ISSN 0002-9920
of the American Mathematical Society March 2001 Volume 48, Number 3
From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE page 294 Experimental Analysis of Algorithms page 304
-,
Perron Trees and Schoenberg Sprouts (see page 303) FOR ALL AMS MEMBERS Add these valuable mathematical volumes to your collection today!
Handbook of Algebra • A multi-volume Handbook
Managing Editor: M. Hazewinkel CWI, Amsterdam, The Netherlands Handbook of Algebra Handbook of Algebra Volume 1 Volume 2 Description Algebra, as we know it today, consists of • 1996 • Hardbound • 2000 • Hardbound many different ideas, concepts and results. • ISBN: 0-444-82212-7 • ISBN: 0-444-50396-X A reasonable estimate of the number of • 936 pages • 840 pages these different "items" would be somewhe re between 50,000 and 200,000. Many of Normal price: Normal price: these have been named and many more USD 200.00 USD 183.50 could (and perhaps should) have a "name" NLG 348.00 NLG 350.00 or a convenient designation. In addition to euro 157.92 euro 158.82 the primary information given in the Discount price: Discount price: Handbook, there are references to relevant USD 150.00 USD 138.00 articles, books or lecture notes to help the NLG 261.00 NLG 262.50 reader. An excellent index has been inclu euro 118.44 euro 119.12 ded which is extensive and not limited to definitions, theorems etc.
SPECIAL OFFER FOR THE HANDBOOK OF ALGEBRA Buy both volumes, and save 30% off each title, a combined price of just ~ USD 268.45 I NLG 488.60 I euro 221 .72!! ~
0 YES! Please send me __copy (ies) of Handbook of Algebra Vol. I Payment Details USD 150.00/ NLG 261.00/ euro 118.44 (ISBN: 0- 444-82212-7) 0 Payment enclosed (make payable to Elsevier Science) 0 Cheque 0 UNESCO Coupons 0 YES! Please send me __copy(ies) of Handbook of Algebra Vol. II USD 138.00/ NLG 262.50/ euro 119.12 (ISBN: 0-444-50396-X) 0 I wish to pay by credit card (accepted up to a maximum value of US$ 5,000.00/NLG 10,000.00). 0 YES! Please send me __copy(ies) of Handbook of Algebra Vol. I & II Your credit card will be debited including VAT USD 268.45/ NLG 488.60/ euro 221.72 (ISBN: 0-444-82212-7 /0-444-50396-X) when applicable.
Dutch Guilder (NLG) prices apply to customers in Europe and Japan. US Dollar IUS$) prices apply to custo 0 American express 0 Eurocard/Mastercard/Access mers in all other countries. If you are a resident of the European Union you should either state your Value 0 Visa Added Tax (VAT) number or add the relevant VAT percentage applicable in your country to the total amount. Card number: Exp. Date: Please Print Clearly Signature: Date: Name: Address: City: Post/Zip Code: Send this form to the Elsevier Science Office nearest you. Countr : For customers in the Americas: Tel: Fax: Elsevier Science, PO Box 880, New York NY 10159-0882, USA E-mail: VAT number: Tel (+ 1) 212 - 633-3730 AMS Membership Number: Toll-free for customers in the USA and Canada 1 - 888- 437 - 4636, 0 If you do not wish to receive information regarding our other publications, Fax (+1) 212 - 633 - 3680 please tick this box. E-Mail [email protected] 0 If you do not wish to receive information from other companies or third parties, please tick this box. For customers in all other locations: Elsevier Science, Marketing Department PO Box 103, 1000 AC Amsterdam For a full listing of Mathematics Handbooks, please visit our web site at: The Netherlands http://www.elsevier.com/locate/puremaths Fax +31 20 485 2425 ELSEVI ER SCIE NCE AMERICAN MATHEMATICAL SOCIETY New Titles from the AMS that enhance consistency processing, fuzzy representations, hybrid Recommended Text approaches involving both constraint programming and integer A Modern Theory of Integration programming, and applications to scheduling problems in domains Robert G. Bartle, Eastern Michigan University, Ypsilanti, and such as sports scheduling and satellite scheduling. University of Illinois, Urbana DIMACS: Series In Discrete Mathematics and Theoretical Computer Science, Volume 57; 2001 ; 175 pages; Hardcover; ISBN 0-8218-2710-3; The theory of integration is one of the twin pillars on which analysis is List $55; Individual member $33; Order code DIMACS/57NT103 built. The first version of integration that students see is the Riemann integral. Later, graduate students learn that the Lebesgue integral is Homotopy Methods in Algebraic Topology "better" because it removes some restrictions on the integrands and J. P. C. Greenlees, University of Sheffield, UK, Editor, with assis the domains over which we integrate. However, there are still draw tance from Robert R. Bruner, Wayne State University, Detroit, backs to Lebesgue integration, for instance, dealing with the Ml, and Nicholas Kuhn, University of Virginia, Charlottesville Fundamental Theorem of Calculus, or with "improper" integrals. This volume presents the proceedings from the AMS-IMS-SIAM This book is an introduction to a relatively new theory of the integral Summer Research Conference on Homotopy Methods in Algebraic (called the "generalized Riemann integral" or the "Henstock-Kurzweil Topology held at the University of Colorado (Boulder). The confer integral") that corrects the defects in the classical Riemann theory Quadraflc Forms ence coincided with the sixtieth birthday of J. Peter May. An article is and Their Applications and both simplifies and extends the Lebesgue theory of integration. included reflecting his wide-ranging and influential contributions to the holot'.. · Auc~~gow Although this integral includes that of Lebesgue, its definition is very subject area. Other articles in the book discuss the ordinary, elliptic -·- close to the Riemann integral that is familiar to students from ---· and real-oriented Adams spectral sequences, mapping class groups, - calculus. One virtue of the new approach is that no measure theory configuration spaces, extended powers, operads, the telescope and virtually no topology is required. Indeed, the book includes a conjecture, p -compact groups, algebraic K theory, stable and study of measure theory as an application of the integral. unstable splittings, the calculus of functors, the E.. tensor product, The text is su~able for a first-year graduate course, although much of and equivariant cohomology theories. The book offers a compen it can be readily mastered by advanced undergraduate students. dious source on modem aspects of homotopy theoretic methods in Included are many examples and a very rich collection of exercises. many algebraic settings. There are partial solutions to approximately one-third of the exercises. Contemporary Mathematics, Volume 271 ; 2001 ; approximately 344 Graduate Studies in Mathematics, Volume 32; 2001; approximately 448 pages; Softcover; ISBN 0-8218-2621-2; List $79; Individual member $47; pages; Hardcover; ISBN 0-8218-0845-1 ; List $59; All AMS members $47; Order code CONM/271 NT1 03 Order code GSM/32NT103 l·*'blJil@ Quadratic Forms and Their Applications Navier-Stokes Equations Eva Bayer-Auckiger, CNRS, Universite de Franche-Comte, Theory and Numerical Analysis Besan90n, France, David Lewis, University College, Dublin, Roger Temam, Indiana University, Bloomington Ireland, and Andrew Ranicki, University of Edinburgh, Scotland, Editors From a review for the First Edition: This volume outlines the proceedings of the conference on This book, in many ways remarkable, gives a detailed account of a "Quadratic Forms and Their Applications" held at University College number of results concerned with the theory and numerical analysis Dublin. It includes survey articles and research papers ranging from of the Navier-Stokes equations of viscous incompressible fluids. applications in topology and geometry to the algebraic theory of -Zentralblatt fur Mathematik quadratic forms and its history. Various aspects of the use of This book was originally published in 1977 and has since been quadratic forms in algebra, analysis, topology, geometry, and number reprinted four times (the last reprint was in 1985). The current volume theory are addressed. Special features include the first published is reprinted and fully retypeset by the AMS. It is very close in content proof of the Conway-Schneeberger Fifteen Theorem on integer to the 1985 edition. The book presents a systematic treatment of valued quadratic forms and the first English-language biography of results on the theory and numerical analysis of the Navier-Stokes Ems! Witt, founder of the theory of quadratic forms. equations for viscous incompressible fluids. Considered are the Contemporary Mathematics, Volume 272; 2000; 311 pages; Softcover; linearized stationary case, the nonlinear stationary case, and the full ISBN 0-8218-2779-0; List $79; Individual member $47; Order code nonlinear time-dependent case. The relevant mathematical tools are CONM/272NT1 03 introduced at each stage. The new material in this book is Appendix Ill, reproducing a survey Constraint Programming and Large Scale article written in 1998. This appendix contains a few aspects not Discrete Optimization addressed in the earlier editions, in particular a short derivation of the Eugene C. Freuder and Richard J. Wallace, University of New Navier-Stokes equations from the basic conservation principles in Hampshire, Durham, Editors continuum mechanics, further historical perspectives, and indications Constraint programming has become an important general approach on new developments in the area. The appendix also surveys some for solving hard combinatorial problems that occur in a number of aspects of the related Euler equations and the compressible Navier application domains, such as scheduling and configuration. This Stokes equations. Readers are advised to peruse this appendix volume contains selected papers from the workshop on Constraint before reading the core of the book. ' Programming and Large Scale Discrete Optimization held at DIMACS. This book presents basic results on the theory of Navier-Stokes It gives a sense of state-of-the-art research in this field, touching on equations and, as such, continues to serve as a comprehensive many of the important issues that are emerging and giving an idea of reference source on the topic. the major current trends. Topics include new strategies for local AMS Chelsea Publishing; 2001 ; approximately 424 pages; Hardcover; search, multithreaded constraint programming, specialized constraints ISBN 0-8218-2737-5; List $59; All AMS members $53; Order code
All prices subject to change. Charges for delivery are $3.00 per order. For optional air delivery outside of the continental U. S., please include $6.50 per item. Prepayment required. Order from : American Mathematical Society, P. 0. Box 845904, Boston, MA 02284-5904, USA. For credit card orders, fax 1-401-455-4046 or call toll free 1-800-321-4AMS (4267) in the U. S. and Canada, 1-401 -455-4000 worldwide. Or place your order through the AMS Bookstore at www.ams.orgAlookstore. Residents of Canada, please include 7% GST. ®.uCI,.,.D£o '' AMS AMERICAN MATHEMATICAL SOCIETY New! Elements of Noncommutative Geometry J.M. Gracla-Bondia, J.C. VRrllly & H. Figueroa, a//, Universidad de Costa Rica, San Jose, Costa Rica An introduction to the language and techniques of non commutative geometry, followed by a unified and compre hensive presentation of the core topics and key research results drawing from several branches of mathematics. The exposition of introductory material is excellent with New! main topics covered repeatedly in the text at gradually Differentiable Manifolds more demanding levels of difficulty. Many applications to diverse fields are presented, including index theory, Second Edition particle physics, and fundamental quantum theory. Rich L. Conlon, Washington University, St louis, MO in proofs, examples, exercises, and solutions, this volume "This textbook, probably the best introduction to differential geometry to be published since Eisenhart's, greatly is an excellent self-study resource and a useful textbook benefits from the author's knowledge of what to avoid, something that a beginner is likely to miss. The presenta for a graduate level course in noncommutative geometry. tion is smooth, the choice oftopics optimal, and the book can be profitably used for self teaching." BIRKHAUSER ADVANCED TEXTS -The Bulletin of Mathematical Books (on the 1st edition) 2000/704 PP./ HARDCOVER/ISBN 0-8176-4124·6 / $59.95 The basics of differentiable manifolds, global calculus, differential geometry, and related topics constitute a core of information essential for the first or second year graduate student preparing for advanced courses and New! seminars in differential topology and geometry. Designed to cover this material in a careful and sufficiently detailed manner, this text requires only a good foundation in general topology, calculus, and modern algebra. Proofs and Fundamentals A First Course in Abstract Mathematics This second edition contains a significant amount of new material, which, in addition to classroom use, will make it a useful reference text. Topics that can be omitted in a first course are clearly marked, making this edition E.D. Bloch, Bard College, Annandale-on-Hudson, NY easier to use for such a course, as well as for private study by non-specialists wishing to survey the field. "... 