly nating series. While considering these examples, Devlin not necessary. Given the conclusions of the article, how believed he was using integral parts of Calculus T /L, can it be true that our non-major calculus students but he was actually using parts of T /L documents which easily use a CAS to help them with the computational are provided as examples of the kinds of activities T /L aspects of calculus? I've spent many hundreds of hours can be used to support. This fact is stated on a card observing and interacting with students as they use that comes in the T /L product box. In fact, these ex­ computer algebra systems to solve problems and look ample documents are advanced examples which assume a semester and a half of experience with Calculus T /L. There is nothing like jumping right into advanced "toing *Dr. J. Douglas Child is the Bush Professor of Computer Science and Mathematics at Rollins College, Winter Park, Florida 32789. His and froing". T/L documents are easily changed. If an email address is [email protected]. instructor doesn't like a document he can easily change it

APRIL 1990, VOLUME 37, NUMBER 4 425 Computers and Mathematics or write his own. The problems mentioned in the article typrmtmg of mathematical expressions. Prettyprinting were ameliorated in the released version of T /L by the everything that Maple produces is a difficult job. The addition of a few comments. No changes to T /L were version currently running on my desk contains none necessary. In most cases instructors will want to at least of these flaws. The testing cycle is quite long so these develop supporting paper documents and worksheets. changes won't appear until an early update of Calculus Pencil and paper activities normally precede computer T/L. activities, especially for novice users. Note that T /L For students who only use T /L to compute deriva­ documents are similar to Mathematica Notebooks. The tives, integrals, ... and draw graphs, it is well worth its trapezoidal rule document simply contains most of the modest price. For instructors who are looking for a pow­ commands necessary to obtain information about its erful tool for implementing student learning activities, it use for a particular definite integral. Students use this may be a real bargain. I wish it were better; with your information as a basis for constructing complete solu­ help it will be. Constructive criticism is welcome. tions/discussions by adding textual comments, graphs, and tables. These are fundamental T /L operations en­ countered in the first or second week. Reviews of Mathematical Software By the time students consider alternating series they will know about how to look for subsequences when considering the behavior of functions near points as well as when investigating the behavior of sequences. The sample documents that deal with limits of functions are FFTLIB by PC Scientific much more detailed than the document used by Devlin Reviewed by Herb Holden because limits are normally considered near the beginning Gonzaga University of calculus. One of these documents was recently used by non-major applied calculus students. No harm was FFTLIB is a library of FORTRAN subroutines for done; no problems or complaints heard. Instead, the performing a variety of discrete Fourier transforms on students are going back to their high schools and telling IBM PCs. The product is distributed on 5 1/4 or 3 1/2 their teachers about T /L. Note that the instructor of diskettes and consists of object code libraries of FFT the course uses weekly student journals as a vehicle for algorithms, a demonstration program, eleven interactive seeing how experimental activities are really received by test programs, seventeen skeleton programs to facilitate students. program development, and a 340 page tutorial and Various comments made throughout the discussion of reference manual. The FORTRAN compilers which are T /L show that Devlin has a different view of T /L than supported are: its author. Statements like "The main instruction method • IBM Professional FORTRAN 1.30 is the familiar 'programmed learning' routine ... ". "But, • Lahey FORTRAN 2.22 this package is supposed to explain the trapezoidal • Microsoft FORTRAN 3.31 or 4.01 rule ... ", and "loss of the mathematics be presented" • Ryan-McFarland FORTRAN 2.42 seem to describe something other than T /L. Where did A hard disk is strongly recommended. A numeric he get such ideas? For one thing he is proud of not coprocessor is desirable as well as a graphics card having "the 'manual' at hand". Calculus T/L comes (Hercules, CGA, EGA, MCGA, or VGA). with a User Guide that begins by describing what The various transforms which may be accommodated T /L is and how it might be effectively used. Chapters are: titled Functions, Graphing, Limits, Derivatives ... follow. One Dimensional Calculus T /L is simply a new kind of tool to be used Real data sequences of arbitrary length; by able instructors. It is not a program that claims to Sine transform of odd real sequences of arbitrary length; teach (and learn?). Most of the sample T/L documents Cosine transform of even real sequences of arbitrary supplement student learning activities; they are never length; intended to be presentations to passive students. Complex data sequences of arbitrary length. The main reason Devlin missed the mark is that he Special subroutines are provided for sequences with asserts that CAS and T /L are not appliances, and then length a power of two, half-wave symmetry, and quarter proceeds to investigate T /L as if it were an appliance, wave symmetry. and hence naturally concludes it is a disaster (as an appliance?). In some sense T /L may be better than an Multi-dimensional appliance; it can be used to create them. Real or complex data of length a power of two in each I agree with Devlin's criticisms about poor cosmetics. dimension; Last summer we wrote code that improves the pret- Real or complex data of arbitrary length;

426 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

Real or complex data stored on disk and length a power PC Scientific, Inc. of two in each dimension. 4 710 Debra Lane This library evolved from the software package FFT­ St. Paul, MN 55126 PACK developed by Paul Swarztrauber at the National Their phone number is 612-830-1232. Other phone Center for Atmospheric Research. The primary algorithm numbers of interest are McGraw Hill at 212-512-2000 used is Stockam's algorithm [1] whose speed increases and HBO at 212-512-1000. when the length of the data sequence has many factors. Some best case and worst case results for the forward References transform of a real data sequence of length n are: [l] Author: Rodrigue, Garry Title: Parallel Computation Powers of two Primes Call #: QA 76.6 P348 1982 [2] Author: Oppeheim, Alan and Shafer, Ronald n sees n sees Title: Digital Signal Processing 64 .16 61 2.47 Call #: TKSI 02.5 0245 128 .44 127 10.77 [3] Author: McClellan, James and Rader, Charles 256 .99 251 42.24 Title: Number Theory in Digital Signal Processing 512 2.31 509 174.06 Call #: TKSl 02.5 M216 1979

These times were obtained on an IBM PS/2 Model 30 286 (10MHz) without a coprocessor. The test program was compiled with Microsoft FORTRAN 4.1 using the Three Programs for the NeXT: compile time options j AL /FPc. (This is the only Groups, Kubik Algebra, and Orbit FORTRAN compiler I have used on a PC. It seems to Reviewed by Suzanne M. Molnar* be an excellent compiler, however, the environment is reminiscent of a punched card system.) Among the first mathematics applications written for the The library subroutines only process real data of type NeXT computer are Groups, Rubik Algebra, and Orbit, REAL which provides six decimal digits of accuracy. written by Charles G. Fleming and Judy D. Halchin Since higher precision real types cannot be used, the of Eastern Illinois University. Since these are the first value of a coprocessor is increased speed only. Any reviews of NeXT software to appear in this column, a increase in accuracy is marginal. few words about the machine, which was developed with The documentation is good and ample examples are the higher education market in mind, are in order. provided. The index and the bibliography are rather The standard NeXT computer comes with 8 megabytes brief. Source code is not provided. of memory, a 256 megabyte erasable optical drive, and This software is marketed as a joint venture between now includes a 40 megabyte "accelerator drive" which PC Scientific and McGraw Hill Book Company. PC serves to speed up response time from that of the orig­ Scientific was most helpful in providing information inal machine. A microphone jack and speaker are built about their product. I called six different offices of the in which provide CO-quality sound. Also included is a McGraw Hill Book Company and could not find anyone 17-inch monochrome, flat screen monitor having a 1120 who had heard of this software. (Just what every software x 832 x 2 resolution, a keyboard, and a two-button developer dreams of.) Each McGraw Hill representative mouse. The computer itself is a one-foot cube which is was sympathetic, provided me with the phone number attached to the adjustable display by a three-meter cord of another McGraw Hill office to try, and wished me allowing the user to place the unit on the floor leaving good luck. (Once I was given the number of HBO in only the small "footprint" display device on the work New York.) My impression of McGraw Hill is that their area. For this review I used a machine with 12 megabytes staff is very courteous. memory and an optional 330 megabyte hard disk. PC Scientific also markets other subroutine libraries. System software includes Mach/UNIX®, which al­ They are: lows for networking and multitasking. The latter is MINPACKJ-LIB for nonlinear systems; reflected in multiple open windows and the ability to FITLIB for curve fitting (reviewed in Notices 3/89); have several programs running at one time. Also in­ SPARSGEM for sparse systems of linear equations; cluded which is of interest to program developers are QUADLIB for numerical integration; Interface Builder™, a programming tool part of the ODELIB for systems of first order ordinary differen­ tial equations. *Department of Mathematical Sciences, College of St. Catherine, St. Paul, MN. A special thanks to the Department of Mathematics and Each package sells for $350. Further information can Computer Science, Macalester College, St. Paul, MN for the use of one be obtained from: of their NeXT computers.

APRIL 1990, VOLUME 37, NUMBER 4 427 Computers and Mathematics

NextStep® object-oriented environment, and the pro­ this column [3]. It has the same features as that program gramming languages Objective®-C (in which these pro­ with the additional ease of use of the NeXT interface. As grams were written) and Allegro CL® Common Lisp. seen in Figure 2 (See next page) the Control Panel allows Among the bundled applications are Mathematica™, the user to manipulate the cube either using buttons or by GNU T£X, WriteNow™, a word-processor, electronic entering one's own sequence of face rotations. 'Turn the mail with voice, and Digital Library, including Webster's cube' turns the entire cube 90 degrees in the direction Ninth New Collegiate Dictionary® and, for those who indicated with respect to the front face. Repeatedly may need it, The Oxford University Press® Edition of selecting 'Undo' takes one backwards through a sequence William Shakespeare: The Complete Works! At a higher of moves. 'Redo' redoes an undo. In order to repeat a education price of $6,495 for the standard configuration sequence of moves one specifies the number of times to of hardware and software outlined above, the NeXT perform the sequence. computer is competitive with similarly configured UNIX The face rotations may be selected by using the systems and the Macintosh IICX. To serve the needs of buttons 'Front', '-Front', etc., where the unsigned word the network user the computer is also available without indicates clockwise and the signed word is counterclock­ the optical disk drive for a cost saving [1]. wise. The same convention is used for entering one's Groups. This program can help students learn about own sequence of moves, with the letters f, b, I, r, u, all of the finite groups of orders up to 20. After making d corresponding to David Singmaster's [4] Front, Back, a selection, a group table is displayed. Figure 1 (See next Left, Right, Up, Down faces, respectively. Figure 2 shows page) illustrates the various options available for the the cube after the sequence -r 2u -f has been repeated dihedral group D4 (the notation used in the program). 3 times. This sequence twists three corners clockwise, Upon selecting 'Inverses' and 'Orders', respectively, ta­ as seen in the Cycle Decomposition window, which lists bles are displayed of the group elements, their inverses corner cycles and edge cycles. and orders. In a similar fashion the 'Conjugacy Classes' The program has its own library of sequences of option displays a list of the conjugacy classes of the moves; it is an easy task to add your own favorite move group. sequences to the library. Help is available for using the The 'Subgroups' option has several sub-options. The 'Control Panel', 'Cycle Decomposition', and 'Library of user may choose to list all of the subgroups or all of Sequences' options. Figure 2 shows the Help window the normal subgroups of the group under consideration. for the Control Panel displayed. For someone using this When more subgroups e?'-ist than fit in the window, a program to 'play with the cube' this is a nice reminder slider can be used to move through them (See Figure to have readily available. What is also extremely useful 1). The 'Cosets' and 'Quotient Group' options display is that you can press 'Unscramble' if you get the cube lists of all subgroups and normal subgroups, respectively. hopelessly out of order - unlike the real thing which has After clicking on a subgroup, the list of cosets is displayed no easy way out for those who are not expert cubists! or the quotient group table is displayed. The 'Center' As with the IBM-compatible version [3] it is best and 'Commutator Subgroup' options list elements of the suited for "individual investigation" for motivating and group, shading in the center and commutator subgroup exploring concepts in group theory. Together with David elements both in that table and in the group table. The Singmaster's book [4, 5] one can learn a lot of group one feature of the program which allows the user to more theory from Rubik's Algebra. The NeXT version shows actively interact with the group is the 'Test a Subest' the cube in white and five shades of grey, some of which option. Here the user may select a subset of the group can be difficult to distinguish. I performed the operations in order to determine if it is a subgroup. The selected on a real cube to convince myself that the cube illustrated elements are shaded in the group and cells flash where in this review has all six colors shown after the sequence closure fails. of moves. This is the only drawback to the program, The NeXT interface with its multiple open windows which is due to the fact that NeXT does not currently displaying the information about a group and the use provide for color, which is to be available this year. of Greek symbols were welcome to this reviewer. In Orbits. This program allows the user to experiment comparison to Exploring Small Groups [2], which allows with the nonlinear dynamical system obtained by iter­ for a more open-ended, flexible, discovery approach, this ating the function f(x) = Ax( I - x), by investigating program is much more static. However, with appropri­ the orbit -x,f(x),f(f(x)), ... - of a point in the unit ately designed exercises, students may find Groups a interval [0, 1]. The system is often described by the useful aid to help them clarify the concepts incorporated finite difference equation Xn+l = Axn(l- Xn). Once an into the program. initial value is selected the behavior of the system is Rubik Algebra. This program has the same name and determined. The program initially presents the window is by the same authors as the one previously reviewed in as seen in Figure 3 (See page 431 ), obtained from the

428 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

Figure 1

lu.f Utl

r right race 1 left race blu \lhl lub u vp t.ce 4 down race. lli.nut Ugn in troat of one or then aearu: coW'Itercloekvin. bru rub Ubr aay indicate a clockviu rotation vith a plus eign in front letter, if you wish. Putt1nq a digit in front of a IIIIIEC'IOLES ¥111 c.un it to be perforud Jl\Lltiple tiaes. BJ.anb; be iabedded in the sequence. rot exuple, tu fd llr ..,.. Jaa-dd a ..ne •rotate the 4o9n race 270 degrees rotate the up race 90 deqnu cl.ockviae, tbe back face 90 clegt:eea c~v1ee, rotate the .right 180 degreee counterclodr.Yiae, and then rotate the down tao degt••• clodwi...

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APRIL 1990, VOLUME 37, NUMBER 4 429 Computers and Mathematics

'Graph the Asymptotic Orbit' option which displays Developers are coming up with various strategies for 1000 terms of the orbit beginning with the IOOlst term. handling the distribution of application software. At One may change parameter A, the initial value for x, a cost of $50 per optical disc this can be expensive and the number of iterations either by using the sliders and inconvenient. To purchase these programs one must or entering specific values. Using the latter method to send the authors an optical disk onto which they will make successive changes to parameter A, then selecting transfer the programs. Address: Halchin and Fleming, the 'Graph the Entire Orbit' button and getting the 2122 Reynolds Drive, Charleston, Illinois 61920. Groups same graphs, I finally realized that although the numbers is available for $20 plus $5 for shipping and handling; change visually, the new values are not active. One must Rubik Algebra is available for the same price plus press Return to register the change, which is indicated shipping and handling. Orbit is free with the purchase of by movement of the slider. At long last I do not press Rubik Algebra. Return at the end of each line of text - old habits of References using the Return key had to be re-learned. [1] NeXT, Inc., 900 Chesapeake Drive, Redwood City, CA Figure 4 (See next page) shows the same orbit as 94063. Figure 3 with the same initial value only using the 'Graph [2] Molnar, Suzanne M. Exploring Small Groups- A Review. the Entire Orbit' button for the first 100 iterations. Figure Notices of the AMS 36, 10 (December 1989), 1358-1361. [3] Sand, Mark. Rubik Algebra. Notices of the AMS 36, 9 4 also shows other options available from the main menu, (November 1989), 1207-1208. namely the 'Parameter Bounds' option for changing the [4] Singmaster, David. Notes on Rubik's 'Magic Cube'. Pub­ bounds on A, and the 'Composition' option which shows lished by the author 1980. a graph of the function f(x) composed with itself a [5] Rubik, Erno, et a!. Rubik 's Cube Compendium. Oxford specified number of times. None of the graphs has the University Press, 1987. axes labeled, which would be useful. [6] Jensen, Roderick V. Classical Chaos. American Scientist As the figures in this review indicate, the program 75 (March-April 1987), 168-181. gives experimental evidence of the chaotic nature of the system for A = 4. It was easy to generate other interesting graphs to illustrate convergence to zero, to Phaser a fixed point different from zero, and to several fixed points, as the parameter A increased from 0 to 4. This Reviewed by Marvin S. Margolis* program provides an easy way to discover the transition from order to chaos of the logistic map, a simple Introduction mathematical model for such phenomena as turbulence Phaser is an animator/simulator for dynamical systems in fluids and evolution of populations. [6] that runs on IBM and compatible personal comput­ The voice output capabilities of the NeXT are used ers. The software is included in a combination manual to tell the user when an error is made upon entering data and illustrated guide to experimental dynamics. The and to describe the error. This can be startling the first manual's formal citation is Differential and Difference time it happens, but I personally prefer the technique Equations through Computer Experiments, Second Edi­ to an obnoxious beep and no indication as to what the tion, by Hliseyin Ko~ak, New York: Springer-Verlag, problem may be. One small irritation is that I could 1989, pp. 224. not resize the main window (Figure 3) without unusual According to an article in Academic Computing, things happening which included the disappearance of a magazine covering computer use in higher educa­ some sliders. When this happened it was best to quit the tion, judges at the third annual EDUCOMJNCRIPTAL program and to start over. Higher Education Software Awards competition rated Phaser the best mathematics software of 1989. The ar­ Conclusion. Each of these programs provide the user ticle's mathematics reviewer praised Phaser as ". . . an with the ability to learn a specific topic in mathematics explorational [educational] tool that is inviting, engaging, with little (if any) need for external documentation. The and instructive." only documentation provided is installation instructions. I also like the program because it motivates the Each program has a 'Help' option in the main menu. If study of dynamical systems. When I first tried Phaser one has used a Macintosh, learning to use the NeXT for it displayed "phase portraits" for differential equations these programs is easy. However, the full potential for that I found puzzling. As a result I began reading the NeXT and mathematics software is in the future. the referenced literature on differential equations and The authors have plans to develop other software using dynamical systems. As I studied the references, I began Mathematica™ with a user friendly interface. Together with the NeXT Interface Builder™ other mathematics *Marvin S. Margolis is an Assistant Professor of Economics at software is sure to follow. Millersville University.

430 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

Figure 3

Figure4

APRIL 1990, VOLUME 37, NUMBER 4 431 Computers and Mathematics to understand dynamical systems better than I ever had into Phaser at a later time for demonstrations or further in the past. Without the program I doubt I would have work. been so motivated. The Manual Hardware Requirements The author divided the 224 page manual into three To run Phaser, one needs an IBM PC, XT, AT, PS/2 parts and two appendices. Part 1 introduces the ideas or compatible, a DOS Version 2.0 or higher operating underlying differential and difference equations and the system, 256K-bytes of memory and an IBM CGA, appropriate terminology. In an overview the author also EGA, or VGA monitor. A color display is best although explains the numerical algorithms he uses to solve the a monochrome monitor will work. With monochrome equations. Phaser provides three algorithms to solve some colors (such as red) may be invisible. equations numerically, an Euler, an Improved Euler, and a fourth-order Runge-Kutta algorithm. How Can One Use Phaser? Part 2 comprises the formal user's manual and begins Lecturers can use Phaser to supplement beginning with a 15 "lesson" tutorial illustrating almost all Phaser's and advanced courses in differential/difference equa­ features. A reference guide follows that contains detailed tions. Phaser presents specific examples of differen­ information about the entries of the three main menus tial/difference equations ranging from the most rudi­ and nine graphical views. Appendix A provides a quick mentary, appropriate for the beginning student, to the reference guide. highly complex, suitable for the research mathematician. Part 3 catalogues the differential/difference equations Used alone it can not substitute for the inherently difficult stored in the permanent library. The author sorts the study of the theory of dynamical systems. Rather, it is a equations by dimension and difference/differential type, complementary tool for doing mathematical experiments determined by solution algorithm. The order of the and illustrating theory with specific examples. differential equations ranges from one to four dimen­ The author of the program and manual, Hiiseyin sions while the difference equations span one to three Kocak, is a professor of mathematics at Brown and dimensions. Appendix B lists all difference/differential Miami Universities. In conceiving Phaser his motivation equation names stored in the library and their brief was to create a ". . . work station for experimental descriptions. dynamics, one where students not only can get a taste The manual contains 108 illustrations. These reduced of the current excitement in the field, but also perhaps sized screen dumps show the results either of various discover new dynamical phenomena in an environment Phaser operations or particular equation solutions. I free from extensive programming efforts." found the figures especially helpful when first learning the program. The references at the end of the manual help also. How Phaser Works Because Phaser only uses menus, it requires no com­ puter programming knowledge and is easy to use. The Difference Equations user first creates, with the help of a menu, a suitable Among mathematically oriented commercial software window (screen) configuration. The configuration may that I have tried, Phaser uniquely solves difference equa­ display a combination of views-phase portraits, texts tions. As with differential equations, Phaser concentrates of equations, Poincare sections, etc. on the geometric ideas associated with the difference Next, the user can specify, from another menu, var­ equation theory. For example, it uses stair step diagrams ious choices in preparation for numerical computations. to analyze one-dimensional difference equations. One can choose, for instance, to study preconfigured The solutions for difference equations involve more equations from a library of several dozen equations. One complexity than the comparable ones for differential quickly appreciates the value of a preconfigured equation equations. For one dimension, the library provides equa­ once one goes through the trial and error process of tion examples with not only "fixed points" and "periodic entering a new equation into the library. Then one can orbits," but also "strange attractors." compute solutions of these equations with different ini­ tial conditions or step sizes, while interactively changing The Library of Equations parameters in the equations. A main feature of Phaser is its library of over 61 pre­ From another menu, the user can manipulate the configured equations. For each dimension, the author solutions graphically, for example, either rotating the orders the equations roughly by increasing level of diffi­ images or taking sections of them. During simulations, culty. One quarter of the library provides examples for one can save the solutions as a ( 1) hardcopy image of the an undergraduate course in differential equations. An­ screen, (2) printed list, or (3) form that one can reload other quarter the author designed for a graduate course.

432 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Computers and Mathematics

The rest of the library reflects his own mathematical Two-Dimensional Difference Equations inclinations. 1. dislin2d - General two-dimensional discrete linear Ifthe library does not contain one's favorite classroom system example or research problem, the program permits one 2. gauss - Fast computation of elliptic integrals to formulate and add a new equation. Phaser has enough 3. quad1 -A quadratic map on the plane space for about forty user-defined equations. The author chose the differential and difference equa­ Third-Degree Difference Equations tions for the library in order to study their qualitative 1. dislin3d - General three-dimensional discrete linear properties. In the first stage in that process, one deter­ system mines an equation's phase portrait by varying the initial 2. act - A three-dimensional cubic map conditions. One would like to know the limiting behavior of all the solutions of an equation. These sets grow in Chaos complexity as the dimension of the equation increases. As one would expect given the role that computers play In the second stage of the qualitative study, one in the new science, Phaser helps illustrate many ideas explores the possible changes in an equation's phase por­ in chaos theory. James Gleick's best-selling book about trait as one varies the equation itself. In applications, for chaos provides historical details about several equation example, many models contain changeable parameters. systems in Phaser's library.* Conversely, Phaser provides By changing the parameters one can investigate the "ro­ concrete examples of some dynamic models discussed in bustness" of the system under small perturbations. "Bi­ Gleick's book. In particular, Chaos and Phaser feature furcation" theory involves studies of qualitative changes the Lorenz, pendulum, and logistic equations. Both also in the phase portraits of dynamical systems as one varies illustrate the idea of strange attractors. parameters. The author designed many examples in the When I first tried Phaser, I was unclear why a "phase library to illustrate "typical" bifurcations. portrait" constituted an important differential equation To give the flavor of the library equations, a list of graph. Phaser's brief manual offered little help. Gleick some early equations in each library category follows: (page 134) explained it most elegantly: One-Dimensional Differential Equations "The strange attractor lives in phase space, one of the 1. Cubicld- General one-dimensional cubic differen­ most powerful inventions of modern science. Phase tial equation space gives a way of turning numbers into pictures, abstracting every bit of essential information from Two-Dimensional Differential Equations a system of moving parts, mechanical or fluid, and 1. linear2d - General two-dimensional linear system making a flexible road map to all its possibilities." 2. pendulum- Nonlinear pendulum on the plane 3. vanderpol - Oscillator of van der Pol: A unique Other Differential Equation Software limit cycle Other mathematically oriented commercial software is 4. prcdprey - Predator-prey equations, competing capable of solving differential equations. As far as I species, etc. know none are as thorough as Phaser nor do any attempt to solve difference equations. For example, Math CAD Three-Dimensional Differential Equations sells an advanced mathematics applications pack as 1. lorenz - The most famous strange attractor an extra cost addition to the regular program. Among 2. linear3d - General three-dimensional linear system other mathematical tools, the math pack features three 3. vibration - Periodically forced linear vibrations differential equations models, solution of a ( 1) first­ order differential equation, (2) second-order differential Four-Dimensional Differential Equations equation, and (3) system of differential equations. 1. harmoscil - A pair of linear harmonic oscillators Matlab 's functions for solving differential equations 2. kepler - Kepler and anisotropic Kepler problems include: ( 1) ode23 - second/third order equations us­ ing the Runge-Kutta solution method (2) ode45 - Library of Difference Equations: fourth/fifth order equations using the Runge-Kutta­ One-Dimensional Difference Equations Fehlberg method. 1. logistic - The logistic map - the one that started it Gauss sells an extra cost addition called SimGauss all that allows one to simulate dynamic models. Since it 2. dislin 1 d - General one-dimensional discrete linear is not menu driven it is probably harder to use than equation 3. discubic- General one-dimensional cubic difference *James Gleick, Chaos: Making a New Science, New York: Viking equation Penguin, Inc., 1987.

APRIL 1990, VOLUME 37, NUMBER 4 433 Computers and Mathematics

Phaser. I do not know whether SimGauss possesses more Conclusions dynamic analyses capability than Phaser. I concur that Phaser is " ... an effective tool for perform­ I am unaware of other mathematically oriented soft­ ing mathematical experiments and for illustrating the ware that solves differential and difference equations. theory of dynamical systems with concrete examples ... " Phaser's easy to learn menus, ready to run equation li­ as the review of the second edition in Academic Comput­ brary, and convenient graphing provide advantages over ing stated. It can help students and probably researchers the above-mentioned competition. also to access a sophisticated and elegant part of mathe­ matics.

The Collected Papers of /R. H. BING/ Sukhjit Singh, Steve Armentrout, Robert J. Daverman, Editors

A powerful mathematician and a great historical background on Bing's life and 1980 Mathematics Subject Classifications: problem solver, R. H. Bing laid the achievements. 57, 54, 26 foundation for a number of areas of ISBN 0-8218-0117-1, LC 88-14445 topology. Many of his papers have This collection will appeal to 2 volumes, 1680 pages (hardcover}, July 1988 continued to serve as a source of major mathematicians in all areas, and especially those in topology, as well as List price $162 , Institutional member $130 , theoretical developments and concrete Individual member $97 applications in recent years. One students, historians, and educators in the To order, please specify COLBING/NA outstanding example was Michael H. mathematical sciences, for it provides Freedman's use of Bing's Shrinking a complete historical summary of the Criterion to solve the four-dimensional mathematical events in the life of the Poincare Conjecture. man and the mathematician, R. H. Bing. All prices subject to change. Shipment Contents: will be made by surface. For air deliv­ ery add, I st book $5, each additional This two-volume set brings together I. R. H. Bing: An introduction; An editorial book $3, maximum $100. Prepayment over one hundred of Bing's research, preface; R. H. Bing: A study of his life, required. Order from American Mathe­ expository, and miscellaneous papers. by S. Singh; A chronology of R. H. Bing; matical Society, P.O. Box 1571, Annex These works range over a great variety of Ph.D. students of R. H. Bing; R. H. Bing: Station, Providence, Rl 02901-1571, or topics in topology, including the topology October 20, 1914-April 28, 1986, by R. D. call toll free 800-321-4AMS (321-4267) Anderson and C. E. Burgess; Abstracts in the U.S. and Canada to charge with of manifolds, decomposition spaces, VISA or MasterCard. continua, metrization, general topology, by R. H. Bing; II. Papers of R. H. Bing; and geometric topology. In addition, III. Classifications of works of R. H. Bing; there are a number of papers in the Publications of R. H. Bing: Classified by the areas of convex functions, linearity, and year; Publications of R. H. Bing: Classified conformal varieties. The introductory by subject matter; Works not included in section in the first volume provides these volumes; Permissions.

434 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Inside the AMS

The Participant's Guide to Planning the January Meetings

Many years ago, the Society recognized the need for each site over the telephone and by mail in order to professional attention to its program of meetings and determine whether or not it deserves a closer look. The conferences, and over the last twenty-five years it has January meetings alternate between cities east of the developed an experienced staff of eight full-time and Mississippi in even-numbered years, and cities in the two part-time Meeting Planners. Meeting Planners are western half of the country in odd-numbered years, but a relatively new breed of professional. It was only a there are a number of factors which influence the JMC's year ago, for example, that the U.S. Department of deliberations, other than general geographic location. Labor approved the title of Meetings and Convention First and foremost, the site must offer facilities adequate Manager for inclusion in the National Directory of Oc­ to house most of the meetings under one roof, with cupational Titles and Codes. It has only been about the remaining facilities within a few minutes' walking five years since Metropolitan State College in Denver distance. Our meetings require an inordinately large began offering a degree in meeting management, and amount of meeting space when compared with similar that colleges and universities which offer degree pro­ size meetings of other associations. A quick comparison grams in hotel/motel management began to include a of the timetable from the New Orleans meetings in number of meeting-planning courses in their curriculum. 1969 with the Phoenix meetings in 1989 reveals that the The Society's Meeting Planners are dedicated first and number of events on the program increased from 103 to foremost to making meetings and conferences as fruitful 185, or by about 80 percent. To say that today's meeting and rewarding for the mathematicians attending them participants are getting more value for their registration as possible, in as unobtrusive a fashion as possible. dollar is no exaggeration! (The number of staff working They are the people you see mostly behind the scenes the meeting, however, increased by only 20 percent, so at meetings, wearing blue badges and a frazzled look, it's understandable that they sometimes seem to be a applying crisis management techniques to the multitude little overwhelmed by it all!) of problems which arise each day in order to ensure that After adequate facilities, the next requirement to they do not interfere with the participants' enjoyment of be satisfied is that there are between 2,000 and 2,400 the meetings. reasonably priced sleeping rooms within easy walking One of the events this staff helps to bring to the distance of the convention center. Given these two membership each year is the Joint Mathematics Meetings nontrivial requirements, the number of cities which can in January, which includes the Annual Meeting of the host the Joint Mathematics Meetings on either side of Society. These meetings are joint with the Mathematical the Mississippi is understandably small. Association of America (MAA), and provide a showcase The third major requirement is affordability, not only for sessions of several other organizations as well. Joint by the participant, but by the Society and the Association Meetings are governed by the AMS-MAA Joint Meetings as well. There are other factors which are considered, of Committee (JMC), composed of the Executive Directors course, such as climate (hence a decided proclivity for of the Society and the Association, the AMS Secretary, locations in the south or far west), easy and affordable and the MAA Associate Secretary for Meetings. The access by mathematicians from all parts of the country, Society's Director of Meetings is the JMC's professional and a minimum number of interesting things to see and consultant. do. The process of putting together the January Joint Once the staff and the JMC are convinced that a Mathematics Meetings begins several years in advance, particular city has the potential to satisfy all three major when the JMC begins to select two or three cities criteria, a site inspection is done and a report and which are likely to have the requisite facilities. The staff recommendation is submitted to the JMC at its next will have already done a preliminary investigation of meeting. If the site and suggested dates are approved,

APRIL 1990, VOLUME 37, NUMBER 4 435 Inside the AMS they are then submitted to the MAA Board of Governors host city by staff, usually the Meeting Coordinator and and the Society's Secretary and Associate Secretaries for the Housing and Registration Coordinator, along with formal approval before being listed on the Calendar of the AMS Associate Secretary for the meetings. This Meetings on the inside front cover of the Notices and in is a crucial visit for the latter since he schedules the Focus. This can take as long as one or two years once the Society's sessions and assigns meeting rooms to them; city initially comes under consideration, but generally, hence, he must be cognizant of the characteristics and these announcements take place four to five years prior to seating capacities of all rooms that will be used. It the meetings. Not much is said about the meetings after is at this time that the final rates are negotiated for that, until the various deadlines begin to be announced, space and sleeping rooms, and the two staff members such as the deadlines for organizers of special sessions, involved are acknowledged to be one of the toughest (and for abstracts for contributed papers, and for abstracts yet fairest) teams most hotel and convention center sales for special sessions. In the meantime, staff is gathering representatives have met, always keeping the membership information from a number of sources and organizations in mind and representing its interests to the fullest. for inclusion in the first announcement of the meetings, Usually formulas for calculation of the hotels' final rates which is input to a TEX file to produce camera copy have been agreed upon during the initial site visit, or for the Notices and Focus. At the same time, work is during later negotiations. If, however, the formula results being done to prepare a city map showing the location in an unfavorable rate at a particular hotel, then that of the various hotels and convention center, to prepare property is given a chance to become more competitive the timetable for the meetings (currently produced in before it is replaced by another hotel extending better Pagemaker 3.02 software on a Macintosh llx), and to rates. Also at this time, the convention center and prepare the preregistration/housing, applicant, employer headquarters hotel will have designated members of and minicourse forms. The first detailed announcement their staffs to serve as their in-house coordinators for the of the meetings, including these forms, is published in meetings, and detailed discussions are held with these the October Notices and Focus in the year preceding individuals on what will be required of these facilities the meetings. This is followed by a brief update in the during the meetings. The Chairman of the LAC is invited November Notices, with the scientific program appearing to join in on the visit, and the role of this committee is in the December issue. more specifically outlined and tailored to fit the particular Sometime during the fall two years preceding the site. meetings, the Secretary of either the Society or the In September of the year preceding the meeting, work Association (they alternate) seeks out a local mathemati­ by staff begins in earnest. A checklist of over a thousand cian willing to chair the Local Arrangements Committee items taking about 3,600 hours to complete has been (LAC) for the meeting. Usually, it is someone recom­ developed in connection with the January meetings, and mended by the Chairman at one of the major universities responsibility for specific parts are assigned to various in the host city as having the attributes of a good orga­ staff members. What follows are synopses of some parts nizer and a reputation for getting the job done on time. of this effort which may provide some insight into just Once the Chairman has agreed to serve, he or she is ap­ what it takes to produce this meeting. pointed by the Presidents of AMS and MAA, and invited The Meeting Coordinator makes arrangements for to attend the meeting of the JMC that following January most general aspects of the meetings, such as the official in order to observe a January meeting in progress, and travel agent, telephones, U.S. mail, office equipment to learn from its Chairman what is involved. The re­ rental, on-site banking, staff travel, freight shipments, mainder of the committee is appointed early in the year layout and design of the registration area, emergency preceding the meetings. The Chairman is provided with procedures, catering, information packets for speakers, a manual, and works closely with the staff during the year obtaining and printing of assorted lecture notes, various following on several aspects of the meetings, primarily signs, on-site labor and clerical help, security, insurance the gathering of local information of various sorts, the and taxes. She also assists the Associate Secretaries of recruiting of student helpers to assist in various ways the AMS and MAA with room assignments for sessions, during the meetings, and the organizing of special local and compiles information from all other coordinators events such as tours. Most of the local information that for inclusion in the Book of Orders for the meeting. The members see in the meeting announcements in Notices Book of Orders contains the details of final arrangements and the program at the meeting has been prepared and for the many aspects of the meeting, and is sent to all written by the Chairman and other members of the LAC. concerned about two weeks prior to the first day of During the meetings, the local information section of the the meetings. When the Meeting Coordinator arrives at registration desk is staffed by members of the LAC and the site, she reconfirms all arrangements included in the other volunteers from the community. Book of Orders, and acts as general trouble-shooter and About one year out, another visit is made to the problem-solver while the meetings are in progress.

436 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Inside the AMS

The Audio-Visual (A-V) Coordinator is responsible for mailing. In the meantime, any hotel or registration for preparing a Request for Proposal for all a-v and changes received are input and new reports generated. computer equipment (plus labor), sending it out to a Winners of the room lottery are drawn and notified. number of previously identified companies in the city Just before the meeting, final reports are run, such as where the meetings will take place, evaluating the bids, the alphabetical list of preregistrants which is posted and making a recommendation as to the choice of at the meeting, and badges and programs are mailed. vendor. Since the total equipment rental/labor bill for When the Housing and Registration Coordinator arrives this item can run as high as $25,000, this must be at the meeting site, she visits all hotels being utilized and rigorously and carefully done. Once the vendor has been reconfirms all arrangements and reservations. During the selected, the A-V Coordinator contacts all speakers on meeting she assists participants with any problems they the program to determine precisely what equipment they may encounter with housing or preregistration. will require in connection with their talks, and begins Over 100 committees meet during the January meet­ to draw up a list of firm requirements for transmission ings. The staff is responsible for determining when to the vendor(s). He is also responsible for preparing committees will meet, assigning suitable rooms, checking detailed descriptions of the setups and layouts for the for scheduling conflicts, and producing both personalized various rooms used for sessions during the meetings, and master schedules for all concerned. Many of these including a-v requirements. When he arrives at the site committees must meet over breakfast, lunch, or dinner, of the meetings, he meets with the various vendors and so catering arrangements must also be made, and ac­ and reconfirms all arrangements. During the meetings, curate counts of those attending must be gathered in time he monitors all sessions, assists speakers and organizers to provide the facilities with the necessary guarantees. with the operation of public address systems and the As abstracts are submitted for presentation at the various pieces of equipment, and works closely with meetings, they are acknowledged and input into a com­ them on producing a successful presentation. puter system which sorts this information in several The Housing and Registration Coordinator and the ways and produces reports which will assist the AMS staff of the Mathematics Meetings Housing Bureau pre­ Associate Secretary in preparing the Society's scientific pare for processing of preregistration and housing for the program for the meetings. (Abstracts for papers on the meetings. This entails initializing the computer system MAA program are handled somewhat differently, and the which handles all record keeping and reports, including MAA Associate Secretary is responsible for its scientific setting up computer codes for all items listed on the pre­ program.) About a week after the abstract deadline, the registration/housing form so that they can be efficiently abstracts and reports are mailed to the Associate Sec­ input to the system. Included in this initialization are retary, who has about four or five working days to do setting up lists of all available room types and rates all the scheduling and room assignments, and return the in all hotels; clarifying procedures with the data entry program to Providence where this additional informa­ clerks, emphasizing special features for the meeting and tion is input in order to produce camera copy for the their handling; preparing for and supervising the mailing Notices and the program mailed or handed out at the of various items, such as registration/housing acknowl­ meetings. Once the final program is known, the Book of edgements, badges and programs; setting up electronic Orders can be completed, packets can be prepared for preregistration forms, acknowledgement letters for fi­ session chairmen, and a number of other last minute nal preregistration, and any other necessary forms and details can be finished in a flurry of activity just before letters; making arrangements with the vendor for produc­ the staff leaves for the meetings. tion of badges, including preparation of camera copy and At last, exhausted and disheveled, your loyal Meeting printing of the various types of badges; ensuring that the Planners arrive in the meeting city, armed with carton various fiscal accounts to be used for the meeting are set slitters, Excedrin and program errata, ready for yet up and ready to go. As the preregistration process begins, another opening day. So, when you see someone wearing copy for on-site registration cards, badges, and any tick­ a blue badge and a great big smile fly by you in the ets required is developed and printed for shipping to the corridor of the convention center, simultaneously yelling meeting. As preregistration/housing forms are received, into a walkie talkie at a houseman to put 50 more chairs payments are recorded and forwarded to the bank, hous­ in that room right now, handing out lecture notes with ing assignments are made, the forms are coded and input one hand and putting a bandaid on a participant's knee to the system, acknowledgements are mailed, financial with another, it's not Superman or Wonder Woman, but reports are sent to the AMS Fiscal Department, and the next best thing - your professional AMS Meeting rooming lists are sent to the various hotels, along with Planner! deposit monies. Badge reports are sent to the vendor, and as completed badges are returned, they are alphabet­ H. Hope Daly ized, proofread against the original form, and prepared Director of Meetings

APRIL 1990, VOLUME 37, NUMBER 4 437 Washington Outlook

science must be our goal. This month's column is written by Hans J. Oser, who is * There must be educational access for all our citizens. a consultant to the Office of Governmental and Public * We must improve the teaching of science and mathe­ Affairs of the Joint Policy Board for Mathematics in matics. Washington, D.C. * We must strengthen the support throughout the educa­ tional system for high quality science and mathemat­ ics instruction and learning. Five months ago the President, speaking with the gov­ Cavazos described his budget proposal for fiscal year ernors of the 50 states in Charlottesville, Virginia, an­ 1991, stressing collaboration with the NSF, both now nounced his plan to improve the Nation's educational and in the future. system. In his State of the Union Message in January, Before questioning him, the committee also heard the President declared that he wanted "American stu­ from Erich Bloch. His main points were: dents to be first in the world in science and mathematics * There is a serious problem at the precollege level. achievement by the year 2000". During the last week * Interest in science and engineering among students has of February, the National Governors Association met in been declining. Only 15 percent of entering college Washington and endorsed six goals for education reform, freshmen planned to major in the natural sciences or fully recognizing that this would be a long and expensive engineering, compared to 20 percent in 1966. process. In his speech in the East Room of the White * Bachelors degrees awarded in these disciplines in House, the President congratulated the assembled gover­ 1988 declined 3 percent from the previous year. In nors for ushering in "a new era of education reform", by computer sciences alone, the decline was 13 percent. endorsing his goals for improving the American schools. * At the graduate level, we are not producing the The House Committee on Science, Space, and Tech­ number of scientists and engineers needed to meet nology saw its patience rewarded on 28 February, when the challenges of a competitive world economy in the the Secretary of Education, Lauro Cavazos presented his years ahead. long-awaited education agenda for the nation. The gover­ * In critical disciplines, the number of doctorates would nors' action provided the opportunity for the committee be much lower if it were not for foreign students, to find out from the Secretary how the administra­ which comprise more than 50 percent of the doctor­ tion expected to proceed with reaching the President's ates in engineering and mathematics. goals. The committee also invited Erich Bloch, the NSF Most of these facts are of course well known by now, director, to state his views on the issues. but they provided the stage for Bloch's discussion of Here are some of the highlights from Cavazos' state­ the NSF's responsibilities in science education and hu­ ment: man resource development at all levels, from precollege * Mathematics and science are key elements, and unless through graduate. He touched upon collaborations with the Nation does far better in this area than it has industry and states and his special concerns about in­ been doing, the President's goals will not be achieved. creasing the participation of underrepresented groups. In * The Department of Education (DOEd) is ready to do responding to the committee's request to address the is­ its part to meet this challenge, and already is working sue of cooperation within the executive branch, he spoke with the National Science Foundation and the White of possible cooperation between the DO Ed and the NSF, House Science Adviser on a plan of action to meet it. through stronger liaison between the two agencies, closer * Science and mathematics must be for all students. and more stable working and funding arrangements, * Science and mathematics literacy is essential for ev­ joint programs with the states and programs targeted eryone. A citizenry that understands and appreciates at urban education, expansion and improvement of na-

438 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Washington Outlook tional assessments of student achievement in the sciences education, DOEd's responsibility is education in general. and mathematics, and increased dissemination of high The fundamental question that dominates the na­ quality projects sponsored by the NSF and expanded tional debate over education reform is the proper role distribution by the DOEd. of the federal government. Bloch said it best, when he Bloch then proceeded to describe the administration's told the committee that the role of the President is to set role as a goal setter and provider of leadership. To goals and to create an environment that permits change. demonstrate the increased awareness of the leadership But reform cannot come from the top, it must be the responsibility, Bloch said that D. Allan Bromley, the concern of everybody in the educational system. President's Science Adviser, will shortly create a new Congressman Ritter (R-PA) came closest to illustrat­ Committee on Human Resources and Education within ing the dilemma of how much federal intervention was the framework of the Federal Coordinating Council on proper, when he rejected "mandates" as an instrument Science, Engineering and Technology (FCCSET). for the Bush administration. He then proceeded to ques­ It became obvious, when the questioning by the tion Cavazos on how he planned to use the fiscal power committee members began, that their strongest concern of the education department to steer the states toward was over the perceived lack of coordination in education accepting alternative certification of teachers, extending policy within the executive branch. The Republicans the school day, the school week, and the school year. did most of the questioning of Cavazos, which went Should we use the federal funding mechanism to leverage something like this: these goals?, he asked. And Cavazos agreed that leverage * What is the difference in the missions between DOEd was a good idea, to use federal dollars as a carrot. and NSF and what have you done to coordinate your Bloch, the practical engineer, came closer to the heart programs? of the matter, when he said market forces would be * What important issues are falling through the cracks extremely important in the educational system. Teacher between the two agencies? salaries are critical in this respect. The universities, he * Does it do any good if Cavazos and Bloch are talking told the committee, had to learn this the hard way a few to one another, if word does not filter down in the years back when they could not find faculty in computer two agencies? science until they began to introduce differential salary * What is your awareness of the education programs in scales. Bloch admitted that he would not object to the Department of Energy? considering leveraging with federal funds the educational * Do you find it shocking that the Secretary of Education programs in the states and in local jurisdictions. has to ask for admission to the newly formed FCCSET The remaining time was mostly spent on outreach committee on education and human resources? to the education community. Why does DOEd have no Both witnesses agreed on the fundamental differences teachers award? Has DOEd invited the winners of the between DOEd and NSF programs. NSF awards all National Teachers Award to meet with the Secretary to grants competitively, and typically for 3 years or longer. listen to what they have to say? Does the National Dif­ DOEd allocates 80 percent of its more than $4 billion fusion Network (a DOEd clearinghouse that distributes budget by various formulas, either to states or to school information on, among others, NSF's curriculum devel­ districts, typically on a year-to-year basis. The remaining opment projects) employ anyone with a science degree? 20 percent, even when competitively awarded, go to the Unfortunately, Secretary Cavazos was forced to leave states which in turn often use non-competitive criteria after two hours, citing an important commitment. The for their allocations. There are more than 200 separate Science, Space and Technology Committee considered programs in the Department of Education. itself lucky to have heard from him after a full year of DOEd does not have as much control over how the trying. money is spent, or how well it is spent, in contrast to the NSF, where peer evaluation is part of the alloca­ Correction tion process, and where renewals are based on previous In the February "Washington Outlook", we failed to performance. On the other hand, in DOEd, some of mention that the Joint Policy Board for Mathematics the (congressionally mandated) formula allocations pro­ was the successor to an earlier joint committee of the duce just the opposite effect: the least successful school three organizations (AMS, MAA, and SIAM) called districts get the preference. Another fundamental differ­ the Joint Projects Committee for Mathematics. That ence between NSF and DOEd is in the approach: NSF committee was formed in 1973 to administer projects of is responsible for mathematics, science and engineering common interest to the three organizations.