'Proofs and Fundamentals' has many strengths. One BIRKHAUSER ADVANCED TEXTS notable strength is its excellent organization. The book 2001/432 PP. /HARDCOVER/ISBN 0-8176-4134-3/ $54.95 (TENT.) begins with a three-part preface which makes its aims very clear. There are large exercise sets throughout the book. The exercises are well-integrated with the text and vary appropriately from easy to hard The topics in Part Ill New! are quite varied, mostly independent from each other, and Symmetry in Mechanics troly dependent on Parts I and ll At the end of the book Forthcoming! there are useful hints to selected exercises. A Gentle, Modem Introduction Dynamic Equations is the author's zeal S.F. Singer, Haverford College, Haverford, PA Perhaps the book's greatest strength on Time Scales and skill for helping students write mathematics better. .. This monograph was written to chip away at the language An Introduction with Applications Throughout the book, theorems are often followed first by barrier between physicists and mathematicians and to informative "scratch work" and only then by proofs. Thus link the abstract constructions of symplectic mechanics M. Bohner, University of Missouri, Rolla, MO & students can see many examples of what they should to concrete, explicitly calculated examples. The context A. C. Peterson, University of Nebraska, Lincoln, NE think, what they should write, and how these are usually is the two-body problem, i.e., the derivation of Kepler's The study of dynamic equations on a measure not the same. " -MAA Online laws of planetary motion from Newton's laws of gravitation. chain (time scale) is a new area of still fairly 2000 I 44B PP., 39 ILLUS./ HARDCOVER After a straightforward and elementary presentation of theoretical exploration in mathematics. ISBN 0·8176·4111·4/ $49.95 this derivation in the language of vector calculus, subse Motivating the subject is the notion that dynamic quent chapters slowly and carefully introduce symplectic equations on measure chains can build bridges manifolds, Hamiltonian flows, Lie group actions, Lie between continuous and discrete mathematics. New! algebras, momentum maps and symplectic reduction, This work is an introduction to measure chain Complex Analysis with many examples, illustrations, exercises and solu theory with particular emphasis on its usefulness tions. The work ends with the derivation it started with, in allowing for the simultaneous development of in One Variable but in the more sophisticated language of symplectic and differential equations and difference equations. Second Edition differential geometry. The study of measure chain theory has also led to R. Narasimhan, University of Chicago, IL & several important applications, e.g., in the study Y. Nievergelt, E. Washington University, Cheney, WA 2001/208 PP., 30 ILLUS./ SOFTCOVER of insect population models, neural networks, ISBN 0-8176-4145-9/$29.95 heat transfer, and epidemic models. "Provides a smooth and unintimidating transition from classical complex analysis in the plane to modem abstract With numerous examples and exercises through New! theory on manifolds ...An excel/en~ carefully written and out, the book may be used in a special topics thematically rich book which does not overwhelm the Elements of Nonlinear seminar at the senior undergraduate or begin reader... Well-suited as a textbook either for sophisticated Analysis ning graduate levels. May also serve as a good beginners or as a sequel to a one-semester introductory reference to stimulate the development of new course." -Jahresbericht der DMV (on the 1/e) M. Chipot, Universitiit Zurich, Switzerland kinds of equations with potentially new applica tions. 2000 /400 PP., 17 ILLUS./ HARDCOVER Explores the vast field of nonlinear analysis by emphasiz ISBN 0-8176-4164-5 I $59.95 ing underlying ideas rather than the sophisticated refine JUNE 2001/ APPROX. 320 PP./ HARDCOVER ments of the theory. 1\vo classical examples from physics, ISBN 0-8176-4225·0I $54.95 (TENT.) elasticity and diffusion, motivate the theoretical parts that TO ORDER : are then applied to various aspects of elliptic and parabol ic problems. Existence, uniqueness, regularity and CALL: 1·800·777·4643 approximation of solutions for quasilinear and monotone www.birkhauser.com FAX: (20 1) 348·4505 problems are studied, as well as some newaspects of the E·MAIL: [email protected] calculus of variations, including Young measures or approximation or minimizing sequences. Prices are valid in North America only and are subject to change without notice. For price and ordering information outside North BIRKHAUSER ADVANCED TEXTS America, please contact BirkMuser Verlag AG, P.O. Box 133, CH-4010 2000 /272 PP. / HARDCOVER /ISBN 0-8176-6406-8/$54.00 Basel, Swilzerland. Phone +41-61-205 0707; Fax +41-61-205 0792 or Birkhiiuser E-mail: [email protected] Boston · Basel · Berlin J.lll Promolion1Y2522 Notices March 2001 of the American Mathematical Society
Feature Articles
294 From Rotating Needles to Stability of Waves: Emerging Connections between Combinatorics, Analysis, and PDE Terence Tao Geometry and arithmetic yield insight on oscillatory integrals and partial differential equations.
304 Experimental Analysis of Algorithms Catherine C. McGeoch How can one predict the efficiency of a computational procedure? New models and methodologies foster better understanding of algorithms.
Communications Commentary 31 2 Presidential Views: Interview with 293 Opinion Hyman Bass 317 The Parrot's Theorem-A Book 316 Thompson and Uhlenbeck Receive Review National Medals of Science Reviewed by QueenaN. Lee-Chua Notices of the American Mathematical Society Departments
EDITOR: Hatold P. Boas Mathematics People ...... 320 ASSOCIATE EDITORS: Rhodes Scholarships Awarded, Diaconis Chosen von Neumann Susanne C. Brenner, Bill Casselman (Covers Editor), Robert J. Daverman, Nathaniel Dean, Rick Durrett, Lecturer, 2000 Maria Mitchell Women in Science Award, Deaths. Susan Friedlander, Robion Kirby, Elliott H. Ueb, Andy Magid, Judith Roitman, Mark Saul, Karen E. Mathematics Opportunities ...... 322 Smith, Audrey Terras, Lisa Traynor Project NExT: New Experiences in Teaching, 2001 Summer Pro- SENIOR WRITER and DEPUTY EDITOR: Allyn Jackson gram for Women in Mathematics, Maria Mitchell Women in Science MANAGING EDITOR: Sandra Frost Award. CONTRIBUTING WRITER: Elaine Kehoe For Your Information ...... 323 PRODUCTION ASSISTANT: Muriel Toupin TIMSS Repeat Study Released, Correction (to February issue cover PRODUCTION: Marcia Almeida, Siulam Fernandes, Thomas J. Kacvinsky, Lori Nero, Donna Salter, image). Deborah Smith, Peter Sykes, Maxine Wolfson ADVERTISING SALES: Leonard Moorehead, lnsidetheAMS ...... 325 Anne Newcomb AMS Epsilon Fund, AMS Establishes Public Awareness Office.
SUBSCRIPTION INFORMATION: Subscription prices Reference and Book List...... 327 for Volume 48 (2001) are $430 list; $272 institutional member; $204 individual member. (The subscription price for members is included in the Mathematics Calendar ...... 333 annual dues.) A late charge of 10% of the subscription price will be imposed upon orders New Publications Offered by the AMS ...... 339 received from nonmembers after January 1 of the subscription year. Add for postage: Surface delivery Publications of Continuing Interest ...... 346 outside the United States and lndia-$17; in lndia $38; expedited delivery to destinations in North America-$35; elsewhere-$70. Subscriptions and Classifieds ...... 348 orders for AMS publications should be addressed to the American Mathematical Society, P.O. Box 845904, Membership Forms ...... 355 Boston, MA 02284-5904. All orders must be prepaid. ADVERTISING: Notices publishes situations wanted Meetings and Conferences Table of Contents ...... 368 and classified advertising, and display advertising for publishers and academic or scientific organizations. Advertising material or questions may be faxed to 401-331-3842 (indicate "Notices advertising" on fax cover sheet). SUBMISSIONS: Articles and letters may be sent to the editor by e-mail at noti ces@math . tamu . edu, by fax at 979-845-6028, or by postal mail at Department of Mathematics, Texas A&M University, College Station, TX 77843-3368. E-mail is preferred. Correspondence with the managing editor may be sent to noti ces@ams. org. For more information, see the section "Reference and Book List". NOTICES ON e-MATH: Most of this publication is available electronically through e-MATH, the Society's resource for delivering electronic products and services. To access the Notices on e-MATH, use the URL: http: //www . ams .org/notices/. Those with VT100-type terminals or without WWW browsing software can connect to e-MATH via Telnet (tel net e-math . ams . org; login and password are "e-math"). From the [NoticesoftheAmericanMathematica/ Society is published monthly except bimonthly in June/ July by the American AMS Secretary Mathematical Society at 201 Charles Street, Providence, Rl 02904-2294. Periodicals postage paid at Providence, Rl, and additional mailing offices. POSTMASTER: Send ad call for Nominat~o.ns for 2002 Leroy P. Steele Prizes ...... 331 dress change notices to Notices of the American Mathe matical Society, P.O. Box 6248, Providence, Rl 02940-6248.] Publication here of the Society's street address and the other information in brackets above is a technical re· quirement of the U.S. Postal Service. Tel: 401-455-4000; e-mail: noti ces@ams . org. ©Copyright 2001 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability. Opinion
component of the MSI focuses on connections between the NSF Mathematical mathematical sciences and other sciences and engineer ing. Mathematics and statistics provide a vocabulary that grows richer as the boundaries between the disciplines be Sdences Initiative come more diffuse. We must nurture and train researchers The National Science Foundation (NSF) is planning a new who are capable of participating in multidisciplinary col Mathematical Sciences Initiative (MSI). Plans call for in laborations. Mathematics and statistics have enabled ex creased funding over the next five years to support fun traordinary advances across the board, yielding new ana damental mathematical sciences research, connections of lytical, statistical, computational, and experimental tools the mathematical sciences to other areas of science and en to tackle a broad range of scientific and technological gineering, and improvements in mathematical sciences ed challenges previously considered intractable. With this ucation. unprecedented expansion of our capacity to discover and The MSI is needed because advances in fundamental its potential for application to societal needs comes a de mathematical sciences, which embrace mathematics and mand for new mathematical and statistical techniques. statistics, are closely intertwined with the discovery process 3. The third component of the initiative advances math in science, engineering, and technology. The mathematical ematical sciences skills in a broad sense. Public literacy in sciences are accelerating progress across the spectrum of mathematics and statistics and appreciation of their roles science and engineering, even in traditionally descriptive in modern life are critical for societal progress. Educa sciences. tion, research, and national work force needs are inextri Why is the MSI needed now? One reason is the "math cably linked: we must attract the best talent from today's ematization" of science. Science is becoming more youth into undergraduate, graduate, and postdoctoral pro mathematical and statistical-not only the physical and in grams in the mathematical sciences and train them for a formation sciences, but also the biological, geophysical, en broad array of careers in academia, industry, and govern vironmental, social, behavioral, and economic sciences. ment. It is important that U.S. students sense the excite There is a vital need for mathematicians and statisticians ment and see the opportunities for careers based on the to collaborate with engineers and scientists to explore the mathematical sciences. The involvement of the research frontiers of discovery, where science and mathematics community in this mathematical sciences education agenda meet and interact. Another reason is that the technical work is indispensable. Providing all young citizens with the force, as well as society at large, needs more mathemati mathematical skills to enjoy productive and fruitful em cal and statistical skills today than ever before. Technol ployment is a national priority. ogy-based industries fuel the growth of the U.S. economy, The NSF has adopted a five-year investment plan for the which, in turn, relies on large numbers of college gradu MSI, starting in fiscal year 2002 (which begins October 1, ates well versed in mathematics, science, and engineering. 2001). Right now, specific plans for the initiative are In our increasingly complex world, the need for broad taking shape. Members of the mathematical sciences mathematical and statistical literacy becomes ever more community are urged to contribute their thoughts and acute. ideas as the initiative moves forward. As plans develop in The MSI has three main components: advancing funda formation will be posted on the Web site of the NSF's mental mathematics; advancing interdisciplinary collabo Division of Mathematical Sciences, http: I /www. nsf. rations among the mathematical sciences, other areas of gov/mps/dms/. science, and engineering; and advancing mathematical skills and mathematical literacy. -Philippe Tondeur, Director 1. The first component of the initiative calls for in NSF Division of Mathematical Sciences creased support for fundamental mathematical research. The mathematical sciences have consistently renewed themselves through synthesis of preceding work and in fusion of new ideas, some of which originate through the application of mathematics in other disciplines. This process of rejuvenation and evolution is indispensable for discovery at the frontiers of the mathematical sciences. 2. Many technological advances-from Doppler radar to magnetic resonance imaging to public key cryptography to the Internet-are rooted in fundamental mathematics and statistics. The concepts and structures developed by fun damental mathematics and statistics often provide just the right framework for the formulation and study of seem ingly unrelated applications. Therefore, the second
MARcH 2001 NOTICES OF THE AMS 293 From Rotating Needles to StabilityofWaves: Emerging Connections between Combinatorics, Analysis, andPDE Terence Tao
Introduction These areas are very active, but despite much In 1917 S. Kakeya posed the Kakeya needle prob recent progress our understanding of the problems lem: What is the smallest area required to rotate and their relationships to each other is far from a unit line segment (a "needle") by 180 degrees in complete. Ideas from other fields may well be the plane? Rotating around the midpoint requires needed to make substantial new breakthroughs. rr /4 units of area, whereas a "three-point U-turn" Kakeya-Type Problems requires rr/8. In 1927 the problem was solved by Besicovitch's solution to the Kakeya needle A. Besicovitch, who gave the surprising answer problem relied on two observations. The first that one could rotate a needle using arbitrarily observation, which is elementary, is that one small area. can translate a needle to any location using At first glance, Kakeya's problem and Besicov arbitrarily small area; see Figure 1. The second itch's resolution appear to be little more than observation is that one can construct open subsets mathematical curiosities. However, in the last three of R 2 of arbitrarily small area which contain a decades it has gradually been realized that this type unit line segment in every direction. A typical way of problem is connected to many other, seemingly to construct such sets (not Besicovitch's original unrelated, problems in number theory, geometric construction) is sketched in Figure 2; for a more combinatorics, arithmetic combinatorics, oscilla detailed construction see [7]. tory integrals, and even the analysis of dispersive For any n ~ 2 define a Besicovitch set to be a sub and wave equations. set of R n which contains a unit line segment in The purpose of this article is to discuss the in every direction. For any such n the construction terconnections between these fields, with an em of Besicovitch shows that such sets can have ar phasis on the connection with oscillatory integrals bitrarily small Lebesgue measure and can even be and PDE. Two previous surveys ((7] and [1]) have made to have measure zero. Intuitively this means focused on the connections between Kakeya-type that it is possible to compress a large number of problems and other problems in discrete combi nonparallel unit line segments into an arbitrarily natorics and number theory. small set. In applications one wishes to obtain more quan Terence Tao is professor of mathematics at UCLA. His titative understanding of this compression effect e-mail address is tao@math. ucla. edu. by introducing a spatial discretization. For
294 NOTICES OF THE AMS VOLUME 48, NUMBER 3 instance, one can replace R unit line segments by 1 x 8 tubes for some 0 < 8 « 1 and ask for the optimal compression of these tubes. Equivalently, one can ask for bounds of the vol- ume of the 8-neighbour hood of a Besicovitch set. Rather surprisingly, these bounds are logarith mic in two dimensions. It is known that the 8-neigh Figure 1. To translate a needle, slide it by R units, rotate by roughly 1 I R, slide it back, bourhood of a Besicovitch and rotate back. This costs 0(1 1R) units of area, where R is arbitrary. set in R 2 must have area at least C/log(1/8);1 this ba \ sically follows from the \ geometric observation that \ \ the area of the intersection \ \ of two 1 x 8 rectangles \ varies inversely with the \ \ angle between the long axes \ of the rectangles. Recently,
U. Keich has shown that \ this bound is sharp. \ This observation can be \ rephrased in terms of the Minkowski dimension of the Besicovitch set. Recall that a bounded set E has Minkowski dimension lX or less if and only if for every 0 < 8 « 1 and 0 < E « 1 one can cover E by at most CE8 - IX+ E balls of radius 8. From the previous discus sion we thus see that Besi covitch sets in the plane must have Minkowski di mension 2. It is unknown if the anal ogous property holds in R n. In one of its principal for Figure 2. The iterative construction of a Besicovitch set. Each stage consists of the mulations, the Kakeya con union of triangles. To pass to the next stage, the triangles are bisected and shifted jecture states that every together to decrease their area. Besicovitch set in Rn has ( n+2 10_10 4n+3) Minkowski dimension n. (There is also a corre max 2+ '7' sponding conjecture for the Hausdorff dimension, but for simplicity we shall not discuss this variant.) although I expect further improvements to follow Equivalently, the Kakeya conjecture asserts that very soon. the volume of the 8-neighbourhood of a Besicov One can discretize the conjecture. Let be a itch set in Rn is bounded below by Cn ,E8E for any n maximal8-separated subset of the sphere sn- 1 (so E > Oand0 < 8 « 1. 1 The Kakeya conjecture is remarkably difficult. that n has cardinality approximately 8 - n), and for It remains open in three and higher dimensions, each w E n let Tw be a 8 x 1 tube oriented in the although rapid progress has been made in the last direction w. The Kakeya conjecture then asserts few years. The best-known lower bound for the logarithmic-type lower bounds on the quantity Minkowski dimension at this time of writing is IUwEnTwl. The above formulation is reminiscent of exist 1 Throughout this article, the letter C denotes a constant ing results in combinatorics concerning the which varies from line to line. number of incidences between lines and points,
MARCH 2001 NOTICES OF THE AMS 295 0 0 G ' ' ' ' ' ' A ' 1 ' ' ' ' ' ' ' ' ' ' ' ' ' 0 0 ' ' ' ' ' ' ' ' ' ' ' ' '~ ' Q • • • A
Figure 3. The left picture depicts six tubes T w pointing in different directions and the three discretized slices Ao, A112• A1. which in this case are three-element sets. The right picture depicts the set G associated to this collection of tubes. Note that the maps (a, b)~ a, (a, b)~ b, (a , b) ~ a+ b have small range (mapping to Ao, A1, and 2A112 respectively), but the map (a, b) ~ a - b is one-to-one. although a formal connection cannot be made be similar flavour (but with more technical formula cause the nature of the intersection of two 6 x 1 tions). For instance, one can define a {3 -set to be a tubes depends on the angle between the tubes, subset of the plane which contains a {3 -dimensional whereas the intersection of two lines is a point subset of a unit line segment in every direction. It regardless of what angle the lines make. However, is then an open problem to determine, for given {3, it is plausible that one can use the ideas from com the smallest p ossible dimension of a {3 -set. Low binatorial incidence geometry to obtain progress dimensional examples of such sets arise in the work on this problem. For instance, it is fairly straight of H. Furstenberg, and it seems that one needs to forward to show that the Minkowski dimension of understand these generalizations of Besicovitch Besicovitch sets is at least (n + 1)/2 purely by using sets in order to fully exploit the connection be the fact that given any two points that are a dis tween Kakeya problems and oscillatory integrals, tance roughly 1 apart, there is essentially only one which we discuss below. Other variants include 6 x 1 tube which can contain them both. replacing line segments by circles or light rays, In the 1990s, work by ]. Bourgain, T. Wolff, considering finite geometry analogues of these W. Schlag, A. Vargas, N. Katz, I. Laba, the author, problems, or replacing the quantity I Uw T w I by the and others pushed these ideas further. For in variant II ~w XTw lip (the relevant conjecture here is stance, the lower bound of (n + 2)/2 for the known as the Kakeya maximal function Minkowski dimension was shown in 1995 by conjecture). Another interesting member of this fam Wolff and relies on the 6-discretized version of the ily is the Falconer distance set conjecture, geometric statements that every nondegenerate which asserts that whenever E is a compact one triangle lies in a unique two-dimensional plane dimensional subset of R2 , the distance set and every such plane contains only a one-parameter {lx - Yl: x,y E £} isaone-dimensionalsubsetofR. set of directions. However, there appears to be a The discrete version of this is the Erdos distance prob limit to what can be achieved purely by applying lem-what is the least number of distances elementary incidence geometry facts and standard determined by n points?- and is also unsolved. For combinatorial tools (such as those from extremal a thorough survey of most of these questions, we refer graph theory). More sophisticated geometric analy the reader to [7]; see also [3]. sis seems to reveal that a counterexample to the Kakeya conjecture, if it exists, must have certain The Connection with Arithmetic rigid structural properties (for instance, the line Combinatorics segments through any given point should all lie in The Kakeya problem looks very geometrical, and a hyperplane). Such ideas have led to a very small it is natural to apply elementary incidence geom recent improvement in the Minkowski bound to etry to bear on this problem. Although this (n + 2)/2 + 10- 10, but they are clearly insufficient approach has had some success, it does not seem to resolve the full conjecture. sufficient to solve the problem. The Kakeya problem is a representative mem In 1998 Bourgain introduced a new type of ber of a much larger family of problems of a argument, based on arithmetic combinatorics (the