APRIL 1990, VOLUME 37, NUMBER 4 439 News and Announcements

Valentine Bargmann stitute for Advanced Study in Prince­ groups of 1953-1954, those on sec­ 1908-1989 ton and was accepted as an assis­ ond quantization of 1946-194 7. Valentine Bargmann, Professor of tant. He was drawn into the work Bargmann's interests in mathe­ Mathematical Physics, Emeritus, died that Albert Einstein was carrying out matical physics were broad. As stu­ of heart failure 20 July 1989 in on unified field theories of gravita­ dents at the University of Berlin, Princeton Hospital, at the age of tion and electromagnetism. For sev­ he and Carl Hempel had a common 81, nine months after the death of eral years, he and Peter Bergmann interest in the philosophical prob­ his wife Sonja. Bargmann exempli­ were Einstein's scientific assistants lems at the foundation of physics, an fied a great European tradition in and coworkers in this enterprise. This interest they shared with Hans Re­ mathematical physics carried on in work continued until 1943, when he ichenbach, then Professor at Berlin. Princeton by H. Weyl, J. von Neu­ undertook war work on shock waves The reader of Reichenbach's book mann and E. Wigner. with John von Neumann. After the Philosophic Foundations of Quan­ Bargmann was born in Berlin, war, he joined von Neumann's com­ tum Mechanics, University of Cal­ Germany, April 6, 1908. He stud­ puter project, working with von Neu­ ifornia Press, 1944, will see some ied at the University of Berlin from mann and Deane Montgomery on typical results of conversations with 1926 to 1933. He moved to ZUrich on the inversion of matrices of large Bargmann, the sharpening and clari­ Hitler's rise to power and wrote his dimension. fication of ideas by the construction doctor's thesis under the guidance of From 1941 on, Bargmann taught of examples and counterexamples. Gregor Wentzel. On the completion graduate courses at Princeton Uni­ His contributions to physics led of his degree, Bargmann emigrated versity, but it was only in 1946 to the creation of several "indus­ to the United States. (That flat state­ that he obtained a regular appoint­ tries". He demonstrated to everyone's ment is correct but it does not evoke ment as a visiting lecturer in physics. surprise that the one-dimensional the temper of the times. Bargmann In the following years he worked Schrodinger equation with two dif­ received a five year German pass­ with E. Wigner on relativistic wave ferent potentials can have identical port in 1931, before the National equations. Together they obtained phase shifts; scattering data does not Socialists came to power, and used it in 1948, the well known Bargmann­ suffice to determine a potential. This to go to Switzerland to study. After Wigner equations for elementary par­ was one of the starting points of the Hitler took office, administrative reg­ ticles of arbitrary spin. inverse scattering method. Bargmann ulations were issued withdrawing the After 1946, apart from one term also was the first to find upper bounds citizenship of persons of the wrong spent at the University of Pittsburgh on the number of bound states in a "race". For that reason, if the Ger­ in 1948, Bargmann was a member of spherically symmetric potential. An­ man government had succeeded in the Faculty at Princeton He taught other important topic in the study finding Bargmann, it would have in­ physics and mathematics to genera­ of the Schrodinger equation is coher­ validated his passport. Nevertheless, tions of graduate and undergraduate ent states. Although Bargmann did the passport was accepted by the students. His lectures were noted for not invent them (they go back to United States government as a valid their clarity and polish. However, for Schrodinger and many others) he basis for an immigration visa. The connoisseurs of the post-war period, showed that they could be viewed passport expired two days after he it was the sets of specialized lec­ as unitary transformations from the reached the United States in 1937.) tures on his own research that were ordinary space of square integrable After receiving helpful sugges­ the gems: those on the Lorentz group functions on the line to a space of tions from American physicists, and its representations of 1948-1949, analytic functions on the complex Bargmann applied for a job at the In- those on ray representations of Lie plane.

440 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News and Announcements

Of all Bargmann's papers, the Professor Perlis, who was Eugene ence and Technology Foundation of most influential for mathematics was Higgins Professor of Computer Sci­ Japan. The two disciplines for the undoubtedly his article on the irre­ ence at Yale University, is probably 1990 awards were technology inte­ ducible unitary representations of the best known for his work in program­ gration and earth science. Lorentz group. He found all of them ming language design. He received in 194 7 and thereby established the his Ph.D. in mathematics from the National Academy of Engineering paradigm for four decades of effort Massachusetts Institute of Technol­ Nominations in representation theory. ogy in 1950. The National Academy of Engineer­ In this work he not only clas­ From 1952 to 1956, Professor ing has announced the election of sified the representations infinites­ Perlis was a mathematics professor eighty new members and seven for­ imally (i.e. the corresponding rep­ and director of the computer science eign associates. Among these are sev­ resentations of the Lie algebra of center at Purdue University, where eral mathematical scientists: James the Lorentz group), but he con­ he developed a digital computer lab­ D. Callen (University of Wiscon­ structed these representations glob­ oratory. He taught at the Carnegie In­ sin, Madison), Michael L. Dertouzos ally and explicitly, and divided them stitute of Technology (now Carnegie­ (Massachusetts Institute of Technol­ into three classes, nowadays known Mellon University) from 1956 to ogy), Alan B. Fowler (IBM Thomas as the "principal series", "discrete 1971. While there, he founded the J. Watson Research Center), George series", and "complementary series". graduate department of computer J. Gleghorn (TRW Space & Technol­ In addition, he systematically studied science and helped develop algebraic ogy Group), Gene H. Golub (Stan­ the entry functions of the represen­ language compilers and assemblers. ford University), Ken Kennedy (Rice tations in terms of the differential He went to Yale in 1971, where he University), Alan S. Manne (Stan­ equations they satisfy, and thereby played a leading role in developing ford University), Bradford W. Parkin­ essentially deduced the "Plancheral Yale's computer science department. son (Stanford University), Ronald formula" for the group. His recent areas of research included L. Rivest (Massachusetts Institute It is not surprising that he was automatic programming and parallel of Technology), David A. Woolhiser elected to the National Academy systems programming. (U.S. Department of Agriculture Re­ of Sciences in the mathematics sec­ Professor Perlis was a member of search Service). tion. Bargmann received other hon­ the American Academy of Arts and ors, such as the Planck medal of Sciences and the National Academy Call for Nominations for Faisal the German Physical Society and the of Engineering. He received the A. Prize Wigner medal, but he was a modest M. Turing Award from the Associ­ The General Secretariat of the King man who sought his satisfaction in ation for Computing Machinery in Faisal International Prize invites uni­ his science, his interaction with his 1966 and the Pioneer Award from versities and research centers through­ friends and his music. the Institute for Electrical and Elec­ out the world to nominate qualified His musical talent was such that tronics Engineers in 1985. candidates for the King Faisal Inter­ he could consider a career as a pro­ national Prize in Science, which will fessional pianist. For many years, he Minsky Wins Japan Prize be awarded in the field of mathemat­ played in small groups delighting his Marvin L. Minsky, professor of elec­ ics in 1991. fellow musicians with the depth of trical engineering at the Massachu­ The prize consists of a certificate, his interpretation. setts Institute of Technology, has re­ a gold medal, and 350,000 Saudi Although Bargmann's writings ceived the Japan Prize of $345,000 Riyals (approximately US $93,333). were influential, his personal influ­ for his work in developing and pop­ The prize, which may be shared by ence was even greater. He and Sonja ularizing the concept of artificial in­ more than one person, will be an­ were pillars of the Princeton intellec­ telligence. nounced in January, 1991. tual community. Our lives will not Professor Minsky received his Nominations should fulfill the fol­ be the same without them. Ph.D. in mathematics from Prince­ lowing criteria: Elliot H. Lieb, Elias M. Stein, ton University in 1954. Known for 1. The nominee must be living and Arthur S. Wightman his research in artificial intelligence, and must have accomplished out­ Princeton University the theory of computation, psychol­ standing work in mathematics, bene­ ogy, and engineering, he has been on fiting mankind and enriching human Alan J. Perlis the MIT faculty since 1958. progress. 1922-1990 The Japan Prize, regarded as the 2. The work submitted must be Alan J. Perlis, a leader in the de­ most prestigious scientific honor given original and published and must not velopment of computer science, died in Japan, is awarded each year in have been previously awarded a prize February 7, 1990 at the age of 67. two scientific disciplines by the Sci- by any international organization.

APRIL 1990, VOLUME 37, NUMBER 4 441 News and Announcements

3. Nominations should be from eluding such criteria as content and and electronic book ordering and recognized educational institutions. design. meetings registration. Nominations by individuals or polit­ JAMS is edited by Michael Artin, ical parties will not be accepted. H. Blaine Lawson, Jr., Richard Mel­ 4. Nominations should include an rose, Wilfried Schmid, and Robert E. NSF Regional Geometry Institute official letter on each nominee, indi­ Tarjan. Those responsible for JAMS During the next three summers, the cating the nominated works; a typed in the AMS office in Providence in­ Department of Mathematics at CV of the nominee's academic back­ clude the production editor, Arlene Boston University will host a Re­ ground, experience, and published O'Sean, as well as the keyboarding, gional Institute in Dynamical Sys­ works; ten copies of the works sub­ TEX-support, and printing staff. tems (RIDS). Funded by the Na­ mitted for nomination; a copy of the tional Science Foundation as part nominee's certificates of education; Woolf Joins Providence Staff of its Regional Geometry Institutes three recent color photos of each William B. Woolf has left his posi­ Program, RIDS will bring together nominee, 4 inches by 6 inches; the tion as Managing Editor of Mathe­ researchers, college and secondary nominee's full address and telephone matical Reviews (MR) to become an teachers, graduate and high school number. Associate Executive Director of the students, to study a range of topics 5. Nomination papers or works AMS in Providence. He will be in in dynamics. will not be returned to the senders, charge of computing services for the RIDS begins with a two-week, whether or not the nominee is awarded Society. intensive short course on the Ge­ the prize. Woolf received his Ph.D. in math­ ometric Theory of Dynamical Sys­ 6. The deadline for nominations ematics from the University of Michi­ tems, to be held July 16-27, 1990, is May 10, 1990. gan in 1959. He served on the math­ at Boston University. Topics will All correspondence should be sent ematics faculty at the University of include chaotic dynamics, complex by registered airmail to: The Gen­ Washington, advancing to the rank dynamics, circle and annulus maps, eral Secretariat, King Faisal Interna­ of associate professor before taking symbolic dynamics, and pedagogical tional Prize, P.O. Box 22476, Riyadh a position as Associate Secretary and issues. Boston University mathemat­ 11495, Kingdom of Saudi Arabia; Director of Administration of the ics faculty members Paul Blanchard, telephone 4652255; telex 404667 American Association of University Robert L. Devaney, David Fried, and PRIZE SJ; fax 4658685; cable Professors in Washington, DC. In G. R. Hall will present the lectures. JAEZAH. 1979 he moved to MR as Managing The short course will prepare par­ In 1987, the King Faisal Prize Editor. His former MR duties not ticipants for the more advanced re­ in Mathematics went to Sir Michael involving computing services will be search lectures in the main part of Atiyah, Oxford University. assumed by MR Associate Executive RIDS, to be held during the sum­ Editor Jane Kister. mers of 1991 and 1992 at Boston AMS Journal Wins Award In his new position with the AMS, University. The sessions will consist The Journal of the American Mathe­ Woolf will oversee the Computer Ser­ of a series of week-long conferences, matical Society (JAMS) has received vices Division of the Society, which each featuring one or two leading the 1989 Award for the Best New has been expanded to include the experts who will speak on such top­ Journal in Science/Technology /Med­ MR Systems Department in Ann Ar­ ics as: renormalization and rigidity, icine from the Association of Amer­ bor. AMS computing activities in­ chaotic dynamics, twist maps, aurae­ ican Publishers (AAP). clude computer typesetting for all tors, computational complexity the­ Award recipients are selected by AMS publications, membership and ory, fractal geometry, complex ana­ a panel of independent judges ap­ sales databases, the MR database, the lytic dynamics, and surface dynamics pointed from the publishing indus­ fiscal system, and routine office ad­ and neural networks. try and the industrial, medical, and ministration. In addition, he will be A major focus of RIDS will be co­ scientific communities by the Pro­ in charge of an AMS electronic com­ ordination with several ongoing pro­ fessional and Scholarly Publishing munications project that will build grams which combine dynamics and Division of AAP. There are two on the Internet electronic network educational issues, including programs categories for entries, Science/Tech­ to produce a complete information­ for high school students and teach­ nology /Medicine and Business/So­ exchange system tailored to the needs ers. Each plenary speaker will deliver cial Science/Humanities. The panel of the mathematical sciences com­ a lecture aimed at introducing high chose JAMS from over 200 profes­ munity. The yet-to-be-named system school students to her or his own sional and scholarly works in the two will provide access to electronic ad­ research specialty. Other invited lec­ categories. The award was made on dresses, conferencing and bulletin turers will discuss the teaching of the basis of all-around excellence, in- board facilities, document delivery, dynamical systems at all levels.

442 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY News and Announcements

It is anticipated that the Na­ sual materials for use on network MathSoft is donating a retail value tional Science Foundation will pro­ and local news programs. of $3 million of software to U.S. high vide travel and lodging expenses SIPI runs a Videotape Referral schools and has developed materials for RIDS participants. Graduate stu­ Service for news directors, producers, to support "Math is Radical." The dents, women, and members of un­ and reporters who are seeking reliable NCTM has recognized the program derrepresented minorities are encour­ sources of quality science footage for as assisting efforts to implement the aged to apply for support. For more use on television news. As a result curriculum and evaluation standards information, write to: Regional In­ of the success of the service, CBS which it set forth last year. To date, stitute in Dynamical Systems, De­ News has asked SIPI for assistance in 180 high schools are participating in partment of Mathematics, Boston producing a weekly, 30-second spot the first year of the program. University, 111 Cummington Street, for the evening news. The segment, Each participating school receives Boston, MA 02215; or send elec­ narrated by anchorman Dan Rather, a "Math is Radical" package, in­ tronic mail to [email protected]. will focus on interesting, cutting-edge cluding MathCAD for IBM PCs or developments in the world of science compatibles, or for Apple Macintosh and technology. computers and classroom teaching New Association Launched in To improve television coverage of tools for MathCAD. Tutorials and France the sciences SIPI would appreciate templates for curriculum develop­ The Henri Poincare Association for receiving quality videotapes, films, ment at the algebra, pre-calculus, and the History and Philosophy of Math­ slides, or photographs of current, ex­ calculus levels are included. There ematics and Physics was created in citing research occurring in univer­ will also be a "Math is Radical" July, 1989 to promote relations sities throughout the country. Con­ contest, in which students submit among mathematicians, physicists, tributing to this project will help to solutions to mathematical and historians and philosophers of problems enhance public understanding of the which the students themselves science. The activities of the Asso­ have importance, fascination, and utility formulated. ciation primarily focus on studies of research in science and mathe­ of the nineteenth and twentieth cen­ matics. Sponsors for turies and seek to define the specific If you have materials you would JPBM Communications Award role that history and philosophy can like to submit, or if you would like The Joint Policy Board for Math­ play in the practice of science. more information about the Science ematics (JPBM) wishes to thank For the 1989-1990 year, the As­ & TV Project or other SIPI programs, the following companies, which co­ sociation sent out to its membership call Lara Trubowitz or Joyce Gramza sponsored the second JPBM Com­ three informational letters and orga­ at 212-661-9110, or write to SIPI at munications Award and the recep­ nized a number of activities asso­ 355 Lexington Avenue, New York, tion in honor of its recipient, Hugh ciated with the various themes cur­ NY 10017. Whitemore, at the Joint Mathematics rently being developed at the Henri SIPI is a national, non-profit or­ Meetings in Louisville (see February Poincare Institute in Paris. ganization dedicated to improving 1990 Notices, page 139): Those on the governing commit­ public understanding of science and Academic Press, Inc. tee of the Association are: Michel technology by bridging the gap be­ The Benjamin/Cummings Pub- Blay, Jean-Luc Chabert, Karine tween science and the media. lishing Company, Inc. Chemla, Thierry Coulhon, Catherine Brooks/Cole Publishing Company Chevalley, Amy Dahan-Dalmedico, John Wiley & Sons Olivier Darrigo!, Dominique Pestre, "Math is Radical" Program Springer-Verlag New York, Inc. Hourya Sinaceur. Membership dues A program designed to improve the for the Association are I 00 FF. For mathematics skills of high school stu­ Workshop for College more information, write to the As­ Faculty dents and offer teachers a tool for Using Graphing Calculators and Com­ sociation at this address: l'Institut enhancing mathematics teaching in puter Graphing to Enhance the Henri Poincare, 11, rue P. et M. Teach­ the classroom has been developed ing Curie, 75005 Paris, France. and Learning ofPrecalculus Math­ by MathSoft, Inc., in conjunction ematics and Calculus. August 12-13, with the National Council of Teach­ 1990. Holiday Inn, Columbus. No Science Visuals for CBS News ers of Mathematics (NCTM). Con­ registration fee. $100 partial travel Sought gressional representatives and state expense awards from Addison-Wesley. The Scientists' Institute for Public departments of education were also For application: F. Demana and B. Information (SIPI) is seeking assis­ involved in announcing the availabil­ Waits, Dept. of Math., The Ohio tance from the scientific and math­ ity of the program entitled "Math is State Univ., 231 W. 18th Avenue, ematical communities to provide vi- Radical". Columbus, Ohio 43210.

APRIL 1990, VOLUME 37, NUMBER 4 443 Funding Information for the Mathematical Sciences

1991-1992 Fulbright Competition pursue study or research. All grants 1990). (See page 488 in this issue of Opens include basic health and accident in­ Notices for an announcement of this The United States Information Agen­ surance. meeting.) cy, the Board of Foreign Scholars, Students currently enrolled in a Graduate students who wish sup­ and the Institute of International Ed­ college or university should contact port may apply to: Michael C. Reed, ucation announce the official opening their on-campus Fulbright Program Department of Mathematics, Duke on May 1, 1990 of the 1991-1992 Adviser for brochures, application University, Durham, NC 27706. The competition for Fulbright grants. forms, and further information. At­ application, which should be received These grants support graduate large applicants should contact the by May 25, 1990, should include a study or research abroad in academic U.S. Student Programs Division at one-page research summary and one fields. Also available are travel-only the Institute for International Educa­ letter from a faculty sponsor. grants to selected countries to sup­ tion, 809 United Nations Plaza, New plement maintenance awards from York, NY 10017; telephone 212-984- another source that do not provide 5327. Fulbright Program Advisers funds for international travel or to establish campus deadlines. At-large Notice of Deadline Change supplement the applicant's personal applicants must submit their com­ The National Science Foundation has funds. pleted applications to the U.S. Stu­ changed the deadline for its under­ Applicants must be U.S. citizens dent Programs Dvision at the above graduate course curriculum develop­ at the time of application and hold address by October 31, 1990. ment program from April 9, 1990 to a bachelor's degree or its equivalent October 15, 1990. The program was by the beginning date of the grant. described in "NSF Expands Curricu­ They also must possess sufficient pro­ Landahl Travel Awards lum Development," in the Funding ficiency in the language of the host The Society for Mathematical Bi­ Information section of the February country to carry out the proposed ology (SMB) has funds for partial 1990 issue of Notices, page 145. The study or research. support of the travel of graduate stu­ program addresses curriculum devel­ Fulbright grants provide round­ dents to meetings co-sponsored by opment for courses other than cal­ trip international travel, maintenance SMB, including the SIAM Meeting in culus; therefore, the deadline change for the tenure of the grant, and tu­ Chicago (July 16-20), and the Inter­ will not affect the calculus curricu­ ition waivers, if applicable. Travel national Union for Pure and Applied lum development program, which grants provide round-trip travel to Biophysics (IUPAB) Meeting in Van­ will likely have a deadline of Febru­ the country where the student will couver, Canada (July 29 - August 3, ary 1, 1991.

444 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1990 AMS Elections Nominations by Petition

Vice-President or Member-at-Large inating Committee, namely: One position of vice-president and member of the Council ex Joan S. Birman Ray A. Kunze officio for a term of two years is to be filled in the election of James E. Humphreys Alan D. Weinstein 1990. The Council intends to nominate two candidates, whose Barbara Lee Keyfitz Robert F. Williams names may be expected to appear in the June issue of Notices, Victor Klee which is scheduled to be mailed by the printer on 23 May. Nom­ The new members will be elected in a preferential ballot. The inations by petition as described in the rules and procedures are President will name five candidates for these three places. The acceptable. names may be expected to appear in the June issue of Notices. Five positions of member-at-large of the Council for a term Nominations by petition, in the manner described in the rules of three years are to be filled in the same election. The Coun­ and procedures, will be accepted. Should the final number of cil intends to nominate seven candidates, whose names may be candidates be less than six, the President will bring it up to six. expected to appear in the June Notices. Nominations by peti­ The name of a candidate for member of the Nominating tion in the manner described in the rules and procedures are Committee may be placed on the ballot by petition. The candi­ acceptable. The Council has stated its intent to have at least ten date's assent and petitions bearing at least 100 valid signatures candidates and will bring the number up to ten if the nomina­ are required for a name to be placed on the ballot. In addition, tions by petition do not do so. several other rules and operational considerations, described be­ Petitions are presented to the Council, which, according to low, should be followed. Section 2 of Article VII of the bylaws, makes the nominations. The Council of 23 January 1979 stated the intent of the Coun­ cil of nominating all persons on whose behalf there were valid Rules and Procedures petitions. Use separate copies of the form for each candidate for vice-president, Prior to presentation to the Council, petitions in aid of a member-at-large, or member of the Nominating and Editorial Boards candidate for the position of vice-president or of member-at­ Committees. large of the Council must have at least 50 valid signatures and 1. To be considered, petitions must be addressed to Robert M. must conform to several rules and operational considerations, Fossum, Secretary, P.O. Box 6248, Providence, Rhode Island 02940, which are described below. and must arrive by 6 July 1990. 2. The name of the candidate must be given as it appears in the Combined Membership List. If the name does not appear in the list, as Editorial Boards Committee in the case of a new member or by error, it must be as it appears in Two places on the Editorial Boards Committee will be filled by the mailing lists, for example on the mailing label of the Notices. If the election. There will be four continuing members of the Editorial name does not identify the candidate uniquely, append the member Boards Committee, namely: code, which may be obtained from the candidate's mailing label or the Linda Keen Barry Simon Providence office. 3. The petition for a single candidate may consist of several sheets Carlos Kenig Daniel Zelinsky each bearing the statement of the petition, including the name of the The new members will be elected in a preferential ballot. The position, and signatures. The name of the candidate must be exactly President will name three candidates for these two places. The the same on all sheets. names may be expected to appear in the June issue of Notices. 4. On the next page is a sample form for petitions. Copies may be Nominations by petition, in the manner described in the rules obtained from the Secretary; however, petitioners may make and use and procedures, will be accepted. Should the final number of photocopies or reasonable facsimiles. candidates be less than four, the President will bring it up to 5. A signature is valid when it is clearly that of the member whose name and address is given in the left-hand column. four. 6. The signature may be in the style chosen by the signer. The name of a candidate for member of the Editorial Boards However, the printed name and address will be checked against the Committee may be placed on the ballot by petition. The candi­ Combined Membership List and the mailing lists. No attempt will be date's assent and petitions bearing at least 100 valid signatures made to match variants of names with the form of name in the CML. are required for a name to be placed on the ballot. In addition, A name neither in the CML nor on the mailing lists is not that of a several other rules and operational considerations, described be­ member. (Example: The name Robert M. Fossum is that of a member. low, should be followed. The name R. Fossum appears not to be.) 7. When a petition meeting these various requirements appears, the Secretary will ask the candidate whether he is willing to have Nominating Committee his name on the ballot. Petitioners can facilitate the procedure by Three places on the Nominating Committee will be filled by accompanying the petitions with a signed statement from the candidate election. There will be seven continuing members of the Nom- giving his consent.

APRIL 1990, VOLUME 37, NUMBER 4 445 NOMINATION PETITION FOR 1990 ELECTION

The undersigned members of the American Mathematical Society propose the name of

as a candidate for the position of (check one): D Vice-President D Member-at-Large of the Council D Member of the Nominating Committee D Member of the Editorial Boards Committee of the American Mathematical Society for a term beginning 1 January, 1991.

Name and Address (printed or typed)

Signature

Signature

Signature

Signature

Signature

Signature

446 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY MEETINGS

Columbus Meetings August 8-11,1990

Preliminary Announcement

The Scientific Program The August 1990 Joint Mathematics Meetings, including the 93rd Summer Meeting of the AMS, the 69th Summer Meeting of the Mathematical Association of America and the celebration of the 75th Anniversary of the founding of the MAA, and the 1990 summer meetings of the Asso­ ciation for Women in Mathematics and Pi Mu Epsilon, will be held August 8- 11 (Wednesday- Saturday), 1990, at The Ohio State University, Columbus, Ohio. Sessions will take place on the campus of the university and the Holiday Inn.

AMS-MAA Invited Address The Society, on the occasion of the MAA's 75th An­ niversary, will cosponsor an invited address by SAUNDERS MAc LANE, University of Chicago, on Algebra as a means of understanding mathematics, at 9:55 a.m. Thursday, August 9.

IMPORTANT DEADLINES AMS Abstracts For Consideration for Special Sessions April 27 Saunders Mac Lane, AMS-MAA Invited Lecturer Of Contributed Papers May 18 MAA Poster Session Presentations May 1 Nominations for MAA Undergraduate Student Papers May 15 WHERE TO FIND IT PAGE MAA Abstracts How to Preregister and Get a Room 460 Of Contributed Papers May 18 Summer List of Applicants June 6 75th Anniversary of the MAA 449 ORDINARY Preregistration and Housing June 6 Summer Meeting of the AMS 448 MAA Minicourse Preregistration June 6 Motions for AMS Business Meeting July 10 AMS Short Course 482 FINAL Preregistration July 11 Other Organizations 458 Housing Changes and Cancellations with Housing Bureau July 16 Other Events of Interest 458 Residence Hall Package Cancellation (90% refund) July 16 Miscellaneous Information 465 MAA Banquet (50% refund) July 31 471 Sock Hop (50% refund) July 31 Timetable 1tJ.lE Banquet (50% refund) July 31 Maps 462 & 467 Other Changes to Preregistration July 31 Preregistration Cancellations (50% refund) August 3

APRIL 1990. VOLUME 37, NUMBER 4 447 Meetings

93rd Summer Meeting of the AMS should submit it by April 27, 1990, three weeks earlier August 8-11, 1990 than the normal deadline for contributed papers, in order Progress in Mathematics Lectures: This series of lectures that it be considered for inclusion. provides a forum for the exposition of mathematical Abstracts should be prepared on the standard AMS topics that have come into prominence in the past form available from the AMS office in Providence or in five years. The members of the Progress in Mathematics departments of mathematics, and should be sent to Ab­ Selection Committee for these lectures are Armand Borel stracts, Editorial Department, American Mathematical (chairman), Paul H. Rabinowitz, Hugo Rossi, John T. Society, Post Office Box 6248, Providence, Rhode Island Tate, and Alan Weinstein. 02940. A charge of $16 is imposed for retyping abstracts The names and affiliations of the speakers, their titles, that are not in camera-ready form. and the days and times they will talk are as follows: Contributed Papers: There will be sessions for con­ JoHN W. MoRGAN, Columbia University, A-trees and tributed papers Thursday morning, Friday morning and their applications, 3:35 p.m. Saturday. afternoon, and Saturday morning and afternoon. MICHAEL G. CRANDALL, University of California, Abstracts should be prepared on the standard AMS Santa Barbara, Viscosity solutions of partial differential form available from the AMS office in Providence or in equations, 3:35p.m. Thursday. departments of mathematics, and should be sent to Ab­ Prizes: The 1990 Leroy P. Steele Prizes and the stracts, Editorial Department, American Mathematical Norbert Wiener Prize in Applied Mathematics will be Society, Post Office Box 6248, Providence, Rhode Island awarded at 11:05 a.m. on Friday, August 10. 02940, so as to arrive by the abstract deadline of May 18, Invited Addresses: By invitation of the AMS Program 1990. A charge of $16 is imposed for retyping abstracts Committee for National Meetings, there will be two fifty­ that are not in camera-ready form. minute invited addresses. The names and affiliations of Late papers will not be accepted. the speakers, their titles, and the days and times they Electronic Submission of Abstracts: This service is will talk are as follows: now available to those who use the TEX typesetting JosEPH G. CoNLON, University of Michigan, Ann system and can be used for abstracts of papers to Arbor, Statistical mechanics of Coulomb systems, 9:55 be presented at this meeting. Requests to obtain the a.m. Friday. package of files may be sent by electronic mail on the MICHAEL E. TAYLOR, University of North Carolina, Internet to [email protected]. Requesting the Chapel Hill, The role ofmicrolocal analysis in PDE, 8:50 files electronically will likely be the fastest and most a.m. Friday. convenient way, but l,lsers may also obtain the package Special Sessions: By invitation of the same commit­ on IBM or Macintosh :diskettes, available free of charge tee, there will be six special sessions of selected twenty­ by writing to: Secretary to Director of Publication, minute papers. The topics of these special sessions and American Mathematical Society, Publications Division, the names and affiliations of the mathematicians arrang­ P.O. Box 6248, Providence, RI 02940. When requesting ing them are: the Abstracts package, users should be sure to specify Combinatorics, THOMAS A. DowLING, DIJEN RAY­ whether they want the plain TEX, A·\1S-TEX, or the ~TEX CHAUDHURI, and NEIL RoBERTSON, The Ohio State package. Again, late papers will not be accepted. University. AMS Committee on Science Policy: A panel discus­ Algebraic geometry, SUSAN JANE COLLEY and GARY sion sponsored by the AMS Committee on Science Policy KENNEDY, Oberlin College. is scheduled for Friday, August 10, at 8:00p.m. Dynamics of biological systems, ZIT A M. DIVIS and DAVID TERMAN, The Ohio State University. Other AMS Events Ring theory, S. K. JAIN, Ohio University, and S. Council Meeting: The Council of the Society will meet TARIQ RIZVI, The Ohio State University. at 2:00p.m. on Tuesday, August 7. Combinatorial games, RICHARD K. GuY, University Business Meeting: The Business Meeting of the So­ of Calgary, and RICHARD J. NOWAKOWSKI, Dalhousie ciety will take place immediately following the award of University. the Steele and Wiener Prizes at 11 :05 a.m. on Friday, Group theory, SURINDER K. SEHGAL and RoNALD August 10. The secretary notes the following resolution SoLOMON, The Ohio State University. of the Council: Each person who attends a Business Most of the papers to be presented at these special Meeting of the Society shall be willing and able to sessions will be by invitation; however, anyone contribut­ identify himself as a member of the Society. In further ing an abstract for the meeting who feels that his or her explanation, it is noted that each person who is to vote at paper would be particularly appropriate for one of these a meeting is thereby identifYing himself as and claiming sessions should indicate this clearly on the abstract, and to be a member of the American Mathematical Society.

448 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

For additional information on the Business Meeting, At the April 1915 meeting of the Council, the Com­ please refer to the box titled Committee on the Agenda mittee reported that by a vote of three to two it had for Business Meetings. decided to recommend that the American Mathematical Society "should not undertake nor become responsible which Other AMS- MAA Joint Sessions for the publication of the Monthly", a decision in to note, however, Evening of Dialogue on Mathematics Education: On the Council concurred. It is interesting resolution: "It is Thursday, August 9, the AMS, MAA and Mathemat­ that the Council did adopt the following Society ical Sciences Education Board are cosponsoring several deemed unwise for the American Mathematical events, beginning at 6:30 p.m. with a gala reception for to enter into the activities of the special field now covered all participants. The reception will be followed at 7:30 by the American Mathematical Monthly; but the Council the p.m. by a session on Mathematics education. ELAINE desires to express its realization of the importance of HARISTON, Acting Chancellor, Ohio Board of Regents, work in this field and its value to mathematical science, deal will preside and will introduce the main speaker for the and to say that should an organization be formed to would entertain evening. LIDA K. BARRETT, President of the MAA, and specifically with this work, the Society good WILLIAM BROWDER, President of the AMS, will act as toward such an organization only feelings of hearty respondents. will and encouragement." So with the committee vote and the decision of the Council to accept the committee's recommendation, the MAA 75th Anniversary stage was set for the forming of a new organization to 69th Summer Meeting provide a home for the Monthly. Slaught sent out a letter August 8- 11, 1990 soliciting interest in a new organization and a surprising The founding of the MAA: Had one vote been cast 450, representing every state in the Union, the District differently, we would not be celebrating in August of 1990 of Columbia and Canada, responded by signing the call the 75th Anniversary of the Mathematical Association of to an organizational meeting. America. There probably would have been no MAA. It all The meeting was held in Room 10 1, Page Hall, on started in 1894 when Benjamin Finkel, who had taught the campus of The Ohio State University in Columbus, high school in Ohio and Tennessee, started publishing December 30-31, 1915. The setting was a meeting of The American Mathematical Monthly, aimed primarily the American Association for the Advancement of Sci­ at a high school audience, but subsequently expanded ence (AAAS). Slaught delivered the opening remarks. He to include undergraduates and their teachers. Finkel, talked about the history of the Monthly and "emphasized who later taught at Drury College in Missouri, solicited the fact that this journal had stood consistently, since for his enterprise some distinguished university faculty: its reorganization, for advancing the interests of math­ G. B. Halsted of Texas, E. H. Moore of Chicago, and ematics in the collegiate and advanced secondary fields, W. E. Byerly of Harvard. The first issue contained and expressed the hope that the new organization might an article by Leonard Eugene Dickson, then a 19-year carry forward these aims with still greater effectiveness, old graduate student at Texas. Eventually he persuaded cooperating, on the one hand, with the various well­ Dickson (by then at Chicago) and Herbert Ellsworth organized secondary associations, and the other hand, Slaught at Chicago to become associate editors, along with the American Mathematical Society in its chosen with G. A. Miller of the University of Illinois. The field of scientific research, but being careful to encroach Monthly continued to appear, though on a rather shaky upon neither of these fields." financial foundation, until 1912, when additional support Hedrick presided at the meeting. There were 104 peo­ was obtained from a number of other colleges and ple attending, of whom 49 were from large universities, universities, primarily in the Midwest. 43 from mainly undergraduate institutions, and six from In an attempt to arrange for more stable and perma­ high schools. W. D. Cairns of Oberlin College was the nent support for the Monthly, Slaught proposed assis­ temporary secretary. All the business of organizing the tance from the American Mathematical Society, which Association was completed in one three-hour session, noted in the minutes of the Council "a communication except for one difficult issue: the choice of a name. from Professor Slaught suggesting the appointment by That took a committee of three to sort through eighteen the Council of a Committee to consider the general proposals. The choice of "Mathematical Association of relation of the Society to the promotion of teaching, America" was unanimous. especially in the collegiate field." Such a committee was Hedrick was chosen as the first President, E. V. appointed and consisted of Thomas Scott Fiske, Henry Huntington of Harvard and G. A. Miller of Illinois Burchard Fine, Earle Raymond Hedrick, William Fogg as Vice Presidents, and Cairns as Secretary-Treasurer. Osgood and Slaught. Hedrick later became the 20th President of the American

APRIL 1990, VOLUME 37, NUMBER 4 449 Meetings

Mathematical Society ( 1929-1930). Cairns was to serve as and reports were duly submitted and distributed. Secretary-Treasurer until 1943, a record for the position, The Association has come a long way from the three but not a match for the record of Finkel, who served sections and the 104 people attending the organizational for 19 years as Editor of the Monthly and another 21 meeting in Columbus to 29 sections with almost 28,000 years as Associate Editor. Cairns became President of members today. With the publication of three journals, the MAA in 1943. a newsletter, and a number of series of books and There was no formal program for that first meeting, monographs, along with national and sectional meetings but L.C. Karpinski of the University of Michigan gave and many other activities, the Association is a widely an illustrated lecture on "The Story of Algebra." Cairns felt force in mathematics and mathematics education. reported that "it is not too much to say that for the space of an hour he both charmed and edified an enthusiastic The 75th Anniversary Celebration: To celebrate the audience of approximately one hundred persons." 75th Anniversary of the founding of MAA, the Joint Three sections had been organized prior to the De­ Mathematics Meetings are being held on The Ohio State cember meeting in Columbus, those of Kansas, Missouri, University campus. and Ohio. Clearly the sectional structure of the Associa­ The first day of the meeting, MAA Day, will be tion was not an afterthought. devoted entirely to activities of the Association. Not Founded at the beginning of the First World War, even committee meetings will be scheduled that day. The the Association claimed 1100 members by the end of the day will begin with Opening Ceremonies. The first invited war three years later. In 1920 the MAA was incorporated address will be given by G. BALEY PRICE, University of in the State of Illinois. Kansas, Past President of the MAA ( 19 57-1 9 58), who The issues before the Association in the days before has been active in the MAA for over fifty years. A short and shortly after its founding do not sound much ceremony will follow marking the installation of two different from many of the issues today. Should a course plaques, one outside room 101 of Page Hall (the very in mathematics be required for graduation? Should room where the MAA was organized in 1915) and the calculus be taught in the freshman year? (W. F. Osgood other near the office of the Department of Mathematics. thought that it should be.) Should the United States All those attending the meetings are invited to visit go on the metric system? Should the same course be Page Hall and the mathematics offices to see the actual taught to future mathematicians and scientists as to settings of these commemorative plaques. The morning other students? Should separate courses in different areas will be completed with an invited address by JuDITH V. be taught, or should they be replaced by an integrated GRABINER, Pitzer College, who is a noted historian of course? Committees were formed to study these questions mathematics.

Page Hall, where MAA was founded in 1915

450 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

In the early afternoon four major thirty-minute ad­ NIVEN, EILEEN L. POIANI, G. BALEY PRICE, JOHN 0. dresses will be given by WADE ELLIS, JR., West Valley RIEDL, KENNETH A. Ross, MARCIA P. SwARD (ex offi­ College; PAUL R. HALMOS, Santa Clara University; PETER cio), and ALFRED B. WILLCOX (ex officio). J. HILTON, State University of New York at Bingham­ ton; and CATHLEEN S. MORAWETZ, Courant Institute of MAA DAY, August 8, 1990: The Opening Ceremonies Mathematical Sciences at New York University. A math­ will take place from 8:30 a.m. to 9:30 a.m. The First ematical circus and group picture taking round out the Invited Address will be given by G. BALEY PRICE, Uni­ afternoon. versity of Kansas, at 9:55 a.m. The title of his address is The program for the remaining three days will more The Seventy-Fifth Anniversary Celebration. At 10:55 a.m. resemble the usual Joint Mathematics Meeting, but with there will be a brief ceremony in Mershon Auditorium some special features. There will be ten Joint Invited marking the dedication of a plaque installed at the site Addresses, each jointly sponsored by the MAA and one where the MAA was organized in December 1915. The other organization. The ten cosponsoring organizations Second Invited Address will be given by JUDITH V. GRA­ (listed in order of founding) are the American Associa­ BINER, Pitzer College, at 11:10 a.m. and is titled Was tion for the Advancement of Science (AAAS), the Amer­ Newton's calculus just a dead end? Maclaurin and the ican Mathematical Society (AMS), Pi Mu Epsilon, Inc. Scottish connection. (PME), National Council of Teachers of Mathematics Beginning at 1:30 p.m. there will be four major thirty­ (NCTM), Canadian Mathematical Society (CMS), Asso­ minute addresses as follows: WADE ELLis, JR., West ciation for Computing Machinery (ACM), Society for Valley College, will speak at 1:30 p.m. on Mathematics Industrial and Applied Mathematics (SIAM), National and computation: Proliferation and fragmentation. PAUL Association of Mathematicians (NAM), Association for R. HALMOS, Santa Clara University, will speak at 2:10 Women in Mathematics (AWM), and American Mathe­ p.m. on Has progress in mathematics slowed down? matical Association of Two-Year Colleges (AMATYC). PETER J. HILTON, State University of New York at The Earle Raymond Hedrick Lectures will be given Binghamton, will speak at 2:50p.m. on The contribution by PHILIP J. DAVIS, Brown University. HELAMAN FER­ of mathematics to education. CATHLEEN S. MORAWETZ, GUSON, mathematical sculptor, will make a thirty-minute Courant Institute for Mathematical Sciences at New presentation on the morning of August 9. York University, will lecture on The last 75 years: Giants The Program Committee coincides with the Commit­ of applied mathematics at 3:30 p.m. tee on the 75th Anniversary: GERALD L. ALEXANDERSON There will be a mathematical circus emphasizing (chair), DAVID W. BALLEW, LEONARD GILLMAN, IVAN juggling and magic from 4:45 p.m. to 5:45 p.m. Just

Mershon Auditorium, site of MAA Day Activities

APRIL 1990, VOLUME 37, NUMBER 4 451 Meetings prior to this, at 4: 15 p.m., an historic group picture will AMS-MAA Address: SAUNDERS MAc LANE, Univer­ be taken. All participants are warmly invited to gather sity of Chicago, Algebra as a means of understanding in the Mirror Lake Hollow Amphitheater. A student mathematics, 9:55 a.m., Thursday. reception is scheduled from 6:00 p.m. to 7:00 p.m. and MAA-NCTM Address: JOHN A. DossEY, Illinois State a banquet open to all participants will be held in the University, Mathematics education- yesterday, today, and evening beginning at 7:00p.m. See the section on Social tomorrow, 11:00 a.m., Thursday. Events for more details. MAA-PME Address: IVAN NIVEN, University of Ore­ Hedrick Lectures: The 38th Earle Raymond Hedrick gon, Problems for all seasons, 2:30 p.m., Thursday. Lectures will be given by PHILIP J. DAVIS of Brown Uni­ CMS-MAA Address: PAULO RIBENBOIM, Queen's Uni­ versity. These lectures are scheduled from 1:15 p.m. to versity, Prime number records, 2:30 p.m., Friday. 2:15p.m. on Thursday, Friday, and Saturday, August 9- MAA-SIAM Address: RICHARD TAPIA, Rice Univer­ 11. The series of lectures is titled Spirals from Theodorus sity, Interior point methods for linear programming: An of Cyrene to meta-chaos. The first lecture is titled Spirals: overview, 3:35 p.m., Friday. Old and new; the second is titled Lessons from Euler; ACM-MAA Address: JuRIS HARTMANIS, Cornell Uni­ and the third is titled Theodorus goes wild. versity, On the computational complexity of doing math­ Other MAA Talks: ARTHUR BENJAMIN of Harvey ematics, 8:50 a.m., Saturday. Mudd College will give a special lecture at 4:30 p.m. on MAA-NAM Address: CARL PRATHER, Virginia Poly­ Thursday, August 9, on The art of mental calculation. technic Institute and State University, Intriguing prob­ Benjamin is noted for his extraordinary ability to do lems about zeros in complex analysis, 9:55 a.m., Saturday. complicated calculations without any external memory AWM-MAA Address: JUDITH ROITMAN, University aids like pencil and paper. HELAMAN FERGUSON, Su­ of Kansas, The uses of set theory, 11:00 a.m., Saturday. percomputing Research Center, will give a thirty-minute AMATYC-MAA Address: KARL J. SMITH, Santa Rosa presentation on sculpturing at 8:00 a.m. on Thursday, Junior College, Crisis in mathematics education: perspec­ August 9. tive from the two-year college, 2:30p.m., Saturday. Joint Invited Addresses: As a special feature of the Abbreviations above are as follows: AAAS = Ameri­ 75th Anniversary Celebration, there will be ten joint can Association for the Advancement of Science; AMS invited fifty-minute addresses as follows: = American Mathematical Society; PME = Pi Mu Ep­ AAAS-MAA Address: RICHARD AsKEY, University of silon, Inc.; NCTM = National Council of Teachers of Wisconsin, Madison, Lost and found mathematics, 8:50 Mathematics; CMS = Canadian Mathematical Society; a.m., Thursday. ACM = Association for Computing Machinery; SIAM = Society for Industrial and Applied Mathematics; NAM = National Association of Mathematicians; AWM = Association for Women in Mathematics; AMATYC = American Mathematical Association of Two-Year Col­ leges. These organizations are listed in the order of their age; AAAS is oldest and AMATYC is youngest. Minicourses: Thirteen Minicourses are being offered by the MAA. The names and affiliations of the organizers, the topics, the dates and times of their meetings, and the enrollment limitations of each are as follows: Minicourse #1: Using metacognitive strategies to im­ prove instruction is being organized by GENEVIEVE KNIGHT, Coppin State College. Part A is scheduled from 8:00 a.m. to 10:00 a.m. on Thursday, August 9, and Part B from 2:30 p.m. to 4:30 p.m. on Thursday, August 9. Enrollment is limited to 30. Are your students using efficient and effective methods to attack exercises and problem situations? If not, why not? Given similar or new situations, students are often unable to implement previously learned knowledge, tools, and strategies in finding results. Missing for the students is the holistic approach of understanding the structure of mathematics and how they intake, process, transfer and use information. Perhaps the answers lie within Philip J. Davis, Hedrick Lecturer

452 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings cognitive science. Metacognition instructional strategies of teaching. Readings and a syllabus with an extensive and techniques enable professors to assist their students bibliography will be sent to participants prior to the in becoming critical thinkers and problem solvers. Using meeting. an interactive mode, the organizer will demonstrate Minicourse #4: A calculus laboratory using Mathemat­ strategies, embedded in a metacognitive framework, ica is being organized by MICHAEL BARRY, BENJAMIN that will assist students in shaping their learning and HAYTOCK and RICHARD McDERMOT, Allegheny College. thinking styles. The course will include i) background Part A is scheduled from 10: 15 a.m. to 12: 15 p.m. on information in metacognition and critical thinking and Thursday, August 9, and Part B from 4:45 p.m. to 6:45 ii) mathematical exercises and examples generated from p.m. on Thursday, August 9. Enrollment is limited to 30. algebra, trigonometry, analytic geometry and calculus. The development of easy-to-use computer software Minicourse #2: Planning, funding, and administering with two and three-dimensional graphics and symbolic teacher enhancement projects is being organized by T. manipulation capabilities is leading to fundamental CHRISTINE STEVENS, St. Louis University. Part A is changes in the way we teach, and the way students learn, scheduled from 8:00 a.m. to 10:00 a.m. on Thursday, undergraduate mathematics. One of the most powerful August 9, and Part B from 2:30 p.m. to 4:30 p.m. on of these software systems is Mathematica, developed by Thursday, August 9. Enrollment is limited to 30. Wolfram Research, Inc. under the direction of Stephen Increasingly, mathematicians are being called upon Wolfram. In this Minicourse participants will receive to join in the national effort to improve the quality hands-on instruction in the use of Mathematica on the of precollege education in mathematics. One way in NeXT computer and in the ways that Mathematica and which mathematicians can become involved is through similar software systems can be used to enhance the teacher enhancement programs. These programs pro­ student's understanding of calculus concepts. Attention vide in-service education for precollege teachers, often will also be given to techniques for the development through summer workshops. Federal funding is available of applications packages and on-line textbook modules to support many of these programs. This Minicourse will using the Mathematica Notebook Facility, and to compar­ focus on questions such as: How do we plan a teacher isons of the features of different versions of Mathematica enhancement project? How do we develop contacts and and of other graphics and symbolic systems. No prior initiate interactions with the schools? What funding experience with computers is required. sources are available? How do we prepare a persuasive Minicourse #5: Using history in teaching calculus funding proposal? Participants will hear presentations by is being organized by V. FREDERICK RICKEY, Bowling successful project directors and by a current or former Green State University and on leave at U.S. Military program officer of a funding agency. Some hands-on Academy. Part A is scheduled from 10:15 a.m. to 12:15 activities in the art of proposal preparation will be p.m. on Thursday, August 9, and Part B from 4:45 included. p.m. to 6:45 p.m. on Thursday, August 9. Enrollment is Minicourse #3: A seminar on women in mathematics limited to 80. is being organized by MIRIAM P. CooNEY esc, Saint Students of the calculus instinctively ask many pene­ Mary's College. Part A is scheduled from 10: 15 a.m. to trating questions: What is the calculus? What good is it? 12:15 p.m. on Thursday, August 9, and Part B from 4:45 Why are the concepts presented the way they are? As the p.m. to 6:45p.m. on Thursday, August 9. Enrollment is calculus reform movement eliminates the computational limited to 30. drudgery to concentrate on the fundamental ideas of the The goal of this Minicourse is to prepare participants calculus, it will be even more imperative to respond to to conduct a seminar that identifies women mathe­ these questions. The answers are inherently historical, maticians (past and present), studies their lives and and so by interjecting a historical vein into our teaching the mathematical times as a context for their work, we can respond to these questions in meaningful and and reveals mathematics as a human pursuit. The con­ inspiring ways. A wide variety of ideas for using the tent of the Minicourse, like the seminar, will include history of the calculus that have been successfully used the history of women mathematicians, gender bias and to motivate students will be presented. Some samples: its historic causes, research on gender differences, al­ The geographical origins of the integral of the secant, ternative teaching/learning styles, and recent research an idea of Fermat for integrating xn, a trick of Eu­ on "women's ways of knowing." Assuming that social­ ler's for max-min problems, and how an analysis of a emotional aspects of learning are important to students wrong proof of Cauchy leads to the definition of uniform of mathematics, the Minicourse will provide strategies convergence. Bibliographies and historical notes will be for creating a seminar that provides a support group to provided. encourage potential mathematics majors, both women Minicourse #6: Writing to learn mathematics is being and men. The format will follow seminar-style discus­ organized by AGNES AzzoLINO, Middlesex County Col­ sions, including consideration of "process" as a mode lege. Part A is scheduled from 8:30 a.m. to 10:30 a.m. on

APRIL 1990, VOLUME 37, NUMBER 4 453 Meetings

Friday, August 10, Part B from 2:30p.m. to 4:30p.m. on Writer, the HP Solve and Plot applications, and the Friday, August 10, and Part C from 8:00 a.m. to 10:00 Matrix Writer. The course will illustrate uses of the 48SX a.m. on Saturday, August 11. Enrollment is limited to in several undergraduate courses: calculus, linear algebra, 50. and differential equations. Participants will be given a The objectives of this Minicourse are to establish a handout that includes several customized programs for community interested in writing to learn mathematics use in these courses. The transfer of data from one 48SX (WTLM) and to broaden the awareness of its mem­ to another and between the 48SX and a microcomputer bers. Participants will consider and discuss ways to use will be demonstrated, and the use of plug-in cards to WTLM; talk to each other, exchange materials, and expand memory - both ROM and RAM - will be consider the next steps for their development of WTLM discussed. An HP-48SX calculator will be loaned to each techniques and assignments. [Participants are encour­ participant in the Minicourse. aged to bring copies of written materials they wish to Minicourse #9: Starting, funding and sustaining math­ share.] The novice will see a broad but cohesive picture ematics laboratories is being organized by JAMES E. of possible writing assignments and will hear about indi­ WHITE, Kenyon College. Part A is scheduled from 8:30 viduals using WTLM. The intermediate will be given an a.m. to 10:30 a.m. on Friday, August 10, and Part B from opportunity to verbalize about successes and problems 2:30p.m. to 4:30p.m. on Friday, August 10. Enrollment and to ask advice of experts. The experts will be afforded is limited to 30. the opportunity to stimulate novices, intermediates, and This Minicourse will familiarize participants with other experts by a method other than the publication of successful examples of the use of computer laborato­ a journal article or presentation of a paper. ries in the undergraduate mathematics curriculum. The Minicourse #7: Exploring mathematics with the NeXT course will feature descriptions of ongoing examples of computer is being organized by CHARLES G. FLEMING such laboratories by three or four faculty who have been and Juov D. HALCHIN, Eastern Illinois University. Part involved in them in a variety of settings. The presenta­ A is scheduled from 8:30 a.m. to 10:30 a.m. on Friday, tions will describe the curricular innovations that have August 10, and Part B from 2:30 p.m. to 4:30 p.m. on been made possible by the availability of a mathematics Friday, August 10. Enrollment is limited to 30. computer laboratory, the software that has been found The goals of the Minicourse are as follows: ( 1) useful, and the means by which the laboratories obtained To examine software currently available for the NeXT their initial funding and continuing support. Part of the computer that can be used in undergraduate mathematics Minicourse will outline sources of funding and methods courses. We will look at software for the NeXT which for increasing the probability of success for proposals is currently available for use in courses such as calculus, for such funding. Participants who want a copy of the differential equations, abstract algebra, and differential instructional software may request it at the course and it geometry, as well as Mathematica packages and front will be sent to them for a nominal fee. ends. (2) To examine the software development tools Minicourse #I 0: CAS laboratory projects for calculus is available on the NeXT computer, including Interface being organized by CARL LEINBACH, Gettysburg College. Builder. We will also look briefly at ways in which other Part A is scheduled from 8:00 a.m. to 10:00 a.m. on programs can communicate easily with Mathematica. (3) Saturday, August 11, and Part B from 1:00 p.m. to 3:00 To demonstrate the hardware capabilities of the NeXT p.m. on Saturday, August 11. Enrollment is limited to computer. In particular, we will demonstrate the sound 30. and high resolution capabilities of the NeXT computer. This course is designed to acquaint participants with Participants will participate in "hands-on" activities a method of presenting calculus as a laboratory course. In with all of the software discussed. While no actual addition to discussing the philosophy and the logistics of programming will be required, participants should have a laboratory calculus course, participants will have hands­ a basic understanding of programming in a high-level on experience working in simulated laboratory sessions, language. Participants will also work with each other and with Minicourse #8: A mathematicians's introduction to the instructor to develop outlines for labs that they can the HP-48SX scientific expandable calculator for first-time present as part of their own calculus course. Laboratories users is being organized by JOHN KENELLY and DoN will be conducted using the DERIVE Computer Algebra LA ToRRE, Clemson University. Part A is scheduled from System available for MS/DOS machines with at least 8:30 a.m. to 10:30 a.m. on Friday, August 10, and Part 512K of internal memory. The labs that will be presented B from 1:00 p.m. to 3:00p.m. on Saturday, August 11. may be easily transferred to a Computer Algebra System Enrollment is limited to 30. having a competent symbolic manipulation package, The Minicourse will be a mathematician's hands-on graphics display, and numerical approximation routines. introduction to the HP-48SX and some of the new Minicourse #11: Producing mathematics courseware features which make it so powerful - like the Equation with Mathematica: Calculus and Mathematica is being

454 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings organized by DoN BROWN, HORACIO PORTA, and JERRY (2) Since many of these students will continue to use UHL, University of Illinois, Urbana. Part A is scheduled spreadsheets in their jobs, the knowledge gained in school from 10: 15 a.m. to 12: 15 p.m. on Saturday, August 11, does not become useless when they finish school. In this and Part B from 3: 15 p.m. to 5: 15 p.m. on Saturday, course, use of spreadsheets in learning the following tra­ August 11. Enrollment is limited to 30. ditional topics for business majors will be demonstrated: Mathematica Notebooks allow fully word-processed Gaussian elimination, matrix operations, solving linear text to be inserted in the middle of active Mathematica programming problems, graphing, limits and continuity, code. This constitutes a new medium of communica­ areas under curves. Necessary computer facilities and tion that combines the advantage of a standard word the software for gaining hands-on experience will be processor, the advantage of an enormously powerful available. Lecture notes will be provided. easy-to-use computer algebra system and superb graphic Participants interested in attending any of the MAA capabilities. With Mathematica Notebooks, the reasons Minicourses should complete the MAA Minicourse Pre­ for an upcoming calculation can be discussed, the cal­ registration Form found at the back of this issue and culation can be executed and the meaning of the result send it directly to the MAA office at the address given can be assessed in one single medium. An electronic on the form so as to arrive prior to the June 6 deadline. text, Calculus & Mathematica, is under development at DO NOT SEND THIS FORM TO PROVIDENCE. Illinois and is running in test form at ten other schools. Please note that these MAA Minicourses are NOT the This Minicourse will consist of a brief introduction AMS Short Course. After the deadline, potential partic­ to Mathematica, the examination of a few Calculus & ipants are encouraged to call the MAA headquarters at Mathematica Notebooks, and an introduction to writing 800-331-1622. new Mathematica Notebooks for use in the classroom. Please note that prepayment is required. Payment can be made by check payable to MAA (Canadian checks must Minicourse #12: Exploring statistics and discrete math­ be marked "in U.S. funds") or VISA or MASTERCARD ematics topics using inexpensive graphing calculators is credit cards. being organized by FRANKLIN DEMAN A and BERT K. The MAA Minicourses are open only to persons who WAITS, Ohio State University. This Minicourse is sched­ register for the Joint Mathematics Meetings and pay uled from 10:15 a.m. to 12:15 p.m. on Saturday, August the Joint Meetings registration fee. If the only reason 11. Enrollment is limited to 40. for registering for the Joint Meetings is to gain admis­ Inexpensive ($75 or less) graphing calculators can sion to a MAA Minicourse, this should be indicated by dramatically change the way we teach (and students checking the appropriate box on the MAA Minicourse learn) "finite" mathematics, precalculus and calculus. Preregistration Form. Then, if the Minicourse is fully Participants will learn how to use "statej of the art" Texas subscribed, full refund can be made of the Joint Meet­ Instruments graphing calculators. Graphing calculators ings preregistration fee. Otherwise, the Joint Meetings are powerful tools that permit the user to make and test preregistration will be processed, and then be subject to generalizations by looking at a large number of examples the 50% refund rule. Participants should take care when in a short period of time, to easily solve large (up to 6 by cancelling Minicourse preregistration to make clear their 6) systems of equations, and to deal with problems and intention as to their Joint Meetings preregistration, since applications that are not contrived. Mathematical topics if no instruction is given, the Joint Meetings registration will include solving systems of equations, matrix algebra, will also be cancelled. PREREGISTRATION FORMS data analysis, and statistical modeling. FOR THE JOINT MEETINGS SHOULD BE MAILED Minicourse #13: Spreadsheet based mathematical top­ TO PROVIDENCE PRIOR TO THE DEADLINE OF ics for nonmathematics majors is being organized by V. JUNE 6. S. RAMAMURTI, University of North Florida. Part A is The registration fee for MAA Minicourses #4, 7, 9, scheduled from 10:15 a.m. to 12:15 p.m. on Saturday, 10, 11, and 13 is $60. The registration fee for Minicourse August 11, and Part B from 3:15 p.m. to 5:15 p.m. on # 12 is $18. The registration fee for all other Minicourses Saturday, August 11. Enrollment is limited to 30. is $36. Business students form a very large clientele for math­ ematics courses nowadays. All these students invariably Contributed Papers: Contributed papers are being learn to use the electronic spreadsheet which is the basic accepted on several topics in collegiate mathematics. The program for financial analysis. Mathematics departments topics, organizers, their affiliations, and the days they can take advantage of this and have these students also will meet are: learn mathematics through the electronic spreadsheet. • Liberal arts mathematics courses, SoLOMON A. GAR­ There are two benefits to this approach: ( 1) These stu­ FUNKEL, Consortium for Mathematics and its Appli­ dents do not need to learn a new programming language. cations (COMAP), Friday, August 10, morning and Spreadsheets are programs they can readily relate to. afternoon.

APRIL 1990, VOLUME 37, NUMBER 4 455 Meetings

This session will be devoted to the mathematical assistance (up to $200) for authors of papers selected content and course design for liberal arts students. have been provided by a grant from the EXXON Ed­ Papers which address these themes in the context of ucation Foundation. Following the mathematical circus long-term literacy goals as well as core undergraduate on Wednesday, there will be a student reception from curriculum issues are solicited. 6:00 p.m. to 7:00 p.m. A breakfast for MAA Chapter • Toward equity and excellence: Efforts to increase the Advisors and Coordinators is scheduled for Thursday number of minorities and women in the profession, morning from 7:30a.m. to 8:30a.m. CAROLYN R. MAHONEY, California State University There will be an MAA Student Chapter Panel Discus­ at San Marcos, Thursday afternoon, August 10, and sion from 9:00a.m. to 10:30 a.m. on Thursday, August Friday morning, August 10. 9, organized by APARNA W. HIGGINS, University of Day­ Papers are welcome discussing precollege interven­ ton. The purpose of the discussion will be to show how tions, college and graduate school seminar workshops, mathematics is used in the "real" world and to indicate a and mentor programs, institutional initiatives as well as variety of career opportunities available to people with a state and national efforts aimed at increasing participa­ Bachelor's degree in mathematics. Mathematicians from tion of underrepresented groups. local Ohio industries will serve on the panel. • The interface between mathematics and operations There will be a session on Modeling from 8:00 research, LINN I. SENNOTT, lllinois State University, a.m. to 9:30 a.m. on Friday, August 10, organized by Normal, Thursday, August 10, morning and after­ BEN A. FusARO, Salisbury State University. Outstanding noon. student teams from the 1990 Mathematics Contest in The session has two purposes: 1) to illustrate the Modeling will present their attempts to handle two contributions of mathematics to the development of op­ unsolved real-world problems. The first requires the erations research via such topics as linear programming, distribution pattern for the diffusion of drugs within queueing theory, etc., and 2) to acquaint mathemati­ the brain. The second asks for an efficient way for two cians with the application of operations research models. vehicles to plow (or sweep) a grid of roads. Papers are solicited in either (or both) of these areas. Poster Session on Symbolic Computation: The Com­ Presentations are normally limited to ten minutes, mittee on the Undergraduate Program in Mathematics although selected contributors may be given up to twenty ( CUPM) Subcommittee on Symbolic Computer Systems minutes. Individuals wishing to submit papers for any of is sponsoring a poster session from 8:00 a.m. to noon on these sessions should send the following information to Thursday, August 9. The particular focus of this session the MAA Washington office at 1529 Eighteenth Street, will be What students learn in the symbolic comput­ NW, Washington, DC 20036 by May 18: ing environment. This will be an opportunity for those 1. Title interested in symbolic computation (including use of 2. Intended session supercalculators) to exchange information and opinions 3. A one-paragraph abstract (for distribution at the informally. Each presenter will have a table to display or meeting) distribute material, and sufficient electrical outlets will 4. A one-page outline of the presentation be available to facilitate computer demonstrations. (Pre­ senters must furnish their own computers, however, and sign a release of liability form which should be requested Other MAA Sessions from the organizer, Joan R. Hundhausen, Department MAA Undergraduate Student Paper Session: The third of Mathematics, Colorado School of Mines, Golden, CO MAA Undergraduate Student Paper Session will take 80401.) place from 10:30 a.m. to 12:20 p.m. on Saturday, A broad range of participation is encouraged, with August 11. The session is sponsored by the MAA in respect to types of institutions, courses, and topics. Since conjunction with Pi Mu Epsilon, the undergraduate only 25 presenters can be accommodated, it may be mathematics honorary society, and the MAA Student necessary to select among potential participants based Chapters. Nominations for 10-15 minute papers from upon descriptions of the projects. Anyone interested in Sections of the MAA, mathematics departments, and participating should write to the organizer prior to May other interested parties, with a brief abstract, should be 1. Please include a brief description of the project or sent to Ronald F. Barnes, Department of Mathematics, topics, numbers of faculty and students (or institutions) University of Houston-Downtown, 1 Main St., Houston, involved, and, if possible, the impact upon the learning TX 77002. Nominations would be appreciated by May of mathematics. 15, 1990. Panel Discussion on Symbolic Computation: The In addition to the Student Paper Session, MAA and CUPM Subcommittee on Symbolic Computer Systems Pi Mu Epsilon will cosponsor a number of other stu­ is sponsoring a panel discussion on The pedagogical im­ dent activities. Funds for these activities and travel pact of computer algebra systems on college mathematics

456 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings curricula. This panel is scheduled from 2:30 p.m. to 4:00 site visit. It will be led by ALAN C. TUCKER, SUNY at p.m. on Thursday, August 9. The panelists are WILLIAM Stony Brook, JoHN W. KENELLY, Clemson University, BoYcE, Rensselaer Polytechnic Institute; JOHN HARVEY, and RICHARD MILLMAN, Wright State University, chair University of Wisconsin, Madison; MICHAEL HENLE, of the Committee on Consultants. The discussion is Oberlin College; and JEANETTE PALMITER, Kenyon Col­ primarily intended for those who are consultants, but lege. The panel organizer is RoBERT J. LoPEZ, Rose­ anyone who wishes to attend is welcome. In particular, Holman Institute of Technology. The panelists, charged mathematicians who are thinking about having a team to be specific, will consider the curricula of the first two of consultants visit their university may want to attend years of college mathematics, the advanced mathematics this session. courses, and courses in disciplines such as engineering Panel Discussion on Quantitative Literacy: There will and physics, addressing the question "Given the avail­ be a panel discussion on Quantitative literacy from 8:50 ability of computer algebra, what mathematics should a.m. to 9:40 a.m. on Thursday, August 9, sponsored (and can) we teach, and how should we teach it?'' by the CUPM Subcommittee on Quantitative Literacy Special Presentation on Symbolic Computation: The Requirements (Linda R. Sons, chair). One of the panelists CUPM Subcommittee on Symbolic Computer Systems is will be JoANNE S. GROWNEY, Bloomsburg University of sponsoring a special presentation by PAUL ZoRN, St. Olaf Pennsylvania. College. He will give a lecture at 4:30 p.m. on Thursday, Calculus Reform Today: An Overview: The CUPM August 9, titled Symbolic computing in undergraduate Subcommittee on Calculus Reform and the First Two mathematics: symbols, pictures, numbers, and insights. Years (CRAFTY) is sponsoring a panel discussion on Panel Discussion on Visualization: The Committee calculus reform from 9:00 a.m. to 10:20 a.m. on Saturday, on Computers in Mathematics Education (CCIME) is August 11. The panelists are members of CRAFTY sponsoring a panel discussion on the Visualization Project (Thomas W. Tucker, chair). With support from the which will address the role of visualization in teaching National Science Foundation, CRAFTY is preparing a and learning mathematics at the undergraduate level. book surveying new calculus projects. The book, which Topics will include the role of computer graphics as should appear in the MAA Notes series in time for the well as non-computer based visualization and aspects of Columbus meeting, includes detailed descriptions of ten visual thinking in a variety of mathematical fields. This projects and abstracts for fifty others. Based on their panel is an outgrowth of the Project on Visualization, experiences in the CRAFTY project, the panelists will a project of CCIME. The panel discussion is scheduled attempt to give an overview of calculus reform: where it from 9:00 a.m. to 10:50 a.m. on Friday, August 10, is today and where it seems to be going. with the discussion expected to last 90 minutes leaving State Mathematics Coalitions: The MAA Science Pol­ 20 minutes available for questions and discussion. The icy Committee is sponsoring a panel discussion on State panel will include WALTER ZIMMERMANN, University Mathematics Coalitions, which are alliances of education, of the Pacific (moderator), HERMAN E. GOLLWITZER, corporate, and public policy leaders working to improve Drexel University, and VALERIE A. MILLER, Georgia mathematics education. The Mathematical Sciences Ed­ State University. ucation Board, through grants from the Exxon Education Panel on Research in Learning Undergraduate Math­ Foundation and the Carnegie Corporation of New York, ematics: The Committee on the Teaching of Under­ is supporting the development of pilot coalitions in 25 graduate Mathematics (CTUM) is sponsoring a panel states. This panel discussion, which is scheduled from discussion on research in learning undergraduate mathe­ 10:30 a.m. to noon on Saturday, August 11, will outline matics from 9:00 a.m. to 10:50 a.m. on Friday, August the goals and the structure of these coalitions and will 10. The moderator will be LIDA K. BARRETT, Missis­ describe particular examples of such coalitions and their sippi State University. The panelists will include JoAN activities. Panelists include MARJORIE ENNEKING, Port­ FERRINI-MUNDY, University of New Hampshire and land State University; RoBERT J. KANSKY, Mathematical Program Director at NSF, and JAMES J. KAPUT, South­ Sciences Education Board; and HARVEY B. KEYNES, Uni­ eastern Massachusetts University. The topics discussed versity of Minnesota, Minneapolis (moderator). will be: survey of current research, relationship between Committee on Participation of Women: The Commit­ research in learning and the use of technology, math tee on the Participation of Women is sponsoring a special anxiety and minorities, and research in undergraduate lecture by JUDY GREEN, Marymount University, titled participation in mathematics. The organizing committee History of women in the MAA. This lecture is scheduled consisted of ED DUBINSKY (chair), JOHN A. DOSSEY, and at 3:35 p.m. on Saturday, August 11. BONNIE GOLD. The MAA Committee on Participation of Women Open Discussion on Consultants: There will be an plans to hold discussion groups following skits about open discussion from 3:50 p.m. to 4:50 p.m. on Friday, mini-inequities at the San Francisco meeting in January, August 10, on what MAA consultants do during a campus 1991. It now seeks volunteer discussion leaders able to

APRIL 1990, VOLUME 37, NUMBER 4 457 Meetings

spend two hours at the Columbus meeting learning how The 7tJl€ Dutch Treat Breakfast will take place on to become an effective discussion leader. Please inform Friday, August 10, at 8:00 a.m. The 7tJl€ Council will Pat Kenschaft, Montclair State College, Upper Montclair meet from 12:15 p.m. to 1:15 p.m. on Friday, August 10. NJ 07043, Committee Chair. 7tJl€ will cosponsor undergraduate student paper ses­ sions with MAA. Further information can be found in Prize Session and Business Meeting: The MAA Prize the MAA section of this announcement. Session and Business Meeting is scheduled from 5:05 Information on the 7tf.lf- banquet can be found in the p.m. to 6:00p.m. on Friday, August 10. The Allendoerfer, Social Events section of this announcement. Ford and P6lya Awards will be presented at this meeting, which is open to all members of the Association. Other Events of Interest Board of Governors: The MAA Board of Governors will meet at 8:30 a.m. on Tuesday, August 7. This Book Sales and Exhibits meeting is open to all members of the Association. AMS Information Booth: All meeting participants are Section Officers: There will be a Section Officers' invited to visit the AMS Information Booth in the exhibit meeting at 4:00 p.m. on Tuesday, August 7. area during the meetings. Complimentary coffee and tea will be served. Carol-Ann Blackwood, Membership Manager of the Society, will be at the booth to answer questions about membership in the Society. Activities of Other Organizations Book Sales: Books published by the AMS and MAA The Association for Women in Mathematics (AWM) is will be sold at discounted prices somewhat below the cost sponsoring a panel discussion on Thursday, August 9, at for the same books purchased by mail. These discounts 9:00a.m. will be available only to registered participants wearing The First Annual Alice T. Schafer Mathematics Prize the official meetings badge. Visa and MasterCard credit will be presented at the A WM Membership Meeting cards will be accepted for book sale purchases at the which will be held at 10:15 a.m. on Thursday, August 9. meetings. The book sales will be open the same days and An open reception is planned for Thursday evening, hours as the exhibits. August 9, at 9:00 p.m. Exhibits: The book and educational media exhibits The Joint Policy Board for Mathematics (JPBM) Com­ are open Wednesday through Saturday, August 8- 11. mittee for Mathematics Department Heads has organized The hours they are open are 1:00 p.m. to 5:00 p.m. on a National Meeting of Department Heads at 8:15 p.m. Wednesday, 9:00 a.m. to 5:00 p.m. Thursday and Friday, on Tuesday, August 7. and 9:00 a.m. to noon on Saturday. All participants are The National Science Foundation (NSF) invites par­ encouraged to visit the exhibits during the meetings. ticipants at the Joint Mathematics Meetings to meet informally with staff members over the lunch hour (noon to 1:00 p.m.) daily, Thursday-Saturday, August 9-11. Other Conferences of Interest Short presentations on proposal writing and processing Participants interested in the Workshop for college faculty and Foundation priorities will be followed by the op­ on using graphing calculators and computer graphing portunity for individual questions. Please bring a lunch to enchance the teaching and learning of precalculus (or not) and join us. The Thursday session will focus on mathematics and calculus which will take place right education, the Friday session will focus on research, and after the Columbus meetings should refer to the news the Saturday session will include discussions of education item in the News and Annoucements section in this issue and research. Friday will also provide an opportunity to of Notices. discuss priorities and processes at other Federal agencies funding mathematics research. Sculpture Exhibit The NSF will also be represented at a booth in the During the week of the meetings, the work of Helaman exhibit area. NSF staff members will be available to Ferguson will be on display in the Gallery of Hopkins provide counsel and information on NSF programs of Hall, 128 North Oval Mall. There will be a reception interest to mathematicians. The booth will be open the with the artist on Thursday, August 9, from 5:00 p.m. to same days and hours as the exhibits. Times that staff will 7:00p.m. in the Gallery. be available will be posted at the booth. The Pi Mu Epsilon (nf.lf-) J. Sutherland Frame Lecture Social Events will be delivered on Thursday, August 9, at 8:30p.m. by MAA Banquet: The culmination of the events on MAA RONALD L. GRAHAM, AT&T Bell Laboratories. Day will be a banquet open to all participants. EILEEN There will be sessions for contributed papers Thurs­ POIANI, Past President of Pi Mu Epsilon, will preside, day afternoon and Friday morning and afternoon. and DAVID P. RosELLE, President of the University of

458 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

Delaware and formerly Secretary of the MAA, will be It is strongly recommended that tickets for this banquet the featured speaker. MAA members attending will be be purchased through preregistration, since only a very recognized by Section, in keeping with the emphasis in limited number of tickets will be available for sale on-site. the Opening Ceremonies on the sectional structure of the Tickets are $10 each for students and $14 for all others; MAA. There will be a lively, anecdotal commentary on the price includes gratuity. The menu includes boneless some high points in the life of the Association. Members breast of chicken with herb mushroom sauce, wild rice are encouraged to come to represent their sections and and pistachios, vegetable, rolls and butter, beverage, and to celebrate this milestone in the MAA's history. Special sherbert. Special meals are available upon request, includ­ 75th Anniversary souvenir tiles will be given to those ing vegetarian. Interested participants should complete attending. the appropriate section of the Preregistration/Housing It is strongly recommended that tickets for this ban­ Form and include payment. In the event of cancellations, quet be purchased through preregistration, since only a a 50% refund of the amount paid for the ticket will be very limited number of tickets will be available for sale made if notification is received in Providence by July on-site. Tickets are $25 each; the price includes gratuity. 31. After that date, no refund can be given. The menu includes chicken coq au vin, whipped potato, MAA Student Chapters/nJLf Reception: All students vegetable, rolls, butter, beverage and strawberries Ro­ are invited to attend a reception cohosted by the MAA manoff. Special meals are available upon request, includ­ Student Chapters and Pi Mu Epsilon on Wednesday, ing vegetarian. Interested participants should complete August 8, at 6:00 p.m. Refreshments will be served. the appropriate section of the Preregistration/Housing Certificates for charter chapters will be awarded. Form and include payment. In the event of cancellations, Sock Hop: Let's go to the hop!! Participants are a 50% refund of the amount paid for the ticket will be invited to don their saddle shoes, poodle skirts, varsity made if notification is received in Providence by July sweaters, and the like and dance (or listen) to the 31. After that date, no refund can be given. music of the SO's and 60's at the Sock Hop on Friday Reception for MAA 25-Year Members: Since there evening, August 10, at 6:30 p.m. Entertainment will be will be a banquet open to all participants on MAA Day provided by a local group called "Frick & Frack" whose (Wednesday, August 8), there will be no MAA Banquet amusing skits will bring back memories of "Happy Days". for 25-year Members at this meeting. However, there Their music will move everyone to cha-cha, twist, stroll, will be a special reception for 25-year members of the jitterbug, and do the limbo, while sipping vanilla and MAA from 6:00 p.m. to 7:30 p.m. on Tuesday, August cherry colas. 7. This reception is open to those individuals who have The hop will take place at the Ramada University been members of the Association for twenty-five years Hotel and Conference Center, located approximately one or more, and will be held in the Atrium at the Ramada mile from the campus on Olentangy River Road. A free University Hotel and Conference Center. The Officers shuttle bus will be provided to and from this event. The of the Association and the members of the Board of menu includes grilled hot dogs and hamburgers, cole Governors will also attend this reception. Champagne slaw, potato salad, brownies, cookies (including Oreos), punch, nonalcoholic punch, and assorted cheeses will be coffee, tea, and punch. For vegetarians, the menu includes provided. No formal program is planned, but President vegetarian lasagne, Italian vegetables, tossed salad, rolls Lida K. Barrett will make some welcoming remarks. with butter, brownies, cookies, coffee, tea, and punch. It is strongly recommended that tickets for this recep­ Participants may indicate their meal preference on the tion be purchased through preregistration, since only a Preregistration/Housing Form. Cherry and vanilla colas very limited number of tickets will be available for sale may be purchased at the bar, as well as the usual selection on-site. Tickets are $5 each; the price includes gratuity. of alcoholic beverages. Interested participants should complete the appropriate It is strongly recommended that tickets for this event section of the Preregistration/Housing Form and include be purchased through preregistration, since only a very payment. In the event of cancellations, a 50% refund of limited number of tickets will be available for sale on­ the amount paid for the ticket will be made if notification site. Tickets are $15 for adults and $6 for children 12 is received in Providence by July 31. After that date, no years of age and under, and may be purchased through refund can be given. preregistration by completing the appropriate section MAA 25-year members are asked to check the box on of the Preregistration/Housing Form, and enclosing the the Preregistration/Housing From so special mention of payment. A 50% refund can be made on Sock Hop tickets their status can be made on their meetings badge. until July 31. After July 31, no refunds are possible. Pi Mu Epsilon Banquet: This banquet will take place on Thursday, August 9, at 6:30 p.m., in the River Den Summer List of Applicants Room of the Drake Union. This building is served by At the direction of the AMS-MAA-SIAM Committee the free campus shuttle bus. on Employment Opportunities, which is charged with

APRIL 1990, VOLUME 37, NUMBER 4 459 Meetings operation of the Employment Register and with the pub­ ORDINARY Preregistration lication of Employment Information in the Mathematical (and Residence Hall Housing) June 6 Sciences, the Society will publish a Summer List of math­ FINAL Preregistration ematical scientists seeking employment for distribution (no Housing) July 11 at the Columbus meetings. Please note that requests for residence hall housing Summer List of Applicants will Copies of the 1990 through the Mathematics Meetings Housing Bureau and regis­ be available at the Transparencies section of the applicant forms for the Summer List of Applicants must may tration desk for $6. Following the meetings, they be received by the ORDINARY deadline of June 6. for be purchased from the AMS office in Providence ORDINARY Preregistration: Those who preregister have $8. This list should prove useful to employers who by the ORDINARY deadline of June 6 will receive their or last-minute openings in the latter part of the summer badge and program in the mail two to three weeks prior in the fall. to the meetings, unless they check the appropriate box to of applicant forms to appear The deadline for receipt the contrary on the Preregistration/Housing Form. 6. in this Summer List is June So, it is extremely important that the mailing address The applicant preregistration resume and instructions given on the Preregistration/Housing Form be one at be found in this issue of Notices. on its completion can which the participant can receive this mailing. There Register at the Summer Instead of an Employment will be a special assistance desk at the meetings to assist for Meetings in Columbus, there will be an opportunity individuals who either do not receive this mailing or posting of both applicant resume forms and employe~s' who have a problem with their badge. Please note that positions in or near the mam announcements of open a $2 replacement fee will be charged for programs and area. There will be no special room meetings registration badges that are mailed but not taken to the meetings. No provisions will be made set aside for interviews. Unfortunately, it will not be possible to make changes arrangements will be the by the Society for interviews; to badges received through the mail before the meetings. employer and applicant. Messages responsibility of each Also, it will not be possible to include any tickets may be left in the message box located in the registration to special events purchased through preregistration in area. the mailing with the badge and program. There will employer forms will be available Special applicant and be a special Tickets section at the Joint Mathematics both at the Transparencies section of the registration desk Meetings Registration Desk where prepurchased tickets for applicants to post resumes and for employers to post to the MAA Banquet, nJ.LE Banquet, MAA Reception for forms announcing positions. 25-year Members, MAA Student Chapter Advisors and an applicant form, but do not Applicants who submit Coordinators Breakfast, and/or the Sock Hop may be plan to attend the meetings, will appear on the printed picked up. made for posting resumes Jist only. There is no provision FINAL Preregistration: Those who preregister by the No for participants who do not attend the meetings. FINAL deadline of July I I will pick up their badge at printed lists of employers or applicants who register and program at the meetings. Unfortunately, it is not the meetings will be available after the meetings. possible to provide FINAL preregistrants with housing or tickets to special events, although the latter may still be available for purchase at the meetings. Please note How to Preregister and that the July 11 deadline is firm and any forms received Get a Room after that date must be returned and full refunds issued. It is essential that the Preregistration/Housing Form How to Preregister (found at the back of this issue) be completed fully and The importance of preregistration cannot be overem­ clearly. In the case of several preregistrations from the phasized. Those who preregister pay fees cons_idera~ly same family, each family member who is preregistering lower than the fe(6s that will be charged for registration should complete a separate copy of the Preregistra­ at the meetings, and receive typeset badges instead of tion/Housing Form, but all preregistrations from one typewritten ones. family may be covered by one payment. Please print or There are two separate preregistration deadlines, each type the information requested, and be sure to complete with its own advantages and benefits. all sections. Absence of information (missing credit card All ORDINARY preregistrants will receive formal numbers, incomplete addresses, etc.) causes a delay in acknowledgements prior to the meetings. FINAL pre­ the processing of preregistration for that person. registrants will receive a letter from the Mathematics Please provide your nickname if you wish this infor­ Meetings Housing Bureau (including receipt of payment) mation to be printed on your badge. Also, it is planned prior to the meetings. to make available at the meetings a list of preregistrants

460 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings by area of interest. If you wish to be included in this list, are accompanied by an amount insufficient to cover the please provide the Mathematical Reviews classification total payments due. We are sorry, but it is not possible number of your major area of interest on the Preregis­ for the Mathematics Meetings Housing Bureau to refund tration/Housing Form. (A list of these numbers appears amounts less than $2. Preregistration forms received well on the back of the AMS abstract form.) The master copy before the deadline of June 6 which are not accompanied of this list will be available for review by participants at by correct payment will be returned to the participant the Message Center section of the registration desk. with a request for resubmission with full payment. This ELECTRONIC Preregistration: Preregistration will, of course, delay the processing of any housing through electronic mail is also available. Anyone wish­ request. ing to preregister through this method should send a An income tax deduction is allowed for education message to [email protected] requesting this expenses, including registration fees, cost of travel, meals service. A message will be sent back within 24 hours with and lodging incurred to (i) maintain or improve skills instructions on how to complete the format required. in one's employment or trade or business or (ii) meet Credit card is the ONLY method of payment which can express requirements of an employer or a law imposed be accepted for electronic preregistration. Forms received as a condition to retention of employment, job status, through this method will be treated in the same man­ or rate of compensation. This is true even for education ner as forms received through U.S. mail. Receipt of that leads to a degree. However, the Tax Reform Act of the Preregistration/Housing Form and payment will be 1986 has introduced significant changes to this area. In acknowledged by the Mathematics Meetings Housing general, the deduction for meals is limited to 80% of the Bureau. Participants are advised to bring a copy of this cost. Unreimbursed employee educational expenses are acknowledgement with them to Columbus. The same subject to a 2% of adjusted gross income floor. However, deadlines apply as for normal preregistration. Please there are exceptions to these rules. Therefore, one should note that forms for the Summer List of Applicants cannot contact one's tax advisor to determine the applicability be sent through electronic mail. Only the form found of these provisions. elsewhere in this announcement can be accepted. There is no extra charge for members of the families Registration Fees: The Joint Meetings registration of registered participants, except that all professional fees at the meetings will be 30% higher than the prereg­ mathematicians who wish to attend sessions must register istration fees listed below. independently. Nonmembers who preregister or register at the meet­ ings and pay the nonmember fee will receive mailings Joint Mathematics Meetings the meetings are over, con­ Member of AMS, Canadian Mathematical from AMS and MAA, after a special membership offer. Society, MAA, DME $ 73 taining information about Emeritus Member of AMS, MAA $ 26 Nonmember $110 Student/Unemployed $ 26 How to Get a Room The use of the services offered by the Mathematics AMS Short Course Meetings Housing Bureau requires preregistration for Student/Unemployed $ 15 the Joint Mathematics Meetings. All reservation requests All Other Participants $ 40 for university accommodations must be received in Emeritus Member of AMS, MAA $ 15 writing and be processed through the Housing Bureau. MAA Minicourses Telephone requests cannot be accepted. Please do not (if openings available) contact the university directly, since they will only refer Minicourses# 1, 2, 3, 5, 6, 8 $ 36 callers back to the Housing Bureau. Minicourses #4, 7, 9, 10, 11, 13 $ 60 For the convenience of participants, a shuttle service Minicourse # 12 $ 18 to and from outlying hotels and professional child care have been arranged. In order to offset the costs of Modes of payment which are acceptable, provided they these services, it has been arranged for the hotels and are payable in U.S. dollars to the order of the American dormitories to collect an additional $3 per room per Mathematical Society, are U.S. Postal Money Orders, night, with the exception of child rates in the dormitories. certified U.S. bank checks, U.S. bank money orders, University Housing: Participants desiring confirmed personal checks drawn on a U.S. bank, or credit card reservations for on-campus housing must preregister (Visa or MasterCard only). and send payment in full for housing to the Mathe­ A $5 charge will be imposed for all invoices pre­ matics Meetings Housing Bureau prior to the June 6, pared when preregistration forms are submitted without 1990 deadline. Participants in the Joint Mathematics accompanying check(s) for the preregistration fee(s) or Meetings may occupy residence hall rooms at The Ohio

APRIL 1990, VOLUME 37, NUMBER 4 461 Meetings

The Ohio State University

1-Denney Hall 11-Mershon Auditorium 2-Drackett Tower 12-North Commons 3-Drake Union 13-North Residence Halls 4-Evans Laboratory 14-0hio Stadium 5-Faculty Club 15-0hio Union e Free Campus Loop Bus Stops 6-Hagerty Hall 16-Smith Laboratory .A. Emergency Telephone Locations ?-Hitchcock Hall 17 -Stillman Hall 8-Hughes Hall 18-Sullivant Hall P Public Parking 9-IceRink 19-Central Classroom RP Garage Parking (Ramp) 10-Mendenhall Laboratory 20-Page Hall (MAA's Birthplace)

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462 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

State University during the period August 5 to August These buildings are equipped with ramps; however, 12 only. All must check out by 10:00 a.m. on August 12. there is a limited number of rooms accessible to the All rooms on campus are offered through a room/board handicapped. The halls and rooms are generally too package ONLY. A very limited number of rooms on narrow for a wheelchair to be maneuvered comfortably. campus will be available for those participants who do All rooms contain two bunk beds, four chest of drawers not preregister but plan on attending the meetings and (including four lockable drawers), and two closets. They registering on-site. . connect to a bathroom as well as a lounge that contains All check-ins and room assignments will be done a telephone, four desks with desk lamps, and four at the University Residential Office (URO), located in chairs. There are very few clothes hangers in the rooms. Drackett Tower, ground floor lobby. The URO will Participants are advised to bring their own alarm clock not accept any payments for housing assigned through and clothes hangers. At check-in, participants will find preregistration. ALL advance payments for housing must two towels and a washcloth in their rooms. They will be sent to the Housing Bureau in Providence. (See also be given a key that unlocks their room and the Preregistration/Housing Form.) ALL balances due on outside door of the building. Rooms will be prepared for preregistration and/or housing must be paid at the occupancy in advance and housekeeping service will be Meetings Registration Desk during the hours registration provided Monday through Friday. There is a daily linen is open. These payments may be made with cash, personal change, including towels. checks, travelers' checks, VISA, or MasterCard. No other Each North Residence Hall has vending machines for credit cards can be accepted. The Meetings Registration soft drinks, candy, and cigarettes located in the basement Desk cannot accept payments for university housing or lobby. They also have laundry facilities equipped that was not obtained through preregistration (walk-in with coin-operated washers and dryers, soap vending room assignments). Payments for rooms assigned after machines, and ironing boards. Firearms, fireworks, pets, preregistration are due at check-in time and must be or open containers of alcohol are not permitted in or made at the URO. Payments at the URO may be made around the residence halls; however alcoholic beverages with cash, traveler's checks, personal checks, VISA or are permitted inside sleeping rooms. There are smoking MasterCard. No other credit cards can be accepted. and nonsmoking lounges; however, smoking is permitted Participants requesting housing on The Ohio State inside sleeping rooms. There are no nonsmoking rooms. University campus will be assigned to a room in one Each hall is equipped with an Early Fire Detection and of the North Residence Halls (see map). The Housing Warning System. All rooms have smoke detectors that Bureau will forward all requests for housing to the URO, are connected to a panel in the URO. who will assign all rooms. The Housing Bureau, therefore, Check-In Locations and Times: All check-ins and room is not responsible for room assignments in the residence assignments will be done in the URO, located in the halls. ground floor lobby of Drackett Tower. The office is open Families with children will be permitted to stay in the 24 hours a day, seven days a week. dormitories. With the exception of two special family Driving directions from the Port Columbus Airport packages, all children over five years of age will be to the URO are as follows: Take a left on Steltzer Road. charged the full adult room and board rate. The two Take ramp onto Route 62. Get off on 5th Avenue West. family packages are: 1) a 27% discount on the room and Take a right on N. 4th Avenue. Take a left on Lane board package for two adults and one child 6-16 years Avenue. Drackett Tower is at the corner of Lane Avenue old staying in the same room, and 2) a 42% discount and Neil Avenue. Directions to specific residence halls on the room and board package for two adults and will be provided by the URO. There will be students two children 6-16 years old staying in the same room. available at the URO to assist with luggage. Participants However, children in a room with just one adult will can park temporarily in the lots labeled "P" (see map) each be charged the per person adult double room and while checking in; however, parking permits for longer board rate. There is no room charge for children under term parking must be purchased at a cost of $1.30 per six years old; however, they will each be charged the full day at the URO. meal rate. There are no cribs or rollaways available. The At the time of check-in, participants assigned rooms maximum number of occupants allowed in one room is through the Mathematics Meetings Housing Bureau will four. (See section on Hotel Accommodations below for be checked against a master list (Housing Bureau ac­ alternate housing for families.) knowlegement may prove useful) and asked to sign a The North Residence Halls consist of both high rise statement to be used solely for the purpose of verify­ and low rise buildings. All buildings are air conditioned. ing the university's billing to the Housing Bureau. Each The low rise buildings have four floors and service person will also receive one room key, a meal card, and elevators that can be used. The high rise buildings have a guest brochure. Those participants being assigned a 12 floors and are equipped with passenger elevators. room on-site by the URO will be required to fill out

APRIL 1990, VOLUME 37, NUMBER.4 463 Meetings a housing form and make payment in order to receive 16. After that, all changes will have to be made directly a room key. Spouses desiring a room key must follow with the URO. The URO will accept changes in packages this procedure also. Please note that, although there is no reserved up until 48 hours prior to check-in. Any re­ deposit required for keys, a penalty of $92 will be imposed quests for exceptions to this policy should be addressed for each key lost or not returned. It is the responsibility to Douglas Kayle, Manager of Conference/Orientation of the Housing Bureau to collect this penalty; therefore, Housing, Royer Student Center, 85 Curl Drive, Colum­ it is requested that proper caution be exercised to avoid bus, OH 43210, 614-292-9725. All daily room-and-board this charge. At checkout, all keys must be returned to the packages include dinner on the night of arrival and URO. Should the clerk not be present, please ensure that breakfast the next day. The last meal of a package will your name is left at the desk with the key. be breakfast on the day of check-out. There can be no Room and Board Rates: The rates found in the chart exceptions to meal plans offered, nor can any refunds be which follows apply for residence hall accommodations issued for meals missed. Meal cards are nonrefundable. at the Ohio State University. Food Services: Residence hall guests will dine in the Please note that a minimum room-and-board pack­ North Commons. It is equipped with ramps for the age would be one night's lodging, one dinner and one handicapped. Serving hours for breakfast are 6:30 a.m. breakfast. The Mathematics Meetings Housing Bureau to 9:00 a.m., Monday-Friday, and 7:00 a.m. to 9:00 will accept changes to paekages reserved up until July p.m., Saturday-Sunday. Serving hours for dinner are

The Ohio State University Room and Board Rates

2 Adults & 2 Adults & Adults* Children* 1 Child 6-16 2 Children 6-16 Children (per person) (per person) (whole package) (whole package) under 6 years **

8/5 $35 single $33 single $65 $77 $9 $21 double $19 double

8/6 $35 single $33 single $65 $77 $9 $21 double $19 double

8/7 $35 single $33 single $65 $77 $9 $21 double $19 double

8/8 $35 single $33 single $65 $77 $9 $21 double $19 double

8/9 $35 single $33 single $65 $77 $9 $21 double $19 double

8/10 $35 single $33 single $65 $77 $9 $21 double $19 double

8/11 $35 single $33 single $65 $77 $9 $21 double $19 double

* There can be a maximum of four adults or children per room. They will EACH be charged the double rate per night. ** There is no room charge for children under six years of age; however, there is a full meal charge (per day) for each such child.