296 NOTICES OF THE AMS VOLUME 48, NUMBER 3 combinatorics of sums and differences), which IA+BI 3 gave improved results on this problem, especially IA -BI:::; IAIIBI. in high dimensions. The connection between Kakeya problems and the combinatorics of addi (A summary of such inequalities can be found in tion can already be seen by considering the [5].) However, these arguments (which are mostly analogy between line segments and arithmetic pro graph-theoretical) do not seem to adapt well to the gressions. (Indeed, the Kakeya conjecture can be Kakeya application, because we are working with reformulated in terms of arithmetic progressions, only a subset G of Ao x A1 rather than all of and this can be used to connect the Kakeya con Ao x A1. jecture to several difficult conjectures in number To overcome this problem, Bourgain adapted a theory, such as the Montgomery conjectures for recent argument ofW. T. Gowers which allows one generic Dirichlet series. We will not discuss this to pass from arithmetic information on a subset connection here, but refer the reader to [1].) of a Cartesian product to arithmetic information Bourgain's argument relies on the following on a full Cartesian product. A typical result is: "three-slice" idea. Let 0 and T w be as in the pre Theorem. Let A, B be finite subsets ofa torsion-free vious section. We may assume that the tubes T w abelian group with cardinality are contained in a fixed ball. Suppose that at most N, and sup pose that there exists a set G c A x B I UwEn Tw I is comparable to bcx for some con of cardinal ity at least aN2 stant a; our objective is to give upper bounds on such that the sum set {a+ b: (a, b) E G} a and eventually to show that a must be zero. has cardinality at most N. Then By choosing an appropriate set of coordinates, there exist subsets A', B' of A, B respectively such that A' - B' has cardinality 13 one can ensure that each of the three slices at most a - Nand A', B' have cardinality at least a 9 N. Xr := {x E R"- 1 : (x,t) E U Tw }. Roughly speaking, this theorem states that if wEn most of A + B is contained in a small set, then by t = 0, 1 I 2, 1, has measure comparable to bcx. Be refining A and B slightly, one can make all of cause of the b-discretized nature of the problem, A - B be contained in a small set also. Such results one can also assume that the discrete set are reminiscent of standard combinatorial theo Ar := Xr n bZ"-1 rems concerning the size of sum and difference sets, but the innovation in Gowers's arguments is has cardinality comparable to bcx+1-n for that the control on A' and B' is polynomial in a. t=0,1/2,1. (Previous combinatorial techniques gave bounds Morally speaking, every tube T w intersects each which were exponential or worse, which is not of the sets Ao, A1;2, A1 in exactly one point. sufficient for Kakeya applications.) Assuming this, we see that every tube T w is asso Recently [2] Katz and the author have obtained ciated with an element of Ao x A1. Because two the bound (4n + 3) / 7 by using control on both the points determine a line, these elements are es sum set Ao + A1 and the variant Ao + 2A1, which sentially disjoint as w varies. Let G denote the set corresponds to the slice A2;3. of all pairs of Ao x A1 obtained this way. Thus G These results have remarkably elementary has cardinality about bl- n. proofs. Apart from some randomization argu The sum set ments, the proofs rely mainly on standard combi natorial tools such as the pigeonhole principle and {a+ b: (a, b) E G} Cauchy-Schwarz inequality, as well as on basic of G is essentially contained inside a dilate of the arithmetic facts such as set A1;2; this reflects the fact that the intersection a+b=c+d a-d=c - b, ofTw with A1 ;2 is essentially the midpoint of the = intersection of T w with Ao and A1. In particular, a - b =(a - b') - (a' - b') +(a' - b), the sum set is quite small, having cardinality only bcx+1- n. On the other hand, the difference set and
{a - b: (a, b) E G} ao + 2bo = a1 + 2b1, bo = bi ==> a1 - bi = 2(ao + bo) - 2b1 - (ao + b of G is quite large, because the tubes Tw all point 0). in different directions. Indeed, this set has the Further progress has been made by pursuing these same cardinality as G, i.e., about b1- n. methods, though it seems that we are still quite Thus, if a is nonzero, there is a large discrep far from a full resolution of the Kakeya problem, ancy in size between the sum set and difference and some new ideas are almost certainly needed. set of G. In principle this should lead to a bound One possibility may be that one would have to on a, especially in view of standard inequalities use combinatorial estimates on product sets in relating the cardinalities of sum sets and difference addition to sum sets and difference sets, since sets, such as one has control of {a+ tb : (a, b) E G} for all
MARCH 2001 NOTICES OF THE AMS 297 Figure 4. Four tubes T;, their shifts T;, and the wave packets «J!;. The interference between the functions S1 «J!; will cause the LP norm to be large when p > 2.
t E [0, 1]. Discrete versions of such estimates exist; problem by Gowers and others may well have further for instance, G. Elekes has recently shown the applications to the Kakeya problem in the future. bound Applications to the Fourier Transform max(JA ·AI, IA+Al);;:: c-1 JAJ 5' 4 Historically, the first applications of the Kakeya for all finite sets of integers A. However, these problem to analysis arose in the study of Fourier bounds do not adapt well to the continuous Kakeya summation in the 1970s. setting because of the difficulty in discretizing If f is a test function on R n, we can define the both addition and multiplication simultaneously. Fourier transform f by A good test problem in this setting is the Erdos ring fc~) := f e-2rrix · ~f(x) dx. problem: Determine whether there exists a (Borel) JR" subring of R with Hausdorff dimension exactly One then has the inversion formula 1/2. This problem is known to be connected with the /3 -set problem and the Falconer distance set f(x) = f e2rrix·~{c~) d~. problem. JR" Interestingly, the Kakeya problem is also Now suppose that f is a more general function, such connected to another aspect of arithmetic combi as a function in the Lebesgue space LP(Rn). The natorics, namely that of locating arithmetic pro Fourier inversion formula still holds true in the gressions in sparse sets. (A famous instance of this sense of distributions, but one is interested in is an old conjecture of Erdos, which is still open, more quantitative convergence statements. that the primes contain infinitely many arithmetic Specifically, we could ask whether the partial progressions of arbitrary length.) This difficulty Fourier integrals arises in the Hausdorff dimension formulation of the Kakeya problem, and also in some more SRf(x) :=I e2rrix · ~f(~) d~ quantitative variants, because of the difficulty in I~I~R selecting a "good" set of three slices in arithmetic converge to fin, say, LP norm. (The pointwise con progression in which to run the above argument. vergence question is also interesting, say for L 2 The combinatorial tools developed for that functions f, but this seems extremely difficult
298 NOTICES OF THE AMS VOLUME 48, NUMBER 3 to show in two and higher dimensions. In one f(x) = 2.: €ye2rriWT"XI./JT(X), dimension this was proven in a famous paper T by L. Carleson.) By the uniform boundedness where ET = ±1 are randomized signs. Using principle, this is equivalent to asking whether Khinchin's inequality (which roughly states that the linear operators SR are bounded on LP(R11 ) one has the formula I LTErfTI- (L,:y lfr12)112 uniformly in R. By scale invariance it suffices to with very high probability), one can eventually show this for sl: compute that 1(1 I) IISlfiiLP(R") ~ CllfiiLP(R")· IISlfllp ~ c-1 K 2 2 -ii llfllp. The operator S1 is known as the ball multiplier Since K is unbounded, we thus see that S1 is or (when n = 2) the disk multiplier, because of the unbounded. formula Fefferman's theorem is an example of how a geometric construction can be used to show the where B is the unit ball in R 11 • In one dimension it unboundedness of various oscillatory integral is a classical result of Riesz that this operator is operators. The point is that while the action of bounded on every [P, 1 < p < oo, and so Fourier these operators on general functions is rather com integrals converge in LP norm. (Indeed, in one di plicated, their action on "wave packets" such as mension S1 is essentially the Hilbert transform.) e2rriwT"xtfJy(x) is fairly easy to analyze. One can In higher dimensions S 1 is bounded in L 2, thanks then generate a large class of functions to test to Plancherel's theorem; however, the behaviour the operator on by superimposing several wave in LP is more subtle. One has an explicit kernel packets together and possibly randomizing the representation which roughly looks like coefficients to simplify the computation. The counterexample provided by the Besicov e±ilx-yl itch construction is very weak (only growing log SI{(x) "" f (1 + lx- y l)(n+ll/2 f(y) dy; arithmically in the scaleR) and can be eliminated if one mollifies the disk multiplier slightly. For to be more precise, one must use Bessel functions instance, the counterexample does not prohibit instead of e±ilx-yl. The kernel is in LP only when the slightly smoother Bochner-Riesz operator Sf, p > ;;+1, so it might seem natural by duality argu defined by ments to conjecture that S1 is bounded when 2 = (1- I~WXB(~)f(~). ;;_:1 < p < 11!!1. In 1971, however, C. Fefferman ~ proved the surprising from being bounded for E > 0, because the analo Theorem.Ifn > 1, thenS1 isunboundedonLP for gous computation gives every p -4 2. ISf(e2rriwT·xi./JT(X))I :::::: R -E
In particular, one does not have [P convergence for all x E f. Indeed, the Bochner-Riesz conjecture for the Fourier inversion formula in higher di asserts that Sf is indeed bounded on LP for all mensions unless p = 2 . E > 0 and 2nJ(n + 1) < p < 2nJ(n- 1). (For other Roughly speaking, the idea is as follows. By values of p one needs E >nit- ~1- ~.)This duality it suffices to consider the case p > 2. Let was proven by L. Carle son and P. Sjolin R be a large number, and let T be a cylindrical conjecture in 1972 in two dimensions, but the higher tube in R" with length R and radius JR and dimensional problem is quite challenging, and oriented in some direction wy. Let 1./JT be a bump only partial progress has been made so far. This function adapted to the tube T, and let f be a conjecture would imply that the partial Fourier shift of Tby 2R units in the Wy direction. Then a integrals will converge in LP if one uses a Cesaro computation shows that summation method (such as the Fejer summation ISI(e2rriwT·xi./Jr(x))l ""1 method, which corresponds toE= 1). The Bochner-Riesz conjecture would be dis for all x E f. proved if one could find a collection of disjoint this computation, one uses the Besi To exploit tubes T for which the compression factor K had to find a collection {T} of covitch construction some power dependence on R as opposed to tubes as above which are disjoint but whose logarithmic, i.e., if K ~ c-1RE for some E > 0. A shifts f have significant overlap. More precisely, more precise statement is known, namely, that we assume that the failure of the Kakeya conjecture would imply the failure of the Bochner-Riesz conjecture. (More IYfl ~k~ ITI succinctly, Bochner-Riesz implies Kakeya.) In 1991 Bourgain introduced a method in which for some K which grows in R (the standard con these types of implications could be reversed, so struction in Figure 2 gives K - log(R) I log log(R) ). that progress on the Kakeya problem would (for Then we consider the function instance) imply progress on the Bochner-Riesz