464 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings

4:30 p.m. to 6:15 p.m., Monday-Sunday. Meal cards hotel in close proximity to the campus. Participants are must be presented at each meal for admission to the also urged to read the "Words to the Wise" in the local dining area. They are not refundable and there is a information insert in the program. replacement charge for lost meal cards. Children 16 years of age and under must be accompanied by parents in the dining area. A typical breakfast is eggs, bacon, Miscellaneous Information hot cereal, cold cereal, toast, griddle cakes with syrup, Audio-Visual Equipment: Standard equipment in all ses­ assorted fresh fruit, juices, etc. A typical dinner offers sion rooms is one overhead projector and screen. (Invited one or two entrees, vegetables, rolls, salad bar, desserts, 50-minute speakers are automatically provided with two fresh fruit, and beverages. Entrees, soups, vegetables, overhead projectors.) Blackboards are available only in breads and desserts are served from a cafeteria line; a some rooms. self-service salad bar and beverage stations are in the AMS speakers requiring additional equipment should dining area. Servings are generous; unlimited seconds are contact the Audio-Visual Coordinator for the meetings, allowed. It is regretted that there are no Kosher meals at the AMS office in Providence at 401-455-4140, or available. electronic mail [email protected] by June 1. A very limited number of meals is available in the MAA speakers requiring additional equipment may dining rooms on a cash basis for guests not staying in the make written request for one additional overhead projec­ residence halls. The approximate cash price for breakfast tor/screen, 35mm carousel slide projector, 16mm sound is $2.50 and for dinner $6. film projector, or VHS video cassette recorder with one There are several eating establishments located in color monitor. Such requests should be addressed to the Ohio Union. They range from Chinese food to the Audio-Visual Coordinator for the meetings who will pizza and are generally open from 10:30 a.m. to 9:30 forward them to the MAA Associate Secretary for Meet­ p.m. The cafeteria, located on the second floor, is open ings for possible approval. These requests should also be from 7:00 a.m. to 10:00 p.m., Monday-Wednesday, to received by June 1. midnight Thursday-Friday, and 2:00p.m. to 10:00 p.m. on Sunday. There is also a sit-down dining area called Camping and RV Facilities: Alton Campground, 6552 the Terrace Dining Room located on the third floor. W. Broad Street, Columbus, OH, 614-878-9127, is the Lunch is served there from 11:30 a.m. to 1:30 p.m; it is closest campground. It is situated 11 miles west of The closed on weekends. Ohio State University Campus (three miles west of the There are also many fast food restaurants in the 270 Outerbelt on US #40 - West side of Columbus). It immediate vicinity of the university, the majority of has full hook-ups and shower house; approximate cost is them located on N. High Street. $12 per night. Hotel/Motel Accommodations: Please see the follow­ Alum Creek State Park, 3615 S. Old State Road, ing page for instructions on how to obtain hotel and Delaware, OH 614-548-4631 or 548-4039 (Campground), motel accommodations. A free shuttle between the ho­ is located approximately 10-15 miles northeast of the tels and the Ohio Union on the campus will be provided campus. Located approximately one mile west of 1-71 according to the following schedule: on State Rt. 36 & 37. Cost is $10 per night with electric hook-up; reservations not accepted. Amenities: shelter, Tuesday, August 7 7:00 a.m. to 11:00 p.m. drinking water, sanitary facilities, picnic tables, parking, Wednesday, August 8 7:00a.m. to 10:00 a.m. and hiking, guides. It has electricity, showers, flush toilets, 4:00 p.m. to 11:00 p.m. fishing, swimming, dump station, and pet camp, and Thursday, August 9 7:00a.m. to 10:00 a.m. and can accommodate 35 foot length RV's. Alum Creek also 4:30 p.m. to 11:30 p.m. has Rent-A-Camp facilities where novice campers may Friday, August 10 7:00 a.m. to 10:00 a.m. and enjoy the camping experience without first purchasing the 4:00 p.m. to 11 :00 p.m. necessary equipment. Participants arrive at the family Saturday, August 11 7:00 a.m. to 10:00 a.m. and camp area to find the 10-by-12-foot lodge-style tent 4:00 p.m. to 6:00 p.m. already set up with a dining canopy. Two cots, sleeping From 5:30p.m. to 9:30p.m. on Friday, August 10, an pads, cooler, propane stove, lantern, broom, dustpan and additional free shuttle will run this route to accommodate welcome mat are all provided as well as a fire ring and participants attending the Sock Hop. picnic table. One Rent-A-Camp unit has been modified Participants should be aware that when major con­ to accommodate individuals who use a wheelchair. ventions occur in any city, additional safety problems are Delaware State Park, State Park, Rt. 1, Delaware, created, especially at night. Those who are attending the OH 43015, 614-369-2761, is located on State Rt. 23, meetings alone, or who are concerned about walking to and 15 miles north of campus. Cost: $5 no electricity; from the meetings after dark, are encouraged to choose a $9 with electricity; reservations not accepted. Amenities:

APRIL 1990, VOLUME 37, NUMBER 4 465

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8; .... Meetings drinking water, sanitary facilities, picnic tables, parking, 2. A copy of any announcement proposed for either hiking. table is to be sent to the Director of Meetings, American Griggs Dam, 2929 Riverside Drive, Columbus, OH Mathematical Society, Post Office Box 6248, Providence, 43221, 614-645-3229, is located on US Rt. 33, northeast Rhode Island 02940 to arrive at least one week before of Columbus, five miles from the campus. Cost: $5 the first day of the scientific sessions. per night. Reservations not accepted. Amenities: shelter, 3. The judgment on the suitability of an announce­ drinking water, sanitary facilities, picnic tables, parking. ment for display rests with the Joint Meetings Commit­ Child Care: Because of the special nature of this meet­ tee. It will make its judgments on a case-by-case basis to ing, free child care for participant's children from infants establish precedents. to twelve years of age will be provided from 7:30a.m. to 4. Announcements of events competing in time or 5:00 p.m. daily, Wednesday through Saturday. This ser­ place with the scheduled scientific program will not be vice will be provided by American Nursing Care, Inc., a accepted. fully licensed, insured, and bonded professional service, 5. Copies of an accepted announcement for either ta­ in Scarlet Suite D in the Ohio Union. This room will ble are to be provided by the proponent. Announcements be of furnished with casual furniture, a crib, a changing are not to be distributed in any other way at the meetings area, some assorted toys and a televison set. Participants (for example, not by posting or personal distribution of should so indicate on the Preregistration/Housing Form handbills). if they anticipate using this service during the meetings. 6. It may be necessary to limit the number of events or Also, a special arrangement has been made for child the quantity of announcements distributed at a meetings. care in the same room from 6:30p.m. to 10:30 p.m. on 7. At the close of registration, both tables will be the night of the MAA Banquet. Please indicate on the swept clean. Therefore, a proponent who wishes the Preregistration/Housing Form whether this service will return of extra copies should remove them before the be used. close of registation. Handicapped: Most (not all) university facilities are Mail: All mail and telegrams for persons attending the accessible to the handicapped. People with special re­ meetings should be addressed as follows: Name of Par­ quirements for campus housing should make these clear ticipant, Joint Mathematics Meetings, cjo Department when submitting preregistration forms. People with spe­ of Mathematics, Ohio State University, Columbus, OH cial questions regarding handicapped access should con­ 43210. Mail and telegrams so addressed may be picked tact Denise Witcher, Department of Mathematics, The up at the mailbox outside the registration area. U.S. mail Ohio State University, 231 West 18th Avenue, Columbus, not picked up will be forwarded after the meetings to the Ohio 43210-1174, or call 614-292-5279. mailing address given on the participant's registration Information Distribution: A table is set up in the record. registration area of Joint Mathematics Meetings for dis­ semination of information of a nonmathematical nature Telephone Messages: A telephone message center is of possible interest to the members. located in the registration area to receive incoming calls A second table is set up in the exhibit area for the for participants. The center is open from August 8 through dissemination of information of a mathematical nature 11, during the hours that the Joint Mathematics Meetings not promoting a product or program for sale. registration desk is open. Messages will be taken and the If a person or group wishes to display information of name of any individual for whom a message has been a mathematical nature promoting a product or program received will be posted until the message has been picked for sale, they may do so in the exhibit area at the Joint up at the message center. Once the registration desk has Books, Journals and Promotional Materials exhibit for a closed for the day there is no mechanism for contacting fee of $30 per item. participants other than calling them directly at their hotel If a person or group would like to display material in or residence hall. The telephone number of the message the exhibit area separate from the Joint Books table, the center is 614-294-1615. proponent must reimburse the AMS and MAA for any Travel: Port Columbus International Airport is lo­ extra furnishings requested (tables, chairs, easels, etc.) in cated about 12 miles east of the campus. Taxi service addition to payment of the $30 per item fee. is available from the airport to the campus area for ap­ The administration of these tables is in the hands proximately $18-$22. There is also an Airport Shuttle of the AMS-MAA Joint Meetings Committee, as are stop outside the lower baggage claim area. Cost is $6 per all arrangements for Joint Mathematics Meetings. The person to the university area. following rules and procedures apply. Rental car agencies are located in the lower level 1. Announcements submitted by participants should baggage claim area. ordinarily be limited to a single sheet no more than Columbus is serviced by Greyhound and Trailways 8!"x 14". bus lines. The bus station is located downtown, a short

468 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings taxi ride from the campus area ($6-$12). For some years now, the AMS-MAA Joint Meetings Committee has engaged a travel agent for the January and August Joint Meetings in an effort to ensure that everyone Committee on the Agenda for Business Meetings attending these meetings is able to obtain the best possible The Society has a Committee on the Agenda for Business airfare. This service is being performed by TRAVCON; Meetings. The purpose is to make Business Meetings orderly their advertisement can be found elsewhere in this and effective. The committee does not have legal or admin­ meeting announcement. Although any travel agent can istrative power. It is intended that the committee consider obtain Supersaver or other such published promotional what may be called "quasi-political" motions. The committee fares, only TRAVCON can obtain the special additional has several possible courses of action on a proposed motion, including but not restricted to and above these fares, and the 45% 5% discount over (a) doing nothing; off regular coach fare. The latter, of course, is financially (b) conferring with supporters and opponents to arrive beneficial only when one does not qualify for one of at a mutually accepted amended version to be circulated in the promotional fares. Participants should pay particular advance of the meeting; attention to the policies stated in the ad. (c) recommending and planning a format for debate to Weather: Columbus has a moderate climate with four suggest to a Business Meeting; distinct seasons. Average temperature for August is 84°F (d) recommending referral to a committee; for the high and 63° for the low. Average precipitation (e) recommending debate followed by referral to a com- (in inches) for the month of August is 2.07. mittee. There is no mechanism that requires automatic submis­ sion of a motion to the committee. However, if a motion has not been submitted through the committee, it may be Local Arrangements Committee thought reasonable by a Business Meeting to refer it rather The members of the Local Arrangements Committee are than to act on it without benefit of the advice of the com­ W. Wistar Comfort (ex-officio), Joseph mittee. Frank Carroll, The committee consists of M. Salah Baouendi, Robert M. C. Ferrar, J. William Friel, William H. Jaco (ex-officio), Fossum (chairman), and Carol L. Walker. James R. Leitzel (chair), Carolyn Mahoney, Kenneth A. In order that a motion for the Business Meeting of August Ross (ex-officio), and Andrew Sterrett. I 0, 1990, receive the service offered by the committee in the most effective manner, it should be in the hands of the secretary by July I 0, 1990. Robert M. Fossum, Secretary

Petition Table At the request of the AMS Committee on Human Rights of Mathe­ maticians, a table will be made available in the meetings registration area at which petitions on behalf of named individual mathemati­ cians suffering from human rights violations may be displayed and signed by meetings participants acting in their individual capacities. Signs of moderate size may be displayed at the table, but must not represent that the case of the individual in question is backed by the Committee on Human Rights unless it has, in fact, so voted. Volunteers may be present at the table to provide information on individual cases, but notice must be sent at least seven (7) days in advance of the meetings to the Director of Meetings in Providence (telephone 401-455-4137). Since space is limited, it may also be necessary to limit the number of volunteers present at the table at any one time. The Committee on Human Rights may delegate a person to be present at the table at any or all times, taking precedence over other volunteers. Any material which is not a petition (e.g., advertisements, resumes) will be removed by the staff. When registration closes, any material on the table will be discarded, so individuals placing petitions on the table should be sure to remove them prior to the close of registration. Orton Hall

APRIL 1990, VOLUME 37, NUMBER 4 469 Meetings

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470 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Timetable (Eastern Daylight Time)

The purpose of this timetable is to provide assistance to preregistrants in the selection of arrival and departure dates. The program, as outlined below, is based on information at press time.

AMS SHORT COURSE SERIES Monday, COMBINATORIAL GAMES August 6 / MORNING AFTERNOON

8:00 a.m. - 2:30 p.m. SHORT COURSE REGISTRATION 2:00p.m.- 3:15p.m. SHORT COURSE LECTURE #3 Outside Conference Theatre, Ohio Union Impartial games Richard K. Guy 9:00a.m. -10:15 a.m. SHORT COURSE LECTURE #1 What is a game? 3:45 p.m. - 5:00 p.m. SHORT COURSE LECTURE #4 Richard K. Guy More ways of combining games John H. Conway 10:45 a.m.- noon SHORT COURSE LECTURE #2 Numbers and games 5:00 p.m. - 5:30 p.m. SHORT COURSE DISCUSSION John H. Conway PERIOD

Tuesday, ~rican /hem-11 ~., August 7 / Ma~::atical Society / As:~i ation of America / Or~~~izations

MORNING

8:30 a.m. - 4:00 p.m. BOARD OF GOVERNORS' MEETING

9:00a.m.- 10:15 a.m. SHORT COURSE LECTURE #5 Hot games, including Go Elwyn R. Berlekamp

10:45 a.m.- noon SHORT COURSE LECTURE #6 Codes and games Vera S. Pless AFTERNOON

2:00 p.m. - 3:15 p.m. SHORT COURSE LECTURE #7 Complexity of games Aviezri S. Fraenkel

2:00 p.m. - 7:00 p.m. COUNCIL MEETING

3:00 p.m. - 7:00 p.m. REGISTRATION FOR JOINT MATHEMATICS MEETINGS Main Lounge, Ohio Union

APRIL 1990, VOLUME 37, NUMBER 4 471 TIMETABLE

Tuesday, /hematloal /., August 7 (cont'd) / Ma~::atical/"'"" Society / As::~~tion of America / Or~~~:zations

AFTERNOON (cont'd)

3:45p.m. - 5:00 p.m. SHORT COURSE LECTURE #8 Welter's game, dots-and-boxes and Sylver Coinage Richard J. Nowakowski

4:00 p.m. - 6:00 p.m. SECTION OFFICERS' MEETING

5:00 p.m. - 5:30 p.m. SHORT COURSE DISCUSSION PERIOD EVENING

6:30 p.m. - 7:30 p.m. RECEPTION FOR 25-YEAR MEMBERS

8:15 p.m. - 10:00 p.m. NATIONAL MEETING OF DEPARTMENT HEADS

Wednesday, /hematical /,er August 8 ~rican / Ma~::atical Society / As::~ation of America / Or~~~izations

MORNING

7:30 a.m. - 4:00p.m. REGISTRATION Main Lounge, Ohio Union

*** MAADAV***

8:30a.m. - 9:30 a.m. OPENING CEREMONIES

9:55a.m.- 10:45 a.m. FIRST INVITED ADDRESS The Seventy-Fifth Anniversary Celebration G. Baley Price

10:55 a.m. - 11 :00 a.m. PLAQUE CEREMONY

11:10 a.m.- noon SECOND INVITED ADDRESS Was Newton's calculus just a dead end? Maclaurin and the Scottish connection Judith V. Grabiner AFTERNOON

1 :00 p.m. - 5:00 p.m. EXHIBIT AND BOOK SALE BOOK SALE

1 :00 p.m. - 5:00 p.m. EXHIBITS

472 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE

Wednesday, /rioao /homatioal /., August 8 (cont'd) /Ma~::atical Society / As::~:tion of America /__ Or~~:zations

AFTERNOON (cont'dl

1:30 p.m. - 2:00 p.m. INVITED ADDRESS Mathematics and computation: Proliferation and fragmentation Wade Ellis, Jr.

2:10 p.m. - 2:40 p.m. INVITED ADDRESS Has progress in mathematics slowed down? Paul R. Halmos

2:50 p.m. - 3:20 p.m. INVITED ADDRESS The contribution of mathematics to education Peter J. Hilton

3:30 p.m. - 4:00 p.m. INVITED ADDRESS The last 75 years: Giants of applied mathematics Cathleen S. Morawetz

4:15 p.m. - 4:30 p.m. TAKING OF GROUP PICTURE

4:45 p.m. - 5:45 p.m. MATHEMATICAL CIRCUS

EVENING MAA STUDENT CHAPTERS/PME 6:00 p.m. - 7:00 p.m. RECEPTION

7:00p.m.- 10:00 p.m. MAA 75TH ANNIVERSARY CELEBRATION BANQUET _l

Thursday, /hematical August 9 ~rican ~er / Ma~::atical Society / As::~ation of America / Or~~~izations

MORNING 7:30 a.m. - 4:00 p.m. REGISTRATION Main Lounge, Ohio Union

8:00 a.m. - 8:30 a.m. SPECIAL PRESENTATION ON SCULPTURING Helaman Ferguson

8:00a.m.- 10:00 a.m. MINICOURSE #1 (Part A) Using metacognitive strategies to improve instruction Genevieve Knight

MINICOURSE #2 (Part A) Planning, funding, and administering teacher enhancement projects T. Christine Stevens

APRIL 1990, VOLUME 37, NUMBER 4 473 TIMETABLE

Thursday /,,., /homat;o• /" August 9 (cont'd) / Ma~:~atical Society / As::~:tion of America / Or~~~;zations

MORNING (cont'd)

8:00 a.m. - noon SPECIAL SESSIONS CUPM SUBCOMMITTEE ON SYMBOLIC COMPUTER SYSTEMS SESSIONS FOR CONTRIBUTED POSTER SESSION PAPERS Symbolic computation Joan R. Hundhausen (organizer)

CONTRIBUTED PAPER SESSION The interface between mathematics and operations research, Part A Linn I. Sennott

8:50 a.m. - 9:40 a.m. AAAS-MAA INVITED ADDRESS Lost and found mathematics Richard Askey

8:50 a.m. - 9:40 a.m. CUPM SUBCOMMITTEE ON QUANTITATIVE LITERACY REQUIRE­ MENTS PANEL DISCUSSION Quantitative literacy JoAnne S. Growney Linda R. Sons (chair)

9:00 a.m. - 10:00 a.m. ASSOCIATION FOR WOMEN IN MATHEMATICS PANEL DISCUSSION

9:00 a.m. - 10:30 a.m. STUDENT CHAPTER PANEL DISCUSSION Aparna W. Higgins (organizer)

9:00 a.m. - 5:00 p.m. EXHIBIT AND BOOK SALE BOOK SALE .------~------~------, 9:00a.m. - 5:00p.m. EXHIBITS

9:55 a.m. - 10:45 a.m. AMS-MAA INVITED ADDRESS Algebra as a means of understanding mathematics Saunders Mac Lane

10:15 a.m.- 12:15 p.m. MINICOURSE #3 (Part A) A seminar on women in mathematics Miriam P. Cooney esc

MINICOURSE #4 (Part A) A calculus laboratory using Mathematica Michael Barry Benjamin Haytock Richard McDermot

MINICOURSE #5 (Part A) Using history in teaching calculus V. Frederick Rickey

10:15 a.m.- 10:45 a.m. AWM MEMBERSHIP MEETING AND PRIZE SESSION

474 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE /, Thursday, /hematlool1 August 9 (cont'd) / M:::atical/''"' Society / As~o: tation of America / Or~~~:zations

MORNING (cont'd)

11 :00 a.m. - 11 :50 a.m. NCTM-MAA INVITED ADDRESS Mathematics education -yesterday, today, and tomorrow John A. Dossey

AFTERNOON

noon - 1 :00 p.m. NATIONAL SCIENCE FOUNDATION I Informal Discussion Group 1:15 p.m.- 2:15p.m. EARLE RAYMOND HEDRICK LECTURE I Spirals from Theodorus of Cyrene to meta-chaos. Spirals: Old and new Philip J. Davis

2:30 p.m. - 3:20 p.m. MAA-PME INVITED ADDRESS Problems for all seasons Ivan Niven

2:30p.m.- 4:00p.m. CUPM SUBCOMMITTEE ON SYMBOLIC COMPUTER SYSTEMS PANEL DISCUSSION The pedagogical impact of computer algebra systems on college mathematics curricula William Boyce John Harvey Michael Henle Robert J. Lopez (organizer) Jeanette Palmiter

2:30 p.m. - 4:30 p.m. MINICOURSE #1 (Part B) Using metacognitive strategies to improve instruction Genevieve Knight

MINICOURSE #2 (Part B) Planning, funding, and administering teacher enhancement projects T. Christine Stevens

2:30 p.m. - 6:00 p.m. CONTRIBUTED PAPER SESSION The interface between mathematics and operations research, Part B Linn I. Sennott

CONTRIBUTED PAPER SESSION Toward equity and excellence: Efforts to increase the number of minorities and women in the profession, Part A Carolyn R. Mahoney

3:30 p.m. - 5:30 p.m. PI MU EPSILON CONTRIBUTED PAPER SESSIONS

APRIL 1990, VOLUME 37, NUMBER 4 475 TIMETABLE

Thursday, ~~mat;o• August 9 (cont'd) ~~'""matical Society ation of America ~~'zations

AFTERNOON (cont'dl

3:35p.m.- 5:05p.m. PROGRESS IN MATHEMATICS LECTURE Viscosity solutions of partial differential equations Michael G. Crandall

4:30p.m.- 5:20p.m. SPECIAL LECTURE The art of mental calculation Arthur Benjamin

4:30p.m.- 5:20p.m. CUPM SUBCOMMITIEE ON SYMBOLIC COMPUTER SYSTEMS SPECIAL PRESENTATION Symbolic computing in undergraduate mathematics: symbols, pictures, numbers, and insights Paul Zorn

4:45p.m.- 6:45p.m. MINICOURSE #3 (Part B) A seminar on women in mathematics Miriam P. Cooney

MINICOURSE #4 (Part B) A calculus laboratory using Mathematica Michael Barry Benjamin Haytock Richard McDermot

MINICOURSE #5 (Part B) Using history in teaching calculus V. Frederick Rickey EVENING

6:30p.m.- 8:15p.m. PME BANQUET

AMS-MAA-MSEB EVENING OF DIALOGUE

6:30p.m.- 7:30p.m. OPEN RECEPTION

7:30p.m.- 9:00p.m. SESSION ON MATHEMATICS EDUCATION Elaine Hariston, presider Main Speaker To Be Announced Lida K. Barrett, respondent William Browder, respondent

8:30p.m.- 9:30p.m. PME J. SUTHERLAND FRAME LECTURE Title to be announced Ronald L. Graham

9:30 p.m. - 11 :00 p.m. AWM RECEPTION

476 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE

Friday, /,,,, /homati~l /, August 10 / Ma~::atical Society / As~:~:tion of America / Or~~~:zations

MORNING

7:30a.m. - 4:00p.m. REGISTRATION Main Lounge, Ohio Union

8:00a.m. - 10:50 a.m. SPECIAL SESSIONS CONTRIBUTED PAPER SESSION Liberal arts mathematics courses, Part A SESSIONS FOR CONTRIBUTED Solomon A. Garfunkel PAPERS CONTRIBUTED PAPER SESSION Toward equity and excellence: Efforts to increase the number of minorities and women in the profession, Part B Carolyn A. Mahoney

8:00a.m.- 9:30a.m. STUDENT CHAPTERS' SESSION Modeling Ben A. Fusaro (organizer)

8:00a.m.- 9:00a.m. PME DUTCH TREAT BREAKFAST

8:30a.m. - 10:30 a.m. MINICOURSE #6 (Part A) Writing to learn mathematics Agnes Azzolino

MINICOURSE #7 (Part A) Exploring mathematics with the NeXT computers Charles G. Fleming Judy D. Halchin

MINICOURSE #8 (Part A) A mathematician's introduction to the HP-48SX scientific expandable calculator for first-time users John Kenelly Don LaTorre

MINICOURSE #9 (Part A) Starting, funding and sustaining mathematics laboratories James E. White

8:50 a.m. - 9:40 a.m. INVITED ADDRESS The role of microlocal analysis in POE Michael E. Taylor

9:00 a.m. - 10:50 a.m. COMMITTEE ON COMPUTERS IN MATHEMATICS EDUCATION PANEL DISCUSSION Visualization project Herman E. Gollwitzer Valerie A. Miller Walter Zimmermann (moderator)

APRIL 1990, VOLUME 37, NUMBER 4 477 TIMETABLE

Friday, /hemahoal /., August 10 (cont'd) /Ma~::atical/''"" Society / As~:~~tion of America / Or~~~:zations

MORNING (cont'd)

OF 9:00 a.m. - 10:50 a.m. COMMITTEE ON THE TEACHING UNDERGRADUATE MATHEMATICS PANEL DISCUSSION Research in learning undergraduate mathematics Lida K. Barrett (moderator) Joan Ferrini-Mundy James J. Kaput

9:00 a.m. - 5:00 p.m. EXHIBIT AND BOOK SALE BOOK SALE 9:00 a.m. - 5:00 p.m. I EXHIBITS PME CONTRIBUTED PAPER 9:15 a.m. - 10:50 a.m. SESSIONS

9:55a.m. - 10:45 a.m. INVITED ADDRESS Statistical mechanics of Coulomb systems Joseph G. Conlon

11:05 a.m. - noon PRIZE SESSION AND BUSINESS MEETING

AFTERNOON

noon - 1 :00 p.m. NATIONAL SCIENCE FOUNDATION I Informal Discussion Group PME COUNCIL 12:15 p.m.- 1:15 p.m.

1 :15 p.m. - 2:15 p.m. EARLE RAYMOND HEDRICK LECTURE II Spirals from Theodorus of Cyrene to meta-chaos. Lessons from Euler Philip J. Davis

INVITED ADDRESS 2:30p.m.- 3:20p.m. CMS-MAA Prime number records Paulo Ribenboim

2:30 p.m. - 4:30 p.m. MINICOURSE #6 (Part B) Writing to learn mathematics Agnes Azzolino

MINICOURSE #7 (Part B) Exploring mathematics with the NeXT computer Charles G. Fleming Judy D. Halchin

MINICOURSE #9 (Part B) Starting, funding and sustaining mathematics laboratories James E. White

478 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE

Friday, /hematloal /., August 10 (cont'd) /Ma~::atical/''""" Society / As::~~tion of America / Or~~~:zations

AFTERNOON (cont'd)

2:30 p.m. - 4:50 p.m. SPECIAL SESSIONS CONTRIBUTED PAPER SESSION PME CONTRIBUTED PAPER Liberal arts mathematics courses, Part B SESSIONS SESSIONS FOR CONTRIBUTED Solomon A. Garfunkel PAPERS

3:35 p.m. - 4:25 p.m. MAA-SIAM INVITED ADDRESS Interior point methods for linear programming: An overview Richard Tapia

3:50 p.m. - 4:50 p.m. OPEN DISCUSSION ON CONSULTANTS John W. Kenelly Richard Millman Alan C. Tucker

5:05 p.m. - 6:00 p.m. PRIZE SESSION AND BUSINESS MEETING

EVENING

6:30 p.m. - 9:00 p.m. SOCK HOP

8:00p.m.- 10:00 p.m. COMMITIEE ON SCIENCE POLICY PANEL DISCUSSION

Saturday, /erican /thematical /her August 11 / Ma~:~atical Society / As~o~iation of America / Or~~~izations

MORNING 7:30 a.m. - 1 :00 p.m. REGISTRATION Main Lounge, Ohio Union

8:00 a.m. - 10:00 a.m. MINICOURSE #6 (Part C) Writing to learn mathematics Agnes Azzolino

MINICOURSE #10 (Part A) CAS laboratory projects for calculus Carl Leinbach

8:00 a.m. - noon SPECIAL SESSIONS

SESSIONS FOR CONTRIBUTED PAPERS

8:50 a.m. - 9:40 a.m. ACM-MAA INVITED ADDRESS On the computational complexity of doing mathematics Juris Hartmanis

APRIL 1990, VOLUME 37, NUMBER 4 479 TIMETABLE

Saturday, /hemati"l /., August 11 (cont'd) / /"""Ma~:~atical Society / As~:~:tion of America / Or~~~:zations

MORNING (cont'd)

9:00 a.m. - 10:20 a.m. CUPM SUBCOMMITTEE ON AND THE FIRST TWO YEARS PANEL DISCUSSION Calculus reform today: an overview Thomas W. Tucker (chair)

9:00a.m.- noon EXHIBIT AND BOOK SALE BOOK SALE

EXHIBITS 9:00 a.m. - noon I

9:55 a.m. - 1 0:45 a.m. MAA-NAM INVITED ADDRESS Intriguing problems about zeros in complex analysis Carl Prather

10:15 a.m. - 12:15 p.m. MINICOURSE #11 (Part A) Producing mathematics courseware with Mathematica: Calculus and Mathematica Don Brown Horacio Porta Jerry Uhl

MINICOURSE #12 Exploring statistics and discrete mathematics topics using inexpensive graphing calculators Franklin Demana Bert K. Waits

MINICOURSE #13 (Part A) Spreadsheet based mathematical topics for nonmathematics majors V.S. Ramamurti

10:30 a.m. - noon SCIENCE POLICY COMMITTEE PANEL DISCUSSION State mathematics coalitions Marjorie Enneking Robert Kansky Harvey B. Keynes (moderator)

10:30 p.m. - 12:20 p.m. MAA STUDENT CHAPTERS/PME CONTRIBUTED PAPER SESSION

11 :00 a.m. - 11 :50 a.m. AWM-MAA INVITED ADDRESS The uses of set theory Judith Roitman

AFTERNOON

noon - 1 :00 p.m. NATIONAL SCIENCE FOUNDATION I Informal Discussion Group I

480 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY TIMETABLE

Saturday, /,,,., /homalical /, August 11 (cont'd) / Ma~::atical Society / As~:~tation of America / O~~zations

MORNING (cont'd)

1 :00 p.m. - 3:20 p.m. SPECIAL SESSIONS

SESSIONS FOR CONTRIBUTED PAPERS

1:00 p.m. - 3:00p.m. MINICOURSE #8 (Part B) A mathematician's introduction to the HP-4BSX scientific expandable calculator for first-time users John Kenelly Don LaTorre

MINICOURSE #1 0 (Part B) CAS laboratory projects for calculus Carl Leinbach

1 :15 p.m.- 2:15p.m. EARLE RAYMOND HEDRICK LECTURE Ill Spirals from Theodorus of Cyrene to meta-chaos. Theodorus goes wild Philip J. Davis

2:30 p.m. - 3:20 p.m. AMATYC-MAA INVITED ADDRESS Crisis in mathematics education: perspective from the two-year college Karl J. Smith

3:15 p.m. - 5:15p.m. MINICOURSE #11 (Part B) Producing mathematics courseware with Mathematica: Calculus and Mathematica Don Brown Horacio Porta Jerry Uhl

MINICOURSE #13 (Part B) Spreadsheet based mathematical topics for nonmathematics majors V. S. Ramamurti

3:35 p.m. - 5:05 p.m. PROGRESS IN MATHEMATICS LECTURE A-trees and their applications John W. Morgan

3:35 p.m. - 4:25 p.m. COMMTTEE ON THE PARTICIPATION OF WOMEN SPECIAL LECTURE History of women in the MAA Judy Green

APRIL 1990, VOLUME 37, NUMBER 4 481 American Mathematical Society Short Course Series Introductory Survey Lectures on Combinatorial Games Columbus, Ohio, August 6-7, 1990

The American Mathematical Society, in conjunction The course will be self-contained but participants with its ninety-third Summer Meeting, will present a two may wish to consult Winning Ways (Berlekamp, Conway, day Short Course entitled "Combinatorial Games" on Guy; Academic Press, 1982) or On Numbers and Games Monday and Tuesday, August 6-7, 1990, at the Ohio (Conway; Academic Press, 1976). State University, Columbus, Ohio. The program is under Advance registration fee: $40 ($15 student/unem­ the direction of Richard K. Guy, The University of ployed). Onsite registration fee: $50 ($20 student/unem­ Calgary. ployed). Registration and housing information can be The course is designed to be a self-contained intro­ found in this issue of Notices, see sections Preregistration, duction to combinatorial games. The subject is both new Housing and Registration at the Meetings. and extensive, and the organizer has been selective in the Those who also plan to attend the Summer Meeting topics treated. The area grew from parts of recreational should take note of a Special Session on Combinatorial mathematics, but has fast developed into a serious area Games, organized by Richard J. Nowakowski, Dalhousie of study, with close connections with other branches of University. For more information, see the Special Session combinatorics - graph theory, coding theory, networks, section of the Columbus meeting announcement in this complexity - and surreal numbers. It is not the clas­ issue of Notices. sical game theory of von Neumann and Morgenstern; The Short Course series is under the direction of the games are completely determined; there is complete the Short Course Subcommittee of the AMS-MAA Com­ information and there are no chance moves. mittee on Employment and Educational Policy (CEEP): Stefan A. Burr (chair), R. Peter DeLong, Lisl Novak Gaal, Robert P. Kurshan, Barbara L. Osofsky, Marjorie L. Stein, and James J. Tattersall. Monday, August 6: RICHARD K. GuY, University of Calgary, What is a Synopses and Reading Lists: game? What is a game? (Richard K. Guy). In this introductory JoHN H. CoNWAY, Princeton University, Numbers and lecture combinatorial games will be contrasted with games classical games. There is complete information and there RICHARD K. GuY, Impartial games are no chance moves. They are two-person games, so that JoHN H. CoNWAY, More ways of combining games there are no coalitions. Positions and options. The ending condition. Normal and misere play. The four outcome Tuesday, August 7: classes. General definition of a (combinatorial) game. ELWYN R. BERLEKAMP, University of California, Berke­ The games born on day one. Some games are numbers. ley, Hot games, including Go The Simplicity Rule. Some games are infinitesimal. Star VERA PLESS, University of Illinois at Chicago, Codes and and Up. The games born on day two. Some games are Games hot. The negative of a game. Sums of games. Simple AVIEZRI S. FRAENKEL, Weizmann Institute of Science, examples of games, chosen from Blue-Red Hackenbush, Complexity of games Toads-and-Frogs, and Domineering. RICHARD J. NowAKOWSKI, Dalhousie University Welter's (1] E. R. Berlekamp, J. H. Conway and R. K. Guy, Winning game, Dots-and-Boxes and Sylver Coinage Ways for your Mathematical Plays, Academic Press, London and New York, 1982; especially Chapters I and 2. WW Synopses of the talks and accompanying reading lists (2] J. H. Conway, On Numbers and Games, Academic Press, follow. Lecture notes will be mailed to those who prereg­ London and New York, 1976; especially Chapter 7. ONAG ister and will be available at the Short Course registration Numbers and Games (John H. Conway). I'll show desk for those registering on site. how games can be suitably construed and combined so

482 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Short Course Series as to form a group, and how the members of a certain Mathematical Go endgames into winning strategies for subgroup of this group are properly called numbers. The the corresponding endgames played and scored according properties of these numbers will then be used to help to conventional rules. you to play some easy games and to think about some An elaborate theory has been developed which facili­ harder ones. tates precise analyses of Mathematical Go endgames. A [II WW, Chapter 2. homomorphism called "chilling" (a special case of "cool­ [2) ONAG, Chapter 8. ing") reduces many Go endgames to familiar numbers Impartial Games (Richard K. Guy). The same options and infinitesimals which occur commonly in many of the are available to both players, regardless of whose turn it games that were presented and solved in Winning Ways. is to move, so that there are only two outcome classes. A generalized "overheating" operator often inverts the The game ofNim. Nim-addition. Poker Nim. Bogus nim­ "chilling" transformation. heaps. The Mex Rule and the Sprague-Grundy theory. To illustrate the power of these techniques, we have Subtraction games. Octal games; Kayles and Dawson's composed an appropriate collection of Go endgame prob­ Kayles. Periodicity of nim-values. Sparse spaces and lems, which can be readily solved by these mathematical common cosets. Misere Nim and an awful warning. methods, but which have proved very difficult for expert [1) WW, Chapters 3 and 4. Go players. The typical such problem on a 19 x 19 [2) ONAG, Chapter II. Go board has about twenty disjoint active areas of play. [3) Richard K. Guy, Fair Game, CO MAP Math. Exploration Typically, the best move within each such area is "obvi­ Series, Arlington MA, 1989. FG ous" to any good Go player, but the selection of which More Ways of Combining Games (John H. Conway). area in which to play next is much more subtle. A well­ Here I'll talk about some ways of combining games that played game starting from one of these conventional Go were not discussed in my first lecture. The associated endgame problems lasts about forty moves, and White theories are usually complete only for impartial games. wins by one point. Victory requires careful and precise Two particular cases are especially interesting. play throughout the endgame. Many excellent Go players Selective compounds of partizan games. It is surpris­ fail to see the distinctions between the many plausible ing that there is a complete theory here-it depends moves; they typically assume that many of these moves on the theory of numbers already discussed and on are equally good, when in fact only a very small number Steinhaus's notion of remoteness. of the plausible moves are good enough to win against a Disjunctive addition of impartial games under misere mathematically proficient gamesman playing Black. play. Here the theory, although complete in an abstract [1) WW, Chapter 6. sense, is so complicated that special techniques are [2) ONAG, Chapter 9. required to apply it successfully. Codes and Games (Vera Pless). The subject of error­ [11 WW, Chapters 9, 10 and 13. correcting codes arose approximately forty years ago in [2) ONAG, Chapters 12 and 14. response to practical problems in the reliable commu­ Hot Games, including Go (Elwyn R. Berlekamp). Go nication of digitally encoded information. Since then is a classical Oriental board game in which Black and a substantial mathematical theory of coding has been White place stones onto the nodes of a board and attempt developed [6]. In this process many unforeseen relation­ to enclose territory and/or capture opposing stones by ships with other parts of mathematics were discovered. surrounding them. In late-stage Go endgames, the board Some coding theorists are accustomed to call on results can often be partitioned into several disjoint areas, such from group theory, the theory of combinatorial designs, that each subsequent move affects only the area within lattices and sphere packings to name a few areas related which it is played. In this sense, a typical late-stage Go to coding theory. It is gratifying also that results and endgame can be naturally decomposed into a sum of constructions from coding theory have proven useful in more elementary games. many related areas. The relations between combinatorial Although the winner of a game of Go is defined as the games and coding theory are relatively new [1] Chapter player with the larger score we are able to define a very 14, and [4]. similar game called Mathematical Go, in which passing We give an introduction to error-correcting codes is illegal and the winner is the player who succeeds in with emphasis on binary codes. We describe in detail making the last legal move. Although the scoring rules three famous, practical codes and their extended codes; for Mathematical Go appear to be very different from the (7, 4, 4) and (8, 4, 4) Hamming codes, the (17, the scoring rules for the various Japanese and Chinese 9, 5) and ( 18, 9, 6) quadratic residue codes and the versions of the game, it turns out that, under appropriate (23, 12, 7) and (24, 12, 8) Golay codes. These codes conditions which occur very often in practice, all of are related to some of Lenstra's coin-turning games. In these different sets of rules agree on who wins. It is fact, the winning moves in these games coincide with the therefore possible to convert winning strategies for many codewords in their corresponding codes. We describe the

APRIL 1990, VOLUME 37, NUMBER 4 483 Short Course Series combinatorial designs contained in these codes and also high complexity is a liability; in games and cryptography the codes' groups. All of these codes can be described it can be an asset. as quadratic residue codes [6]. The group of the famous [7] C. Berge, Graphs, North-Holland, Amsterdam 1985 (24, 12, 8) code Cis the Mathieu group. Codewords in (Ch.l4). C, and hence winning moves in 24 coin Mogul, can be [8] A. S. Fraenkel and D. Lichtenstein, Computing a per­ found by means of the MOG [5]. fect strategy for n x n chess requires time exponential in n, J. vol. 31 ( 1981 ), pp. 199-214. All of the codes given above are lexicodes. These are Combinatorial Theory (Ser.A), [9] A. S. Fraenkel and Y. Yesha, The generalized Sprague­ a new class of codes whose theory is closely related to Grundy function and its invariance under certain mappings, J. the theory of certain impartial games [4]. They are con­ Combinatorial Theory (Ser.A), vol. 43 (1986), pp. 165-177. structed by a greedy algorithm similar to the calculation [10] M. R. Garey and D. S. Johnson, Computers and In­ of nim-values. Constant weight binary lexicodes with tractability: A Guide to the Theory of NP-Completeness, Free­ minimal distance 4 are related to Welter's games. man, San Francisco 1979 (Appendix A8). [1) WW, Chapter 14. [11] D. S. Johnson, Games and puzzles (The NP-completeness [2)- column: an ongoing guide, 9th edition), J. of Algorithms, vol. 4 [3] FG, pages 59-70. (1983), pp. 397-411. [4] J. H. Conway and N.J. A. Sloane, Lexicographic codes: [12] S. Reisch, Hex ist PSPACE-vollstiindig, Acta Informat­ error-correcting codes from game theory, IEEE Trans. Info. The­ ica, vol. 15 (1981), pp. 167-191. ory, IT-32, No. 3 ( 1986), 337-348. [13] L. J. Stockmeyer and A. K. Chandra, Provably difficult [5] R. T. Curtis, A new combinatorial approach to M24, combinatorial games, SIAM J. Computing, vol. 8 ( 1979), pp. Math. Proc. Comb. Phil. Soc., (1976), 25-42. 151-174. [6] V. Pless, Introduction to the Theory of Error-Correcting Welter's Game, Dots-and-Boxes and Sylver Coinage Codes, second edition, John Wiley and Sons, New York, 1989. (Richard J. Nowakowski). An overview of three surpris­ Complexity of Games (Aviezri S. Fraenkel). We re­ ingly complicated two-player games. Welter's Game is strict attention to the class of 2-player games with perfect played on a semi-infinite strip of squares. There is a information (no hidden information as in some card finite number of coins on the strip, with at most one per games) and no chance moves (no dice), with outcome re­ square. A legal move for either player is to move a coin stricted to Lose, Win and Draw. The complexity level of to an empty square to its left. A player who cannot move even this restricted class varies greatly: while Nim has a loses. The game is like Nim - each occupied square trivial winning strategy, Chess, another family-member, corresponds to a pile of counters but now no two piles has been challenging people's minds for centuries. In can have the same number. The analysis required to Nim, a finite number of piles of tokens is given. The determine a winning move leads naturally to considering two players alternate in selecting a pile, removing from frieze patterns with nim-addition. Dots and Boxes is it any positive number of tokens. The player making played on a rectangular array of dots. Players take turns the last move wins. Features of Chess not present in in joining either two horizontally or two vertically adja­ Nim include: cycles (repeated positions), capture rules cent dots. When a player completes a unit square (box) of tokens and partizanship of moves (White can't move she initials it and must draw another line. The person at a Black piece). Introducing these features one by one the end of the game with the most boxes wins. The game on top of Nim rather than simultaneously, leads to a can be played at various levels of sophistication. The spectrum of new games of increasing complexity, span­ third level (after 'greedy' and 'not-so-greedy') depends ning the gap between Nim-type and Chess-type games. It on analyzing 2 by n arrays with the last-player-to-move­ also serves to fix some points on the games complexity wins rule. In Sylver Coinage the players take turns at map, especially on a number of scenic highways leading choosing positive integers that are not positive linear from Nim to Chess. The talk will review the classical combinations of some previously chosen numbers. The Sprague-Grundy theory, governing Nim-type games, and game is finite but the number of moves is unbounded. then sketch its generalization to permit handling cycles Choosing 5 is a winning first move but what do you (potential draws). Even adding special capture rules on choose if your opponent chooses 100000001? There are top of cycles preserves polynomiality, but many capture many open questions! rules and partizan moves lead to Pspace-hard and even [1] WW, Chapters 15, 16 and 18. Extime-complete and Exspace-complete games. In exis­ [2] ONAG, Chapter 13. tential problems such as the Traveling Salesman problem, [3] FG, pages 71-80.

484 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Invited Speakers and Special Sessions

Invited Speakers the timetable for announcing the existence of Special at AMS Meetings Sessions. The individuals listed below have accepted invitations to address the Society at the times and places indicated. August 1990 Meeting in Columbus, Ohio For some meetings, the list of speakers is incomplete. Associate Secretary: W. Wistar Comfort Deadline for organizers: Expired Deadline for consideration: April 27, 1990 Columbus, OH, August 1990 Susan Jane Colley and Gary Kennedy, Algebraic geometry Joseph G. Conlon John Morgan Zita M. Divis and David Terman, Dynamics ofbiological Michael G. Crandall (Progress in systems (Progress in Mathematics Lecture) Thomas A. Dowling, Dijen Ray-Chaudhuri and Neil Mathematics Lecture) Michael E. Taylor Robertson, Combinatorics Saunders Mac Lane (AMS-MAA) Richard K. Guy and Richard J. Nowakowski, Combina­ torial games Amherst, MA, October 1990 S. K. Jain and S. Tariq Rizvi, Ring theory Christopher B. Croke John J. Mallet-Paret Surinder K. Sehgal and Ronald Solomon, Group theory William M. Goldman Henry P. McKean, Jr. October 1990 Meeting in Amherst, Massachusetts Eastern Section Denton, TX, November 1990 Associate Secretary: W. Wistar Comfort Avner D. Ash John Luecke Deadline for organizers: Expired PeterS. Constantin Clarence W. Wilkerson Deadline for consideration: July 16, 1990 Colin C. Adams, Hyperbolic manifolds M. K. Bennett and Garrett Birkhoff, Lattices, geometry, San Francisco, CA, January 1991 and combinatorics Kenneth A. Ribet Michael F. Atiyah Melvyn S. Berger and Robert A. Gardner, Non-linear (Gibbs Lecturer) Hector J. Sussmann dynamics in mathematics and science Maria M. Klawe Haskell Cohen, Semigroups Invited addresses at Sectional Meetings are selected William M. Goldman and Bernard Maskit, Discrete by the Section Program Committee, usually twelve to groups and geometric structures in 2, 3 and 4 dimen­ eighteen months in advance of a meeting. Members sions wishing to nominate candidates for invited addresses James E. Humphreys and Ivan Mirkovic, Lie groups and should send the relevant information to the Associate algebraic groups Secretary for the Section who will forward it to the Chjan C. Lim, Algebraic graph theory Section Program Committee. V. S. Prasad, Ergodic theory Charles Radin and Marjorie Senechal, Aperiodicity and Organizers and Topics order of Special Sessions The list below contains all the information about Spe­ November 1990 Meeting in Denton, Texas Central Section cial Sessions at meetings of the Society available at the Associate Secretary: Andy Roy Magid time this issue of Notices went to the printer. The sec­ Deadline for organizers: Expired tion below entitled Information for Organizers describes Deadline for consideration: July 16, 1990

APRIL 1990, VOLUME 37, NUMBER 4 485 Invited Speakers and Special Sessions

Avner D. Ash and MarkS. Reeder, Arithmetic groups January 1992 Meeting in Baltimore, Maryland Elizabeth M. Bator, Russell G. Bilyeu and Paul W. Lewis, Associate Secretary: W. Wistar Comfort Banach spaces-functional analysis Deadline for organizers: April 8, 1991 Deadline for consideration: To be announced Ilya Bakelman, Geometric inequalities and convex bodies Scott T. Chapman and Nick H. Vaughan, Commutative March 1992 Meeting in Springfield, Missouri algebra Central Section DanielS. Freed, Robert F. Williams and Michael Wolf, Associate Secretary: Andy Roy Magid Texas topology and geometry Deadline for organizers: June 26, 1991 Anant P. Godbole, The probablility theory of patterns Deadline for consideration: To be announced and runs January 1993 Meeting in San Antonio, Texas John Luecke and Robert Myers, Low dimensional topol­ Associate Secretary: Lance W. Small ogy Deadline for organizers: April 13, 1992 Lisa Mantini and Roger C. Zierau, Representation theory Deadline for consideration: To be announced of Lie groups John W. Neuberger and Henry A. Warchall, Differential August 1993 Meeting in Vancouver, British Columbia, equations Canada Peter F. Stiller, Algebraic geometry Associate Secretary: Andy Roy Magid Emil J. Straube, Several complex variables Deadline for organizers: November 11, 1992 Deadline for considertion: To be announced

November 1990 Meeting in Irvine, California January 1994 Meeting in Cincinnati, Ohio Far Western Section Associate Secretary: Joseph A. Cima Associate Secretary: Lance W. Small Deadline for organizers: April 5, 1993 Deadline for organizers: Expired Deadline for consideration: To be announced Deadline for consideration: July 16, 1990 January 1996 Meeting in Orlando, Florida W. Wistar Comfort January 1991 Meeting in San Francisco, California Associate Secretary: Roy Magid Deadline for organizers: April 12, 1995 Associate Secretary: Andy Deadline for consideration: To be announced Deadline for organizers: April 16, 1990 Deadline for consideration: September 19, 1990 Information for Organizers Special Sessions at Annual and Summer Meetings are March 1991 Meeting in South Bend, Indiana Central Section held under the supervision of the Program Committee Associate Secretary: Andy Roy Magid for National Meetings (PCNM). They are administered Deadline for organizers: June 16, 1990 by the Associate Secretary in charge of that meeting Deadline for consideration: To be announced with staff assistance from the Meetings and Editorial Departments in the Society office in Providence. March 1991 Meeting in Tampa, Florida According to the "Rules for Special Sessions" of the Southeastern Section Society, Special Sessions are selected by the PCNM from A. Cima Associate Secretary: Joseph a list of proposed Special Sessions in essentially the same Deadline for organizers: June 22, 1990 Deadline for consideration: To be announced manner as Invited Speakers are selected. The number of Special Sessions at a Summer or Annual Meeting is limited. The algorithm that determines the number August 1991 Meeting in Orono, Maine of Special Sessions allowed at a given meeting, while W. Small Associate Secretary: Lance here, but can be found in "Rules Deadline for organizers: November 15, 1990 simple, is not repeated Deadline for consideration: To be announced for Special Sessions" on page 614 in the April1988 issue of Notices. October 1991 Meeting in Philadelphia, Pennsylvania Each Invited Speaker is invited to generate a Special Eastern Section Session, either by personally organizing one or by having Associate Secretary: W. Wistar Comfort a Special Session organized by others. Proposals to Deadline for organizers: January 10, 1991 organize a Special Session are sometimes requested Deadline for consideration: To be announced either by the PCNM or by the Associate Secretary. Other proposals to organize a Special Session may be submitted October 1991 Meeting in Fargo, North Dakota to the Associate Secretary in charge of that meeting (who Central Section Associate Secretary: Andy Roy Magid is an ex-officio member of the committee and whose Deadline for organizers: January 25, 1991 address may be found below). These proposals must Deadline for consideration: To be announced be in the hands of the PCNM well in advance of the

486 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Invited Speakers and Special Sessions

meeting and, in any case, at least nine months prior to Eastern Section the meeting at which the Special Session is to be held in W. Wistar Comfort, Associate Secretary order that the committee may consider all the proposals Department of Mathematics for Special Sessions simultaneously. Proposals that are Wesleyan University Middletown, CT 06457 sent to the Providence office of the Society, to Notices, Electronic mail: [email protected] or directed to anyone other than the Associate Secretary (Telephone 203-347-9411) will have to be forwarded and may not be received in Southeastern Section time to be considered for acceptance. Joseph A. Cima, Associate Secretary It should be noted that Special Sessions must be Department of Mathematics announced in Notices in such a timely fashion that any University of North Carolina, Chapel Hill member of the Society who so wishes may submit an Chapel Hill, NC 27599-3902 Electronic mail: [email protected] abstract for consideration for presentation in the Special (Telephone 919- 962- I 050) Session before the deadline for such consideration. This As a general rule, members who anticipate organtzmg Special deadline is usually three weeks before the Deadline for Sessions at AMS meetings are advised to seek approval at least nine Abstracts for the meeting in question. months prior to the scheduled date of the meeting. No Special Sessions Special Sessions are very effective at Sectional Meet­ can be approved too late to provide adequate advance notice to ings and can usually be accommodated. They are selected members who wish to participate. by the Section Program Committee. The processing of proposals for Special Sessions for Sectional Meetings Information for Speakers is handled by the Associate Secretary for the Section, A great many of the papers presented in Special Sessions who then forwards the proposals to the Section Program at meetings of the Society are invited papers, but any Committee, which makes the final selection of the pro­ member of the Society who wishes to do so may submit posals. Each Invited Speaker at a Sectional Meeting is an abstract for consideration for presentation in a Special invited to organize a Special Session. Just as for national Session, provided it is received in Providence prior to meetings, no Special Session at a Sectional Meeting may the special early deadline announced above and in the be approved so late that its announcement appears past announcements of the meeting at which the Special the deadline after which members can no longer send ab­ Session has been scheduled. Contributors should know stracts for consideration for presentation in that Special that there is a limitation in size of a single Special Session. Session, so that it is sometimes true that all places are The Society reserves the right of first refusal for the filled by invitation. Papers not accepted for a Special publication of proceedings of any Special Session. These Session are considered as ten-minute contributed papers. proceedings appear in the book series Contemporary Abstracts of papers submitted for consideration for Mathematics. presentation at a Special Session must be received by More precise details concerning proposals for and the Providence office (Editorial Department, American organizing of Special Sessions may be found in the Mathematical Society, P. 0. Box 6248, Providence, RI "Rules for Special Sessions" or may be obtained from 02940) by the special deadline for Special Sessions, any Associate Secretary. which is usually three weeks earlier than the deadline for contributed papers for the same meeting. The Council has Proposals for Special Sessions to the decreed that no paper, whether invited or contributed, Associate Secretaries may be listed in the program of a meeting of the Society The programs of Sectional Meetings are arranged by the Associate unless an abstract of the paper has been received in Secretary for the section in question: Providence prior to the deadline. Far Western Section (Pacific and Mountain) Electronic submission of abstracts is now available to Lance W. Small, Associate Secretary those who use the TEX typesetting system. Requests to Department of Mathematics obtain the package of files may be sent electronically via University of California, San Diego La Jolla, CA 92093 the Internet to [email protected]. Requesting Electronic mail: [email protected] the files electronically will likely be the fastest and (Telephone 619- 534- 3590) most convenient way, but users may also obtain the Central Section package on IBM or Macintosh diskettes, available free Andy Roy Magid, Associate Secretary of charge by writing to: Electronic Abstracts, American Department of Mathematics Mathematical Society, Publications Division, P.O. Box University of Oklahoma 6248, Providence, RI 02940, USA. When requesting 60 I Elm PHSC 423 Norman, OK 73019 the Abstracts package, users should be sure to specify Electronic mail: [email protected] whether they want the plain TEX. A.MS-TEX. or the IHEX (Telephone 405- 325- 6 711) package.

APRIL 1990, VOLUME 37, NUMBER 4 487 Invited Speakers and Special Sessions

Number of Papers Presented Site Selection for Sectional Meetings Joint Authorship Sectional Meeting sites are recommended by the As­ Although an individual may present only one ten-minute sociate Secretary for the Section and approved by the contributed paper at a meeting, any combination of joint Committee of Associate Secretaries and Secretary. Rec­ authorship may be accepted, provided no individual ommendations are usually made eighteen to twenty-four speaks more than once. An author can speak by invitation months in advance. Host departments supply local in­ in more than one Special Session at the same meeting. formation, ten to twelve rooms with overhead projectors An individual may contribute only one abstract by for contributed paper sessions and special sessions, an title in any one issue of Abstracts, but joint authors auditorium with twin overhead projectors for invited are treated as a separate category. Thus, in addition to addresses, and registration clerks. The Society partially abstracts from two individual authors, one joint abstract reimburses for the rental of facilities and equipment, and by them may also be accepted for an issue. for staffing the registration desk. Most host departments volunteer; to do so, or for more information, contact the Associate Secretary for the Section.

Symposium on Some Mathematical Questions in Biology Neural Networks Vancouver, Canada, August 2-3

The twenty-fourth annual Symposium on Some Mathe­ Program matical Questions in Biology on Neural networks will be held on Thursday and Friday, August 2-3, 1990, during Thursday, August 2, 9:30 a.m. the Congress of the International Union for Pure and Chairman: Robert Miura, University of British Columbia Applied Biophysics (IUPAB), July 29-August 3, 1990. The symposium is sponsored by the American Mathe­ Neurodynamics. JACK D. CowAN, University of matical Society, the Society for Industrial and Applied Chicago Mathematics (SIAM), and the Society for Mathematical Learning, networks, and approximation theory. T. A. Biology (SMB). Poamo, Massachusetts Institute of Technology The AMS-SIAM-SMB Committee on Mathematics in Neural networks, information theory, and perception the Life Sciences serves as the Organizing Committee in animals and machines. RALPH LINSKER, IBM T. J. for the symposium. The committee consisted of Jack D. Watson Research Center Cowan (organizer), Michael C. Mackey, Marc Mangel, Hans G. Othmer, Richard E. Plant (chairman), and John Rinzel. The theme of the symposium is Neural networks. Friday, August 3, 9:30 a.m. There will be two morning sessions on Thursday and Chairman: John Rinzel, National Institutes of Health Friday, August 2 and 3, each including three one-hour lectures. Learning and generalization in multi-layered neural For further information, contact the Symposium Con­ networks: A statistical analysis. D. RUMELHART, Stanford ference Coordinator, AMS, P.O. Box 6248, Providence, University RI 02940 or by electronic mail: [email protected]. Growing and pruning neural networks: relation to sta­ See the Funding Information section (Landahl Travel tistical mechanics. ALAN LAPEDES, Los Alamos National Awards) of this issue for information on travel support Laboratory from the SMB for graduate students to attend this Perspectives in computational neurobiology. TERRENCE meeting. J. SEJNOWSKI, Salk Institute

488 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Call For Topics For 1992 Conferences

Suggestions are invited from mathematicians, either 1992 AMS Symposium in Pure Mathematics singly or in groups, for topics of the various confer­ The Symposium in Pure Mathematics has traditionally ences that will be organized by the Society in 1992. been conducted in the spring of even-numbered years in The deadlines for receipt of these suggestions, as well conjunction with a sectional meeting. The symposium as some relevant information about each of the con­ can be held independently of a sectional meeting and ferences, are given below. An application form to be serves to honor great accomplishments in mathematics. used when submitting suggested topic(s) for any of these Proceedings are normally published by the Society as conferences (except the Short Course Series) may be ob­ volumes in the series Proceedings of Symposia in Pure tained by writing to the Meetings Department, American Mathematics. Mathematical Society, P.O. Box 6248, Providence, RI Topics in recent years have been: 02940, or telephoning 401-455-4146, or sending E-mail 1984-Pseudodifferential operators and Fourier integral to [email protected]. operators with applications to partial differential equa­ Individuals willing to serve as organizers should tions, organized by FRANCOIS TREVES of Rutgers Univer­ be aware that the professional meetings staff in the sity Society's Providence office will provide full support 1987- The mathematical heritage of Herman Weyl, or­ and assistance before, during, and after each of these ganized by R. 0. WELLS, JR. of Rice University. conferences. Organizers should also note that for all 1989- Complex geometry and Lie theory, organized by conferences, except Summer Research Conferences, it is JAMES CARLSON and C. HERBERT CLEMENS, University required that the proceedings be published by the Society, of Utah. and that proceedings of Summer Research Conferences Deadline For Suggestions: September 1, 1990 are frequently published. A member of the Organizing Committee must be willing to serve as editor of the proceedings. All suggestions must include ( 1) the names and affil­ 1992 AMS Summer Institute Summer Institutes are intended to provide an under­ iations of proposed members and the chairman of the presentation of the state of the art in an active Organizing Committee; (2) a one- to two-page descrip­ standable research in pure mathematics and usually extend tion addressing the focus of the topic, including the field of over a three-week period. Dates for a Summer Institute importance and timeliness of the topic, and estimated must not overlap those of the Society's summer meeting, attendance; (3) a list of the recent conferences in the the time of this printing have not yet been same or closely related areas; (4) a tentative list of names which at There should be a period of at least one and affiliations of the proposed principal speakers; and determined. them. Proceedings are published by the (5) a list of likely candidates who would be invited to week between Society as volumes in the series Proceedings of Symposia participate and their current affiliations. Any suggestions as to sites and dates should be made as early as possible in Pure Mathematics. Current and recent topics: in order to allow adequate time for planning. However, 1988-Operator theory /Operator algebras and applica­ proposers of conferences should know that, by action of tions, organized by WILLIAM B. ARVESON of University the AMS Board of Trustees, the Meetings Department of California, Berkeley, and RONALD G. DOUGLAS of of the Society is responsible for the final selection of State University of New York at Stony Brook. the site for each conference and for all negotiations with complex variables and complex geome­ the host institution. Individuals submitting suggestions 1989-Several try, organized by STEVEN G. KRANTZ of Washington for the conferences listed below are requested to recom­ University. mend sites or geographic areas which would assist the geometry, organized by RoBERT E. Meetings Department in their search for an appropriate 1990-Differential of University of California, Los Angeles, and site. In the case of Joint Summer Research Conferences GREENE SHING-TUNG YAU of Harvard University. in the Mathematical Sciences, a one-, two-, or three-week conference may be proposed. Deadline For Suggestions: September 1, 1990

APRIL 1990, VOLUME 37, NUMBER 4 489 Call for Topics

1992 AMS-SIAM-SMB Symposium 1992 AMS-IMS-SIAM Joint Summer Some Mathematical Questions in Biology Research Conferences in the This one-day symposium, sponsored jointly by the AMS, Mathematical Sciences the Society for Industrial and Applied Mathematics, and These conferences, jointly sponsored by the AMS, the the Society for Mathematical Biology, is usually held Institute for Mathematical Statistics, and the Society in conjunction with the annual meeting of a biological for Industrial and Applied Mathematics, are similar in society closely associated with the topic. Papers from the structure to those held at Oberwolfach and represent symposia are published by the AMS as volumes in the diverse areas of mathematical activity, with emphasis on series Lectures on Mathematics in the Life Sciences. areas currently especially active. Careful attention is paid Current and recent topics: to subjects in which there is important interdisciplinary 1987-Models in population biology, organized by ALAN activity at present. Topics for the seventh series of one­ HAsTINGS of the University of California, Davis. week conferences, being held in 1990, are: Probability 1988-Dynamics of excitable media, organized by HANS models and statistical analysis for ranking data; Inverse G. 0THMER of the University of Utah. scattering on the line; Deformation theory of algebras 1989 -Sex allocation and sex change: Experiments and and quantization with applications to physics; Strategies models, organized by MARC MANGEL of the University for sequential search and selection in real time; Schottky of California, Davis. Problems; and Logic, local fields, and subanalytic sets. 1990-Neural Networks, organized by JACK D. CowAN If proceedings are published by the AMS, they appear of the University of Chicago. as volumes in the series Contemporary Mathematics. Deadline For Suggestions: September 1, 1990 Deadline For Suggestions: February 1, 1991

Call for Topics for 1992 AMS-SIAM Summer Seminar 1992 AMS Short Course Series The goal of the Summer Seminar, sponsored jointly by The AMS Short Courses consist of a series of introduc­ the AMS and the Society for Industrial and Applied tory survey lectures and discussions which take place Mathematics, is to provide an environment and program over a period of one-and-one-half days during the Joint in applied mathematics in which experts can exchange Mathematics Meetings held in January and August each the latest ideas and newcomers can learn about the field. year. Each theme is a specific area of applied mathe­ Proceedings are published by the AMS as volumes in the matics or mathematics used in the study of a specific series Lectures in Applied Mathematics. subject or collection of problems in one of the physical, Current and recent topics: biological, or social sciences, technology, or business. 1987- Computational Aspects of VLSI Design with an Current and recent topics: Emphasis on Semiconductor Device Simulation, orga­ Combinatorial games (August 1990), Mathematical nized by RANDOLPH BANK of the University of Califor­ questions in robotics (January 1990), Cryptology and nia, San Diego. computational number theory (August 1989), Matrix the­ 1988- Computational solution of nonlinear systems of ory and applications (January 1989), Chaos and fractals equations, organized by EuGENE ALLGOWER of Colorado (August 1988), Computational Complexity Theory (Jan­ State University. uary 1988). Proceedings are published by the Society 1989- The mathematics of random media, organized as volumes in the series Proceedings of Symposia in by WERNER KOHLER of Virginia Polytechnic Institute Applied Mathematics, with the approval of the Editorial and BENJAMIN WHITE of Exxon Research & Engineering Committee. Company. Deadline for Suggestions: Suggestions for the January 1990- Vortex dynamics and vortex methods, organized 1992 course should be submitted by July 1, 1990; sug­ by CLAUDE GREENGARD of IBM T. J. Watson Research gestions for the August 1992 course should be submitted Center and CHRISTOPHER R. ANDERSON of University of by December 1, 1990. California, Los Angeles. Submit suggestions to: James W. Maxwell, AMS, P.O. Deadline For Suggestions: September 1, 1990 Box 6248, Providence, RI 02940.

490 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1989-1990. Academic Year Devoted to Hyperbolic Geometry and Quasiconformal Mathematical Sciences Mappings, Mittag-Leffler Institute, Djur­ sholm, Sweden. (Dec. 1988, p. 1584) Meetings and Conferences 1989-1990. Special Year in Geometry, University of Maryland, College Park, MD. (Dec. 1989, p. 1432)

Spring Term 1990. Aldo Andreotti Spe­ cial Semester in Pisa, Pisa, Italy. (Mar. 1990, p. 324) 1990. IMACS International Workshop THIS SECTION contains announcements of meetings and conferences of interest to Massively Parallel Methods in Com­ on some segment of the mathematical public, including ad hoc, local, or regional meetings, putational Physics, Boulder, Colorado. and meetings or symposia devoted to specialized topics, as well as announcements (Sep. 1989, p. 914) of regularly scheduled meetings of national or international mathematical organizations. I 990. IMACS Conference on Computer (Information on meetings of the Society, and on meetings sponsored by the Society, will Aided Design, Yugoslavia. (Sep. 1989, be found inside the front cover.) p. 914) AN ANNOUNCEMENT will be published in Notices if it contains a call for papers, and specifies the place, date, subject (when applicable), and the speakers; a second 1990. CWI-IMACS Symposia on Parallel full announcement will be published only if there are changes or necessary additional Scientific Computing, Amsterdam, The information. Once an announcement has appeared, the event will be briefly noted in each Netherlands. (Feb. I 990, p. 216) issue until it has been held and a reference will be given in parentheses to the month, * 1990. Concentration Year on Stochastic year, and page of the issue in which the complete information appeared. Asterisks (*) Models, Statistical Methods, and Algo­ mark those announcements containing new or revised information. rithms in Image Analysis, Rome, Italy. IN GENERAL, announcements of meetings and conferences held in North America carry only date, title of meeting, place of meeting, names of speakers (or sometimes a general PuRPOSE: The aims of this year of statement on the program), deadlines for abstracts or contributed papers, and source study are: I). To enhance the in­ of further information. Meetings held outside the North American area may carry more teraction and collaboration between detailed information. In any case, if there is any application deadline with respect to mathematicians and scientists from participation in the meeting, this fact should be noted. All communications on meetings other disciplines (in particular: biol­ and conferences in the mathematical sciences should be sent to the Editor of Notices, Providence. ogy, physics, engineering, and medi­ care of the American Mathematical Society in DEADLINES for entries in this section are listed on the inside front cover of each issue. In cine); 2). To stimulate the formal­ order to allow participants to arrange their travel plans, organizers of meetings are urged of mathematical ization and study to submit information for these listings early enough to allow them to appear in more than problems suggested from the applied one issue of Notices prior to the meeting in question. To achieve this, listings should be sciences; 3 ). To facilitate the collabo­ received in Providence SIX MONTHS prior to the scheduled date of the meeting. ration between mathematicians com­ EFFECTIVE with the 1990 volume of Notices, the complete list of Mathematical Sciences ing from different backgrounds and Meetings and Conferences will be published only in the September issue. In all other researchers from industry or other issues, only meetings and conferences for the twelve-month period following the month productive activities in projects of of that issue will appear. As new information is received for meetings and conferences common applied interest. that will occur later than the twelve-month period, it will be announced at the end of the will not be repeated until the date of ORGANIZERS: P. Baldi, D. Geman, U. listing in the next possible issue. That information the meeting or conference falls within the twelve-month period. Grenander, M. Piccioni, E. Presutti, B. Ripley, Y. Vardi. INVITED SPEAKERs: Y. Amit, R.G. Azencott, M. Barnsley, 0. Catoni, INFORMATION: Secretariat: G. Castel­ T.S. Chiang, Y. Chow, P. Clifford, lini or G. Silveri, lAC, Viale del Poli­ April1990 clinico 137, 1-00161 Roma (ph. (39 F. Comets, B. Derrida, P. Diaconis, 29-May 5. Gruppentheorie (Pro-Endliche 6) 88470-213 or 214 and Fax n. (39 D.J. Geman, S. Geman, B. Gidas, F. Gruppen), Oberwolfach, Federal Repub­ 6) 88470228). Godliebsen, C. Graffigne, P.J. Greene, lic of Germany. (Jul./ Aug. 1989, p. 766) U. Grenander, X. Guyon, B. Hajek, 1990-1991. Academic Year Devoted to 30-May 4. Mathematiques pour Ia Robo­ R. Holley, C.R. Hwang, N. Karssemei­ Operator Theory and Complex Analysis, tique, Marseille, France. (Jan. 1990, p. 54) jer, R.L. Kashyap, J. Kay, J.T. Kent, Mittag-Leffler Institute, Djursholm, Swe­ D. Lee, T.M. Liggett, S.K. Mitter, R. den. (Dec. 1989, p. 1432) Molina, B.D. Ripley, R. Schonman, May 1990 L.A. Shepp, A.D. Sokal, H. Spohn, D. Pittsburgh Con­ Tjostheim, A. Trouve, Y. Vardi, G. 3-4. Twenty-first Annual Wahba, A.S. Willsky, L. Younes. ference on Modeling and Simulation, Uni-

APRIL 1990, VOLUME 37, NUMBER 4 491 Meetings and Conferences

versity of Pittsburgh, Pittsburgh, PA. many. (Jul./ Aug. 1989, p. 766) * 21-25. Domain Decomposition Meth­ (Sep. 1989, p. 916) 14-15. Finite Element Applications in ods for Partial Differential Equations, 3-4. Automation and Information En­ Computational Mechanics, Champaign, Moscow, USSR. gineering Annual Research Review Con­ IL. (Mar. 1990, p. 326) PROGRAM: This is the fourth interna­ ference, College Park, MD. (Mar. 1990, 14-15. Regional Workshop on Nonlinear tional symposium on domain decom­ p. 326) Conservation Laws, Stony Brook, NY. position methods for POE's. The pro­ 4. Second Conference on Mathematical (Jan. 1990, p. 55) gram will focus almost exclusively on Biology, Stony Brook, NY. (Mar. 1990, 14-18. Conference on Nonlinear Anal­ domain decomposition methods, but p. 326) ysis and Partial Differential Equations, talks on multigrid methods might also 5-6. Pacific Northwest Geometry Semi­ Rutgers University, New Brunswick, NJ. be scheduled. Thirteen main speakers nar, University of Oregon, Eugene, OR. (Mar. 1990, p. 326) have been invited. There will also be (Jan. 1990, p. 55) 14-18. Workshop on K-Theory, Mathe­ a number of shorter contributed talks. 5-6. Midwest Partial Differential Equa­ matical Sciences Research Institute, Berke­ All papers will be published in Rus­ tions Seminar, Northwestern Univ., ley, CA. (Jan. 1990, p. 55) sian by the USSR Academy and in Evanston, IL. (Mar. 1990, p. 326) 14-18. Singularities et Theorie de Hodge, English by SIAM. 6-9. Computer Algebra and Differential Marseille, France. (Jan. 1990, p. 55) INFORMATION: 0. Widlund,Program Committee, cfo Courant Institute of Equations (CADE-90), Cornell Univer­ 14-18. Surfaces, Submanifolds, and their Mathematical Sciences, 251 Mercer sity, Ithaca, NY. (Feb. I 990, p. 218) Applications, Univ. of Leeds, Leeds, Eng­ St., Room 530, New York, NY 10012; 6-12. Geschichte der Mathematik, Ober­ land. (Mar. 1990, p. 326) (212)998-331 0. wolfach, Federal Republic of Germany. 17-19. Interface '90 (formerly Computer (Jul./ Aug. 1989, p. 766) Science and Statistics: Symposium on In­ * 22-24. Third International Symposium 7-9. 1990 IEEE Symposium on Research terface), East Lansing, MI. (Sep. 1989, of Numerical Analysis, Madrid, Spain. in Security and Privacy, Oakland, CA. p. 916) INVITED SPEAKERS: Professors Samar­ (Oct. 1989, p. 1095) 17-19. Colloquium: Computer Graphics ski, Tihomirov, and Mihailov (USSR); 7-9. Geometric Modeling and Design, in Pure Mathematics, University of Iowa, Douglas (USA); Ortiz (Great Britain); Wayne State Univ., Detroit, MI. (Mar. Iowa City, lA. (Dec. 1989, p. 1435) Ansorge, Niethammer, and Werner 1990, p. 326) 18-20. Nineteenth Annual State of Jef­ (Germany); Ciarlet, Pironneau, and 7-I 0. SIAM Conference on Applications ferson Mathematics Congress, Whiskey­ Raviart (France); Axelsson (The of Dynamical Systems, Orlando, FL. (Sep. town, CA. (Feb. 1990, p. 219) Netherlands); and others. 1989, p. 916) 20-23. 1990 International Industrial En­ INFORMATION: I.S.N.A. 3, C. Vega, 7-II. Recent Advances in Regression, gineering Conference, San Francisco, CA. Univ. Politecnica de Madrid, Ramiro Montreal, Canada. (Jan. 1990, p. 55) (Mar. 1990, p. 327) de Maeztu, sfn, 28040 Madrid, Spain. 7-11. Algorithme et Programmation, 20-25. NSF /CBMS Conference on Op­ * 22-25. Computer Simulations Statisti­ Marseille, France. (Jan. 1990, p. 55) erator Algebras, Texas Christian Univer­ cal Methods and Applications, Portofino, 7-11. Statistics Week, Universite de sity, Fort Worth, TX. (Jan. 1990, p. 55) Italy. Montreal, Canada. (Feb. 1990, p. 218) 20-26. The Schrodinger Equation and INFORMATION: F. Marchetti, 7-June I. College on Recent Develop­ Its Classical Counterparts, Oberwolfach, Univ. di Genova, Dipartimento di Matemat­ ments and Applications in Mathematics Federal Republic of Germany. (Jul./ Aug. ica, v L.B. Alberti 4, 1-16132 Genova, and Computer Science, International Cen­ 1989, p. 767) Italy; ( +39-10)353-8717. tre for Theoretical Physics, Trieste, Italy. 21-24. The Simulation of Random Pro­ (MayfJun. 1989, p. 601) cesses and Fields - Mathematics and 23-25. 1990 International Symposium * 9-ll. Applications of Mathematical/Sta­ Applications, Portofino, Italy. (Sep. 1989, on Multiple-Valued Logic, Charlotte, NC. tistical Libraries and Problem-Solving Sys­ p. 916) (Apr. 1989, p. 496) tems, Monterey, CA. 21-25. NSF-CBMS Regional Confer­ 23-25. Workshop on Viscous and Nu­ ence on The Interface between Analytic merical Approximation of Shock Waves, INFORMATION: D. Mar, IMSL User Number Theory and Harmonic Analysis, North Carolina State University, Raleigh, Group North America, Naval Post­ Manhattan, KS. (Jan. 1990, p. 55) NC. (Feb. 1990, p. 219) graduate School, 1332 Lincoln Av­ 21-25. Eleventh United States National 23-27. Azumaya Algebras, Group Ac­ enue, Pacific Grove, CA 93950; 408- Congress of Applied Mechanics, Tucson, tions, and Modules: A Conference in 646-2672. AZ. (Nov. 1988, p. 1389) Honor of Goro Azumaya's 70th Birth­ 21-25. Nonlinear Analysis, Function day, Indiana University, Bloomington, 9-12. Computer Algebra and Parallelism Spaces and Applications IV, Czechoslo­ IN. (Feb. 1990, P. 219) (CAP-90), Cornell University, Ithaca, NY. vakia. (Jan. 1990, p. 56) 24. Second Conference on Lagrange Cal­ (Feb. 1990, p. 218) 21-25. DIMACS Workshop: Algebraic culus, Community College of Philadel­ 13-l 9. Abstrakte Konvexe Analysis, Issues in Geometric Computations, New phia, Philadelphia, PA. (Feb. 1990, p. 219) Oberwolfach, Federal Republic of Ger- Brunswick, NJ. (Mar. 1990, p. 327) 24-25. Twelfth Symposium on Mathe-

492 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences

matical Programming with Data Perturba­ ORGANIZING CoMMITTEE: D. Gar­ 4-12. Recent Developments in Geomet­ tions, George Washington Univ., Wash­ ity, Oregon State Univ.; D. Wright, ric Topology and Related Topics, Villa ington, DC. (Nov. 1989, p. 1250) Brigham Young Univ.; J. Henderson "La Querceta", Montecatini Terme. (Feb. 24-26. Conference on Probability Mod­ and F. Tinsley, Colorado College. 1990, p. 220) els in Mathematical Physics, Colorado PRINCIPLE SPEAKER: R. Daverman, 4-15. Analyse Harmonique sur Les Springs, CO. (Jan. 1990, p. 56) Univ. of Tennessee. Groupes Reductifs P-Adiques, Marseille, 25-31. Tenth International Conference INFORMATION: D. Garity, Dept. of France. (Jan. 1990, p. 56) on Pattern Recognition, Resorts Hotel, Math., St. Cloud State Univ., St. 4-28. Supercomputing Program for Un­ Atlantic City, NJ. (Mar. 1988, p. 466) Cloud, MN 56301. dergraduate Research, Cornell National 27-June 2. Lyapunov-Exponents, Ober­ 31-J une 3. Percolation Models of Ma­ Supercomputing Facility, Ithaca, NY. wolfach, Federal Republic of Germany. terial Failure, Cornell University, Ithaca, (Feb. 1990, p. 220) (Jul./ Aug. 19 89, p. 767) NY. (Dec. 1989, p. 1436) 6-8. First IFIP Conference on Fractals, 28-June I. Twenty-second Annual Con­ Lisbon, Portugal. (Jan. 1990, p. 57) ference on Statistics, Tours, France. (Jan. 6-9. Fifth Annual Conference of the 1990, p. 56) European Consortium for Mathematics June 1990 28-June I. Tenth International Confer­ in Industry, Lahti, Finland. {Apr. 1989, ence on Distributed Computing Systems, June-July 1990. International IMACS p. 496) Paris, France. (Jan. 1990, p. 56) Conference on Mathematical Modelling 6-12. 1990 Barcelona Conference on 28-June I. Mecanique Celeste et Sys­ and Applied Mathematics, Vilnius, USSR. Algebraic Topology, Centre de Recerca temes Hamiltoniens, Marseille, France. (Sep. 1989, p. 917) Matematica, Barcelona, Spain. (Sept. (Jan. 1990, p. 56) 1-8. Third International Symposium on 1988, p. 1060) 28-June 1. Workshop on Set Theoretic Orthogonal Polynomials and Their Ap­ 6-15. Third Logical Biennial (in honour Methods in Algebra, Baylor University, plications, Erice-Trapani (Sicily), Italy. of S.C. Kleene), Chaika (near Varna), Waco, Texas. (Feb. 1990, p. 219) (Dec. 1989, p. 1436) Bulgaria. (Oct. 1989, p. I 096) 28-June 2. Geometry of Complex Projec­ 1-10. Fourth Annual Meeting of the 7-July 4. 1990 Joint Summer Research tive Varieties, Cetraro, Italy. (Nov. 1989, International Workshop in Analysis and Conferences in the Mathematical Sci­ p. 1250) its Applications, Dubrovnik-Kupari, Yu­ ences, University of Massachusetts at 29-30. Algebraic Logic Conference in goslavia. (Oct.l989,p. 1096) Amherst, MA. Honour of Professor Don Monk, Boulder, 3-6. 1990 Annual Meeting of the Sta­ IN FORMA noN: C. Kohan ski, AMS, Colorado. (Nov. 1989, p. 1250) tistical Society of Canada, St. John's, P.O. Box 6248, Providence, RI 02940. Canada. (Sep. 1989, 29-31. GAMM/IFIP Workshop: "Sto­ Newfoundland, Conference chastic Optimization: Numerical Meth­ p. 917) 10-14. Sixth Haifa Matrix (Feb. 1990, ods and Technical Applications", Neu­ 3-6. Symposium on Chaos in Biolog­ Technion City, Haifa, Israel. biberg, Federal Republic of Germany. ical and Agricultural Systems, Lincoln, p. 220) (Jan. 1990, p. 56) Nebraska. (Jan. 1990; p. 56) 10-16. Reelle Algebraische Geometrie, Oberwolfach, Federal Republic of Ger­ 29-June I. Eleventh Annual Conference 3-9. Graphentheorie, Oberwolfach, Fed­ 767) of the Canadian Applied Mathematics eral Republic of Germany. (Jul./ Aug. many. (Jul./ Aug. 1989, p. Society, Halifax, Nova Scotia. (Oct. 1989, 1989, p. 767) 10-16. Fourth Czechoslovak Symposium p. 1096) 4-7. Fifth Annual IEEE Symposium on on Combinatorics, Prachtice, Czechoslo­ 29-June 2. Dynamical Theories of Turbu­ Logic in Computer Science, Philadelphia, vakia. (Nov. 1989, p. 1251) lence in Fluid Flows, Minneapolis, MN. PA. (Sep. 1989, p. 917) * 10-16. International Workshop on Algo­ (Nov. 1989, p. 1250) 4-8. Workshop on Model Theory, Berke­ rithms and Parallel VLSI Architectures, 29-June 2. Workshop on Dynamical Sys­ ley, CA. (Sep. 1989, p. 917) Pont-li-Mousson, France. tems in Fluid Mechanics, Minneapolis, 4-8. Nonlinear Phenomena in Atmo­ INFORMATION: A.-J. van der Veen, MN. (Nov. 1989, p. 1251) spheric and Oceanic Sciences, Minneapo­ Dept. of Electrical Engineering, Delft 30-31. Conference on Algebraic Logic, lis, MN. (Nov. 1989, p. 1251) Univ. of Technology, 2628 CD Delft, Boulder, Colorado. (Dec. 1989, p. 1436) 4-8. International Conference on Approx­ The Netherlands; +3115781442. 31-June I. Annual Meeting of the Cana­ imation Interpolation and Summability in Honor of A. Jakimovski, Tel Aviv, Israel. 11-14. Fourteenth Rolf Nevanlinna Col­ dian Society for History and Philosophy loquium, University of Helsinki, Helsinki, of Victoria, (Dec. 1989, p. 1436) of Mathematics, University Finland. (Jul./ Aug. 1989, p. 767) British Columbia. (Feb. 1990, p. 220) 4-8. International Conference on Boot­ strapping and Related Techniques, Trier, 11-14. Fifth SIAM Conference on Dis­ *31-June 2. Seventh Annual Western crete Mathematics, Atlanta, GA. (Sep. Oregon Federal Republic of Germany. (Jan. 1990, Geometric Topology Workshop, 1989, p. 917) State University. p. 56) 4-8. The Mathematics of Neural Nets, 11-15. World Organization of Systems SPONSOR: National Science Founda­ Salisbury State Univ., Salisbury, MD. and Cybernetics Eighth International Con­ tion and Oregon State Univ. (Mar. 1990, p. 328) gress, New York, NY. (Please note date

APRIL 1990. VOLUME 37, NUMBER 4 493 Meetings and Conferences

change from Mar. 1989, p. 315) Complex Analysis, Oberwolfach, Federal Math., North Dakota State Univ., SU 11-15. Chaotic Processes in the Geolog­ Republic of Germany. (Jul./ Aug. 1989, Station, P.O. Box 5075, Fargo, ND ical Sciences, Minneapolis, MN. (Nov. p. 767) 58105; 701-237-7490; email: 1989, p 1251) 18-20. Joint WNAR-IMS Regional Meet­ [email protected]. Montana State University, Bozeman, 11-15. Third International Conference on ing, 24-30. Mathematische Probleme in der (Mar. 1989, p. 315) Hyperbolic Problems, Uppsala, Sweden. MT. Nichtlinearen Elastizitiit, Oberwolfach, (Jan. 1990, p. 57) 18-21. A Conference on Partial Dif­ Federal Republic of Germany. (Jul./ Aug. 11-15. Rigorous Results in Quantum Dy­ ferential Equations, in Honor of Shmuel 1989, p. 767) Agmon, lnst. of Math., Hebrew Univ., namics, Liblice Castle, Czechoslovakia. 25-29. International Symposium on Israel. (Feb. 1990, p. 221) (May/Jun. 1989, p. 602) Jerusalem, Fuzzy Approach to Reasoning and De­ 11-15. NSF/CBMS Conference on 18-22. Fourteenth Mathematical Sci­ cision Making, Bochyne, Czechoslovakia. Wavelets, University of Lowell, Lowell, ences Lecture Series on Partially Ordered (Oct. MA. (Nov. 1989, p. 1251) Sets, Johns Hopkins University, Balti­ 1989, p. 1096) more, Maryland. (Oct. 1989, p. 1096) 11-15. IMACS First International Con­ 25-29. Logique et Informatique, Mar­ ference on Computational Physics, Boul­ 18-22. Approximations Diophantiennes seille, France. (Feb. 1990, p. 221) Nombres Transcendants, Marseille, der, CO. (Jan. 1990, p. 57) et 25-July 13. SMS-NATO ASI: Shape France. (Jan. 1990, p. 57) 11-15. The Mathematics of Computer Optimization and Free Boundaries, Uni­ Graphics, Salisbury State Univ., Salis­ 18-23. The Fifth International Con­ versite de Montreal, Montreal, Canada. bury, MD. (Mar. 1990, p. 328) ference on Topology and its Applications, (Dec. 1989, p. 1437) Yugoslavia. (Mar. 1990, 11-15. Undergraduate Faculty Enhance­ Dubrovnik, 27-29. ACM Conference on Lisp and p. 329) ment "Workshop on Computational Num­ Functional Programming, Nice, France. ber Theory", Univ. of Wisconsin­ 18-26. Recent Developments in Hoo Con­ (Jan. 1990, p. 58) Theory, Villa Olmo, Como. (Feb. Whitewater, WI. (Mar. 1990, p. 328) trol 27-30. Fourth International Congress 1990, p. 221) 12-15. Ninth International Conference on Algebraic Hyperstructures and Ap­ on Analysis and Optimization of Systems, 18-29. Radar/Sonar, Minneapolis, MN. plications, Xanthi, Greece. (Apr. 1989, Antibes, France. (Jan. 1990, p. 57) (Nov. 1989, p. 1251) p. 496) 12-15. Workshop on Spectral and Scat­ 18-29. AMS-SIAM Summer Seminar on tering Theory of Partial Differential Op­ Vortex Dynamics and Vortex Methods, University of Washington, Seattle, WA. erators, Institute of Mathematics, He­ July 1990 brew Univ., Jerusalem, Israel. (Feb. 1990, INFORMATION: B. Verducci, AMS, P.O. p. 220) Box 6248, Providence, RI 02940. July 1990. AMS Summer Research Insti­ 13-15. Seventh Annual Quality and Pro­ tute on Differential Geometry, University ductivity Research Conference, Madison, 20-22. Sixteenth International Workshop of California, Los Angeles, 'CA. Theoretic Concepts in Computer WI. (Mar. 1989, p. 315) on Graph- INFORMATION: W. Drady, AMS, P.O. Berlin, Federal Republic of Ger­ 13-16. Function Estimation and Sta­ Science, Box 6248, Providence, RI 02940. p. 57) tistical Applications, Cornell University, many. (Jan. 1990, Ithaca, NY. (Feb. 1990, p. 221) 20-23. Second Symposium on Chaotic 1-7. Modulfunktionen In Mehreren Vari­ Center ablen, Oberwolfach, Federal Republic of 13-22. Free Boundary Problems: Theory Dynamical Systems, Conference (near Utrecht), The Germany. (Apr. 1989, p. 497) and Applications, Centre de Recherches "Woudschoten" 1990, p. 32,9) Mathematiques, Universite de Montreal, Netherlands. (Mar. 1-15. International Symposium on Alge­ Canada. (Jul./ Aug. 1989, p. 767) * 21-27. Curves and Surfaces, Chamonix- braic Topology - Adams Memorial Sympo­ sium, University of Manchester, England. 14-16. Sixth Summer Conference on Mont Blanc, France. (Sep. 1989, p. 918) General Topology and Applications, Long INFORMATION: C. Rivet, Curves and Island Univ. (C.W. Post Campus), Surfaces TIM3-IMAG, Univ. Joseph 1-18. Twentieth Summer Session on Brookville, NY. (Feb. 1990, p. 221) Fourier BP 53X, 38041, Grenoble Probability Theory, Saint-Flour (Cantal), France. (Mar. 1989, p. 315) 14-16. Fifth Southeast Asian Conference Cedex, France. on Mathematical Education (SEACME 2-6. Tenth Australian Statistical Con­ Regional Confer­ 5), Gadong, Brunei Darussalam. (Nov. * 24-28. NSF /CBMS ference/Second Pacific Statistical Con­ Theory and Combinato­ 1989, p. 1251) ence on Ergodic gress, Sydney, Australia. (Jul./ Aug. 1989, rial Number Theory, North Dakota State Statis­ p. 768) 14-16. Function Estimation and Univ., Fargo, ND. tical Applications, Cornell Univ., Ithaca, 2-6. The Jonsson Symposium, Laugar­ NY. (Mar. 1990, p. 328) PROGRAM: H. FUrstenberg will de­ vatn, Iceland. (Sep. 1989, p. 918) 15-20. Global Differential Geometry and liver ten lectures on the recent de­ 2-6. Thirty-fourth Annual Meeting of the Global Analysis, Berlin, Federal Republic velopments in topics related to the Australian Mathematical Society, Towns­ of Germany. (Jan. 1990, p. 57) conference title. ville, Queensland, Australia. (Oct. 1989, ORGANIZER: D. Comez. p. 1097) 17-23. Partial Differential Equations in INFORMATION: D. Comez, Dept. of 2-6. Fifteenth International Biomet-

494 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences ric Conference, Budapest, Hungary. (Jan. 16-20. NSF-CBMS Regional Conference their Applications, Wake Forest Univer­ 1990, p. 58) "The Polynomial Identities and Invari­ sity, Winston Salem, NC. (Nov. 1989, 2-7. Groupes Ordonnes et Groupes ants of n x m Matrices", DePaul Univ., p. 1252) de Permutation, Marseille, France. (Jan. Chicago, IL. (Mar. 1990, p. 330) 31-August 2. Dynamics of Numerics 1990, p. 58) 16-21. Design and Analysis of Scien­ and the Numerics of Dynamics, Bristol, 2-1 0. Continua with Microstructures, tific Experiments, Cambridge, MA. (Mar. England. (Nov. 1989, p. 1252) Villa "La Querceta", Montecatini Terme. 1990, p. 330) (Feb. 1990, p. 222) 16-2 7. Regional Institute in Dynami­ 2-31. Time Series, Minneapolis, MN. cal Systems, Boston Univ., Boston, MA. August 1990 (No~ 1989, ~ 1252) (Mar. 1990, p. 330) 2-3. AMS-SIAM-SMB Symposium on 2-August I 0. Representations des * 16-August 3. Conference on "Enriching Some Mathematical Questions in Biology, Groupes et des Algebres de Lie, Uni­ Discrete Mathematics Courses with Re­ Neural Networks, Vancouver, Canada. versite de Montreal, Canada. (Feb. 1990, cent Developments", Univ. of Wyoming, (Mar. 1990, p. 330) p. 222) Laramie, WY. 5-9. From Topology to Computation: 3-6. Eleventh Dundee Conference on Or­ PROGRAM: Mornings will be devoted Unity and Diversity in the Mathematical dinary and Partial Differential Equations, to lectures by the speakers which will Sciences (in Honor of Professor Stephen Dundee, Scotland. (Sep. 1989, p. 918) address selected important and recent Smale's 60th Birthday), Berkeley, CA. 5-7. Lattice Path Combinatorics and Ap­ developments in discrete mathemat­ (Jan. 1990, p. 58) plications, McMaster University, Hamil­ ics. Afternoons will be devoted to dis­ 5-11. Mathematical Methods in Tomog­ ton, Ontario, Canada. (Feb. 1990, p. 222) cussions, work sessions and/or talks raphy, Oberwolfach, Federal Republic of 6-7. International Colloquium on Appli­ which will assist in the incorporation Germany. {Apr. 1989, p. 497) cations of Mathematics (on the occasion of these ideas and concepts into col­ 5-11. International Conference on Ap­ of the 80th birthday of Lothar Collatz), lege courses at all levels. Funds to proximation Theory, Hungary. (Oct. 1989, Hamburg, West Germany. (Feb. 1990, cover living costs and a small stipend p. 1097) p. 222) are available. 6-7. AMS Short Course on Combinato­ The main speaker 8-14. Variationsrechnung, Oberwolfach, INVITED SPEAKERS: rial Games, Columbus, Ohio. will be Wiiliam F. Lucas, The Clare­ Federal Republic of Germany. (Apr. who will lec­ INFORMATION: M. Foulkes, AMS, P.O. 1989, p. 497) mont Graduate School, ture each day for the three week Box 6248, Providence, RI 02940. 9-11. "Universita'di Genova - The Ohio period. Carl W. Lee, The University State University Joint Conference" on of Kentucky, will lecture for at least 6-9. 1990 Joint Statistical Meetings, New Trends in Systems Theory, Genoa, two weeks. Anaheim, CA. (Mar. 1988, p. 466) Italy. (Jul./ Aug. 1989, p. 768) INFORMATION: A.D. Porter, Mathe­ 6-10. Singularities, Honolulu, Hawaii. 9-14. 4eme Colloque International de matics Department, P.O. Box 3036 (Feb. 1990, p. 223) Theorie des Graphes et de Combinatoire, University Station, University of 8-11. Joint Mathematics Meetings, Ohio Marseille-Luminy, C.I.R.M., France. (Jan. Wyoming, Laramie, WY 82071-3036. State University, Columbus, OH. (includ­ 1990, p. 58) ing the summer meetings of the AMS, 9-20. Geometry and Topology of Four­ 22-28. Konvexgeometrie, Oberwolfach, AWM, MAA and PME). This is the 75th Manifolds, McMaster University, Hamil­ Federal Republic of Germany. (Apr. Anniversary of the MAA. ton, Ontario, Canada. (MayfJun. 1989, 1989, p. 497) INFORMATION: H. Daly, AMS, P.O. p. 602) 23-27. CADE 10 - Tenth Interna­ Box 6248, Providence, RI 02940. 15-20. SIAM Annual Meeting, Chicago, tional Conference on Automated Deduc­ IL. (Mar. 1990, p. 329) tion, West Germany. (Feb. 1990, p. 223) 8-11. The Fifth KIT Mathematics Work­ 15-21. Stochastic Image Models and Al­ 23-28. Fourth International Congress on shop, Korea Institute of Technology, Tae­ gorithms, Oberwolfach, Federal Republic Computational and Applied Mathematics, jon, Korea. (Mar. 1990, p. 330) of Germany. (Apr. 1989, p. 497) Leuven, Belgium. (Jan. 1990, p. 58) * 11-14. Crypto '90, University of Cali- 15-22. 1990 European Summer Meet­ 23-August 4. Third Workshop on Stochas­ fornia, Santa Barbara, CA. ing (Logic Colloquium '90), University of tic Analysis, Silivri, Istanbul-Turkey. (Feb. 1990, p. 223) SPONSORs: International Association Helsinki, Finland. (Feb. 1990, p. 222) for Cryptological Research; Univ. of 15-23. Colloquium in Honor of Roland 26-29. International Conference on New California, Santa Barbara Computer Theory Fraisse, Centre International de Recon­ Trends in Geometric Function Security Dept.; IEEE Computer Soci­ of Madras, tres Mathematiques, Luminy, France. and Applications, University ety Technical Committee on Security p. 918) (MayfJun. 1989, p. 602) Madras, India. (Sep. 1989, and Privacy. 16-20. SIAM Annual Meeting, Chicago, 29-August 4. Mechanik Und Algebrais­ CALL FOR PAPERS: Papers are SO­ IL. (Nov. 1988, p. 1389) che Geometrie, Oberwolfach, Federal Re­ licited on all topics related to current public of Germany. (Apr. 1989, p. 497) 16-20. Symposium Fraisse, Marseille, work in the theory and application France. (Jan. 1990, p. 58) 30-August 4. The Fourth International of cryptographic techniques. Authors Conference on Fibonacci Numbers and are requested to submit 12 copies of