MARCH 2001 NOTICES OF TilE AMS 299 conjecture. The key observation is that every func bilinear and multilinear variants, etc. There are tion can be decomposed into a linear combination some additional rescaling arguments available in of wave packets by applying standard cutoffs both the bilinear case, as well as some L 2-based esti in physical space (by pointwise multiplication) and mates, but apart from this there are few effective in frequency space (using the Fourier transform). tools known outside of Bourgain's wave packet After applying the Bochner-Riesz operator to the analysis to attack these types of problems. wave packets individually, one has to reassemble One variant of the above problems comes from the wave packets and obtain estimates for the replacing Euclidean space by a curved manifold. sum. Kakeya estimates play an important role in There are some interesting three-dimensional ex this, since the wave packets are essentially sup amples of C. Sogge and W. Minicozzi showing that ported on tubes; however, this is not the full story, the Kakeya conjecture can fail on such manifolds, since these packets also carry some oscillation, and which then implies the corresponding failure of one must develop tools to deal with the possible oscillatory integral conjectures such as the natural cancellation between wave packets. The known analogue of Bochner-Riesz. This may shed some techniques to deal with this cancellation, mostly light on the robustness of Kakeya estimates and based on L 2 methods, are imperfect, so that even their applications in variable coefficient situations. if one had a complete solution to the Kakeya con Certainly the arithmetic and geometric techniques jecture, one could not then completely solve the used currently to attack Kakeya problems do not Bochner-Riesz conjecture. Nevertheless, the adapt well to curved space. best-known results onBochner-Riesz (e.g., in three dimensions the conjecture is known [6] for Applications to the Wave Equation p > 26/7 and for p < 26/19) have been obtained In the previous section we saw how Kakeya-type by utilizing the best-known quantitative estimates problems are related to oscillatory integrals. There of Kakeya type. is also a similar, and in some sense more natural, These techniques apply to a wide range of os connection between Kakeya problems and linear cillatory integrals. A typical question, the (adjoint) evolution equations such as the free Schrodinger, restriction problem, concerns Fourier transforms wave, and Airy equations. of measures. Let dcr be surface measure on, say, For brevity of exposition we shall restrict our 1 the unit sphere sn- . The Fourier transform d(; attention to the solutions of the free wave equa of this measure can be computed explicitly using tion Bessel functions and decays like lxl-(n-1>12 at in finity. In particular, it is in the class LP(R n) for all Utt(t, x) = ilu(t, x); u(O, x) = f(x), ur(O, x) = 0 p > 2n/(n- 1). The restriction conjecture asserts with initial position f and initial velocity zero. that the same statement holds if d(; is replaced However, much of our discussion has analogues by (ti;; for any bounded function f on the sphere. for other linear evolution equations such as the This question originally arose from studying the Schrodinger equation. restriction phenomenon (that a Fourier transform One can solve for u explicitly using the formula of a rough function can be meaningfully restricted to a curved surface such as a sphere, but not u(t) = cos(tr-i.)f, to a flat surface like a hyperplane); it is also related but this does not reveal much information about to the question of obtaining LP estimates on the size and distribution of u. On the other hand, eigenfunctions of the Laplacian on the torus this formula does show that the wave equation is (although the eigenfunction problem is far more connected to the oscillatory integral problems difficult due to number theoretic issues), as well mentioned earlier. For instance, the disk multi as LP estimates on solutions to dispersive PDE, as plier S1 can be rewritten as we shall see below. The restriction conjecture is logically implied by SI{ = X[-2rr,2rrj(r-i.)f, the Bochner-Riesz conjecture and is slightly easier since S1 preserves those Fourier modes e2rrix·'l; to deal with technically. It has essentially the same amount of progress as Bochner-Riesz; for instance, which are (generalized) eigenfunctions of ~ it is completely solved in two dimensions and is with eigenvalue in [-2rr, 2rr] and eliminates all known to be true [6] for p > 26/7 in three dimen others. In particular, we have the Fourier repre sions. One uses the same techniques, namely wave sentation packet decomposition of the initial function f, 2 2 S1f = J"" sin( rrt) cos(tH)f dt Kakeya information, and L estimates to handle the -oo rrt cancellation, in order to obtain these results. There is an endless set of permutations on these = J"" sin(2rrt) u(t) dt -oo rrt types of oscillatory integral problems: more general phases and weights, square function and of the disk multiplier in terms of wave evolution maximal estimates, more exotic function spaces, operators.
300 NOTICES OF 1HE AMS VOLUME 48, NUMBER 3 t
Figure 5. A schematic depiction of the Wolff example at time zero. Because of the zero initial velocity, the wave trains will move in two opposite directions. A general class of problems is the following: a focus point (which would generate a large LP given size and regularity conditions on the initial norm for p > 2) for any length of time. This data f, what type of size and regularity control does phenomenon is known as local smoothing. A very one obtain on the solution? useful class of local smoothing estimates is From integration by parts (or from the above known collectively as Strichartz estimates. A explicit formula), one has energy conservation typical example of a Strichartz estimate is
J ~IV'u(t,x)l 2 + ~1Ut(t,x)l 2 dx = J ~IV'f(x)l 2 llullrt,
MARCH 2001 NOTICES OF THE AMS 301 Bochner-Riesz conjecture; the idea is to write the Bochner-Riesz multiplier Sf in terms of wave operators cos(t.J="K) in the manner briefly dis cussed earlier. However, the conjecture is far from settled; even in two dimensions the conjecture is completely proven for only p > 74 (due toT. Wolff), and at the critical exponent p = 4 the conjecture is known only for E > 1/8- 1/88 [6], [8]. These estimates are all proven via Kakeya methods. Briefly, the idea is to decompose the initial data f (and hence the solution u by linearity) into pieces which are localized in both space and frequency. This decomposes the solution u into wave pack ets, which are oscillatory functions which travel along light rays. One then uses Kakeya-type argu ments to control how many times these light rays intersect each other, together with orthogonality arguments to control to what extent the oscillations of each wave packet reinforce each other. A recently developed, but apparently quite powerful, tech nique here is induction on scales, in which one assumes that the desired estimate is already proven at smaller scales and uses this hypothesis together Figure 6. The Wolff example at a time 1 5 t 5 2; only the with the Kakeya strategy just discussed to obtain incoming wave trains are shown. the same estimate at higher scales. There are several other wave equation estimates tubes Twe place a "wave train", which is basically which are related to those discussed here. An eiRx·wT times a bump function adapted toT. Let f active area of research is to obtain good bilinear be the sum of all these wave trains (although or even multilinear estimates on solutions to the we may randomize the signs of these trains to wave equation, as opposed to the linear estimates simplify computations). described here; these estimates have direct appli At time zero, the function f has low L"" norm. cation to nonlinear wave equations, since one However, as time evolves, each wave train T splits can often use techniques such as the method of as the superposition of two pulses, one moving in Taylor series to write the solution of nonlinear the direction WT and the other in the direction wave equations as a series of multilinear expres -WT. For times 1 5 t 52 a large portion of the sions of solutions to the linear wave equation. wave train at T now lives in the shifted tube f. There are some tantalizing hints that these Kakeya Because of the large overlap of these tubes, the techniques could also be used to handle nonlinear LP norm of u is large for all 1 5 t 5 2 ; as with ~e 1> wave equations directly (or wave equations with Fefferman's argument, it is about K2 z--p . By rough metrics, potential terms, etc.), but these letting R - oo, one can make K unbounded, and ideas are still in their infancy. this gives the theorem. Since u can be written in terms of circular Because the Besicovitch construction has a averages of f, there is also a close relationship logarithmic compression rate, one could get between wave equation estimates and estimates around this obstruction by requiring an epsilon of for circular means. (Such circular means estimates regularity on the initial data. The local smoothing can then be used, for instance, to make progress conjecture of C. Sogge asserts that no further loss on the Falconer distance problem mentioned of regularity occurs or, more precisely, that earlier.) There is also an extremely strong square lluliLP([l,2]xR") 5 Cp,E 11(1 + vCLl)EfiiLP(R") function estimate conjectured for the wave equation which, if true, would imply the local for all E > 0, n(~- f,>- ~. and 2 5 p 5 oo. This smoothing, Bochner-Riesz, restriction, and conjecture is easy when p = 2 or p = oo ; the most Kakeya conjectures. It would also give estimates interesting case is when p = 2n/(n- 1). for other seemingly unrelated objects such The local smoothing conjecture is extremely as the helix convolution operator f ..... f * d(J", strong and would imply many of the known where d(J" is arclength measure on the estimates on the wave equation. It implies the helix {(cos t, sin t, t): 0 5 t 5 2rr} in R3 . (The Kakeya conjecture, for a counterexample to the connection arises because the Fourier transform Kakeya conjecture could be used to strengthen of d(J" is concentrated near the light cone.) Wolff's argument to disprove the local smoothing These estimates are quite difficult, and the conjecture. This conjecture also implies the partial progress which has been made on them
302 NOTICES OF TilE AMS VOLUME 48, NUMBER 3 Figure 7. A schematic depiction of how a wave (such as the one drawn on the left) can be written as a superposition of wave packets or "photons". These objects are localized in space and also have a localized direction, and different wave packets are essentially orthogonal. There is no canonical way to perform this decomposition, but one usually uses a combination of spatial cutoffs and cutoffs in Fourier space. has proceeded via Kakeya estimates. Although these deep wave equation estimates have not yet About the Cover found significant applications, I am confident that Around 1926 A. S. Besicovitch showed that there exist planar they will do so in the near future. regions of arbitrarily small area in which one could rotate a seg ment of fixed length, thus solving what is now known as the Acknowledgments Kakeya needle problem. As Terry Tao's I thank Nets Katz for some helpful references and article explains, even nowadays Besicov the referees for very helpful comments. The author itch's Theorem and variations on it are a is a Clay long-term Prize Fellow and is supported fruitful source of analysis. by grants from the Sloan and Packard Founda X A One of the principal steps in tions. This article is based on a talk given at the Besicovitch's proof was the construction Clay millennium event. of certain complicated regions of arbi trarily small area in the plane containing References needles of a fixed length and a range of [1) ]. BoURGAIN, Harmonic analysis and combinatorics: directions. The original construction of How much may they contribute to each other?, Math such regions was quite complicated. ematics: Frontiers and Perspectives, IMU/ Amer. Math. Already in the next year's volume of the Soc., 2000, pp. 13-32. Mathematische Zeitscrift, 0. Perron ex [2) N. KATZ and T. TAO, Bounds on arithmetic projections, hibited a simpler one, involving what are now called "Perron and applications to the Kakeya conjecture, Math. Res. trees", and in the early 1960s I. J Schoenberg simplified Perron's Lett. 6 (1999), 625-630. construction in turn, showing how these trees could be con [3) __ , Some connections between the Falconer and structed recursively by what he called "sprouting". The cover Furstenberg conjectures, in preparation. illustrates Schoenberg's construction and demonstrates visu [4) S. KLAINERMAN and S. SELBERG, Bilinear estimates and ally that the ratio of the areas of the regions in the third and applications to non-linear wave equations, preprint. fourth columns has limit 0. [5) I. RuzsA, Sums of finite sets, Number Theory: New The Mathematical Association of America received a grant York Seminar (D. V. Chudnovsky, G. V. Chudnovsky, and M. B. Nathanson, eds.), Springer-Verlag, 1996. from the NSF around 1960 to make a film about this topic. What [6) T. TAo and A. VARGAS, A bilinear approach to cone was essentially a transcript of the film appeared in an often cited multipliers I, II, GAFA 10 (2000), 185-258. article by Besicovitch ("The Kakeya problem", volume 70 of the [7) T. WOLFF, Recent work connected with the Kakeya American Mathematical Monthly, 1963, pp. 697-706). The film problem, Prospects in Mathematics (Princeton, NJ, itself may have been the first professionally produced mathe 1996), Amer. Math. Soc., Providence, RI, 1999, pp. matical animation. Are there any viewable copies of the film left? 129-162. Is there anybody still around who took part in that project? [8) T. H. WOLFF, A sharp bilinear cone restriction esti -Bill Casselman (covers@ams. org) mate, Ann. of Math., to appear.