APRIL 1990, VOLUME 37, NUMBER 4 495 Meetings and Conferences

an extended abstract or full paper by 14-18. Zeta Functions in Geometry, 21-29. The International Congress of April 16, 1990, sent to: S.A. Van­ Tokyo Institute of Technology, Tokyo, Mathematicians 1990, Kyoto, Japan. stone, Program Chair, Dept. of Com­ Japan. (Feb. 1990, p. 223) (Nov. 1988, p. 1389) binatorics and Optimization, Univ. of 15-1 9. International Conference on Knot 23-September 1. Twenty-eigth Interna­ Waterloo, Waterloo, Ontario, Canada Theory and Related Topics, International tional Symposium on Functional Equa­ N2L 3Gl. House, Osaka, Japan. (Apr. 1989, p. 497) tions, Graz-Mariatrost, Austria. (Dec. INFORMATION: S.S. McMahan, Gen­ 15-1 9. International Conference on Com­ 1989, p. 1438) eral Chair, Cylink, 110 S. Wolfe Rd., mutative Algebra and Combinatorics, 26-September 1. Komplexe Analysis, Sunnyvale, CA 94086. Nagoya University, Nagoya, Japan. (Feb. Oberwolfach, Federal Republic of Ger­ 12-13. Workshop on Using Technology 1990, p. 224) many. (Apr. 1989, p. 497) to Enhance the Teaching and Learning 15-20. Conference on Gaussian Random 27-31. Eleventh Prague Conference on of Calculus and Precalculus Mathematics, Fields (The Third Nagoya Levy Seminar), Information Theory, Statistical Decision The Ohio State Univ., Columbus, OH. Nagoya University, Nagoya, Japan. (Nov. Functions and Random Processes, Praha, (Mar. 1990, p. 330) 1989, p. 1253) Czechoslovakia. (Jan. 1990, p. 59) 12-18. Algebraische Zahlentheorie, Ober­ 16-18. SIGAL International Symposium 28-30. IMACS European Simulation wolfach, Federal Republic of Germany. on Algorithms, Tokyo, Japan. (Oct. 1989, Meeting on Problem Solving by Simula­ (Apr. 1989, p. 497) p. 1097) tion, Esztergom, Hungary. (Mar. 1989, 12-18. Pre-Congress Topology Confer­ 16-1 9. Current Topics in Operator Alge­ p. 316) ence, University of Hawaii, Honolulu, bras, Nara Ken-New Public Hall, Nara, 28-31. Operations Research 1990, Inter­ HI. (Feb. 1990, p. 223) Japan. (Feb. 1990, p. 224) national Conference Operations Research, 13-16. Alaska Conference, Quo Vadis, 18-22. The Second International Confer­ Vienna, Austria. (Jul./Aug. 1989, p. 768) Graph Theory?, University of Alaska, ence on Graph Theory, Kanagawa, Japan. 29-31. International Colloquium on Fairbanks, AK. (Oct. 1989, p. 1097) (Feb. 1990, p. 224) Words, Languages, and Combinatorics, 13-1 7. Fifth International Conference 19-20. Inverse Problems in Engineering Kyoto Sangyo University, Kyoto, Japan. on Hadronic Mechanics and Nonpoten­ Sciences, Osaka Institute of Technology, (Feb. 1990, p. 224) tial Interactions, University of Northern Osaka, Japan. (Feb. 1990, p. 224) 30-September 1. International Sympo­ Iowa, Cedar Falls, Iowa. (Jul./ Aug. 1989, 19-25. Mathematische Methoden Des sium on the Semigroup Theory and its p. 768) VLSI-Entwurfs Und Des Distributed Com­ Related Fields, Ritsumeikan University, 13-17. Eleventh IFAC World Congress, putings, Oberwolfach, Federal Republic Kyoto, Japan. (Feb. 1990, p. 224) Tallin, USSR. (Sep. 1989, p. 918) of Germany. (Apr. 1989, p. 497) 30-September 2. International Sympo­ 13-1 7. Algebraic Geometry and Ana­ 20-24. Second International Joint Con­ sium on Functional Differential Equa­ lytic Geometry, Tokyo, Japan. (Sep. 1989, ference of the ISSAC-90 and the AAECC- tions and Related Topics, Kyoto Shi­ p. 919) 8, Nihon University, Tokyo, Japan. (Feb. gaku Kaikan (YOUANDI), Kyoto, Japan. (Feb. 1990, p. 224) 13-1 7. 1990 International Conference on 1990, p. 224) Parallel Processing, Pennsylvania State * 20-24. International Conference on Com- 30-September 4. International Confer­ University, University Park, PA. (Dec. binatorics (ICC '90), Hcfei, Anhui, China. ence on Potential Theory, Nagoya, Japan. (MayjJun. 1989, p. 602) 1989, p. 1438) PROGRAM: The technical program will 30-Septcmber 4. International Sym­ 13-17. NSF /CBMS Conference on Ran­ include a variety of scheduled and posium on Computational Mathematics, dom Number Generation and Quasi-Monte submitted talks related to the gen­ Matsuyama, Japan. (Feb. 1990, p. 224) Carlo Methods, Univ. of Alaska, Fair­ eral themes of the conference. The banks, AK. (Mar. 1990, p. 331) language of the conference will be 31-September 1. Tokyo History of Math­ 13-18. Institute of Mathematical Statis­ English. The conference will provide ematics Symposium 1990, University of tics Fifty-third Annual Meeting(jointly a timely opportunity for researchers Tokyo, Tokyo,Japan.(Feb. 1990,p.224) with the Second World Congress of the and students to exchange ideas in for­ 31-September 2. Conference on Repre­ Bernoulli Society), Uppsala, Sweden. (Sep. mal and informal dialogue. It is hoped sentation Theories of Lie Groups and Lie 1989, p. 919) that ICC '90 can serve as a focus for Algebras, Lake-Kawaguchi, Yamanashi, 13-18. Tsukuba International Confer­ ongoing research in these areas as well Japan. (Feb. 1990, p. 224) ence on Representations of Algebras and as for modern applications in other 31-September 4. International Sympo­ Related Topics, University of Tsukuba, disciplines. sium on Functional Analysis and Re­ Japan. (Nov. 1989, p. 1253) INFORMATION: K. Tung-Hsin, Hefei lated Topics, Sapporo, Japan. (Feb. 1990, 14-18. The Asian Mathematical Con­ Branch of Academia Sinica, P.O. p. 225) ference 1990, Hong Kong, China. (Sep. Box 10, Hefei, Anhui, 230031, China 31-September 4. General Topology and 1989, p. 919) (PRC). Geometric Topology Symposium, Uni­ versity of Tsukuba, Japan. (Feb. 1990, 14-18. Harmonic Analysis, Sendai 1990, 20-25. Fifth Conference on Numerical p. 225) Tohoku University, Scndai, Japan. (Feb. Methods, Miskolc, Hungary. (Jan. 1990, 1990, p. 223) p. 59)

496 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences

for Theoretical Physics, Trieste, Italy. Symposium on Computer Arithmetic, Sci­ (May/Jun. 1989, p. 602) entific Computation and Mathematical September 1990 16-22. Risikotheorie, Oberwolfach, Fed­ Modelling - SCAN 1990, Albena (near September/October 1990. IMACS­ eral Republic of Germany. (Apr. 1989, Varna), Bulgaria. (Feb. 1990, p. 225) GAMM Conference on Computer Arith­ p. 498) 24-29. Mathematical Modelling of In­ metic, Scientific Computation and 17-21. Atelier International de Theorie dustrial Processes, Tecnopolis, Bari. (Feb. Mathematical Modelling, Bulgaria. (Sep. des Ensembles, Marseille, France. (Jan. 1990, p. 225) 1989, p. 919) 1990, p. 60) * 24-30. Ibero-American Conference, IMACS Symposium on Modelling and 17-22. DMV-Jahrestagung 1990, Bre­ Seville, Spain. of Electrical Machines, EN­ Simulation men, Federal Republic of Germany. INFORMATION: U. D'Ambrosio, In­ - Nancy, France. (May/ Jun. 1989, SEM (Jul./ Aug. 1989, p. 769) stitute of Math, Univ. Estadual de p. 602) * 21-2 7. International Functional Analysis Campinas, CP 6063 ( 13081) Camp­ 2-7. Twelfth International Conference on Meeting on the Occasion of the Sixti­ inas (SP), Brazil. Nonlinear Oscillations, Cracow, Poland. eth Birthday of Professor M. Valdivia, 25-29. International Symposium on (Sep. 1989, p. 919) Peiilscola (Castellon), Spain. * 2-7. International Conference on Integral Structures in Mathematical Theories, San Equations and Boundary Value Problems, SPONSORs: Univ. of Valencia and Sebastian, Spain. (Please note changes Yantai University, Shandong, People's Polytechnical Univ. of Valencia. from Jan. 1990, p. 60) Republic of China. (Nov. 1989, p. 1253) ORGANIZING CoMMITTEE: K.D. Bier­ INVITED SPEAKERS: A. Arhangelsii, 2-8. Topologie, Oberwolfach, Federal stedt, Univ. Paderborn, W. Germany; Moscow; A. Barabashev, Moscow; E. Republic of Germany. (Apr. 1989, p. 497) J. Bonet, Univ. Politecnica de Valen­ Bencivenga, Irvine; R. Bkouche, Lille; cia; J. Horvath, Univ. of Maryland; 3-6. Fourth Asian Logic Conference, H. Breger, Hannover; A. Bundy, Edin­ M. Maestre, Univ. de Valencia. Tokyo, Japan. (Mar. 1989, p. 316) burgh; M. Burgin and V. Kuznetsov, CONFERENCE TOPICS: Topics expected 3-7. IMACS Symposium on Intelligent Kiev; N. da Costa, Sao Paulo; M.L. to be emphasized include locally con­ Bran­ Models in Systems Simulation, Brussels, dalla Chiara, Firenze; L. Daston, vex spaces, Frechet spaces, Banach Dress, Belgium. (Mar. 1989, p. 316) deis; J.W. Dauben, NY; A. space theory, function spaces, holo­ Bielefeld; J. Echeverria, San Sebas­ des Groupes et Anal­ 3-7. Representation morphy, toplogical tensor products tian; A.W.F. Edwards, Cambridge; S. Complexe, Marseille, France. yse and operator ideals. Feferman, Stanford; J. Flum, Frei­ 1989, p. 768) (Jul./ Aug. INVITED SPEAKERS: Altomare, Anse­ burg; D. Gillies, London; G. Giorello, 3-7. International Conference on Dynam­ mil, Aron, Bessaga, Bomba!, Cobos, Milan; I. Grattan-Guinness, Middle­ ical Systems and Related Topics, Nagoya A. Defant, S. Dierolf, Diestel, Dineen, sex; J. Hernandez, Madrid; A. Ibarra, University, Nagoya, Japan. (Feb. 1990, Florencio, Floret, Godefroy, Isidro, San Sebastian; H.J. Jahnke, Biele­ p. 225) Jarchow, H. Konig, Llavona, Laursen, feld; E. Knobloch, Berlin; L. Kruger, 3-8. Physical Interpretations of Rela­ Meise, Moscatelli, Mujica, Nachbin, Gottin- tivity Theory, Imperial College, London, Orihuela, Pelczynski, Pietsch, Pisier, gen; B. Lawvere, NY; M. Loi, Paris; England. (Mar. 1990, p. 331) Schachermayer, Schmets, Taskinen, G. Ludwig, Marburg; S. Mac Lane, 8-12. Neuronet-90: IMACS International D. Vogt. Chicago; M.S. Mahoney, Princeton; J. a Symposium on Neural Nets and Neu­ CALL FOR PAPERS: There will be Mosterin, Barcelona; C. U. Moulines, ral Computers, Prague, Czechoslovakia. limited number of short communi­ Berlin; I. Niiniluoto, Helsinki; M. (Please note change from May/Jun. 1989, cations (of 15 minutes). Interested Otte, Bielefeld; L. Pepe, Ferrara; M. p. 602) contributors are invited to submit an Przelecki, Warszawa; V. Rantala, Tam­ abstract (to J. Bonet) as soon as pos­ 9-15. Surgery and L-Theory, Oberwol­ perc; M. Resnik, N. Carolina; E. sible. The abstracts will be refereed. It fach, Federal Republic of Germany. (Apr. Scheibe, Heidelberg; H.J. Schmidt, is planned to publish the Proceedings 1989, p. 498) Osnablick; J.D. Sneed, Colorado; P. of the Conference. Suppes, Stanford; R. Thorn, Paris; R. I 0-12. Second International Workshop INFORMATION: J. Bonet, Dept. of Torretti, Puerto Rico; J. Wolenski, on Advances in Robot Kinematics, Linz, Matematica Aplicada, E.T.S. Arqui­ Krakow. Austria. (Mar. 1990, p. 332) tectura, Univ. Politecnica de Valen­ I 0-14. Mathematiker-Kongress, Dres­ cia, C. de Vera, E-46071 Valencia, * 28-29. Linear Algebra and its Applica- den, German Democratic Republic. Spain; 346-3615051 ext. 408 or 402; tions, Miami University, Oxford, OH. (Jul./ Aug. 1989, p. 769) email: maestre@evalun !!.bitnet. INVITED SPEAKERS: C. Curtis, U niv. I 0-14. Greco Calcul Formel, Marseille, of Oregon; G. Strang, MIT; H. Zassen­ France. (Jan. 1990, p. 60) 23-29. Random Graphs and Combinator­ haus, Ohio State Univ. 10-14. Real Analysis and Measure The­ ical Structures, Oberwolfach, Federal Re­ CALL FOR PAPERS: Abstracts for con­ ory, Capri, Italy. (Mar. 1990, p. 332) public of Germany. (Apr. 1989, p. 498) tributed papers should be sent by June I 0-0ctober 5. School on Qualitative 24-28. Structure Galoisienne Arithme­ I, 1990, to J. Skillings, Dept. of Math. Aspects and Applications of Nonlinear tique, Marseille, France. (Jan. 1990, p. 60) and Stat., Miami Univ., Oxford, OH Evolution Equations, International Centre 24-28. IMACS-GAMM International 45056.

APRIL 1990, VOLUME 37. NUMBER 4 497 Meetings and Conferences

INFORMATION: J.H. Skillings, Miami 21-27. Mathematische Methoden In Der placing to much trust in computers Univ., Dept. of Math. and Stat., Bach­ Robotik, Oberwolfach, Federal Republic to handle complexities that we do elor Hall, Oxford, OH 45056; 513- of Germany. (Apr. 1989, p. 498) not fully understand and to model 529-5818. 21-27. Arithmetik der Korper, Oberwol­ realities that defy our attempts at fach, Federal Republic of Germany. (Oct. simplification. Leaders from science, 30-0ctober 6. Diophantische Approxi­ 1989, p. 1098) engineering, business, economics, and mationen, Oberwolfach, Federal Republic government will try to map out bet­ 21-27. International Functional Analysis of Germany. (Apr. 1989, p. 498) ter ways to deal with the risks and Meeting on the Occasion of the Sixti­ enhance the benefits of computer sys­ eth Birthday of Professor M. Valdivia, tems. The goal of the conference will Peiiiscola, Spain. (Oct. 1989, p. 1098) be to reach a consensus on the major 22-25. Fifth Jerusalem Conference on In- October 1990 problems and develop an agenda for formation Technology (JCIT-5), action. 1-5. Organisations et Theorie des Jeux, Jerusalem, Israel. (Jan. 1990, p. 60) ORGANIZERs: K. Duncan and W. Carl­ Marseille, France. (Jan. 1990, p. 60) 26-27. Statistical Mechanics at the 45th son. 1-5. Third Joint Europe/U.S. Short Parallel: Fourth Annual Meeting, Univer­ INFORMATION: J. Adams, Association Course in Hypersonics, RWTH Aachen site de Montreal, Canada. (Feb. 1990, for Computing Machinery, II West Univ. of Technology, Federal Republic p. 226) 42nd St., New York, NY 10036; 212- of Germany. (Mar. 1990, p. 332) 28-November 3. Mathematical Eco­ 869-7440. 5-6. Math-History Conference, LaCrosse, nomics, Oberwolfach, Federal Republic Annual Conference on Tech­ WI. (Jan. 1990, p. 60) of Germany. (Apr. 1989, p. 498) 9-11. Third nology in Collegiate Mathematics, The Differential 29-November 2. Trieste Conference on 5-7. Workshop on Partial Ohio State Univ., Columbus, OH. (Mar. Ithaca, NY. Integrable Systems, Trieste, Italy. (Jan. Equations, Cornell Univ., 1990, p. 333) (Mar. 1990, p. 332) 1990, p. 61) * I 0. Differential Geometry Day, Eastern Mit Aktuellem 29-November 2. Algorithme Parallele et 7-13. Arbeitsgemeinschaft Illinois University, Charleston, IL. Thema, Oberwolfach, Federal Republic Architectures Nouvelles, Marseille, France. of Germany. (Apr. 1989, p. 498) (Jan. 1990, p. 61) INVITED SPEAKERS: R. Bishop, Univ. 8-12. Congres Franco-Sovietique de Pro­ 29-November 2. The International Con­ of Illinois; H. Donnelly, Purdue Univ.; grammation Mathematique, Marseille, ference "D-Modules and Microlocal Ge­ C. Gordon, Washington Univ.; F. France. (Jan. 1990, p. 60) ometry", Lisbon, Portugal. (Mar. 1990, Xavier, Univ. of Notre Dame; K. Yang, Arkansas State Univ. 14-20. Geometrie, Oberwolfach, Fed­ p. 333) INFORMATION: J.F. Glazebrook, Dept. eral Republic of Germany. (Apr. 1989, 29-November 16. Workshop on Mathe­ of Mathematics, Eastern Illinois Uni­ p. 498) matical Ecology, Trieste, Italy. (Jan. 1990, p. 61) versity, Charleston, IL 61920; 217- 15-1 9. Modeles pour L'Analyse des Don­ 581-6275. nees Multidimensionnelles, Marseille, 31-November 3. Latinamerican Semi­ France. (Jan. 1990, p. 60) nar on Applications of Mathematics and 10-11. Far Western Section, University 15-1 9. Tercer Congreso N acional de Computer Science to Biology, La Habana, of California, Irvine, CA. Cuba. (Feb. 1990, p. 226) Matematicas, San Jose, Costa Rica. (Feb. INFORMATION: W. Drady, American 1990, p. 225) Mathematical Society, P.O. Box 6248, * 19-20. Nineteenth Midwest Conference Providence, Rl 02940. on Differential and Integral Equations, November 1990 Univ. of Missouri-Rolla, Rolla, MO. 12-16. Supercomputing '90, New York, 2-3. Central Section Meeting of the NY. (Sep. 1989, p. 920) INVITED SPEAKERs: T.A. Burton, C.C. AMS, University of North Texas, Den­ 12-16. Workshop on Representations Chicone, P.W. Eloe, A.M. Fink, D.B. ton, TX. of Reductive Groups over Finite Fields, Hinton, J.L. Henderson, W.G. Kelley, INFORMATION: W. Drady, AMS, P.O. Mathematical Sciences Research Insti­ Y. Sibuya. Box 6248, Providence, RI 02940. tute, Berkeley, CA. (Jan. 1990, p. 61) INFORMATION: L.M. Hall, Dept. of 18-24. Komplexitatstheorie, Oberwol­ Math. and Stat., Univ. of Missouri­ 4-10. WahrscheinlichkeitsmaBe auf Grup­ fach, Federal Republic of Germany. (Apr. Rolla, Rolla, MO 65401; 314-341- pen, Oberwolfach, Federal Republic of 1989, p. 498) 4911; [email protected]. Dead­ Germany. (Oct. 1989, p. 1098) of contributed pa­ 25-December I. Stochastische Approx­ line for abstracts 5-7. Second SIAM Confernce on Linear 17, 1990. imation Und Optimierungsprobleme In pers: August Algebra in Signals, Systems & Controls, Der Statistik, Oberwolfach, Federal Re­ San Francisco, CA. (Sep. 1989, p. 920) 20-21. Eastern Section, University of public of Germany. (Apr. 1989, p. 498) MA. * 6-7. 1990 ACM Conference on Critical Massachusetts at Amherst, Amherst, 25-December 1. Lineare Modelle und Issues, Arlington, VA. INFORMATION: W. Drady, American Multivariate Statistische Verfahren, Ober­ Mathematical Society, P.O. Box 6248, PURPOsE: Many scientists fear that wolfach, Federal Republic of Germany. Providence, RI 02940. we are courting costly failures by (Jul./ Aug. 1989, p. 769)

498 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences

Algebra, Brussels Free Univ., Brussels, sonal invitation. Without a personal Belgium. (Mar. 1990, p. 334) invitation participation is not possi­ December 1990 ble. Mathematicians interested in a 2-8. Multigrid Methods, Oberwolfach, meeting should contact the Institute." Federal Republic of Germany. (Apr. 27-February 2. Harmonische Analyse 1989, p. 498) * January 1991 und Darstellungstheorie Topologischer 3-5. First International Symposium on * 6-12. Automorphe Formen und Anwen­ Gruppen, Oberwolfach, Federal Repub­ Uncertainty and Analysis: Fuzzy Reason­ dungen, Oberwolfach, Federal Republic lic of Germany. ing, Probabilistic Methods and Risk Man­ of Germany. agement, College Park, Maryland. (Oct. CHAIRMEN: R.E. Howe, New Haven; 1989, p. I 098) CHAIRMEN: S.S. Kudla, College Park; E. Kaniuth, Paderborn. For­ 3-7. Sixteenth Australasian Conference J. Schwermer, Eichstiitt. INFORMATION: Mathematisches Ge­ on Combinatorial Mathematics and Com­ INFORMATION: Mathematisches For­ schungsinstitut Oberwolfach, 24, D-7800 binatorial Computing, Palmerston North, schungsinstitut Oberwolfach, Ge­ schaftsstelle: Albertstrabe at New Zealand. (Feb. 1990, p. 226) schaftsstelle: Albertstrabe 24, D-7800 Freiburg im Breisgau. "Participants "Participants at Oberwolfach meetings do need a per­ 3-7. SINO-JAPANESE Joint Seminar Freiburg im Breisgau. do need a per­ sonal invitation. Without a personal on Nonlinear PDEs with Emphasis on Oberwolfach meetings Without a personal invitation participation is not possi­ Reaction-Diffusion Aspects., Taipei, Tai­ sonal invitation. is not possi­ ble. Mathematicians interested in a wan. (Jan. 1990, p. 61) invitation participation ble. Mathematicians interested in a meeting should contact the Institute." 3-7. Workshop on General Group Repre­ meeting should contact the Institute." sentation Theory, Mathematical Sciences Research Institute, Berkeley, CA. (Jan. 7-10. Sixth Caribbean Conference in February 1991 1990, p. 61) Combinatorics and Computing, Univer­ 9-15. Allgemeine Ungleichungen, Ober­ sity of the West Indies, St. Augustine, * 3-9. Konstruktive Methoden in der Kom­ wolfach, Federal Republic of Germany. Trinidad. (Jan. 1990, p. 61) plexen Analysis, Oberwolfach, Federal (Apr. 1989, p. 498) * 13-1 9. Combinatorical Optimization, Republic of Germany. of Ger­ * 15-19. Curves and Surfaces: An Algorith­ Oberwolfach, Federal Republic CHAIRMEN: H.-P. Blatt, Eichstiitt; D. mic Viewpoint, Kent State Univ., Kent, many. Gaier, GieBen; R.S. Varga, Kent. OH. CHAIRMAN: R. Burkard, Graz; M. INFORMATION: Mathematisches For­ INVITED SPEAKER: C. A. Michelli, T.J. Grotschel, Augsburg. schungsinstitut Oberwolfach, Ge­ Watson Research Center (I.B.M.). INFORMATION: Mathematisches For­ schaftsstelle: Albertstrabe 24, D-7800 INFORMATION: A.S. Cavaretta, Dept. schungsinstitut Oberwolfach, Ge­ Freiburg im Breisgau. "Participants at of Math. Sciences, Kent State Univ., schaftsstelle: Albertstrabe 24, D-7800 Oberwolfach meetings do need a per­ Kent, OH 44242; 216-672-2696; In­ Freiburg im Breisgau. "Participants at sonal invitation. Without a personal ternet: [email protected] .. Oberwolfach meetings do need a per­ invitation participation is not possi­ sonal invitation. Without a personal ble. Mathematicians interested in a 16-22. Mathematische Logik, Oberwol­ invitation participation is not possi­ meeting should contact the Institute." fach, Federal Republic of Germany. (Apr. ble. Mathematicians interested in a 10-16. Endlichdimensionale Lie-Algcbrcn, 1989, p. 498) meeting should contact the Institute." * Republic of Ger­ * 17-21. Non-linear Dispersive Wave Sys­ Oberwolfach, Federal tems, Univ. of Central Florida, Orlando, 16-19. Joint Mathematics Meetings, many. the an­ FL. San Francisco, CA. (including CHAIRMEN: M. Goze, Muhlhouse; O.H. nual meetings of the AMS, AWM, MAA, INVITED SPEAKER: T. Brooke Ben­ Kegel, Freiburg. and NAM) For­ jamin. INFORMATION: Mathematisches Ge­ INFORMATION: L. Debnath, Univ. of INFORMATION: H. Daly, AMS, P.O. schungsinstitut Oberwolfach, 24, D-7800 Central Florida, Department of Math­ Box 6248, Providence, RI 02940. schaftsstelle: Albertstrabe "Participants at ematics, Orlando, FL 32816; 407-275- Freiburg im Breisgau. * 20-26. Spektraltheorie Singu1lirer do need a per­ 2478. Oberwolfach meetings Gewohnlicher Differentialoperatoren, Ober­ sonal invitation. Without a personal 25-January 1. Lineare Modclle Und Mul­ wolfach, Federal Republic of Germany. invitation participation is no possi­ tivariate Statistische Verfahren, Oberwol­ CHAIRMEN: H.D. NieBen, Essen; A. ble. Mathematicians interested in a fach, Federal Republic of Germany. (Apr. Schneider, Dortmund; J. Weidmann, meeting should contact the Institute." 1989, p. 498) Frankfurt. * I 0-16. Affine Differentialgeometrie, Ober- For­ INFORMATION: Mathematisches wolfach, Federal Republic of Germany. schungsinstitut Oberwolfach, Ge­ schiiftsstelle: Albertstrabe 24, D-7800 CHAIRMEN: K. Nomizu, Providence; Spring 1991. IMACS International Sym­ Freiburg im Breisgau. "Participants at U. Pinkall, Berlin; U. Simon, Berlin. posium on Iterative Methods in Linear Oberwolfach meetings do need a per- INFORMATION: Mathematisches For-

APRIL 1990, VOLUME 37, NUMBER 4 499 Meetings and Conferences

schungsinstitut Oberwolfach, Ge­ meeting should contact the Institute." schliftsstelle: Albertstrabe 24, D-7800 schliftsstelle: Albertstrabe 24, D-7800 Freiburg im Breisgau. "Participants at Freiburg im Breisgau. "Participants at 16-1 7. Central Section, Indiana U niver­ Oberwolfach meetings do need a per­ Oberwolfach meetings do need a per­ sity, South Bend, IN. sonal invitation. Without a personal sonal invitation. Without a personal INFORMATION: W. Drady, American invitation participation is not possi­ invitation participation is not possi­ Mathematical Society, P.O. Box 6248, ble. Mathematicians interested in a ble. Mathematicians interested in a Providence, RI 02940. meeting should contact the Institute." meeting should contact the Institute." * 17-23. Elementare und Analytische * 24-March 2. Medical Statistics: Statisti­ Zahlentheorie, Oberwolfach, Federal April1991 cal Models for Longitudinal Data, Ober­ Republic of Germany. * 7-13. Algebraische Gruppen, Oberwol- wolfach, Federal Republic of Germany. CHAIRMEN: H.-E. Richert, Vim; W. fach, Federal Republic of Germany. CHAIRMEN: N. Breslow, Seattle; J. Schwarz, Frankfurt; E. Wirsing, Vim. CHAIRMEN: P. Slodowy, Stuttgart; T.A. Mau, Dusseldorf; M. Schumacher, INFORMATION: Mathematisches For­ Springer, Utrecht; J. Tits, Paris. Freiburg. schungsinstitut Oberwolfach, Ge­ INFORMATION: Mathematisches For­ INFORMATION: Mathematisches For­ schliftsstelle: Albertstrabe 24, D-7800 schungsinstitut Oberwolfach, Ge­ schungsinstitut Oberwolfach, Ge­ Freiburg im Breisgau. "Participants at schliftsstelle: Albertstrabe 24, D-7 800 schliftsstelle: Albertstrabe 24, D-7800 Oberwolfach meetings do need a per­ Freiburg im Breisgau. "Participants at Freiburg im Breisgau. "Participants at sonal invitation. Without a personal Oberwolfach meetings do need a per­ Oberwolfach meetings do need a per­ invitation participation is not possi­ sonal invitation. Without a personal sonal invitation. Without a personal ble. Mathematicians interested in a invitation participation is not possi­ invitation participation is not possi­ meeting should contact the Institute." ble. Mathematicians interested in a ble. Mathematicians interested in a meeting should contact the Institute." meeting should contact the Institute." 21-23. Sixth S.E.A. Meeting, South­ eastern Approximation Theorists Annual 8-12. Seventh International Conference 25-March l. IEEE Computer Society * Meeting, Memphis State Univ., Mem­ on Data Engineering, Kobe, Japan. COMPCON Spring '91, San Francisco, phis, TN. (Mar. 1990, p. 334) CA. (Jan. 1990, p. 62) 22-23. Southeastern Section, University PURPOSE: The purpose of this confer­ of South Florida, Tampa, FL. ence is to provide a forum for the sharing of practical experiences and March 1991 INFORMATION: W. Drady, American research advances from an engineer­ Mathematical Society, P.O. Box 6248, * 3-9. Partielle Differentialgleichungen, ing point of view among those inter­ Providence, RI 02940. Oberwolfach, Federal Republic of Ger­ ested in automated data and knowl­ edge management. many. 22-24. Fifth SIAM Conference on Par­ CONFERENCE TOPICS: AI and knowl­ allel Processing for Scientific Computing, CHAIRMEN: J. Bruning, Augsburg; L. edge based systems, applications and Houston, TX. (Mar. 1990, p. 334) Hormander, Lund; W. von Wahl, application systems, benchmarks and Bayreuth. * 24-30. Gewohnliche Differentialgleichun­ performance evaluation, design and INFORMATION: Mathematisches For­ gen, Oberwolfach, Federal Republic of human interfaces, data engineering schungsinstitut Oberwolfach, Ge­ Germany. tools and techniques, database design schliftsstelle: Albertstrabe 24, D-7800 CHAIRMEN: H.W. Knobloch, Wurz­ and modeling, database management Freiburg im Breisgau. "Participants at burg; J. Mawhin, Louvain-la-Neuve; and structure, deductive and extensive Oberwolfach meetings do need a per­ K. Schmitt, Salt Lake City. databases, distributed database con­ sonal invitation. Without a personal INFORMATION: Mathematisches For­ trol, distributed database systems, in­ invitation participation is not possi­ schungsinstitut Oberwolfach, Ge­ tegrity and security techniques, learn­ ble. Mathematicians interested in a schliftsstelle: Albertstrabe 24, D-7800 ing and discovery in databases, object­ meeting should contact the Institute." Freiburg im Breisgau. "Participants at oriented database systems, query lan­ guages and processing, scientific data­ * 10-16. Mathematische Stochastik, Ober- Oberwolfach meetings do need a per­ bases, supercomputer databases. wolfach, Federal Republic of Germany. sonal invitation. Without a personal invitation participation is not possi­ CALL FOR PAPERS: Each papers length CHAIRMEN: P.L. Davies, Essen; B.W. ble. Mathematicians interested in a should be limited to 8 proceedings Silverman, Bath. meeting should contact the Institute." pages, which is about 5000 words, INFORMATION: Mathematisches For­ or 25 double spaced typed pages. schungsinstitut Oberwolfach, Ge­ * 31-April 6. Arbeitsgemeinschaft mit Ak­ Five copies of completed papers or schliftsstelle: Albertstrabe 24, D-7800 tuellum Thema, Oberwolfach, Federal Re­ panel proposals should be mailed be­ Freiburg im Breisgau. "Participants at public of Germany. fore July 1, 1990 to: N.J. Cercone, Oberwolfach meetings do need a per­ CHAIRMEN: N.N. Center for Systems Science, Simon sonal invitation. Without a personal INFORMATION: Mathematisches For­ Fraser Univ., Burnaby, British Co­ invitation participation is not possi­ schungsinstitut Oberwolfach, Ge- lumbia, Canada V5A 1S6; (604)291- ble. Mathematicians interested in a 3229; [email protected].

500 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Meetings and Conferences

* 14-20. Brauer Groups and Representa­ Oberwolfach meetings do need a per­ CHAIRMEN: L. Danzer, Dortmund; tion Theory of Finite Groups, Oberwol­ sonal invitation. Without a personal G.C. Shephard, Norwich. fach, Federal Republic of Germany. invitation participation is not possi­ INFORMATION: Mathematisches For­ ble. Mathematicians interested in a schungsinstitut Oberwolfach, Ge­ CHAIRMEN: H. Opolka, Gottingen; F. meeting should contact the Institute." schaftsstelle: Albertstrabe 24, D-7 800 Van Oystaeyen, Wilrijk; W. Scharlau, Freiburg im Breisgau. "Participants at Mi.inster. * 22-24. Second International Conference Oberwolfach meetings do need a per­ INFORMATION: Mathematisches For­ on Algebraic Methodology and Software sonal invitation. Without a personal schungsinstitut Oberwolfach, Ge­ Technology, (AMAST), Iowa City, lA. invitation participation is not possi­ schaftsstelle: Albertstrabe 24, D-7800 ble. Mathematicians interested in a Freiburg im Breisgau. "Participants at PURPOSE: The goal of the conference meeting should contact the Institute." Oberwolfach meetings do need a per­ is to consolidate the trend of using al­ gebraic methodology as a foundation sonal invitation. Without a personal * 9-15. Singulare Storungsrechnung, Ober- for software technology showing that invitation participation is not possi­ wolfach, Federal Republic of Germany. ble. Mathematicians interested in a universal algebra provides a practical meeting should contact the Institute." mathematical alternative to ad hoc CHAIRMEN: W. Jager, Heidelberg; N.N. approaches used in software develop­ INFORMATION: Mathematisches For­ * 28-May 4. Deductive Systems, Oberwol- ment. schungsinstitut Oberwolfach, Ge­ fach, Federal Republic of Germany. CALL FOR PAPERS: Talks reporting schaftsstelle: Albertstrabe 24, D-7800 research in algebra suitable as a foun­ Freiburg im Breisgau. "Participants at CHAIRMEN: W.W. Bledsoe, Austin; G. dation for software technology as well Oberwolfach meetings do need a per­ Jager, Bern; M.M. Richter, Kaiser­ as software technologies developed by sonal invitation. Without a personal slautern. means of algebraic methodologies are invitation participation is not possi­ INFORMATION: Mathematisches For­ welcome. Submit a two page abstract ble. Mathematicians interested in a schungsinstitut Oberwolfach, Ge­ (including a few citations of rele­ meeting should contact the Institute." schaftsstelle: Albertstrabe 24, D-7800 vant work) of your talk to: AMAST Freiburg im Breisgau. "Participants at Conference, Computer Science Dept., * 30-July 6. Elliptische Operatoren auf Oberwolfach meetings do need a per­ The University of Iowa, Iowa City, Singularen und Nichtkompakten Man­ sonal invitation. Without a personal lA 52242 by January I, 1991. nigfaltigkeiten, Oberwolfach, Federal Re­ invitation participation is not possi­ INFORMATION: In Canada: T. Miildner, public of Germany. ble. Mathematicians interested in a Acadia Univ., Jodrey School ofComp. meeting should contact the Institute." CHAIRMEN: J. Bruning, Augsburg; R. Sci., Wolfville, Nova Scotia, Canada Melrose, Cambridge. BOP IXO; [email protected]; In INFORMATION: Mathematisches For­ Europe: M. Nivat, Univ. Paris 7, 2, May 1991 schungsinstitut Oberwolfach, Ge­ Place Jussieu, 75005 Paris, France; schaftsstelle: Albertstrabe 24, D-7800 * 12-18. Nichtlineare Evolutionsgleichun­ (I) 43 25 98 74; In US: T. Rus, Univ. Freiburg im Breisgau. "Participants at gen, Oberwolfach, Federal Republic of of Iowa, Dept. of Comp. Sci., Iowa Oberwolfach meetings do need a per­ Germany. City, lA 52242; (319)335-0694; email: sonal invitation. Without a personal rus@herky .cs. uiowa.edu. CHAIRMEN: S. Klainerman, Princeton; invitation participation is not possi­ ble. Mathematicians interested in a M. Struwe, Zi.irich. * 26-June I. Optimalsteuerung und Varia­ meeting should contact the Institute." INFORMATION: Mathematisches For­ tionsrechnung-Optimal Control, Ober­ schungsinstitut Oberwolfach, Ge­ wolfach, Federal Republic of Germany. schaftsstelle: Albertstrabe 24, D-7800 CHAIRMEN: R. Bulirsch, Mi.inchen; A. Freiburg im Breisgau. "Participants at July 1991 Oberwolfach meetings do need a per­ Miele, Houston; J. Stoer, Wi.irzburg; sonal invitation. Without a personal K. Well, Oberpfaffenhofen. * 8-12. Second International Conference invitation participation is not possi­ INFORMATION: Mathematisches For­ on Industrial and Applied Mathematics, ble. Mathematicians interested in a schungsinstitut Oberwolfach, Ge­ Washington, DC. (Please note changes meeting should contact the Institute." schaftsstelle: Albertstrabe 24, D-7800 from Nov. 1988, p. 1389) Freiburg im Breisgau. "Participants at PROGRAM: The conference organizers * 19-25. Differentialgeometrie im Grossen, Oberwolfach meetings do need a per­ are developing a program that will fo­ Oberwolfach, Federal Republic of Ger­ sonal invitation. Without a personal cus worldwide attention on the impor­ many. invitation participation is not possi­ tance of mathematical and computa­ ble. Mathematicians interested in a CHAIRMEN: W. Ballmann, Bonn; J.-P. tional methods in the solution of real meeting should contact the Institute. Bourguignon, Palaiseau; W. Klingen­ world problems. A major exposition berg, Bonn. of computer hardware and software INFORMATION: Mathematisches For­ exhibits and demonstrations that will June 1991 schungsinstitut Oberwolfach, Ge­ enable you to explore state-of-the-art schaftsstelle: Albertstrabe 24, D-7 800 * 2-8. Diskrete Geometrie, Oberwolfach, technology is also planned. Freiburg im Breisgau. Participants at Federal Republic of Germany. CONFERENCE TOPICS: Presentations

APRIL 1990, VOLUME 37, NUMBER 4 501 Meetings and Conferences

are solicited in all areas of applied and DEADLINEs: Proposals to organize a ORGANIZER: R. Brualdi, Univ. of computational mathematics, compu­ session: As soon as possible; Phase Wisconsin-Madison. ter science, applied probability and I submissions due: August l, 1990; INFORMATION: SIAM, Conference Co­ statistics, scientific computing, and Phase II submissions due: October I, ordinator, Dept. CC0190, 3600 Uni­ applications in engineering and the 1990; Late submissions: December l, versity City Science Center, Philadel­ biological, chemical, and physical sci­ 1990. phia, PA 19104-2688; tel 215-382- ences. 9800; Fax 215-386-7999; email: INVITED SPEAKERS: J.M. Ball, G.I. * 28-August 3. Gruppen und Geometrien, [email protected]. Barenblatt, M. Brady, A.R. Conn, W. Oberwolfach, Federal Republic of Ger­ Eckhaus, A. Fasano, G.H. Golub, M. many. * 23-29. Sixth Symposium on Classical Analysis, Kazimierz Dolny, Poland. Groetschel, T.J.R. Hughes, N.K. Kar­ CHAIRMEN: M. Aschbacher, Pasadena; markar, K. W. Kirchgassner, P.-L. W.M. Kantor, Eugene; F. Timmes­ PROGRAM: Discussions will take place Lions, Y.F. Meyer, M. Mimura, J.D. feld, GieBen. on the results and problems in such Murray, G. Ruget, E. Tardos, D.J. INFORMATION: Mathematisches For­ fields as: several complex variables Wallace, M.F. Wheeler, H. Yserentant. schungsinstitut Oberwolfach, Ge­ (especially L2-methods), Riemannian CALL FOR PAPERS: Authors will have schaftsstellc: Albertstrabe 24, D-7 800 and Hermitian geometry, spectral the­ approximately 15 minutes to present Freiburg im Breisgau. "Participants at ory in Hilbert space, probability, ap­ contributed papers, with an additional Oberwolfach meetings do need a per­ plications in the mathematical physics. five minutes for questions. To con­ sonal invitation. Without a personal Particular consideration will be given tribute, you must submit a summary invitation participation is not possi­ to the interrelation of ideas from dif­ not exceeding 75 words on an ICIAM ble. Mathematicians interested in a ferent areas. 91 contributed paper /poster presenta­ meeting should contact the Institute." INFORMATION: T. Mazur, Technical tion form. Deadline for submission is Univ., Dept. of Math., Malczewskiego September 30, 1990. 29, 26-600 Radom, Poland. INFORMATION: ICIAM 91 Conference Manager, cjo SIAM, 3600 University City Science Center, Philadelphia, PA 191 04-2688; email: [email protected]; FAX: 215- August 1991 386-7999; tel: 215-382-9800. * 3-7. Interamerican Conference on Math­ * 8-14. ICOR '91 International Conference ematics Education, Univ. of Miami, Coral October 1991 on Radicals, Szekszard, Hungary. Gables, FL. 12-13. Eastern Section, Temple Univer­ INFORMATION: L. Marki or R. Wie­ INFORMATION: U. D'Ambrosio, In­ sity, Philadelphia, PA. gandt, Mathematical Institute, Hun­ stitute of Math., Univ. Estadual de INFORMATION: W. Drady, American garian Academy of Science, Pf.l2 7, Campinas, CP 6063 ( 13081) Camp­ Mathematical Society, P.O. Box 6248, H-1364 Budapest, Hungary. inas (SP), Brazil. Providence, RI 02940. * 14-20. Dynamische Systeme, Oberwol- 25-26. Central Section, North Dakota fach, Federal Republic of Germany. State University, Fargo, NO. CHAIRMEN: J. Moser, ZUrich; E. Zehn­ INFORMATION: W. Drady, American der, ZUrich. Mathematical Society, P.O. Box 6248, INFORMATION: Mathematisches For­ September 1991 Providence, RI 02940. schungsinstitut Oberwolfach, Ge­ schaftsstelle: Albertstrabe 24, D-7 800 * 10-13. IFAC/IMACS Symposium on Freiburg im Breisgau. "Participants at Fault Detection, Supervision and Safety Oberwolfach meetings do need a per­ for Technical Processes-SAFEPROCESS '91, Baden-Baden, Federal Republic of sonal invitation. Without a personal Germany. invitation participation is not possi­ ble. Mathematicians interested in a INFORMATION: Dipl. lng. H. Wiefels, meeting should contact the Institute." VDI/VDE-GMA, Postfach 1139, D- March 1992 4000 Dusseldorf l, Federal Republic 27-28. Central Section, Southwest Mis­ * 22-26. Thirteenth IMACS World Con­ of Germany. souri State University, Springfield, MO. gress on Computation and Applied Math­ ematics, Trinity College, Dublin Uni­ * 16-19. Fourth SIAM Conference on Ap­ INFORMATION: W. Drady, American versity, Dublin, Ireland. (Please note plied Linear Algebra, Univ. of Minnesota, Mathematical Society, P.O. Box 6248, changes from Nov. 1989, p. 1254) Minneapolis, MN. Providence, RI 02940.