MARCH 2001 NOTICES OF TilE AMS 303 Experimental Analysis of Algorithms Catherine C. McGeoch
As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality. -Albert Einstein
In theory there is no difference between theory and practice. In practice there is. -Yogi Berra
he algorithm and the program. The given applications. Another effort focuses on using abstract and the physical. If you want to experiments to extend and improve the kinds of understand the fundamental and uni results obtained in traditional algorithm analysis. Tversal properties of a computational That is, rather than having a goal of measuring pro process, study the abstract algorithm grams, we develop experiments in order to better and prove a theorem about it. If you want to know understand algorithms, which are abstract math how the process really works, implement the ematical objects. In this article we concentrate on algorithm as a program and measure the running examples from this latter type of research in ex time (or another quantity of interest). perimental algorithmics. This distinction between the abstract model It is natural to wonder whether such an effort and the physical artifact exists in the study of is likely to bear fruit. If the ultimate goal of algo computational processes just as in every other rithm analysis is to produce better programs, area of mathematical modeling. But algorithmic wouldn't we be better off studying programs in problems have some unusual features. For exam their natural habitats (computers) rather than per ple, we usually build models to serve as handy rep forming experiments on their abstractions? Con resentations of natural phenomena that cannot versely, if the goal is to advance knowledge in an be observed or manipulated directly. But programs area of mathematical research (algorithm analysis), and computers are completely accessible to there are we wise to study abstract objects using im searcher and are far more manipulable than, say, perfect, finite, and ever-changing measurement weather patterns. They are also much easier to tools? understand: hypothetically, one could obtain com One answer to the first question is that reliable plete information about the behavior of a program program measurement is not as easy as it sounds. by consulting technical documents and code. And Running times, for example, depend on complex finally, algorithms are usually invented before pro interactions between a variety of products that grams are implemented, not the other way around. comprise the programming environment, includ This article surveys problems and opportunities ing the computer chip (perhaps Intel Inside), mem that lie in the interface between theory and prac ory sizes and configurations, the operating system tice in a relatively new research area that has been (such as Windows 98 or Unix), the programming called experimental algorithmics, experimental language (maybe Java or C), and the brand of com analysis of algorithms, or algorithm engineering. piler (like CodeWarrior).l These products are so Much work in this field is directed at finding and phisticated, varied in design, and, especially when evaluating state-of-the-art implementations for lintel Inside™ is a registered trademark of the Intel Catherine C. McGeoch is an associate professor of Corporation. WindowsTM 98 is a registered trademark of computer science at Amherst College. She has been Microsoft Corporation. Unix™ is a registered trademark active in the development of an experimental of The Open Group. java™ is a registered trademark of tradition in algorithmic studies. Her e-mail address is Sun Microsystems, Inc. C is not a trademark. CodeWar [email protected]. rior™ is a trademark of Metrowerks, Inc.
304 NOTICES OF TilE AMS VOLUME 48, NUMBER 3 1. Array A[1 ... n] contains n distinct numbers. We want the rth smallest. Set lo- 1 and hi - n. 2. Partition. Set x- A[lo]. Rearrange the numbers in A[lo ... hi] into three groups around some index p such that: a. For lo ~ i < p, we have A[i] < x. b. A[p] =X. c. For p < j .:5 hi, we have A[j] > x. 3. Check Indices. If p = r, stop and report the element A[p]. Otherwise, if p < r, set lo-p+ 1 and go to Step 2. Otherwise, p > r, so set hi-p- 1 and go to Step 2.
Figure 1: Selection Algorithm S. The algorithm reports the rth.smallest number from array A[1 ... n]. used in combination, extremely difficult to model. ment of new mathematical techniques (mostly There is no known general method for making ac combinatorial), and, not infrequently, efficient al curate predictions of performance in one pro gorithms that can have substantial impact on gramming environment based on observations of practice. It is clear, however, that our analytical running times in another. Experimental analysis of techniques are far too weak to answer all our ques the abstraction allows us to have more control tions about algorithms. Computational experi over the trade-off between generality and accuracy ments are being used to suggest new directions for when making predictions about performance. research, to support or disprove conjectures, and An answer to the second question is straight to guide the development of new analytical tools from the mathematician: algorithms and their and proof strategies. analyses are beautiful and fundamental, and they This article presents examples from two broad deserve study by any means available, including research efforts in experimental algorithmics: first, experimentation. to develop accurate models of computation that Certainly algorithms existed long before the allow closer predictions of performance, and sec first computing machine was a gleam in Charles ond, to extend abstract analyses beyond tradi Babbage's eye. For example, Euclid's Elements, circa tional questions and assumptions. We start with 300 B.C., contains an algorithm for finding the a short tutorial on the notations and typical results greatest common divisor of two numbers. While obtained in algorithm analysis. many algorithms have been discovered and pub lished over the centuries, it is only with the advent A Tutorial on Algorithm Analysis of computers that the notion of analyzing algo The selection problem is to report the rth_smallest rithmic efficiency has been formalized. number in a collection of n numbers. For example, Perhaps the first real surprise in algorithm r = 1 refers to the minimum of the collection, and analysis occurred with Strassen's discovery in 1968 r = (n + 1)/2 is the median when n is odd. For con of a new method for multiplying two n x n ma venience we assume that no duplicate numbers ap trices. While the classic algorithm we all learned pear in the collection. in high school requires 2n3 - n2 scalar arithmetic Figure 1 presents a well-known selection algo operations, Strassen's algorithm uses fewer than rithm. The n numbers are placed in an array called 7n10g2 7 - 6n2 operations, where log2 7 < 2.808. A in no particular order. Each number has some Therefore, this new algorithm uses fewer opera position from 1 to n in the array: the notation A[i] tions than the classic method when n is greater refers to the number in position i, and i is called than 654. Strassen's discovery touched off an in an index. The notation .fJ - lo means "set .fJ equal tensive search for asymptotically better matrix to the value of lo". multiplication algorithms. The current champion The main operation of algorithmS is to choose requires no more than cn2·376 scalar operations for a partition element x from a contiguous subarray a known constant c. It is an open question whether of A defined by indices lo and hi and to partition better algorithms exist. (See any algorithms text the subarray by rearranging its contents so that book, such as [1], for more about matrix multipli numbers smaller than x are to its left and num cation.) These fancy algorithms are not much use bers larger are to its right. This puts x at some lo in practice, however. It would be a daunting cation p; that is, A[p] = x. After partitioning, we prospect even to write an error-free program for know that x is the pth_smallest number in A. We one of them, and the extra computation costs im are looking for the rth_smallest number: depend posed by their complexity make them unlikely to ing on the relationship of p tor, the algorithm ei outperform the classic method in typical scenar ther stops or repeats the process on one side or ios. the other of p. Algorithm analysis is a vigorous and vital sub To analyze the algorithm, we derive a function discipline of computer science, providing deep that relates input size to the cost of the compu new insights into the fundamental power and lim tational resources used by the algorithm. The pre itations of computation, fodder for the develop- cise meanings of "input size" and "cost" depend
MARCH 2001 NOTICES OF TilE AMS 305 upon the model of computation being assumed. Definition: f(n) E O(g(n)) if there exist positive con Here we shall use the simple RAM (random access stants c and no such that f(n) :$ c · g(n), V n ;:::: no. machine) model under which all scalar numbers Definition: f(n) E O(g(n)) iff g(n) E O(f(n)). have unit size and all basic operations on scalar values-such as arithmetic, comparison, and copy The statements below are typical of the kinds ing of values-have unit cost. Therefore the input of results obtained in classic algorithm analysis. size is n. • Any correct selection algorithm must at least For our purposes it is sufficient to define cost examine every element of A. Therefore, any al as the number of times x must be compared to el gorithm that solves the selection problem must ements from A[lo ... hi] during the partitioning have a worst-case cost function in Q(n). This step. A partitioning method is known (described is called a lower bound on the problem of se later in this article) that uses hi -lo comparisons lection. to partition subarray A[lo ... hi] of size hi - l o + 1. • It is easy to see that Cw(n) E O(n2). We have The only problem remaining is to count up total a complexity gap between the O(n2) worst costs. case bound and the O(n) lower bound on the Obviously, the total cost depends on which par problem. Complexity gaps generate research tition element x = A[lo] is used each time: we may questions: Does an O(n) algorithm exist? Or get lucky and find p = rafter just one partitioning should the lower bound be raised to, say, operation, or we may have to repeat the process O(n2), because selection is fundamentally several times. The worst-case cost, Cw(n), is the harder than the first lower bound indicates? maximal number of comparisons over all arrays of Or is the lower bound somewhere in between, size n and all r: a worst-case scenario holds, for perhaps at O(n log n)? • 1n fact, a variation on algorithmS is known that example, when r =nand p becomes 1, 2, ... , n in successive partitioning stages. Letting t denote has O(n) worst-case cost, thus closing the complexity gap. The variation uses an elabo the cost of a single comparison of x to an array el rate strategy for choosing the partition element ement, we have Cw(n) = t(n - i). 2:?= 1 at each stage and is generally considered too The average-case cost, Ca(n, r), is found by av complicated and expensive to be useful in eraging over some probability distribution on ar practice. rays of size n. Assume here that every number in The goal of the asymptotic analyses above is to the collection is equally likely to be in position lo place cost functions into complexity classes, while and selected as the partition element x. Then we the goal of exact analysis is to find closed forms have: that retain constant factors in the leading terms. Ca(l, 1) = 0 A closed form for Cw is easy to obtain. Also, it can Ca(n, r) = t(n - 1) be shown that: r- 1 Ca(n, r) = t [2 [(n + 1)Hn - (n + 2- r)Hn+1- r + ~ ( L Ca(n - p, r- p) + p=1 -(r + l)Hr + n + 5/3] - Orn/3- Or1/3- 20rn0r1f3], I Ca(n - p, r)) . p=r+1 where Hn is the nth harmonic number and Orn is the Kronecker delta function. This implies that Note that establishing the correctness of this re Ca(n, r) E O(n) (see Knuth [5], Exercise 5.2.2.31, for cursive formula requires a proof (which exists) details). that the partitioning operation preserves the prop erty that each number in the subset is equally Memory Sensitive Models of Computation likely to end up in position A[l o] and be chosen as In the good old days (say around 1970) it was pos the partition element at a later stage. sible to obtain quite accurate predictions of pro Note also that the cost of this, and any, algorithm gram running time by inserting appropriate time is described by a collection of cost functions that units for the constants (like t above) in an exact correspond to different scenarios. Another cost analysis using the RAM model of computation. function might be obtained with other probabilis Nowadays, however, the RAM model is inadequate tic assumptions, or a different definition of cost to the task. might be used under some other model of com For example, consider the seemingly innocu putation. ous instruction x - A[l o] in Step 2 of our exam The first goal in algorithm analysis is the ple algorithm. The actual work to carry out this classification of cost functions according to com instruction is performed in a computer by the Cen plexity classes as defined below. Throughout, we tral Processing Unit (CPU). The instruction involves assume that f(n) and g(n) map nonnegative inte about three CPU operations: (1) calculate the mem gers to positive real numbers. ory address of A[lo], (2) fetch the value from that
306 NOTICES OF THE AMS VOLUME 48, NUMBER 3 memory address, and then (3) store the value at If A[lo] is not in a register, it may be in the the memory address associated with x. (Normally, Level I or Level II cache. Typical cache access times array addresses need to be calculated, but scalar on workstations are around 5 to 10 nanoseconds. addresses do not.) Cache sizes and eviction policies are part of the A fast modern CPU is capable of performing overall design of the computer. If not in a cache, each operation in about a nanosecond. But the A[lo] is likely to be in main memory, with access CPU may be forced to wait upon a slow memory, times around 50 to 100 nanoseconds. But it is pos and the additional time needed for the fetch and sible that A[l o] has been placed in secondary mem store operations can vary by enormous factors, de ory, which may have access times greater than pending on exactly where A[lo] and x reside in the 106 nanoseconds. Main-to-secondary eviction poli memory hierarchy. Figure 2 gives a simplified di cies are incorporated into the operating system. agram of a memory hierarchy ranging from a small, Whereas the abstract RAM model assigns unit fast, and expensive register set, to a huge, slow, and cost to one access of A[lo], true memory access inexpensive secondary memory. times can vary by factors as large as a million. Without a decent model of the memory hierarchy, we cannot predict whether a given program will take one minute or two years to run. (Of course, this worst-case scenario is not typical, but pre dictions about program running times that are off by three or more orders of magnitude are not un usual.) Several threads of research in experimental al gorithmics are concerned with developing new models of computation that incorporate aspects of the memory hierarchy, especially with respect to caches and to policies at the main-to-secondary Secondary Memory memory levels. In 1996, for example, LaMarca and Ladner in Figure 2: The memory hierarchy, simplified. troduced a RAM model with a two-layer memory This drawing is not to scale. (cache and main memory). They and several other researchers have since been able to reanalyze clas The idea is to keep values that the CPU needs sic algorithms and data structures under the new close to it in the fastest memory, but of course not model. These results, obtained through a combi everything will fit. Therefore, memories are de nation of analytical and experimental approaches, signed to retain values that will be needed soon and have produced new theorems and new analysis to evict unneeded values by moving them back to techniques, better predictions of program perfor slower memory. It is not always possible to pre mance, faster programs, and clear indications that dict which values will be needed next, so most our "common sense" understanding of what con memory levels operate under some kind of prin tributes to program efficiency is due for revision. ciple of locality, which says that values that have To get a feeling for the type of mathematics in been recently accessed, or that are near values re volved, consider reanalyzing our selection algo cently accessed, are likely to be needed soon. There rithm under the new two-layer model. We need to may be a different version of this principle at each be more explicit about the partition step: Figure 3 level of the hierarchy, and wide variations exist in shows one well-known partitioning method that computing environments with respect to number performs hi -lo array-element comparisons for of levels, sizes of memories, and the eviction poli subarray A[lo . .. hi] of size hi - lo + 1. cies adopted. The new model organizes main memory and the Consider now the cost of fetching the value cache into contiguous blocks of scalar values. Main A[lo]. If that value has been used recently, it may memory contains M blocks named mo through be in a register, and the fetching cost is negligible. mM - l, and the cache holds C blocks named co to Decisions about which elements get to stay in reg cc -1, with C < < M. Suppose an operation accesses isters are primarily made by the compiler, a trans element i which resides in block mx. Then i and lation program that converts the original program the entire block containing it are transferred to to machine-readable language. It is an NP-hard cache block cy, where y =x mod C. This transfer problem (about which more later) to make optimal incurs a cost of tout. but any subsequent access of decisions about placing variables in registers; there an element in block cy will have smaller cost tin· fore efficient optimal algorithms are not likely to This block will be evicted from the cache and be found. Modern compilers incorporate heuristic sent back to main memory when another element strategies for deciding which values are to be is accessed from some block mz such that stored in registers. y =z mod C.
MARCH 2001 NOTICES OF TilE AMS 307 Partition. 2.1. Set x ~ A[lo]. Set./! ~ lo + 1 and h ~ hi. Iterate the following three steps. 1. Use./! to scan up from the left, comparing each A[./!] to x. Stop when either A[./!]> x or./!= h. 2. Use h to scan down from the right, comparing each number A[h] to x. Stop when A[h] < x or h = ./!. 3. If./!< h, swap (exchange) the numbers A[./!] and A[h] and go to Partition Step 2.1. Otherwise, if./!= h, go to Step 2.2 below. 2.2 . The above process maintains the invariant that all numbers in A[lo + 1 ... ./!- 1] are less than x and numbers in A[h + 1 ... hi] are greater than x. At this point we also have -1! =h. If A[-/!] < x, swap A[lo] and A[-/!] and set p ~ -/!; otherwise, swap A[lo] and A[./!- 1] and set p ~ ./!- 1. Partitioning is now complete.
Figure 3. Partitioning A[lo ... hi] around x = A[lo].
Now suppose subarray A[i . . . j] resides in mem from several weeks to a few hours have been re ory block mx. The access cost for a particular array ported in the literature. element A[k], with i :5 k ::s; j, is therefore • tout if this is the first access of A[k] or any of Heuristics for NP-Hard Problems its block neighbors A[i ... j]. We now turn to another research thread in exper • tout if an array element from any blocky such imental algorithmics. The most interesting ques that y =x mod C has been accessed more re tion about any solvable computational problem is cently than any element from block x (caus whether it is tractable or intractable. A problem is ing an eviction). tractable if it can be solved in polynomial time; that • tin otherwise. is, there exists an algorithm for the problem that Homework Problem. Write new formulas for has worst-case cost in O(nk) for some constant k. Cw(n) and Ca(n, r) under this model. Perform an A problem is intractable if no such algorithm ex exact analysis on both formulas. Would an alter ists. For convenience we define intractable prob native partitioning method, say one that traverses lems to be those having exponential lower bounds array elements from left-to-right rather than both in O(cn) for some constant c > 1 and ignore the ends-toward-the-middle, give a lower memory ac fact that there exist functions like nlogn that are cess cost? neither polynomial nor exponential. Clearly even this simple two-level model greatly Perhaps the most important technical idea to complicates the analysis task. But the tighter analy arise from the study of algorithms and problem sis does appear to be well worth the effort in many complexity is the identification of problems that cases. Jon Bentley (personal communication) are NP-complete and NP-hard. Hundreds of prac reports that simple program modifications can re tical problems that arise naturally in all spheres duce cache effects and improve running times by of industry, commerce, science, and government factors as large as 16. For a nice introduction to work have been identified as NP-complete and/or memory-sensitive models of computation, see NP-hard. The technical definitions of these two LaMarca and Ladner's [6] analysis of four sorting problem classes are rather too involved to fit into algorithms under the cacheing model. They draw this article, but the point is this: it is an open ques conclusions about efficiency that flatly contradict tion whether these problems are tractable or in a common lore based on traditional analyses and tractable. That is, for every problem in these classes obtain much tighter predictions of program run there is a huge complexity gap between an expo ning times than had been possible before. nential worst-case bound on the best algorithm Another research effort in memory-sensitive known, and a polynomial lower bound on the prob analysis concerns algorithms for data sets that lem complexity. Do efficient algorithms exist for must reside in secondary memory because they are these problems, or are they fundamentally too too large to fit in main memory. Important appli hard to be solved in polynomial time? cations for such algorithms include Web search en For technical reasons, "NP-complete" refers to de gines that peruse directories containing hundreds cision problems, which are always phrased as yes of millions of entries, many-body simulation al or-no questions (Does input instance I have prop gorithms for research problems in the natural sci erty P?). "NP-hard" can refer to more general types ences, and algorithms for image processing and sci of problems, such as optimization problems that in entific visualization. The RAM model can also be volve minimizing or maximizing some quantity as extended to models that account for data trans sociated with the desired solution. The two classes fers between main and secondary memory (which are related as follows: Let X and Ybe NP-complete follow different rules from caches). New algorithms problems, and let Z be NP-hard but not a decision and analyses of old algorithms under these new problem. Then (1) a polynomial time algorithm for models can result in huge improvements to pro X exists if and only if one exists for Y; (2) if a poly gram running times. Reductions of running times nomial time algorithm to produce optimal solutions
308 NOTICES OF THE AMS VOLUME 48, NUMBER 3 for Z exists, then polynomial algorithms exist for Graph-coloring is a generalization of the famous X andY. map-coloring problem to nonplanar graphs. Thus, discovery of a polynomial-time algorithm Graph-coloring is of interest to a university reg for any NP-complete or NP-hard problem would istrar who assigns times (colors) to courses (nodes), have profound implications about the complexity under the constraint that no two courses with the of several hundred other problems of great prac same professor can meet at the same time. It is of tical interest, besides earning the discoverer a cool interest to anyone who needs to create work sched million dollars. (To learn more about the million ules or timetables that avoid certain kinds of con dollar Millennium Prize Problems announced by the flicts. And graph-coloring arises in the context of Clay Mathematics Institute, visit www. cl aymath. the compiler task, mentioned earlier in this arti o rg/ p ri ze_p rob 1 ems I.) Proof of an exponential cle, of assigning values to registers so as to mini lower bound for any NP-complete problem would mize memory access costs. reverberate similarly. To learn more about the the Recall that the CPU can work only on values that ory of NP-completeness, see the classic text by are in registers. Memory costs are incurred when Garey and Johnson [3]. Three famous NP-hard a value must be transferred from another memory problems are described below. to a register (here we assume that this transfer cost Traveling Salesperson. You are given a graph is constant). Consider the sequence of instructions that oc or digraph G = (V, E) with positive-valued weights curs at the beginning of our selection algorithmS: on its edges. A Hamiltonian tour of G is a closed path over vertices and edges that visits each ver (1) lo 1 tex of G exactly once. The cost of the tour is the (2) hi n sum of the weights of edges traversed on the path. (3) X A[lo] The problem is to find a minimum-cost Hamil (4) lo+1 tonian tour of any given graph G. .e This problem is of interest to traveling sales (5) h hi persons who wish to tour all cities in a sales re gion while minimizing total travel cost. It is also Two values interfere if one must be accessed be of interest to any organization that must sched tween two accesses of the other. When two values ule regular tours of delivery trucks, like the