502 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

New Journal MATRIX THEORY AND APPLICATIONS Charles R. Johnson, Editor LENINGRAD MATHEMATICAL JOURNAL (Proceedings of Symposia in Applied Mathematics, Volume 40) The Society is pleased to announce the Leningrad This volume contains the lecture notes prepared for the Mathematical Journal, a cover-to-cover translation into English AMS Short Course on Matrix Theory and Applications, held of Anre6pa 111 aHaJIJ/13. With research papers, expository in Phoenix in January, 1989. Matrix theory continues to enjoy surveys, and book reviews, the Leningrad Mathematical a renaissance that has accelerated in the past decade, in Journal contains contributions by some of the most prominent part because of stimulation from a variety of applications Soviet mathematical scientists. Readers will appreciate and considerable interplay with other parts of mathematics. this opportunity to keep abreast of Soviet mathematical In addition, the great increase in the number and vitality of developments in all areas of mathematics. specialists in the field has dispelled the popular misconception The mathematics section of the Academy of Sciences of the that the subject has been fully researched. USSR began publishing the Russian-language journal in 1989 The purpose of the Short Course, which attracted via the Leningrad branch of ··Nauka··. The topics explored in approximately 140 participants, was to present a sample the first issues of the Leningrad Mathematical Journal range of the ways in which modern matrix theory is stimulated over a wide array of mathematical areas. A brief sampling by its interplay with other subjects, such as combinatorics, includes: probability methods in the theory of conformal probability theory, statistics, operator theory and control mappings, quantization of Lie groups and Lie algebras, almost theory, algebraic coding theory, partial differential equations, cocommutative Hopf algebras, valence bonds in quasicrystals, and analytic function theory. Among the themes in this the dynamics of analytic transformations, algebraic K-theory, volume are the notion of majorization, the trend away from and compact four-dimensional exotica. "basis-free·· point of view, problem-dependent symmetries, Members of the Editorial Board are: A. N. Andrianov and the synergy between matrix theory and systems theory. (Deputy Editor-in-Chief), V. M. Babich, M. Sh. Birman, 0. Ya. The immense variety of tools and problems in this area Viro, S. P. Merkur'ev, N. K. Nikol'skiT (Deputy Editor-in-Chief), illustrates one reason for using the term ··matrix theory" A. P. Oskolkov (Executive Secretary), A. N. Parshin, V. N. or ··matrix analysis" instead of ··linear algebra··; a large Popov (Deputy Editor-in-Chief), M.A. Semenov-Tyan-ShanskiT, portion of current work is neither primarily linear nor primarily A. 0. Slisenko, A. A. Suslin, N. N. Ural'tseva, D. K. Faddeev algebraic in nature. The remarkable breadth of interest in this (Editor-in-Chief), and V. P. Khavin. subject seems to point to future developments as fruitful and The publication of the Leningrad Mathematical Journal is unexpected as in the past. larger effort by the AMS to stimulate interactions part of a Contents between American mathematicians and their Soviet Richard A. Brualdi, The many facets of combinatorial matrix theory; colleagues. In this spirit, the Society is proud to offer this new Persi Diaconis, Patterned matrices; Joseph A. Ball, Israel Gohberg, journal. and Leiba Rodman, Tangential interpolation problems for rational ISSN 1048-9924 matrix functions; Roger A. Horn, The Hadamard product; Charles R. Six issues per year, Volume 1 , May 1990 Johnson, Matrix completion problems: A survey; Arunava Mukherjea, Individual member $720*, List price $720*, The role of nonnegative idempotent matrices in certain problems Institutional member $576* in probability; Ingram Olkin, Interface between statistics and linear To order, please specify 90LMJ/N algebra. *NOTE: Please add for postage: surface delivery to destinations outside the U.S. 1980 Mathematics Subject Classification: 15-02 and lndia-$13; to lndia-$23. Expedited delivery to destinations in North America-$16; 0160-7634 elsewhere-$30. Subscriptions to AMS journals are sold only on a calendar year basis ISBN 0-8218-0154-6, LC 90-30584, ISSN (January-December). Split and multiple year subscription orders will not be accepted. 260 pages (hardcover), April1990 Cancellation refunds are computed by deducting an $8 cancellation fee, and the price of Individual member $34, List price $57, each issue already shipped, from the price paid. Institutional member $46 To order, please specify PSAPM/40N

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APRIL 1990, VOLUME 37, NUMBER 4 503 New AMS Publications

CLASSICAL ASPHERICAL MANIFOLDS LA DICHOTOMIE F. Thomas Farrell and Lowell Edwin Jones ELLIPTIQUE-HYPERBOLIQUE EN (CBMS Regional Conference Series, Number 75 HOMOTOPIE RATIONNELLE Supported by the National Science Foundation) Y. Felix Aspherical manifolds-those whose universal covers are (Asterisque, Number 176) contractible-arise classically in many areas of mathematics. The homotopy groups 1r;(X) of a finite 1-connected C.W. They occur in Lie group theory as certain double coset spaces complex X are finitely generated abelian groups: Therefore, and in synthetic geometry as the space forms preserving the 1r;(X) = zn; EB T;, where T; is a finite group. There are then geometry. two possibilities: either all the n;, except a finite number, are This volume contains lectures delivered by the first author zero, or there exists an infinite sequence of nonzero n;. In at an NSF-CBMS Regional Conference on K-Theory and the first case, the space is called elliptic: the Euler-Poincare Dynamics, held in Gainesville, Florida in January, 1989. characteristic is nonnegative and the cohomology satisfies The lectures were primarily concerned with the problem .of Poincare duality. In the second case, the space is called topologically characterizing classical aspherical manifolds. hyperbolic and the sequence 2:::~= n; has an exponential This problem has for the most part been solved, but the 1 growth. In the text, the study of the dichotomy is then 3- and 4-dimensional cases remain the most important extended to the structure of the Lie algebra 1r*(nX) ® Q. open questions; Poincare's conjecture is closely related to the 3-dimensional problem. One of the main results is Contents that a closed aspherical manifold (of dimension =f; 3 or 4) Definition de Ia categorie de Lusternik-Schnirelmann; Espaces is a hyperbolic space if and only if its fundamental group rationnels et modeles minimaux; Le mapping theorem; cat0 et les is isomorphic to a discrete, cocompact subgroup of the modeles minimaux de Sullivan; Espaces 1r-finis; La croissance Lie group O(n, 1; IR). One of the book's themes is how exponentielle; L'algebre de Lie 7r*(QX) ® Q; Cohomologie d'une algebra de Lie graduee; Operation d'holonomie d'une fibration; the dynamics of the geodesic flow can be combined with Categorie d'une application; Profondeur des algebras de Lie; topological control theory to study properly discontinuous Croissance des ideaux de 1r * (QX) ® Q; Profondeur un; La group actions on R". dichotomie; Bibliographie; Index terminologique; Index des symboles. Some of the more technical topics of the lectures have been deleted, and some additional results obtained since the ISSN 0303-1179 conference are discussed in an epilogue. The book requires 188 pages (sottcover), 1989 some familiarity with the material contained in a basic, Individual AMS or SMF member $16, List price $23 graduate-level course in algebraic and differential topology, as To order, please specify AST/176N well as some elementary differential geometry. Contents The structure of manifolds from a historical perspective; Flat Riemannian manifolds and infrasolvmanifolds; The algebraic K -theory of hyperbolic manifolds; Locally symmetric spaces of noncompact type; Existence of hyperbolic structures; Epilogue. SEMINAIRE BOURBAKI, VOLUME 1988/89, 1980 Mathematics Subject Classifications: 57R55, 18F25, 22E40, EXPOSES 700-714 53C20 (Asterisque, Number 177-178) ISBN 0-8218-0726-9, LC 90-39, ISSN 0160-7642 64 pages (sottcover), Apri11990 Comme les precedents volumes de ce Seminaire, celui-ci All individuals $14, List price $24 contient des exposes de synthese sur des sujets d'actualite: To order, please specify CBMS/75N quatre sur les Groupes de Lie et leurs representations, deux de Geometrie riemannienne, deux sur les Feuilletages, un de Geometrie algebrique, un de Geometrie arithmetique, un de SOCIETE MATHEMATIQUE DE FRANCE, ASTERISQUE Geometrie symplectique, un de Theorie des nombres, un de Logique et un de Physique theorique. The AMS distributes Asterisque only in the U.S., Canada, and Mexico. On y fait, entre autres, le point sur !'invariant de Godbillon­ Orders from other countries should be sent to the SMF, B.P. 126-05, Vey, les feuilletages riemanniens, les fonctions Lp-adiques, 75226 Paris Cedex 05, France, or to OFFILIB, 48 rue Gay-Lussac, les groupes associes aux algebras de Kac-Moody, les 75240 Paris Cedex 05, France. Individual members of either AMS or SMF are entitled to the member price. (ISSN 0303-1179) modeles minimaux des variates algebriques de dimension 3 et Ia theorie d'Arakelov.

504 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New AMS Publications

Contents Jacques Tits, Groupes associes aux algebres de Kac-Moody; John Coates, On p-adic L-functions; Laurent Clozel, Nombres de Tamagawa des groupes semi-simples [d'apres Kottwitz]; Yves Colin de Verdiere, Distribution de points sur une sphere [d'apres Lubotzky, Phillips et Sarnak]; Krzysztof Gawtdzki, Conformal field theory; Pierre Berard, Variates Riemanniennes isospectrales non isometriques; Etienne THE BEAUTY AND Ghys, L 'invariant de Godbillon-Vey; Andre Haefliger, Feuilletages Reimanniens; Christian Kassel, Le residu non commutatif [d'apres M. Wodzicki]; Gerard Laumon, Faisceaux caracteres [d'apres Lusztig]; COMPLEXITY Patrick Dehornoy, La determimjtion projective [d'apres Martin, Steel et Woodin]; Guy Henniart, Formes de Maass et representations OF THE MANDELBROT SET galoisiennes [d'apres Blasius, Gloze/, Harris, Ramakrishnan et Taylor]; Janos Kollar, Minimal models of algebraic threefolds: Mori's program; University Edition Christophe Soule, Geometrie d'Arakelov des surfaces arithmetiques; with John Hubbard Viterbo, Capacites symplectiques et applications [d'apres Claude A Science Television Production Ekeland-Hofer, Gromov];

ISSN 0303-1179 388 pages (softcover), 1989 Individual AMS or SMF member $33, List price $47 Providing an accessible introduction to the basics of To order, please specify AST/177-78N fractals, this videotape presents an appealing balance of the theoretical and aesthetic aspects of the Mandelbrot set. Viewers will appreciate the clarity of exposition as John THEORIE DE HODGE Hubbard uses a combination of lecture, boardwork, (Asterisque, Number 179-180) Macintosh computer demonstrations, and colorful com­ This volume contains several contributions of a conference puter-generated films and pictures to bring the concepts to on Hodge Theory held in Luminy in June 1987. life. There is an exposition of the recent quite spectacular results on intersection cohomology with values in a polarized Part I focuses on iteration and Julia sets, while Part II variation of Hodge structure, and with constant values on addresses Mandelbrot sets. Part III examines a way of locally symmetric spaces, on mixed Hodge modules and perverse sheaves. Several subjects at the border line of using the concept of electric field lines to understand Hodge Theory are presented as well: extensions of variations these fractal sets. The concluding remarks round out the of Hodge structure, applications to Hodge Theory to vanishing lecture by pointing to a philosophical framework that theorems and to singularities. relate these sets to phenomena occurring in the natural Contents world. Requiring only a background in calculus, this Daniel Barlet, Asymptotic filtration and poles of fx lfi2>.D; James videotape will provide a useful tool in classrooms and A. Carlson and Richard M. Hain, Extensions of variations of mixed would be an excellent addition to a videotape library. Hodge structure; Eduardo Cattani and Aroldo Kaplan, Degenerating variations of Hodge structure; Helene Esnault and Eckart Viehweg, Vanishing and non vanishing theorems; Helmut A. Hamm, Cohomology with coefficients in Z(p) and de Rham cohomology: examples; Morihiko Saito, "Introduction to mixed modules"; Joseph H. M. 1980 Mathematics Subject Classification: 58 Steenbrink, The spectrum of hypersurface singularities; Sampei Usui, VHS format, approx. one hour, U. S. standard Type I degeneration of Kunev surfaces; Steve Zucker, L 2-cohomo/ogy Price $59 and intersection homology of locally symmetric varieties, Ill. To order, please specify VIDHUBBARD/NA 1980 Mathematics Subject Classification: 14 ISSN 0303-1179 278 pages (softcover), 1989 Individual AMS or SMF member $23, List price $33 Order from the American Mathematical Society, P. 0. Box 1571, To order, please specify AST /179-80N Annex Station, Providence, RI 02901-1571 or call800-321-4AMS (321-4AMS) in the U.S. and Canada to use VISA or MasterCard. Prepayment required. Videotapes are not returnable. Prices subject to change without notice. Free shipment by surface; for air delivery, please add $5 first video, $3 each additional, $100 max.

APRIL 1990, VOLUME 37, NUMBER 4 505 AMS Reports and Communications

Recent Appointments AMS-MAA Invited Addresses. By for National Meetings, there were invitation of the AMS-MAA Joint six fifty-minute invited addresses. Committee members' terms of of­ Program Committee (Hugh L. Mont­ The names of the speakers, their fice on standing committees expire gomery, David P. Roselle, Mary Ellen affiliations, and the titles of their on December 31 of the year given Rudin, and Peter Sarnak, Chair­ talks are as follows: SuN-YuNG AL­ in parentheses following their man), four speakers addressed the ICE CHANG, University of California, names, unless otherwise specified. AMS and MAA on the history and Los Angeles, Geometric spectral and development of mathematics. The inverse spectral problems; IsRAEL C. Haynes Miller has been appointed names of the speakers, their affil­ GoRBERG, Tel Aviv University, Is­ chair of the Editorial Boards Com­ iations, and the titles of their talks rael, Linear operators, matrix func­ mittee by President William Brow­ were as follows: JoHN BAR WISE, Stan­ tions and control; MIKE HOPKINS, der. Continuing members of the com­ ford University, Non-wei/founded sets Massachusetts Institute of Technol­ mittee are Linda Keen ( 1992), Carlos and their applications; CHARLES W. ogy, Stable homotopy theory; HEN­ Kenig ( 1991 ), Richard M. Schoen CURTIS, University of Oregon, A cen­ RYK IWANIEC, Rutgers University, ( 1990), Barry Simon ( 1992), and tury of representation theory of fi­ Problems and methods in analytic Daniel Zelinsky ( 1991). nite groups; BARRY SIMON, Califor­ number theory; JANOS KoLLAR, Uni­ Marc A. Rieffel ( 1991 ), by recom­ nia Institute of Technology, Fifty versity of Utah, The structure ofalge­ mendation of the Editorial Boards years of eigenvalue perturbation the­ braic threefolds; I. M. SIGAL, Univer­ Committee, has been elected by the ory; NOLAN R. WALLACH, Rutgers sity of Toronto and the University of Council to the Mathematical Sur­ University, Some applications of California, Irvine, Quantum many­ veys and Monographs Editorial Com­ group representations. body problem. mittee. Continuing members of the The four speakers were intro­ The six speakers were introduced committee are Victor W. Guillemin duced by Sun-Yung Alice Chang, by Peter Perry, Joseph Ball, Ethan ( 1992), DavidS. Kinderlehrer ( 1991 ), Ramesh A. Gangolli, C. Ward Hen­ Devinatz, Hugh L. Montgomery, and M. Susan Montgomery ( 1990), son, and Gerald J. Janusz, respec­ Charles H. Clemens, and Joseph Con­ chair. tively. lon, respectively. Sixty-third Josiah Willard Gibbs Special Sessions. By invitation of Lecture. The 1990 Gibbs Lecture was the same committee, there were eigh­ The Annual Meeting presented by George B. Dantzig of teen special sessions of selected in Louisville Stanford University. The title of twenty-minute papers. The topics of The January 1990 Joint Mathematics his lecture was The wide world of the sessions, and the names and af­ Meetings, including the 96th Annual pure mathematics that goes by other filiations of the organizers are listed Meeting of the American Mathemati­ names. Professor Dantzig was intro­ below. Linear operators, matrix func­ cal Society, the 73rd Annual Meeting duced by William Browder, President tions and control, JosEPH BALL, Vir­ of the Mathematical Association of of the Society. ginia Polytechnic Institute and State America, and the 1990 annual meet­ Colloquium Lectures. A set of University, and ISRAEL G. GORBERG, ings of the Association for Women three lectures was given by Schlomo Tel Aviv University. in Mathematics and the National As­ Sternberg, Harvard University. The The Shrodinger equation, JosEPH sociation for Mathematicians, were title of his lectures was Some thoughts CONLON, University of Michigan. held January 17-20 (Wednesday­ on the interaction between group the­ Multivariable operator theory, Saturday), 1990, in Louisville, Ken­ ory and physics. The presiders at RAUL E. CuRTO and PAuLS. MULHY, tucky. Scientific sessions took place these three lectures were William University of Iowa. in the Louisville Convention Center Browder, Sun-Yung Alice Chang, and Homotopy theory, ETHAN DEV­ and the Hyatt Regency Louisville. William P. Thurston. INATZ and MIKE HOPKINS, Univer­ There were 3, 14 7 registrants, includ­ Invited Addresses. By invitation sity of Chicago. ing 2,240 members of the Society. of the AMS Program Committee

506 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY AMS Reports and Communications

Group representations and opera­ Joyati Debnath, George T. Gilbert, Former Secretary tor algebras, ROBERTS. DORAN, Texas Maria Girardi, Robert Gordon, T. Everett Pitcher 1/89-12/90 Christian University. Hesterberg, Charles S. Holmes, Mel­ Associate Secretaries Functional equations and their ap­ vin Holmes, Rodney Hood, Diane Joseph A. Cima 1/89-12/90 plications, BRUCE R. EBANKS, Uni­ L. Johnson, Gene D. Johnson, San­ W. Wistar Comfort 1/83-12/90 versity of Louisville. dria N. Kerr, Alexandra Kurepa, Lee Andy Roy Magid 1/88-12/91 History ofmathematics, FLORENCE Larson, Bruce M. Landman, John Lance W. Small 1/88-12/91 D. FASANELLI, National Science Foun­ A. Morrison, M. Nourimagahadam, Treasurer dation, and VICTOR J. KATZ, Univer­ Timothy Peil, Margaret L. Reese, Franklin P. Peterson 8/73-12/90 sity of the District of Columbia. Emilio Roxin, John M. Russell, Erin Control of infinite dimensional M. Schram, Donald R. Snow, Monty Associate Treasurer systems, ROBERT E. FENNEL, Clem­ Strauss, Karel Stroethoff, G. Rubin Steve Armentrout 7/77-12/90 son University, and SUZANNE MARIE Thomas, Jerry E. Vaughn, Denis A. *As a result of his election to the LENHART, University of Tennessee. White, and Eric Wingler. Executive Committee, he will be a Geometric function theory in one AMS-MAA Science and Govern­ Member-at-Large in 1990 and 1991. and several complex variables, CARL ment Speakers. The presidents of the H. FITZGERALD, University of Cali­ AMS and the MAA, William Brow­ MEMBERS-AT-LARGE fornia, San Diego, and TED J. SuP­ der and Lida K. Barrett, invited All terms are for 3 years and expire FRIDGE, University of Kentucky. Luther Williams, Senior Science Ad­ on December 31 of the given year. Oscillation theory in ordinary dif­ visor, National Science Foundation, 1989 Secretary of ferential equations, GARY D. JONEs, and Alvin Trivelpiece, H. Blaine Lawson, Jr. one-hour addresses Murray State University. Energy, to give Yiannis N. Moschovakis in Louisville. Topology in computer graphics to the Joint Meetings Linda A. Ness EFIM D. KHAL­ The title of Luther Williams' talk and image processing, Marc A. Rieffel Keystone of mod­ IMSKY, College of Staten Island, T. was Mathematics: William A. Veech** and ern science and technology. The title YuNG KoNG, Queens College, CarolS. Wood RALPH KOPPERMAN, City College, of Alvin Trivelpiece's talk was Re­ CUNY. vitalizing mathematics education: A 1990 Algebraic geometry, JANOS KoL­ national imperative. Richard K. Guy LAR, University of Utah. Rhonda J. Hughes Semi-group theory, INESSA LEVI Officers of the Society Robion C. Kirby and W. WILEY WILIAMS, University 1989 and 1990 Irwin Kra** of Louisville. Except for the Members-at-Large of Albert Marden Function theoretic methods in dif­ the Council, the month and year Harold M. Stark ferential equations, PETER McCoY, of the first term and the end of the 1991 United States Naval Academy. present term are given. For Members­ Jonathan L. Alperin Optimization and nonlinear anal­ at-Large of the Council, the last year Fan R. K. Chung ysis, LYNN McLINDEN, University of of the present term is listed. Lawrence J. Corwin Illinois, Urbana, and JAYS. TREIMAN, Michael C. Reed Western Michigan University. COUNCIL Hugo Rossi Analytic number theory, HUGH L. President William P. Thurston** MONTGOMERY, Univ. of Michigan. William Browder 1/89-12/90 1992 Geometric spectral and inverse Ex-President Sheldon Axler spectral problems, PETER A. PERRY, G. D. Mostow 1/89-12/89 Joan S. Birman University of Kentucky. President-Elect Charles Herbert Clemens Markov processes and stochastic Michael Artin 1/90-12/90 Carl Pomerance analysis, PHILIP E. PROTTER, Purdue Shing-Tung Yau Vice Presidents University. **Members-at-large, as provided for Contributed papers. There were Lenore Blum 1/90-12/91 Sun-Yung Alice Chang 1/89-12/90 in Article 7, Section 4 (last sentence) twenty-nine AMS sessions for con­ of the Bylaws of the Society. tributed 10-minute papers. The fol­ Barry Simon 1/88-12/89 lowing mathematicians served as pre­ Dennis P. Sullivan 1/90-12/91 Publications and Communications siding officers for these sessions: J. William P. Thurston 1/88-12/89* Committees Marshall Ash, Scott Chapman, Shan­ Secretary Bulletin Editorial Committee non S. Cobb, Robin Cunningham, Robert M. Fossum 1/89-12/90 Morris W. Hirsch 1/88-12/89

APRIL 1990, VOLUME 37, NUMBER 4 507 AMS Reports and Communications

Roger E. Howe 1/88-12/90 Chair, Committee on Transactions and Memoirs RichardS. Palais 1/90-12/92 Science Policy Editorial Committee Colloquium Editorial Committee Ronald G. Douglas 1/87-12/89 Ronald L. Graham 1/88-12/89 Raoul H. Bott 1/85-12/90 Michael C. Reed 1/90-12/92 David J. Saltman 1/90-12/90 Chair, Committee to Monitor Journal of the AMS BOARD OF TRUSTEES Michael Artin 1/88-12/92 Problems in Communication Marian B. Pour-El 1/85-12/89 Steve Armentrout Reviews Editorial Mathematical RichardS. Palais 1/89-12/90 (ex officio) 7/77-12/90 Committee William Browder Proceedings Editorial Committee Melvin Hochster 1/84-12/89 (ex officio) 1/89-12/90 J. Davis 1/88-12/91 B. A. Taylor 1/90-12/92 William Ramesh A. Gangoli 1/85-12/89 Andrew M. Odlyzko 1/84-12/89 Mathematical Surveys and Frederick W. Gehring 1/83-12/92 Monographs Editorial Committee Representatives on American Ronald L. Graham 1/82-12/91 Journal of Mathematics M. Susan Montgomery 1/85-12/90 M. Susan Montgomery 1/86-12/90 M. Salah Baouendi 1/88-12/92 Franklin P. Peterson Mathematics of Computation (ex officio) 8/73-12/90 Editorial Committee John C. Polking 1/90-12/94 Walter Gautschi 1/84-12/92 Paul J. Sally, Jr. 1/84-12/93

Miscellaneous

Personal Items Michio Kuga, of SUNY at Stony ber 27, 1989, at the age of 70. He Andrew M. Odlyzko, the head of Brook, died on February 14, 1990, at was a member of the Society for 10 AT&T Bell Laboratories' Communi­ the age of 61. He was a member of years. cation and Computer Systems De­ the Society for 16 years. Irving Sussman, Professor Emer­ partment, delivered the eighth Leon­ Sim Lasher, Associate Professor itus of Santa Clara University, died idas Alaoglu Memorial Lecture in Emeritus of the University of Illinois on February 18, 1990, at the age of Mathematics at the California Insti­ at Chicago, died on January 5, 1990, 82. He was a member of the Society tute of Technology. at the age of 73. He was a member for 45 years. of the Society for 51 years. Deaths Donald H. Rock, of St. Paul, Min­ Melvin Bloom, Professor Emeritus of nesota, died on November 4, 1989, Visiting Mathematicians Miami University, died on Septem­ at the age of 78. He was a member Luis Ribes (Carleton University, Ot­ ber 11, 1989, at the age of 85. He of the Society for 50 years. tawa, Canada), Universidad Aut6no­ was a member of the Society for 42 Nicholas Solimene, of Wood ma de Madrid, Madrid, Spain, Au­ years. Haven, New York, died on Decem- gust 1990 to July 1991, algebra.

508 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New Members of the AMS

ORDINARY MEMBERS Wende Chen, Academia Sinica, Dadang Hendirman, Bandung, Wojciech Kryszewski, Univ of Poland Rex L Agacy, Townsville, Beijing, People's Republic of Indonesia Lodz, Ya V Kurylev, Leningrad, USSR Australia China Martin Hermann, Jena, German V V Kvaratskheliya, Tbilisi, Ahmad Fawzi Alameddine, King Zu Chi Chen, Univ of Science & Democratic Republic Fahd Univ, Dhahran, Saudi Technology, Hefei, People's Salvador Hernandez, Valencia, USSR V F Lazutkin, Leningrad, USSR Arabia Republic of China Spain Robyn L Lebiecki, La Crosse, WI Natalia Alexandrov, Rice Univ, Chou-Hsieng Joe Chin, National Norimichi Hirano, Yokohama Huazhong Houston, TX Chiao Tung Univ, Hsin Chu, National Univ, Japan Zhaojun Liang, Normal Univ, Hubei, Jesus Araujo, Univ of Oviedo, Taiwan Ladislav Hlavaty, Institute People's Republic of China Gijon, Spain Sinisa Crvenkovic, Univ of Novi of Physics, Prague, Gui Zhen Liu, Shandong Univ, Konstantin Athanassopoulos, Sad, Yugoslavia Czechoslovakia Jinan, People's Republic of Freie Univ Berlin, Federal Endre Csaki, Budapest, Hungary Jia Xing Hong, Shanghai, China Republic of Germany Guo Ren Dai, Sichuan Univ, People's Republic of China Doron S Lubinsky, Univ Alexander M Barg, Institute­ People's Republic of China Ming You Huang, Changchun, of Witwatersrand, Problems of Information Ingrid Daubechies, American People's Republic of China Johannesburg, Republic of Transportation, Moscow, Tel & Tel Bell Laboratories, Masataka Ishikawa, Berkeley, CA South Africa USSR Murray Hill, NJ Haydeh Jarahery, Ferdowsi Univ V Mackevicius, Vilnius, USSR Urs W Barmettler, Unteraegeri, Ian M Davies, Univ College of of Mashhad, Iran Behroz Mashayekhy, Ferdowsi Switzerland Swansea, England Marek Jarnicki, Krakow, Poland Univ of Mashhad, Iran VA Baskakov, Moscow, USSR Marlene G DeTienne, Clinton, Zbigniew Jelonek, Jagiellonian Meier, Zurich, Muharrem Berisha, Pristina, MA Univ, Krakow, Poland David Switzerland Yugoslavia Jozsef Denes, Budapest, Hungary Guo Ying Jiang, Fudan Univ, Ruth I Michler, Berkeley, CA Gennady Berman, Siberian Jesus Esquinas, Center for the Shanghai, People's Republic Noguera Miquel, Terrassa, Spain Branch Academy of Science, Mathematical Sciences, of China Oscar Moreno, Univ of Puerto Krasnoyarsk, USSR Madison, WI Boleslaw Kacewicz, Univ of Rio Piedras Sergei L Bezrukov, Institute Laura Fainsilber, Berkeley, CA Warsaw, Poland Rico, Andrey Morozov, Novosibirsk, Problems of Information Reinhard 0 W Franz, Univ of Vadim A Kaimanovich, Transportation, Moscow, Bielefeld, Federal Republic Leningrad Shipbuilding USSR Leonid Yu Motylev, Moscow, USSR of Germany Institute, USSR Janos Bognar, Budapest, Robert Fuller, Eotvos Lorand Ji Ding Kang, Southwest China USSR Thandwa Zizwe Mthembu, Hungary Univ, Budapest, Hungary Teacher's Univ, Sichuan, Univ of Witwatersrand, Alexander A Bolonkin, Brooklyn, Vladimir Georgiev, Bulgarian People's Republic of China Republic of NY Academy of Science, Sofia S D Karakozov, Pedagogichesky Johannesburg, Africa Nikolai Bozhinov, Bulgarian Vladimir Stefanov Gerdjikov, Institute, Barnaul, USSR South A Yu Muravitskii, Kishinev, Academy of Science, Sofia Sofia, Bulgaria Victor N Kasyanov, Academy of Elizabeth Bradley, Massachusetts V L Girko, Kiev State Univ, Science, Novosibirsk, USSR USSR Hagen Neidhardt, Joint Institute Institute of Technology, USSR Sergei V Ketov, Academy of Physics, Cambridge Richard Robert Gold, Palos Science, Tomsk, USSR of Theoretical USSR Maxim Bruckheimer, Weizmann Verdes, CA Saeed Keyvanfar, Ferdowsi Moscow, Alexander A Novikov, Steklov Institute of Science, V V Gorlov, Minsk, USSR Univ, Mashad, Iran Mathematical Institute, Rehovot, Israel Seth A Greenblatt, Sorites Group Kazem Khashyarmanesh, USSR Bair V Budaev, Leningard, USSR Incorporated, Springfield, Mashhad, Iran Moscow, Shmuel Onn, Cornell Univ, Alexandru Buium, Bucharest, VA G Khudaiberganov, Tashkent NY Romania Zdzislaw Grodzki, Lublin State Univ, USSR Ithaca, Pekka Orponen, Univ of Svetlana Buzasi, Debrecen, Technology Univ, Poland V S Kiryakova, Bulgarian Toronto, Ontario, Canada Hungary Juan Mateos Guilarte, Salamanca Academy of Science, Sofia Pare, Dalhousie Univ, Eddy Chakurov, Higher Institute Univ, Spain Yuri Kivshar, Institute of Low Robert Halifax, Nova Scotia, Canada of Electronic & Mech Ben-Yu Guo, Shanghai, People's Temperature Physics & Arkady S Pikovsky, Gorky, Engineering, Varna, Bulgaria Republic of China Engineering, Kharkov, USSR Peng Nian Chen, Xiamen Univ, Lian Fa He, Hebei Normal Stacy A Klamkin, Springer­ USSR M M Popov, Zaporozhye, USSR People's Republic of China Univ, Shijiazhuang, People's Verlag, New York, NY Putinar, Bucharest, Sui Yang Chen, Lanzhou Univ, Republic of China John W Klay, Pittsburgh, PA Mihai Romania People's Republic of China Gabor Hegedus, Budapest, Irena L Krolikiewicz, Wieliczka, Hungary Poland

APRIL 1990, VOLUME 37, NUMBER 4 509 New Members of the AMS

Meng Zhao Qin, Academia Zeng Kun Xu, Zhejiang Normal Kaori lmai Ota Roberta Gayle Fothergill Sinica, Beijing, People's Univ, People's Republic of Shuichi Sato Kevin John Harrington Republic of China China Polskie Towarzystwo Tara G Sheehan L Yampolskii, M I Rakhimberdiev, Alma-Ata, Alexandre Matematyczne Michael M Stark Kharkov State Univ, USSR USSR Lech Maligranda Brandeis University Remseth, Ostenstad, Mao Dong Ye, Zhejiang Univ, Biorn B Jones Norway People's Republic of China Real Sociedad Matemdtica Lewis Sofyadi Roezin, PT Caltex Hong Bing Yu, Univ of Science Espanola Brown University Pacific Indonesia, Jakarta & Technology, Hefei, Hernandez Gregorio Fatihcan M Atay People's Republic of China Vladimir Rogalewicz, Research Societe Mathematique Suisse California State University of Plant Production, Anatoliib Yuditskii, Institute of Institute Boas Erez Hayward Prague, Czechoslovakia Control Sciences, Moscow, Ann M Ritchey Marek Rutkowski, Technical USSR Jurg R Husler Univ of Warsaw, Poland Societe Mathematique de Belgique California State University Los Longin Rybinski, Higher College Jean-Pierre Tignol Angeles of Engineering, Zielona Societe Mathematique de France Theodore W Myer Gora, Poland Fran~ois Zara California State University Albert Sachs, Munchen, Federal RECIPROCITY MEMBERS Societe de Mathematiques Sacramento Republic of Germany Allahabad Mathematical Society Appliquees et lndustrielles Virginia S Puett Pawel Sadowski, Bialystok S P Gupta Branch of Warsaw Univ, Frederic Abergel Rebecca E Tripp Asociaci6n Matemdtica Espanola Poland Societal Catalana de Caltech Univ of Pisa, Jesus Hernandez Antonino Salibra, Matematiques Dana D Hobson Italy Joan Tarres College Emanuel Christian Savin, lasi, Angel Jorba Calvin Australian Mathematical Society Francisco Marques Alan John DeVries Romania Grant Cairns Giuseppe Scollo, Univ of Grau Miquel Brenda L Genet Twente, Enschede, Leslie Stephen Jennings Marc Noy Lisa A Kamps Peter G Taylor Netherlands Carles Rafels i Pallarola Central Michigan University Chun Li Shen, East China Dansk Matematisk Forening Carles Romero Chesa Timothy L Allen Normal Univ, Shanghai, Henning Haahr Andersen Jose L Ruiz Peter A Carlson People's Republic of China Jesper M Moller Sanjay Chawla Feodor Smirnov, Steklov Svenska Matematikersamfundet Institute of Technology, Deutsche Mathematiker- Lars Hakan Eliasson Alecia Teres Devantier Leningrad, USSR Vereinigung e. V. Tony G Elmroth Jennifer M Dollar Andrzej Sosnowski, Technical Hans Fiillmer Union Matemdtica Argentina Andrew J McFarland Univ of Warsaw, Poland W Haack Pablo Miguel Jacovkis Bradley S Thomas Laszlo Szekelyhidi, Debrecen, B Heinrich Matzat Unione Matematica ltaliana City College (CUNY) Hungary Hans E Porst Grace 0 Dike Szyszkowski, Marie Margherita Galbiati lreneusz Helmut Spath Elizabeth Finlay Curie-Sklodowska Univ, Mimmo Iannelli Michelle Volker Strehl Carlos A Lopez Lublin, Poland Wiskundig Genootschap Dietmar Vogt Wei-Jian Mei Dimitris Terzakis, TEl of Jan Karel Lenstra Edinburgh Mathematical Society Deborah H Seidner Izakloiu, Crete, Greece Eduard J N Looijenga Michael Tkacenko, Kalinin, Kenneth J Brown Kostas Stroumbakis Peter Stevenhagen USSR Jack Carr Dora P Trujillo J A Van Gelderen V Totik, Szeged, Hungary Gesellschaft fiir Angewandte Cleveland State University H Treder, Babelsberg, German Erik A van Doorn Mathematics und Mechanik Zing Chen Democratic Republic 6sterreichische Mathematische Edwin J Kreuzer Joseph M Ellis Marius Tucxnak, National Gesellschafl Roy W Gardiner Institute of Science & Iranian Mathematical Society Peter Schmitt Technology, Bucharest, Ali Vahidian Kamyad Edward A Kamposek Romania Israel Mathematical Union John Scott Ryan Villeneuve de Pierre Villalongue, Eli L Turkel Colgate University Ia Raho, France Korean Mathematical Society Michael T Houser David T Wang, New Jersey NOMINEE MEMBERS Institute of Technology, Kee-Young Shin College of William f3 Mary Newark Byung Keun Sohn Adelphi University Xiong Gao Purushothaman Masilamani Yu Wang, Shanghai Institute of London Mathematical Society Beth Goldbach Quarles Education, People's Republic Angelos S Pouleas Richard D Ambrose Xueli Zou of China Auburn University Concordia University Ann A Watkins, New Milford, David J H Garling Robert Lee Russell Da Yuan NJ Trevor 0 Hawkes Hong Elisabeth Werner, Case Western James L Hindmarsh Baylor University Eastern New Mexico University Reserve Univ, Cleveland, E A Whelan Walden Henry Lewis George L Baldwin Ohio Mathematical Society of Japan Hongmin Lu Florida State University Soemantri Widagdo, Fort Lee, Ryutaro Horiuchi Ronald B McCright Amelia E Rogers NJ James Woodrow Miller Hajime Ishihara George Washington University Micha Wolfshtein, Technion- Colleen H O'Connor Norio Iwase William P Miller Israel Institute of Shelle A Palaski Technology, Haifa Kazuo Kido Diane Marie Pappert Dae Yeon Won, Albany, CA Toshihisa Kumazawa Boise State University Sita Ramamurti Hiroo Naitoh Kai Tai Chan

510 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY New Members of the AMS

Gary Keith Schwartz North Carolina State University Ehab Bassily B J 0 Kahn Umamaheshwari S Albert K Germany Bruce L Bauslaugh Sudha Krishnaswami Shivapuram Walter Cyrus McCarter Ying Cai Michelle A Northsheild Goucher College Martin J O'Connell Gemai Chen Jeong Eun Park Karen D Cohen North Georgia College Wen-jun Chen Texas Tech University Lara J Petze Jami Lynn Boyle Xiu-Qing Chen Jeffry M Baker Kossi Delali Edoh Hobart €4 William Smith College Laura E Chadwick Stephen I Black He-Zhi Fan D Vincent Cassano Stephen Edward Stubbs Kent W Easley Qiang Lan Majed Omar El-Qasas Idaho State University Northeastern University Barry Lee Randolph P Flowe Scott Stephen Searcy Zhendi Yan Jiashun Liu Tralissa F Griffin Northwestern University Indiana University at Lixin Liu Arauind Kancherla Anne Marie Brett Bloomington Yun Yun Liu Brian George Kreeger Nikolaos A Askitas Occidental College Casey McConill Chunhsin Li Donald K Berry Brian Winthrop Adams Yuhe Song Zhaojue Li Victoria Bush Danial D Daley Aluysiyus Sutjijana Hongwei Liu George G Flexman Brooke Mitchell Jun Wu Xuehua Lu Anthony Gonsiorowski Ohio University South Dakota State University Herman Lukito Nahm Woo Hahm Theresa A Marner Scott E Black B Neelakantan Fuqing Huang Raza M Syed Lois M Buzzard Danald G Ryan William H Johnson Oklahoma State University, Joe A Christensen Desiree Sage Gie-Whan Kim Stillwater James C Finney Dharshana Weerasinghe Raymond Chiu-Lun Lai Tamera C Hamby JongWoo Joo Lihua Xie Ih-Ren Lan Ravinder Kumar Steven Young Peihuan Meng Pennsylvania State University, University Park Jeri V Kurtz Kesheng Yu Frank E Nyikos Jr Yue Ye David Edward Meel Qiu Shi Zheng John P Saxon Southern Illinois U, Carbondale Xian Zhu Thomas David Smotzer Portland State University Robert W Anderson Ying Zhu Mark B Sokolich Al-Moez Iqbal Alimohamed Virginia J Appuhn University de Montreal Thaddeus Tarpey Michael M Ball Abdelrahim M Barham Fram;ois F F Filion Wann Roslini Wan Yusoff Alexis Dimitriadis Maria Luz Gamier Bay Hui-Hsien Wang Sandra K Kralovec University of California, Irvine Philippe J Bossaers Tony A Werckman John R Neil DanS Cheng Lisa Carol Gariepy Mei-Qin Zhan Rhonda Anne Nordstrom Jared E Derksen Tomas M Shuell Cnythia Dawn Haack David A Short Lake Forest College Bernadette M Hare Richard T Weber Timothy P Ellsworth Purdue University, Calumet Campus Min He University of California, Los Marshall University Fung-Kwan Carmen Lam Marcia A Karnes Angeles Suzeth A Bond Haiping Lin John E Miller Christopher K Hall Queen's University Munir Mahmood Helene Berard University of Cincinnati McMaster University Mark L Miller Lingbo Cao Yvonne M Goetz Boris Brauckmann Jack Mummert Debbie J Dupuis Andrew W Jergens Ping Shing Chan Niko S Nikolaidis Sarah H Franklin MiLin Suyun Chen Craig F Pedersen Wenjiang Fu Yinghui Liu Spiro Paul Daoussis Lisa G Pemberton Ma Gertrudes Paraiso Guhit Wancang Ma John C M Doma Theresa A Pohlmann Perry Y C Lee Charles Alan Mueller Sean M Milosevic-Hill Stephen L Powers Ke Ning Albert H Tyson Malibo Don Lepati William C Schuh Gary R Miller Sei-Qwon Oh Jalaja Varadan Desmond Sheeran Carlos Parra Luqi Wang Sugilar Lawrence W Stark Fenguo Peng Wei Tang Gerald T Stewart Michigan State University Wojciech Szmanski Mingguang Yin Qian Wu Jongsig Bae Robin H Vandivier Paul Eric Becker Rensselaer Polytech Inst Feng Zheng Christopher J Progler Zhangyong Wan Ismail Kocayusufoglu Southwest Texas State University Ping Wang SUNY at Binghamton Jeanne E Kropp Stephen A Hassler Yanhui Zhang Natasa Jonoska Ping Zhang Syracuse University Suzanne X Zheng SUNY at Stony Brook Mississippi State University Eric Grossman University of Delaware Carlos E Duran Anuradha Vishwanathan May F Hamdan Maria B Grosso Anthony J Rossini Murray State University Shou-Chang Lee Chi Lap Ly SUNY, College at Brockport Colleen K McGraw Glen D Arterburn University of Florida Paul F Montanaro Meng-Shiou Shieh Kenneth Lynn McReynolds Denise M Farnan Xiaobin Yuan Lisa H Swacker San Jose State University James E Keesling Jr New Mexico State University, Las Bernadette E Moise Temple University Penelope Ann Kirby Cruces Sarah F Wada Charvi H Bakshi Zhiqiang Liu Benedict Kuplah Nmah Simon Fraser University Clifford A Johnston Jay A Rosen

APRIL 1990, VOLUME 37, NUMBER 4 511 New Members ofthe AMS

Gregory J Smith University of N Cat Chapel Hill Kevin L Coskey Nancy Ann Neudauer University of Hartford Bruce G Bridgman Jeffrey E Eldridge Carlos Eduardo Ortiz Laura E Nolte James Guszcza Hany M Farag Semra Oztiirk Abraham Benjamin Ginsberg David G Radcliffe University of Illinois at University of North Dakota, Grand Chris Hillman Stephen E Schwalm Urbana-Champaign Forks David A Hubbard Hichem Sellami Eric E Hjelmfelt Jay H Achcubach Birgit Jander Huyun Sha Christopher Lee Oliver Karen L Bednarz Fred Craig Kuczmarski Neeza Thandi Kyungmee Park Tina R Melby Christina H Lamont Robert J Tickle Chiung-Hui Tseng Kee K Tan Stephen Wick Jeffrey Cheong Kee Lim Jose Fernando Toledo University of Manitoba Robert J Sterioff Henri V Tran University of Notre Dame Jianwei Li William T Webber Heiko Von Der Mosel Padmini Kamath University of Maryland Baltimore Dave Zick John Calvin Williams University of Oregon County of Wise, Madison Dag E Wold George C Palen University Kalyanbrata Ghosh Jack Asavanant University of Wise, Milwaukee University of Maryland, College University of Rhode Island Mohamed Naceur Azaiez Nikolaos Kodogianidis Park Richard C DeVault Gregory Patrick Bevis Nihal Jagath Siriwardana Tony N Eleftherakis Donna A Passman Dresden Barney Douglas Ward University of Massachusetts, University of Rochester Chunhui Chen Villanova University Amherst Songjie Ren Shaun Cooper Gregory B Burton Keith Hartt University of Southern California Zhan Deng Virginia Commonwealth University of Michigan Bhoomaiah Alishetti Bradbury Franklin Jr University Douglas J Shaw John EBanks Christopher Gomes Kathryn D Harden Ramani Natarajan Jeffrey David Horn University of Minnesota- William W Miles Semerano Chung-Yan Hung Minneapolis Antonella Steven H Mills Shamit S Shah Kenneth Jewell Rui Antonio Loja Fernandes Kathryn A Wallo Mark A Welle Daniel Juan-Pineda Wesleyan University University of Missouri, Columbia Shuguang Xu Scott N Kersey Qingshou Kong Marilyn M Simmons Sang Dong Kim University of Texas at Austin Oscar E Masaveu University of Missouri, Rolla Chang Ock Lee John Michael Adams Sakthivel Shanmugam Shaohsin Chen Jongwoo Lee Alex Kolesnik Tamirisa Linda Marie Fitzgerald Ki-Suk Lee Uma Alison A Miller Allan C deCamp Callie Jo Harmon Michael J Mossinghoff Steven D Leonhardi York University Robert M Jackson Vi K Nguyen Alan L Letarte Claire Kari David Bryan Porter PaulK Swets Sysh-Cherng Liang David T Richardson Lin-Tian Luh Elliot M Pearl University of Virginia Rickey W Richardson Liliane A Maia Mariusz Rabus Pedraic A Murphy Elizabeth Anne Wood Judith J McDonald James Wang Kathy Jean Yuille University of Washington Stephen Paul Mellendorf Xu-Jiong Weng Susan Elizabeth Abe University of Missouri, St Louis Aaron G Montgomery Timothy Mark Chinowsky Mark S Fisher Seong-~yun Nam

Some Mathematical Questions in Biology: I . MODELS IN POPULATION BIOLOGY Alan Hastings, Editor difference equations form a common theme. The topics (Lectures on Mathematics in the Life Sciences, Volume 20) covered are cultural evolution, multilocus population genetics, spatially structured population genetics, chaos and the and the dynamics of ecological Population biology has had a long history of mathematical dynamics of epidemics, modeling. The 1920s and 1930s saw major strides with communities. the work of Latka and Volterra in ecology and Fisher, Haldane, and Wright in genetics. In recent years, much more 1980 Mathematics Subject Classifications: 92A 10, 92A 15 techniques have been brought ISBN 0-8218-1170-3, LC 89-15119 sophisticated mathematical ISSN 0075-8523 to bear on questions in population biology. Simultaneously, 136 pages ( softcover), September 1989 advances in experimental and field work have produced a Individual member $17, List price $28, wealth of new data. While this growth has tended to fragment Institutional member $22 the field, one unifying theme is that similar mathematical To order, please specify LLSCI/20NA questions arise in a range of biological contexts. All prices subject to change. Shipment will be made by surface. This volume contains the proceedings of a symposium on For air delivery add, 1st book $5, each additional book $3, max­ Some Mathematical Questions in Biology, held in Chicago in imum $100. Prepayment required. Order from American Math­ 1987. The papers all deal with different aspects of population ematical Society, P.O. Box 1571, Annex Station, Providence, RI biology, but there are overlaps in the mathematical techniques 02901-1571, or call toll free 800-321-4AMS (321-4267) in the used; for example, dynamics of nonlinear differential and U.S. and Canada to charge with VISA or MasterCard.

512 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY Classified Advertisements

SUGGESTED USES for classified advertising are positions available, books or lecture notes ILLINOIS for sale, books being sought, exchange or rental of houses, and typing services. SOUTHERN ILLINOIS UNIVERSITY THE 1990 RATE IS $50.00 per inch on a single column (one-inch minimum), calculated at EDWARDSVILLE from the top of the type; $22 for each additional Ifz inch or fraction thereof. No discounts Mathematics and Statistics for multiple ads or the same ad in consecutive issues. For an additional $10 charge, Edwardsville, IL 62026-1653 announcements can be placed anonymously. Correspondence will be forwarded. Advertisements in the "Positions Available" classified section will be set with a minimum SlUE, a state university 20 miles from down­ one-line headline, consisting of the institution name above body copy, unless additional town St. Louis, MO., a major cultural and headline copy is specified by the advertiser. Advertisements in other sections of the classified educational center, invites applications for pages will be set according to the advertisement insertion. Headlines will be centered in one tenure-track and one visiting position boldface at no extra charge. Classified rates are calculated from top of type in headline to with rank open beginning Sept. 1990. Only bottom of type in body copy, including lines and spaces within. Any fractional text will be applicants who have a doctorate, or equivalent charged at the next I!z inch rate. Ads will appear in the language in which they are submitted. experience, or will complete Ph.D. require­ Prepayment is required of individuals but not of institutions. There are no member ments by Sept. 1, 1990 will be considered. discounts for classified ads. Dictation over the telephone will not be accepted for classified We seek applicants with excellent research advertising. accomplishments/potential and a strong com­ DEADLINES are listed on the inside front cover or may be obtained from the AMS mitment to teaching. Salary is competitive Advertising Department. and based on qualifications and experience. U. S. LAWS PROHIBIT discrimination in employment on the basis of color, age, sex, race, Direct inquiries to Search Committee Depart­ religion or national origin. "Positions Available" advertisements from institutions outside the ment of Mathematics and Statistics, SlUE, U. S. cannot be published unless they are accompanied by a statement that the institution Edwardsville, IL 62026-1653. SlUE is an does not discriminate on these grounds whether or not it is subject to U. S. laws. Details AA/EEO employer. and specific wording may be found near the Classified Advertisements in the February and July/August issues of the Notices. SITUATIONS WANTED ADVERTISEMENTS from involuntarily unemployed mathematicians are accepted under certain conditions for free publication. Call toll-free 800-321-4AMS MASSACHUSETTS (321-4267) in the U.S. and Canada for further information. SEND AD AND CHECK TO: Advertising Department, AMS, P.O. Box 6248, Providence, BOSTON UNIVERSITY Rhode Island 02940. AMS location for express delivery packages is 201 Charles Street, Mathematics Providence, Rhode Island 02904. Individuals are requested to pay in advance, institutions are not required to do so. AMS FAX 401-331-3842. Mathematics, Assistant Professor, Ph.D. or Ed.D. required. Boston University College of Basic Studies is instituting a new general education mathematics program for fresh­ POSITIONS AVAILABLE STAFFING at the above address. men; tenure-track teaching positions available UC Irvine is an Affirmative Action/Equal September, 1990. Salary negotiable. Deadline Opportunity Employer. for applications 20 May 1990 but early appli­ CALIFORNIA cations are encouraged and will be acted on as soon as all recommendations are received. UNIVERSITY OF CALIFORNIA COLORADO IRVINE Send resume and three recommendations to: Department of Mathematics UNITED STATES AIR FORCE ACADEMY Professor William E. Davis, Jr., Chairman, Irvine, CA 92717 Department of Mathematical Division of Science and Mathematics, Col­ Sciences lege of Basic Studies, Boston University, 871 The University of California at Irvine has made Visiting Professor Commonwealth Avenue, Boston, MA 02215. a significant commitment for the development An Equal Opportunity, Affirmative Action Em­ of the mathematics department. A number of The Department of mathematical Sciences of ployer. faculty appointments will be made over the the United States Air Force Academy invites next few years. These include: nominations and applications for a Visiting 1. At least five full time positions at any Professor position. We seek a Professor with of the tenured professorial levels. The De­ extensive experience teaching undergraduate partment is particularly interested in areas mathematics, statistics or operations research SMITH COLLEGE of Algebra, Analysis, Applied Mathematics, and a strong record of scholarly activity. Du­ Assistant Professor Geometry, Mathematical Physics, Numeri­ ties will include reviewing our academic pro­ Mathematics Department cal Analysis-Scientific Computing, Probability, grams, teaching undergraduate courses and and Topology. Selection will be based on promoting our research programs. Applicants Smith College will have a one-year visiting po­ research experience and teaching ability. should have a demonstrated commitment to sition in mathematics for 1990-91 at the rank 2. At least one full time tenure track As­ undergraduate research and education. The of entry-level Assistant Professor. Teaching sistant Professor position. Candidates must appointment is usually for one year and will load will be five courses. Ph.D. preferred. have a Ph.D. and a research record either in begin in July 1991. Inquiries are welcome for Applications will be accepted until the posi­ Mathematical Physics or Differential Geome­ Visiting Professor positions for subsequent tion is filled. Please send vitae and letters try. years. Salary is commensurate with qualifi­ addressing both teaching and research to: Applications must include curriculum vitae, cations. To apply, please send nominations Department of Mathematics Search Commit­ bibliography and three letters of reference. (to include resume and references) by 1 May tee, Clark Science Center, Smith College, The COMMITTEE ON STAFFING will solicit 1990 to: Chairman, Department of Mathe­ Northampton, MA 01063. Smith College is an supplementary letters of reference. Please matical Sciences, United States Air Force EOE/AA Employer. Minorities and women are send applications to the COMMITTEE ON Academy, CO 80840-5701. encouraged to apply.

APRIL 1990, VOLUME 37, NUMBER 4 513 Classified Advertisements

WORCESTER POLYTECHNIC INSTITUTE mathematics and, desirably, basic computer NEW YORK science. Responsibilities will also include con­ The Department of Mathematical Sciences POTSDAM COLLEGE tinuing research, scholarly activities, and di­ will have several tenure track positions at Chair, Department of Computer and visional service. Salary commensurate with all levels for fall of 1990. These positions Information Sciences qualifications. Screening will begin March 1, require a strong research record or potential 1990. and evidence of quality teaching. Fields of Applications are sought for the position of Send letter of application with curriculum interest are numerical analysis, computational Chair of the Department of Computer and vita and letters of recommendation to Dr. fluid mechanics, nonlinear PDE, optimiza­ Information Sciences at Potsdam College of Thomas J. Thomas; Chairman of Division tion, control theory, optimal design, dynam­ the State University of New York. of Mathematics and Science; Notre Dame ical systems, applied discrete mathematics, The Department is one of seventeen in the College; St. John's University; 300 Howard operations research, and statistics/applied School of Liberal Studies, the largest of the Avenue; Staten Island, N.Y. 10301. St. John's probability. three schools of Potsdam College. Currently is EO/AA employer. WPI, the nation's third oldest college of sci­ there are 10 full time faculty and 200 majors ence and engineering, offers degrees through in the Department. the Ph.D. The Mathematical Sciences De­ Preference will be given to individuals pos­ partment currently offers an undergraduate sessing a Ph.D. in Computer Science though and master's degree in applied mathemat­ applicants with Ph.D.'s in closely related fields OKLAHOMA ics. Worcester, Massachusetts is the second and substantial graduate preparation in com­ THE UNIVERSITY OF OKLAHOMA largest city in New England, approximately 40 puter science (at least a Master's degree) will Department of Mathematics miles west of Boston. also be considered. Some industrial experi­ Interested applicants should send a cur­ ence would be welcome. Rank is negotiable, Applications are invited for one or more riculum vita to: Samuel M. Rankin, Ill, Head, salary and fringe benefits are very compet­ positions at the Assistant Professor level Department of Mathematical Sciences, 100 itive. Responsibilities are for the academic (or higher) in Mathematics beginning Fall Institute Rd., Worcester, MA 01609. Applica­ year. 1990. Candidates must have a Ph.D. de­ tions will be accepted until the positions are The successful candidate should have a gree, demonstrated excellence in research, filled. EOE/ AA. clear vision of an undergraduate computer and potential for high-quality teaching. Strong science program appropriate to a selective candidates in all areas will be considered, liberal arts college whose primary mission is with preference given to research interests teaching and should possess the leadership compatible with those of our current faculty. and administrative skills needed to make Duties include research, normally teaching six NEVADA this vision a reality. A strong record of credit hours per semester, and Departmental UNIVERSITY OF NEVADA successful undergraduate teaching and active and University service appropriate to rank. Tenured Professor and Chairman in scholarship are expected. Salary and rank will be commensurate with Mathematical Sciences Applicants should provide a letter dis­ qualifications and experience. There may also cussing how their education and background be visiting positions. Applicants should send The Department of Mathematical Sciences at have prepared them to fulfill the responsibil­ their vita and have at least three letters of ref­ the University of Nevada, Las Vegas has a va­ ities of this position as described, a current erence sent to Dr. Ruediger Landes, Search cancy for a tenured professor and chairman. resume, the names, addresses and telephone Committee Chair, Department of Mathemat­ Responsibilities include all aspects of de­ numbers of three to five references to ics, University of Oklahoma, 601 Elm, Room partmental administration, teaching, research Dr. Richard J. Del Guidice 423, Norman, Oklahoma 73019-0315. Initial and development, to include strengthening Dean screening begins December 15, 1989 and both undergraduate and graduate programs. School of Liberal Studies every two weeks thereafter. Applications will Qualifications include a Ph.D. in Mathemat­ Potsdam College be accepted until the position(s) are filled. ics, statistics or related areas as well as a Potsdam, N.Y. 13676 The University of Oklahoma is an Affirmative successful record in teaching and research Deliberations will begin 15 April and continue Action/Equal Opportunity Employer. are required. Past experience, or a strong until the position is filled. Potsdam College interest in administration is necessary. Salary actively seeks applications from women and will be commensurate with the candidate's minority candidates. AA/EOE. experience and qualifications. THE UNIVERSITY OF OKLAHOMA The University of Nevada, Las Vegas is a Applied Non-Linear Analysis growing, urban university with an enrollment Applied Mathematics of approximately 16,000 students located in Position (CAPS) Related metropolitan Las Vegas. The University will be housing a supercomputer center. The University of Oklahoma seeks a (tenure Interested applicants should submit a letter track) Assistant Professor (or higher) with a Non-Linear Analysis. A of application, current resume, photocopies ST. JOHN'S UNIVERSITY speciality in Applied of transcripts, and five letters of reference to NOTRE DAME COLLEGE Ph.D. in Mathematics is required. Expertise the Search Committee for Chairman, Depart­ Division of Mathematics and Science in fluid dynamics and numerical and com­ ment of Mathematical Sciences, University putational experience are desirable. Potential of Nevada, Las Vegas, Las Vegas, Nevada Applications are invited for a tenure-track for excellence in mathematics teaching and 89154. Completed applications will be re­ position at the Assistant/ Associate Professor research is required. Competitive Salary. viewed when received. The search will con­ level beginning September 1990. This position is expected to contribute to tinue until the position is filled. Eligibility for Applicants must hold the Ph.D. degree the mathematical support of the Center for the U.S. Employment will be required prior to in Mathematics and will be expected to Analysis and Prediction of Storms, A Science employment. UNLV is an EEO/AA employer. teach a variety of undergraduate courses in and Technology Center at the University of

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Oklahoma funded by the National Science Salary per Board of Trustee approved salary Desired background: Unix, C, hyperbolic ge­ Foundation. schedule. Starting date: September 12, 1990. ometry, (combinatorial) group theory, basic Applicants should send a letter of ap­ A letter of application, resume, college tran­ computer science. plication, a complete vita, and have three scripts (unofficial), and three current relevant Salary on the RA 1A scale: £10458-£16665 letters of reference sent to: Andy R. Magid, letters of recommendation must be received pa. Chair, Applied Analysis Search Committee, by 4:30pm, May 7, 1990. All application mate­ Application forms and further particulars Department of Mathematics, University of rials become the property of Olympic College. from the Personnel Office, University of War­ Oklahoma, 601 Elm, Room 423, Norman, Apply: Bernice Vonnegut, Olympic College wick, Coventry CV4 7AL, England (0203 Oklahoma 73019-0315. Closing date for ap­ Personnel, 16th and Chester, Bremerton, WA 523627). Closing date 22 May 1990. Please plications is December 20, 1989 and every 98310-1699. (206) 478-4980. Equal Opportu­ quote Ref No 30/3A/89 (please mark clearly two weeks thereafter until the position is nity Employer. on envelope). filled. The University of Oklahoma is an Equal AN EQUAL OPPORTUNITIES EMPLOYER Opportunity 1Affirmative Action Employer. GREECE

TEXAS UNIVERSITY OF CRETE, GREECE Department of Mathematics LAMAR UNIVERSITY Beaumont, Texas Permanent and visiting positions at all lev­ els and areas of Mathematics are available. The Department of Mathematics seeks appli­ Expressions of interest are invited from math­ PUBLICATIONS WANTED cations for a tenure-track Assistant/ Associate ematicians holding the Ph.D. degree. (Greek Professor position beginning Fall 1990. Appli­ citizenship is required by law, for the per­ cants must hold an earned Ph.D., should be manent positions. Knowledge of Greek is Wanted: Mathematical books, journals, re­ active researchers in analysis andfor applied required in both cases, except for a very prints, ephemera. Contact R. K. Dennis, mathematics, and have a strong commitment restricted number of visiting positions.) Write Math. Dept., White Hall, Cornell U., Ithaca, to teaching. to Prof. Susanna Papadopoulou, Chairper­ NY 14853-7901. Tel: 607-255-4027, FAX: 607- Lamar is a state supported educational and son, Mathematics Department, University of 255-7149. e-mail: research institution of approximately 12,000 Crete, lraklion, Greece, including vita and [email protected] students. The Department, which is located representative research work. within the College of Engineering, offers the B.A., B.S., and M.S. degrees in Mathematics and has 16 full-time faculty. For the approximately 250,000 people in TAIWAN the Beaumont area, there are eight museums, NATIONAL CHUNG-HSING UNIVERSITY a symphony orchestra, ballet, opera, and Department of Applied Mathematics other theatrical productions. Other facilities Taichung, Taiwan and events are found in Houston 85 miles SITUATIONS WANTED west. Nearby outdoor recreational opportu­ The Department of Applied Mathematics nities include two of the largest freshwater seeks applicants for 5-6 positions at Pro­ TEACHING MATHEMATICIAN. PH.D lakes in Texas, the Big Thicket National fessor or Associate professor level in numer­ 1982. AGE 38 Preserve, four national forests, the Gulf of ical analysis, differential equations, computer Mexico, and a large inland saltwater lake. science, statistics or applied mechanics. Com­ Speciality: Differential Equations. 5 published Salary and rank are commensurate with petitive salary with good benefits and low tax articles. 15 years experience in teaching and qualifications and experience. Send a resume rate. The department has a Ph.D. program. research in U.S.A. and abroad. Excellent and three letters of recommendation to: Dr. Write to: Dr. L. Y. Lin, Head, Department of students and peer evaluations of classes. John R. Cannon, Chair; Department of Math­ Applied Mathematics, National Chung-Hsing References and resume available upon re­ ematics; Lamar University; P.O. Box 10047; University, Taichung, Taiwan. quest. Available immediately. M. Abbassi, 500 Beaumont, TX 77710. Full consideration will Tulip Rd., State College, PA 16801 be given to applications received before May 1, 1990. Lamar is an EEO/AA employer. UNITED KINGDOM

UNIVERSITY OF WARWICK WASHINGTON Research Fellow Mathematics Institute OLYMPIC COLLEGE Mathematics Applications are invited for one or more SERC funded posts, starting 1 October 1990, to work ERRATA Teaching faculty member, full-time, tenure­ with the Geometry Group led by Professor D track position. Provide instruction in Math­ B A Epstein. The Geometry Group consists of In the Positions Available section in the March ematics, assist in curriculum planning and pure mathematicians and computer scientists issue of Notices, page 359, the application development and student advising. May in­ at Warwick University, and is associated with closing date for the position advertised by clude evening instruction. A Master's Degree the NSF supported Geometry Supercomputer the University of Texas at San Antonio was in Mathematics or Mathematics Education is Project based in Minneapolis. Its objectives incorrectly published as March 15, 1990. The required and successful teaching of Math­ are to use computers to understand phenom­ correct closing date should have been April ematics at a community college preferred. ena in geometry, mainly hyperbolic geometry. 1, 1990.

APRIL 1990, VOLUME 37, NUMBER 4 515 At NSA,We Use Simple Tools To Solve Complicated Problems. Oddly enough, the one thing most mathematicians de- agency charged with providing foreign intelligence infor- sire from a career is the one thing most careers can't offer them. mation, safeguarding government communications, and The chance to use pure math in a real-world setting. securing computer systems for the Department of Defense. At the National Security Agency, we're different. We So when you're ready to apply some of what you've hire the best mathematicians, then let them do what they learned, apply to the do best. This includes creating and applying a host of National Security Agency advanced concepts-from Galois theory and combinatorics first. Send your resume to National to probability theory and astrodynamics. the address below, or con- Security Of course, the rewards don't come just when the prob- tact your campus place- Agency lem's solved. By working for NSA, you're joining a federal ment office. The Opportunities of a Lifetime ATTN: M322 (ACF), Ft. Meade, MD 20755-6000 An equal opportunity employer. U.S. dtizenship required for applicant and immediate family members. Topology of 4-Manifolds Michael Freedman and Frank Quinn One of the great achievements of contemporary mathematics is the new understanding of four dimensions. Michael Freedman and Frank Quinn have been the principals in the geometric and topo­ logical development of this subject, proving the Poincare and An­ nulus conjectures respectively. Recognition for this work includes the award of the Fields Medal of the International Congress of Mathematicians to Freedman in 1986. In Topology of 4-Manifolds these authors have collaborated to give a complete and accessible account of the current state of knowledge in this field. The basic material has been considerably simplified from the original publications. The advanced material goes well beyond the literature; nearly one-third of the book is new. This work is indispensable for any topologist whose work in­ cludes four dimensions. Princeton Mathematical Series, 39 79 illustrations. Analytic Pseudodifferential Cloth: $49.50 ISBN 0-691-08577-3 Operators for the Heisenberg In Japan order from United Publishers Services Group and Local Solvability Spin Geometry Daryl Geller Many of the operators one meets in several complex variables, H. Blaine Lawson, }r., and Marie-Louise such as the famous lewy operator, are not locally solvable. Never­ Michelsohn theless, such an operator L can be thoroughly studied if one can This book offers a systematic and comprehensive presentation of find a suitable relative parametrix-an operator K such that LK is the concepts of a spin manifold, spinor fields, Dirac operators, essentially the orthogonal projection onto the range of L. The anal­ and A-genera, which, over the last two decades, have come to ysis is by far most decisive if one is able to work in the real ana­ play a significant role in many areas of modern mathematics. lytic, as opposed to the smooth, setting. With this motivation, the Since the deeper applications of these ideas require various gen­ author develops an analytic calculus for the Heisenberg group. eral forms of the Atiyah-Singer Index Theorem, the theorems and Mathematical Notes, 37 their proofs, together with all prerequisite material, are examined Paper: $29.50 ISBN 0-691-08564-1 here in detail. The exposition is richly embroidered with examples and applications to a wide spectrum of problems in differential ge­ ometry, topology, and mathematical physics. Instabilities and Fronts Princeton Mathematical Series, 38 in Extended Systems Cloth: $55.00 ISBN 0-691-08S42-0 In Japan order from United Publishers Services Pierre Collet and Jean-Pierre Eckmann The book examines in detail a number of model equations from physics. The mathematical developments of the subject are based Casson's Invariant for on bifurcation theory and on the theory of invariant manifolds. These are combined to give a coherent description of several prob­ Oriented Homology lems in which instabilities occur, notably the Eckhaus instability 3-Spheres and the formation of fronts in the Swift-Hohenberg equation. These phenomena can appear only in infinite systems, and this An Exposition book breaks new ground as a systematic account of the mathemat­ Selman Akbulut and john D. McCarthy ics connected with infinite space domains. In the spring of 1985, A. Casson announced an interesting invar­ Princeton Series in Physics iant of homology 3-spheres via constructions on representation Philip W. Anderson, ArthurS. Wightman, and Sam B. Treiman, Editors spaces. This invariant generalizes the Rohlin invariant and gives Cloth: $29.50 ISBN 0-691-08568-4 surprising corollaries in low-dimensional topology. In the fall of that same year, Selman Akbulut and John McCarthy held a semi­ nar on this invariant. These notes grew out of that seminar. The authors have tried to remain close to Casson's original out­ line and proceed by giving needed details, including an exposi­ tion of Newstead's results. They have often chosen classical con­ crete approaches over general methods. For example, they did not AT YOUR BOOKSTORE OR attempt to give gauge theory explanations for the results of Newstead; instead they followed his original techniques. Princeton University Press Mathematical Notes, 36 I 41 WILLIAM ST. • PRINCETON. NJ 08540 • (6fn) 258-4900 Paper $19.50 ISBN 0-691-08S63-3 ORDERS 800-PRS-ISBN (n7-4726) ...----Cambridge University Press-----. The Ergodic Theory of Discrete Groups Numbers, Groups and Codes P.J. Nicllolls J.F. Humphreys ancl M. Y. Prest This volume aims to provide the novice with a rigorous introduction from This textbook aims to introduce the concepts of algebra through many first principles to some of the major aspects of modern ergodic theory. specific examples, and to relate them to their applications, particularly London Mathematical Society Lecture Note Series 143 in computer science. 1990 I 350 pp. I 37674-2 I Paper $34.50 19891288 pp. 135084-0 I Hardcover $59.50 35938-41 Paper $19.95 Analysis at Urbana Volume 1: Analysis in Function Spaces Volume 2: Analysis in Abstract Spaces Lectures on the Asymptotic Theory of Ideals Edited by E. Berkson, T. Peck, and J. Uhl D.Rees London Mathematical Society Lecture Note Series 137 and 138 Introduces and proves some of the main results of the asymptotic theory Vol 1: 19891400pp.l36436-11 Paper$39.50 of ideals. The author attempts to prove his Valuation Theorem, Strong Vol 2: 19891368 pp. 136437-X I Paper $34.50 Valuation Theorem, and Degree Formula, and to develop their consequences. Advances in Homotopy Theory London Mathematical Society Lecture Note Series 113 Papers in Honour of I.M. James, Cortona 1988 19881200pp.l31127-61 Paper$24.95 Edited by S. Salamon, B. Steer, and W. Sutherland The present volume contains papers from internationally distinguished research workers that reflect recent exciting breakthroughs in An Introduction to Mathematical homotopy theory. Physiology and Biology London Mathematical Society Lecture Note Series 139 J.Mazumrlar 19901177 pp.l37907-51 Paper$27.95 Written as an explanation of basic biology and physiology, this textbook presents the important mathematical techniques needed to An Introduction to Harmonic Analysis appreciate mathematical modeling in biology and medicine. on Semisim.Pie Lie Groups Australian Mathematical Society Lecture Series 4 V.S. Varadara1an 19891212 pp.l37002-71 Hardcover$49.50 This advanced-level textbook is an introduction to the representation 37901-61 Paper $16.95 theory of semisimple Lie groups. It provides an account of compact groups and discusses the Harish-Chandra modules of SL2 (R) and SL2 (C). A Course in Mathematics Cambridge Studies In Advanced Mathematics 16 for Students of Physics 19891350 pp. 134156-61 Hardcover $69.50 Volumes 1 and 2 Paul Bamberg and Shlomo Sternberg Solitons: An Introduction Review of Volume One P.B. Drazin and R.S. Johnson ••• "The authors have drawn on their wide teaching experience to produce This successful book discusses the theory of solitons and its diverse a book whose tone is authoritative, whose vision is fresh, and whose applications to nonlinear systems that arise in the physical sciences. Cambridge Texts in AppUed Mathematics 3 whole feel is originai."-The London Times 19891240 pp. I 33389-X I Hardcover $59.50 This two-volume text covers the theory and physical applications of 33655-41 Paper$19.95 linear algebra and of the calculus of several variables, particularly the exterior calculus. Volume Inequalities in the Geometry Vol. 1: 1988 403 pp.I25017-X I Hardcover $49.50 of Banach Spaces Vol. 2: 1990 432 pp. 133245-11 Hardcover about $54.50 The Volume of Convex Bodies and Banach Space Geometry Gilles Pisier Numerical Recipes The author provides a self-contained presentation of a number of recent The Art of Scientific Computing (FORTRAN Version) results which relate the volume of convex bodies in n-dimensional William H. Press, Br1an P. Flannery, Euclidean space and the geometry of the corresponding Saul A. Teukolsky, finite-dimensional normed spaces. and William T. Vetterling Cambridge Tracts in Mathematics 94 The original Numerical Recipes was published in the FORTRAN 19891 250 pp. I 36465-51 Hardcover $49.50 computer language with all the programs translated into Pascal as an extensive appendix. With the new Numerical Recipes in Pascal, the Algebraic Homotopy Pascal index has been removed from the original Numerical Recipes. Hans Joachim Baues Therefore, this book has a new subtitle, FORTRAN Version, and Presenting work that has never before appeared in book form, the has been assigned a new ISBN to reflect the absence of the author describes powerful new tools for the homotopy classification Pascal programs. FORTRAN: 19891720 pp. 138330-71 Hardcover $47.50 problems, particularly for the classification of homotopy types and for Pascal: 19891762 pp. 137516-91 Hardcover $47.50 the computation of the group of homotopy equivalences. C: 19881768 pp. 135465-X I Hardcover $47.50 Cambridge Studies in Advanced Mathematics 15 19891480 pp. 133376-81 Hardcover $89.50 Number Theory and Dynamical Systems Now in paperback ... Edited by M. Dodson and J. Vickers Close and unexpected connections between number theory and dynamical systems emerge from selected contributions from a meeting Ergodic Theory held at the University of York in 1987. Karl Petersen London Mathematical Society Lecture Note Series 134 Cambridge Studies In Advanced Mathematics 19891172 pp.l36919-31 Paper$24.95 19831352 pp. 138997-61 Paper $29.95

At bookstores or from CAMBRIDGE UNIVERSITY PRESS 40 West 20th Street, New York, NY 10011. Call toll-free 800-872-7423 outside NY Stale. 800-227.0247, NY Stale only. MasterCardNISA accepted. Prices subject to change. NATIONAL UNIVERSITY OF SINGAPORE

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Applications are invited for teaching appointments from candidates who are able to teach in one or more of the following areas: Pure Mathematics Applied Mathematics Operational Research Statistics Candidates should possess a PhD degree in Mathematics or its equivalent.

Gross annual emoluments range as follows: Lecturer S$50,390- 64,200 Senior Lecturer S$58,680- 100,310 Associate Professor S$88,650 - 122,870 (US $1.00 = S$1.96 approximately) The commencing salary will depend on the candidate's qualifications, experience, and the level ofappointment offered.

Leave and medical benefits will be provided. Depending on the type of contract offered, other benefits may include: provident fund benefits or an end-of-contractgratuity, a settling-in allowance ofS$1 ,000 or S$2,000, subsidised housing at nominal rentals ranging from S$1 00 to S$216 p.m., education allowance for up to three children subject to a maximum ofS$10,000 per annum per child, passage assistance and baggage allowance for the transportation of personal effects to Singapore. Staff members may undertake consultation work, subject to the approval of the University, and retain consultation fees up to a maximum of60% of their gross annual emoluments in a calendar year.

The Department ofMathematics is a department in the Faculty ofScience. There are 8 faculties in the National University ofSingapore with a current student enrolment of some 14,000. All departments are well-equipped with a wide range of facilities for teaching and research.

All academic staff have access to the following computer and telecommunication resources: an individual microcomputer (an IBM AT-compatible or Apple Macintosh); an IBM mainframe computer with 16 MIPS of computing power; departmental laser printers; a wide spectrum of mainframe and microcomputer software; voice-mail; BITNET to access academic institutions world-wide. In addition, a proposed campus network based on state-of-the-art optical fibre technology will be installed by 1990 to facilitate resource sharing and electronic communication for the academic community.

Application forms and further information on terms and conditions of service may be obtained from: The Director The Director Personnel Department North America Office National University of Singapore National University of Singapore 10 Kent Ridge Crescent 780 Third Avenue, Suite 2403 Singapore 0511 New York, NY 10017, U.S.A. Tel: (212) 751-0331

Enquiries may also be sent through BITNET to: PERSDEPT @ NUSVM, or through Telefu:: (65) 7783948 Deakin University Deakin Geelong Victoria University Australia 3217 Professor of Mathematics

Applications are invited for appointment to the foundation Chair of Mathematics in the Department of Computing and Mathematics within the School of Sciences. The University has grown rapidly since its establishment in 1977 and this presents challenging opportunities for educational innovation and development.

The University is seeking a Professor to provide academic leadership in mathematics and/or statistics, and to play a major role in the expansion of research and graduate studies, in the development of off-campus programs, in the growth of external funding initiatives, and in academic development.

The successful applicant is likely to hold a Ph.D. and will have strong research interests preferably in applied mathematics, statistics or operations research together with a commitment to link theoretical development and industrial application.

The appointment, at a salary of $A65,837 is for a fixed term of five years which may be renewed for further five year terms. Additional establishment benefits to facilitate the Professor's and the Department's research will be negotiated. The University reserves the right to fill the post by invitation or not to make an appointment.

Enquiries of an academic nature should be directed to the Chair of the Department, Dr. I. Collings, telephone (61 52) 471324. Further information about the University, conditions of employment, and method of application may be obtained from the Personnel Manager, Deakin University, Geelong, Victoria, Australia, 3217, with whom applications close on 18 May 1990.

Equal Opportunity Is University Policy. ann31147

USER GUIDE Second Edition This is the second edition of the MathSci User Guide, which provides support and documentation for users of MathSci. the online database covering the world's literature in all areas of mathematics, computer science, and statistics. This handy reference will prove invaluable to the many users who now access MathSci online, from tapes, or on compact disc. The MathSci User Guide provides complete instructions on the use of MathSci. Instructions on getting started, descriptions of aspects of the database, sample searches, information on source publications, explanation of the classification schemes, lists of subject words and keywords, and institution codes and addresses are all in this comprehensive manual. In addition, there is a description of how to use the computer typesetting system TEX to typeset MathSci records directly from the database, and appendices on translitera­ tion of the Cyrillic alphabet and Chinese names, on mnemonics representing mathematical symbols, and on TEX codes used in MathSci. The MathSci database is derived from six printed publications, Mathematical Reviews, Current Mathe­ matical Publications, Current Index to Statistics, Index to Statistics and Probability, Computing Reviews, and the ACM Guide to Computing Literature. Making this broad range of information available, the MathSci User Guide will prove useful to researchers as well as librarians.

1980 Mathematics Subject Classification: 00 All prices subject to change. Shipment will be made by surface. For air delivery ISBN 0-8218-0233-X add, 1st book $5, each additional book $3, maximum $100. Prepayment 624 pages (hardcover), February1990 required. Order from American Mathematical Society. P.O. Box 1571. Annex AMS/ASA/IMS member $68, List price $85 Station, Providence, Rl 02901-1571, or call toll free 800-321-4AMS (321-4267) in To order, please specify USERSGUIDE/NA the U.S. and Canada to charge with VISA or MasterCard. Essential Results Leo Tames TEX With Leo you can write mathematics and see it on your PC's screen as you type. Leo displays of Functional Analysis special symbols, fractions, matrices, square roots, Robert J. Zimmer super and sub scripts, fonts, accents. Matrices, parentheses, braces, etc. automatically adjust to This slim book concisely presents fundamental analysis, providing essential the required size. Leo reads and writes 1E;X files. ideas of functional results from the many domains of mathematics and Print with Leo draft printer, or with your 1E;X mathematical physics to which functional analysis program. Convenient menu and control key oper­ makes contributions. Zimmer summarizes measure ation. Requires IBM PC or compatible with 512k. theory and the elementary theory of Banach and Specify AMS'IE;X or LA'IE;X. $150. MC/VISA or Hilbert spaces, then discusses topological vector check. spaces, seminorms defining them, and natural classes of linear operators. He then presents basic results for a wide range of topics: convexity and Draft printing matches screen display: fixed point theorems, compact operators, compact groups and their representations, spectral theory of bounded operators, ergodic theory, commutative [~:: ~=: l C*-algebras, Fourier transforms, Sobolev embed­ ding theorems, distributions, and elliptic differential operators. ABK Software Paper $14.95 168 pages 4495 Ottawa Place Library cloth edition $34.95 Boulder, CO 80303 Chicago Lectures in Mathematics series (303)-494-4872 ~ The University of Chicago Press ~~ 5801 South Ellis Avenue Chicago, /L 60637

FACULTY POSITION IN MATHEMATICS CHAIRMAN DEPARTMENT OF BIOMATHEMATICAL The Department of Science and Mathematics at SCIENCES Coker College is seeking a mathematician, Ph.D. The Mount Sinai School of Medicine invites required, to teach all levels of calculus, differential applications or nominations for the position of Chairman of the Department of Biomathemat­ equations, and statistics. The ability to teach com­ teal Sciences. We are seeking an energetic, inno­ puter science and/or discrete mathematics is highly vative scientist to develop new strengths andes­ desirable. This is a tenure-track position starting tablish the Department as a leading center in August 1990, but because of the lateness in the year modern Biomathematics. Candidates must have program, a successful record will be for one year with the pos­ an active research this appointment in graduate teaching, administrative experience, siblity of conversion to tenure-track. Rank and and demonstrated leadership abilities. salary commensurate with experience. Coker Col­ The Chairman is expected to strengthen the re­ lege is a private, four-year liberal arts college in search efforts of the Department, enhance grant and Hartsville, SC. Heavily endowed, no courses have support, develop and maintain doctoral postdoctoral educational programs, and foster more than 20 students and many upper level courses collaborations within the Institution. have only three or four students. Independent study Send curriculum vitae, names of 3 references, a courses are encouraged. Mathematics and math statement of research interests and goals for the education are among the majors offered. April 27, Department to: 1990 application deadline. Send letter of applica­ Chainnan, Biomathematical Sciences Search Committee, Box 1102, Mount Sinai School of tion, vita, and names of three references to: Profes­ Medicine, One Gustave L. Levy Place, New York, sor Kaye Crook, Chair, Mathematician Search Com­ NY 10029. mittee, Department of Science and Mathematics, An A.ffinnative Action/ Equal Opportunity Coker College, Hartsville, SC 29550. AAEO Employer. CBMS CONFERENCE BOARD OF THE MATHEMATICAL SCIENCES Issues in Mathematics Education

The AMS is pleased to announce a new book series: The Conference Board of the Mathematical Sciences (CBMS): Issues in Mathematics Education Series published in cooperation with the Mathematical Association of America. The purpose of this new series is to stimulate the flow of information among mathe­ matical scientists, mathematics educators, and mathematics teachers about innovative efforts to revitalize the teaching of mathematics and statistics at al.llevels. The inaugural volume of CBMS Issues in Mathematics Education is described below. Standing orders are accepted for any book series published by the Society. Proforma invoices are sent to standing order customers prior to the publication of each new _volume. Shipment is made upon receipt of payment and publication. To begin a standing order for this new series or for any other AMS series, please contact Customer Services. MATHEMATICIANS AND EDUCATION REFORM Naomi Fisher, Harvey Keynes, and Philip Wagreich, Editors (CBMS Issues in Mathematics Education, Volume 1)

Educational issues are receiving unprecedented attention in the broad mathematical sciences commu­ nity, as mathematicians and other scientists have become concerned about the quality of instruction in the nation's schools, colleges, and universities. A mathematically literate population is crucial to supporting our increasingly technological society. In addition, the mathematical sciences community faces the challenge of increasing the ·number of students who are prepared to pursue a career in mathematics, science, or engineering. This challenge requires not only raising the quality of mathematics education, but also showing students the beauty and useful­ ness of the subject. In these ways, mathematical scientists can make crucial contributions to educational reform. In response to these concerns, the Conference Board of the Mathematical Sciences has launched a new book series published by the American Mathematical Society in cooperation with the Mathematical Association of America entitled Issues in Mathematics Education. The purpose of this new series is to stimulate the flow of infor­ mation among mathematical scientists, mathematics educators, and mathematics teachers about innovative efforts to revitalize the teaching of mathematics and statistics at all levels. The present volume, Mathematicians and Education Reform, the first in this new series, contains the proceedings of the Mathematicians and Education Reform workshop held in July 1988, at the University of Illinois at Chicago. The workshop provided an opportunity for participants to share ideas about the various ongoing pre­ college projects organized and directed by mathematicians and to reflect on the most effective ways that mathe­ maticians can contribute to educational reform. The major part of the proceedings is devoted to in-depth articles that explore the process of designing an educational project. A section on issues and reactions presents a forum for exchanging ideas on more general issues. From practical information about organizing a program to exploration of the intellectual issues of educa­ tional reform, this volume presents a range of views on various aspects of the involvement of mathematicians in educational change. While it will prove especially useful for those considering involvement in an educational program, this book is also important reading for the entire community, for the issues explored here will be of increasing importance for the future of the mathematical sciences. 1980 Mathematics Subject Classification: 00 ISBN 0-8218-3500-9, LC 89-18601, ISSN 1047-398X 217 pages (softcover), January 1990 All Individuals $21, List price $35, To order, please specify CBMATH/1NA

All prices subject to change. Shipment will be made by surface. For air delivery add, 1st book $5, each additional book $3, maximum $100. Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station, Providence, AI 02901-1571, or call toll free 800-321-4AMS (321-4267) in the U.S. and Canada to charge with VISA or MasterCard. Application for Membership 1990

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Fasten PAYMENT securely Computational Aspects of VLSI Design with an Emphasis on Semiconductor Device Si1nulation Randolph E. Bank, Editor (Lectures in Applied Mathematics, Volume 25)

Numerical simulation is rapidly becoming an important part of the VLSI design process, allowing the engineer to test, evaluate, and optimize various aspects of chip design without resorting to the costly and time-consuming process of fabricating prototypes. This procedure not only accelerates the design process, but also improves the end product, since it is economically feasible to numerically simulate many more options than might otherwise be considered. With the enhanced computing power of today's computers, more sophisticated models are now being developed.

This volume contains the proceedings of the AMS-SIAM Summer Seminar on Computational Aspects of VLSI Design, held at the Institute for Mathematics and Its Applications at the University of Minnesota, in the spring of 1987. The seminar featured presentations by some of the top experts working in this area. Their contributions to this volume form an excellent overview of the mathematical and computational problems arising in this area.

1980 Mathematics Subject Classification: 35G, 65L, 65M, 65N ISBN 0-8218-1132-0, LC 89-18355, ISSN 0075-8485 200 pages (hardcover), February 1990 Individual member $32, List price $53, Institutional member $42 To order, please specify LAM/25NA

All prices subject to change. Shipment will be made by surface. For air delivery add, 1st book $5, each additional book $3, maximum $100. Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station, Providence, RI 02901-1571, or call toll free 800-321-4AMS (321-4267) in the U.S. and Canada to charge with VISA or MasterCard. NEW SERIES University Lecture Series

The AMS is pleased to announce the University Lecture Series. This new book series provides a way for excellent, and sometimes inspired, lecture series to reach an audience beyond those able to attend the live lectures. Presented by the outstanding mathematicians of our day, these lectures will be impor­ tant for their mathematical insight and depth, as well as for their historical and archival value. The inaugural volume in the University Lecture Series is described below.

Selected Applications of Geometry to Low-Dimensional Topology Michael H. Freedman and Feng Luo (University Lecture Series, Volume 1) This book, the inaugural volume in the new Univer­ and proceed to the topology and geometry of foliated sity Lecture Series, is based on lectures presented at 3-manifolds. They also explain, in terms of general Pennsylvania State University in February 1987. position, why four-dimensional space has special The Lectures attempt to give a taste of the accom­ attributes, and they examine the insight Donaldson plishments of manifold topology over the last 30 theory brings. The book ends with a chapter on exotic years. By the late 1950s, algebra and topology had structures on R\ with a discussion of the two compet­ produced a successful and beautiful fusion. Geo­ ing theories of four-dimensional manifolds, one topo­ metric methods and insight, now vitally important logical and one smooth. in topology, encompass analytic objects such as instantons and minimal surfaces, as well as .. Background material was added to clarify the discus­ nondifferentiable constructions. _ _ sions in the lectures, and references for more detailed - ... r.. study are included. Suitable for graduate students and Keeping technical details to a minimum, the authors researchers in mathematics and the physical sciences, lead the reader on a fascinating exploration of the book requires only background in undergraduate several developments in geometric topology. They mathematics. It should prove valuable for those begin with the notions of manifold and smooth 1 wishing a not-too-technical introduction to this vital structures and the Gauss-Bonnet theorem, area of current research.

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All prices subject to change. Shipment will be made by surface. For air delivery add, 1st book $5, each additional book $3, 1980 Mathematics Subject Classifications: 57, 53,54 maximum $100. Prepayment required. Order from American ISBN 0-8218-7000-9, LC 89-18287 Mathematical Society, P.O. Box 1571, Annex Station, 79 pages (softcover), December 1989 Providence, RI 02901-1571, or call toll free 800-321-4AMS Individual member $20, List price $33, (321-4267) in the U.S. and Canada to charge with VISA Institutional member $26 or MasterCard. To order, please specify ULECT/1NA COLORING THEORIES Steve Fisk (Contemporary Mathematics, Volume 103)

The focus of this work is the study of global properties of various kinds of colorings and maps of simplicial complexes. In addition to the usual sorts of coloring, the author studies colorings determined by groups, colorings based on regular polyhedra, and continuous colorings in finitely and infinitely many colors. The emphasis is on how all the colorings fit together, rather than on the existence of colorings or the number of colorings. Beginning with some fun­ damental properties of simplicial complexes and colorings, the author shows how colorings relate to various aspects of group theory, geometry, graph theory, and topology.

CONTENTS Properties of the combinatorial category; The symmetric group complex Sn; Complexes arising from geometry; Graphs; Complexes with a structure group; Reflexive and self dual complexes; Continuous colorings; Coloring with arbitrary complexes.

1980 Mathematics Subject Classification: 05 ISBN 0-8218-5109-8 • LC 89-27623 • ISSN 0271-4132 164 pages (softcover) • December 1989 Individual member $19 • List price $31 • Institutional member $25 To order, please specify CONM/103NA

All prices subject to change. Shipment will be made by surface. For air delivery add, 1st book $5, each additional book $3, maximum $100. Prepayment required. Order from American Mathematical Society, P.O. Box 1571, Annex Station, Providence, Rl 02901-1571, or call toll free 800-321-4AMS (321-4267) in the U.S. and Canada to charge with VISA or MasterCard. Summer List of Applicants THEORY OF FUNCTIONS AND Instructions for Applicant Form on facing page RELATED QUESTIONS OF ANALYSIS The form. Forms submitted by job applicants who at­ L. D. Kudryavtsev, Editor tend the August meetings in Columbus will be posted. The first impression a prospective employer has of an (Proceedings of the Steklov Institute, Volume 180) applicant may be based on the appearance of this form. The forms should be carefully typed using a fresh This volume contains the proceedings of the All-Union black ribbon. The best results are obtained with a Conference on the Theory of Functions and Related carbon-coated polyethylene film ribbon, but satisfac­ Questions of Analysis, held in Dnepropetrovsk, U.S.S.R., in tory results may be obtained using a ribbon made of the summer of 1985. The conference was held in honor of nylon or other woven fabric if suitable care is exer­ the 80th birthday of Sergei Mikhallovich Nikol 'sku. cised. It is important that the keys be clean and make a sharp, clear impression. Use a correcting typewriter or correction tape or fluid if necessary. Submit the The collection contains a survey of the work of Nikol'skil original typed version only. Hand lettered forms are as well as papers on the theory of approximation of func­ acceptable if prepared carefully. tions, the metric theory of functions, the theory of spaces of The summary strip. Information provided here will differentiable functions of several variables, and applica­ be used to prepare a printed list of applicants for dis­ tions of these ideas. tribution to employers. Please supply all information requested, and confine your characters to the boxes 1980 Mathematics Subject Classifications: 01, II, 26, 28, 30, provided. Use the codes below. Circled letters iden­ 31, 34, 35, 39, 40, 41, 42, 44, 45, 46, 47 ,53, 54, 65 and others tify corresponding items on the form and the strip. ISBN 0-8218-3218-3, LC 89-18114, ISSN 0081-5438 300 pages (hardcover), December 1989 Address forms to the Mathematics Meetings Hous­ Individual Member $79, List price $132, ing Bureau, P. 0. Box 6887, Providence, RI 02940. Institutional member $106 The deadline for receipt is June 1, 1990. To order please specify STEKL0/180NA

All prices subject to change. Shipment will be made by surface. For air delivery add, I st book $5, @ Specialties each additional book $3, maximum $100. Prepayment required. Order from American Mathe­ matical Society, P.O. Box 1571, Annex Station, Providence, RI 02901-1571, or call toll free 800- 321-4AMS (321-4267) in the U. S. and Canada 10 charge with VISA or MasterCard. AL =Algebra AN= Analysis BI = Biomathematics BS = Biostatistics CB = Combinatorics CM = Communication CN = Control CS = Computer Science CT = Circuits DE = Differential Equations EC = Economics ED = Mathematical Education ------FA = Functional Analysis FI = Financial Mathematics Recent Developments in Geometry FL = Fluid Mechanics GE = Geometry ------HM = History of Math LO = Logic MB = Mathematical Biology ME = Mechanics 5.-V. Cheng, H.Choi, and Robert E. Greene, Editors MO =Modelling MP =Mathematical Physics (Contemporary Mathematics, Volume 101) MS =Management Science NA = Numerical Analysis This volume is the outgrowth of a Special Session on Geometry, NT = Number Theory OR = Operations Research held at the November 1987 meeting of the AMS at the PR = Probability SA = Systems Analysis University of California at Los Angeles. The unusually well­ ST = Statistics TO = Topology attended session attracted more than sixty participants and outstanding featured over forty addresses by some of the day's @ Career Objectives geometers. By common consent. it was decided that the papers to be collected in the present volume should be surveys of relatively broad areas of geometry, rather than detailed AR = Academic Research AT = Academic Teaching presentations of new research results. A comprehensive survey NR =Nonacademic R&D NC = Nonacad. Consulting of the field is beyond the scope of a volume such as this. NS =Nonacademic Supervision Nonetheless, the editors have sought to provide all geometers, whatever their specialties, with some Insight into recent developments in a variety of topics in this active area of @ CD Duties research. 1980 Mathematics Subject Classifications: 53C20. 53C42, 53C55. 53A 10, 58E20, 58G25. 32930, 32H20 T =Teaching U =Undergraduate ISBN 0-8218-51 07-1 , LC 89-18039 G =Graduate R =Research ISSN 0271-4132 C = Consulting A = Administration 352 pages (softcover). December 1989 S = Supervision IND = Industry Individual Member $23, List price $39, GOV = Government DP = Data Processing Institutional member $31 To order please specify CONM/1 OlNA All prices subjectto change. Shipment will be made by surface. For air delivery add, 1st Location book $5, each additional book $3. maximum $100. Prepayment required. Order from American Mathematical Society. P.O. Box 1571. Annex Station, Providence, Rl 02901- 1571. or call toll free 800-321-4AMS (321-4267) in the U.s. and Canada to charge with S = South VISA or MasterCard. E =East C =Central M =Mountain W =West 0 =Outside U.S. I = Indifferent Summer List of Applicants Return form to: Mathematical Sciences Employment Register, P.O. Box 6887, Providence, RI 02940 (Please type. See instructions on facing page.)

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1. Using metacognitive strategies to improve instruction Genevieve Knight $36 2. Planning, funding, and administering teacher enhancement projects T. Christine Stevens & John Thorpe $36 3. A seminar on women in mathematics Miriam P. Cooney $36 4. A calculus laboratory using Mathematica Michael Barry, Benjamin Haytock & $60 Richard McDermot 5. Using history in teaching calculus V. Frederick Rickey $36 6. Writing to learn mathematics Agnes Azzolino $36 7. Exploring mathematics with the NeXT computer Charles G. Fleming & Judy D. Halchin $60 8. A mathematician's introduction to the HP-48SX scientific expandable Don LaTorre & John Kennelly $36 calculator for first-time users 9. Starting, funding and sustaining mathematics laboratories James E. White $60 10. CAS laboratory projects for Calculus Carl Leinbach $60 11. Producing mathematics courseware with Mathematica: Don Brown, Horacia Porta & Jerry Uhl $60 Calculus & Mathematica 12. Exploring statistics and discrete mathematics topics Bert K. Waits & Franklin Demana $18 13. Spreadsheet based mathematical topics for non-mathematics majors V. S. Ramamurti $60 n I plan on preregistering for the Columbus, Ohio meetings ONLY in order to attend the MAA Minicourse(s} indicated above. It is my understanding tha.t, should the course(s) of my choice be filled, full refund of the Columbus meetings preregistration fee will be made. Preregistration/Housing Form, Columbus, Ohio August 8-11, 1990 Must Be Received in Providence No Later Than June 6, 1990 Please complete this form and return it with your payment to Mathematics Meetings Housing Bureau P.O. Box 6887, Providence, Rhode Island 02940- Telephone: (401) 455-4143-Telex: 797192

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D I am not staying in the residence halls and will need to purchase a parking sticker for The Ohio State University campus at the meeting. A number of additions have been made in this 1990 xiv + 356 pages revised edition. A new section presents Price: US $102.50/Dfl. 200.00 Ehrenpreis' "fundamental principle" in full. Also ISBN 0-444-88488-2 added is a discussion of the theorem of Siu on The first part of this monograph introduces the Lelong numbers of plurisubharmonic the theory of Fnkhet algebras. The problem of finding analytic structure in the spectrum of a ced Modules over Group Frechet algebra is the subject of the second part. Particular attention is paid to function algebraic 1990 472 pages s characterizations of certain Stein algebras within Price: US $112. 75/Dfl. the class of Frechet algebras. ISBN 0-444-88593-5 Almost Free Modules Mathematics Studies, 161) This new edition has Set-Theoretic Methods with textual changes; in the addition of a new by P.C. Eklof and A. H. Mekler

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