U.S. interfere, we want to place them in different reg Post Office and your favorite grocery store chain. isters so as to avoid the cost of evicting one to make lo hi Bin Packing. Given a list of n weights, all from room for the other. For example, and inter the real number range (0, 1), the problem is to or fere because accesses of hi on line (2) come between ganize the weights into unit-capacity bins so as to accesses of loon lines (1) and (3) and also because lo is accessed between lines (2) and (5). If both lo minimize the number of bins used. For example, and hi were assigned to register R 1 , then these val the weight list containing 0.4, 0.3, 0.6, 0.5 could be ues would incur the following memory transfer grouped into (0.4, 0.3), (0.6), (0.5), which uses three costs: (1) fetch lo; (2) evict Io, fetch hi; (3) evict hi, bins, or into (0.4, 0.6), (0.3, 0.5), which uses only fetch lo; (5) evict lo, fetch hi; for a total of seven two bins. memory transfers. These two values could coex This problem is of interest to anyone walking ist peacefully if placed in registers R 1 and R2, al into a lumberyard with a list of board lengths though this might cause interference difficulties needed for a construction project who must group with other values. the lengths so as to minimize the number of stan More generally, given a sequence of instruc dard 8-foot boards that must be purchased. It is tions in a program, we can build an interference of interest to any construction supply firm that graph G where nodes represent program values must cut stock from unit-sized pieces. The prob and an edge is placed between two nodes if their lem is also of interest to anyone who wants to values interfere. The problem is to assign a mini back up files by copying onto unit-sized floppy mum number of registers (colors) to values (nodes) disks while minimizing the total number of disks such that interfering values receive distinct regis needed. A two-dimensional version of the problem ter assignments. (where weights come in (x, y) pairs) is of interest As is the case with all NP-hard problems, no to any printer or pattern cutter who must lay out polynomial-time algorithm for finding minimal parts on unit-sized rectangular sheets so as to colorings is known, and it is not known whether minimize the total number of sheets needed. such an algorithm exists. While the theoreticians Graph-Coloring. Given a graph G, a coloring of struggle with the general problem, the compiler G is an assignment of colors to vertices under the writers must settle for coloring algorithms that use constraint that no two vertices sharing an edge can a "small number" of colors, if not the absolute have the same color. The graph-coloring problem minimum. Dozens of algorithms have been pro is, for any given graph G, to find a coloring that posed, which fall into some broad categories, minimizes the total number of distinct colors used. broadly sketched in the list below. (Colors are
MARCH 2001 NOTICES OF THE AMS 309 assumed to be numbered 1 through k, for some best in which kinds of scenarios. appropriate k .) Indeed, graph-coloring has been the focus of Greedy Algorithms. A greedy algorithm may what might be termed a new mode of research in iterate over nodes, assigning to each the least algorithmic studies: a coordinated, multiparticipant numbered color that does not violate any edge effort to identify the state-of-the-art in algorithmic constraint. Or an algorithm may iterate over col performance for some given problem. Since 1992 ors, assigning each color to a large independent set the DIMACS Center (an NSF- and corporate-funded (finding a maximum independent set is an NP center for research in discrete mathematics and hard problem). Greedy algorithms are often re theoretical computer science, located at Rutgers peated with different node or color orderings each University) has sponsored an annual Implementa time, and the best solution found is saved. These tion Challenge to encourage and promote experi algorithms are fast but may produce colorings mental research on algorithms for specific prob using many more colors than necessary. lems. The focus on a particular problem allows a Branch-and-Bound. Systematically generate col great deal more coordination and interaction than orings of G, avoiding redundant colorings and un might otherwise be possible by independent re productive starts, and save the best one found searchers. within a given time limit. Note that even on mod The second DIMACS Challenge (1993) [4] con erate-sized graphs only a tiny fraction of colorings cerned algorithms for three NP-hard problems: can be generated from the enormous space of pos graph-coloring, finding large cliques in graphs, sibilities; various strategies for generating the best and finding satisfying truth assignments for ones first may be considered. Boolean formulas. While a great deal of progress Heuristic Search Methods. Start with a valid col was made, one conclusion drawn at that work oring. "Step" to another valid coloring by randomly shop was that huge gaps remain in our under modifying the current coloring in some small way standing of the "best algorithms" for particular ap (perhaps by changing the color of one node). Con plications. Much more work remains to be done. tinue stepping until a good coloring is found. Steps Besides the DIMACS Challenges, several other toward better colorings are of course preferred, but venues for experimental research on algorithms it is important occasionally to allow steps toward have appeared in the past decade. Two annual worse colorings to avoid being trapped in a local conferences (WAE in Europe and ALENEX in North minimum. A rich variety of stepping rules and America), the electronic journal on Experimental strategies for controlling the stepping process Algorithmics, and numerous small workshops have have been proposed. all been developed to provide forums for re These general strategies can be modified to fit searchers to report on their experimental obser many varieties of NP-hard problems, including vations. Traveling Salesperson and Bin Packing. ln some cases (like these two), strategies that exploit Questions of Methodology special problem structures are known that out Research in experimental algorithmics produces perform the general techniques. Note that the new understanding of models, problems, algo analysis of algorithms for NP-hard problems usu rithms, and program efficiency. Another important ally involves finding functions and complexity component of experimental algorithmics has been classes for two interesting quantities: algorithm ef the development of methods and techniques for ficiency and the quality of solution produced by performing experiments on algorithms. As sug the algorithm. gested in the beginning of this article, the rela ln the case of heuristic algorithms like those tionship between algorithms and programs and the sketched above, average-case analyses are usual types of questions asked about algorithms extremely difficult to obtain even under the sim are not typical of those that arise in mathematical plest computational and probabilistic models. An modeling. Consequently, new problems of de alytical statements about how a coloring algorithm signing and conducting experiments and of de performs on, say, random graphs with edge prob veloping appropriate statistical tools can be iden ability p, are rare, and statements about perfor tified. Some research questions concerning mance on "graph structures typical of register methodologies for experimental analysis of algo interference problems" are far beyond our ana rithms are sketched below. See [7] for more dis lytical capabilities. cussion. Computational experiments can be used to en How to Escape the Artifact? We study a perfect rich our understanding of these heuristic algo abstract object (the algorithm) using an imperfect rithms for NP-hard problems. Experiments have measuring device (the computer). Difficulties been used to direct the discovery of new algo associated with numerical precision, nonrandom rithms and new analyses, to allow prediction of ness in pseudo-random number generators, and algorithmic responses to variations in input inaccurate measuring tools can produce spurious properties, and to sort out which approaches work and erroneous results. Tools for measuring pro-
310 NOTICES OF THE AMS VOLUME 48, NUMBER 3 gram running times are notoriously quirky, and in while the O(n) implementation with huge coeffi some contexts it can be quite difficult to find a de cients took only 32 minutes. finition of "cost" that is useful, measurable, and This and other experiences reported in the lit generalizable. New techniques are needed for iden erature suggest some rough guidelines about the tifying and minimizing the impact of artifact on relative importance of algorithmic and environ observation. mental factors: In typical scenarios changes in en Sampling from Large and Infinite Spaces. It is vironment (such as switching computers, compiler not possible to measure the performance of a optimization level, or operating system; or im given graph-coloring algorithm over the space of proving programmer skills) can affect running all graphs of a given size n, nor is it always feasi times by factors in ranges between 2 and 100. Im ble to sample from the space of all colorings of a provements on a larger scale can be obtained given graph. We need good methods for defining through asymptotic improvements to algorithms, and generating input classes and input samples to by factors of O(n) or more, and algorithmic im support observations that can be generalized out provements by factors below, say, O(log n) are not likely to have much real impact on performance. side the realm of observation. For example, very lit Continuing research in the new tle is known about methods for extrapolating from and rapidly de veloping field of algorithm experimentation will a finite data set to obtain the kind of asymptotic allow us to refine these guidelines, to produce bet statements about complexity classes (e.g., does ter understanding of environmental effects, and to 2 the data come from a function that is O(n )?) that generate better algorithms and tighter analyses. are of interest to computer scientists. Acknowledgment. I thank the editor, Harold New Data Analysis Techniques. Computational Boas, and an anonymous reviewer for several sug experiments on algorithms can produce enormous gestions for improving the clarity of this article. data sets. Some sets are too massive to be amenable to standard statistical techniques, so we must References either make do with fewer measurements or [1] S. BAASE and A. VAN GEUJER, Computer Algorithms: In develop faster techniques. Computational experi troduction to Design and Analysis, Third Edition, Ad ments can be very interactive and iterative, and it dison-Wesley, 2000. A very readable undergraduate seems likely that new methods of data analysis can textbook. be developed to take advantage of these [2] ]. L. BENTIEY, Programming Pearls, Second Edition, properties. ACM Press and Addison-Wesley, 2000. Short articles about the interface between theory and practice in computer programming. How Important Are Algorithms? [3] M. R. GAREY and D. S. JoHNSON, Computers and In As the discussion in this article suggests, suffi tractability: A Guide to the Theory of NP-Complete ciently precise results for all but the simplest al ness, W. H. Freeman and Company, 1979. A classic gorithms, input models, and models of computa introduction to the field. tion are quite difficult to obtain by purely analytical [4] D. S. JoHNSON and M. TRICK, eds., Cliques, Coloring, methods. With our current techniques, asymptotic and Satisfiability: Second DIMACS Implementation Challenge, DIMACS Series in Discrete Mathematics analyses of algorithms are just barely adequate for and Theoretical Computer Science, vol. 26, Amer. making very rough predictions about program run Math. Soc., 1996. Papers presented at the workshop. ning times. Some researchers have wondered if [5] D. E.KNurn, TheArto(ComputerProgramming, Vol we would be better off discarding our analytical ume 3: Sorting and Searching, Addison-Wesley, 1973. models and building new empirical models based Classic results in algorithm analysis. entirely on observations. [6] A. LAMARcA and R. E. LADNER, The influence of caches In an experiment to test the relevance ofbig-oh on the performance of sorting, Proceedings of the 8th algorithm analysis to practice, Bentley [2] describes Annual ACM-SIAM Symposium on Discrete Algo rithms, ACM, 1997, pp. 370-379. A very readable a race between two algorithms implemented under technical paper. extreme conditions. He implemented an O(n) al [7] C. McGEOCH, Towards an experimental method for gorithm (under the RAM model) using 1980's algorithm simulation, INFORMS]. Comput. 8 (Win low-end technology, which required 19.5n ter 1995). A survey of methodological issues, with milliseconds to run, and an O(n 3) algorithm in an extensive bibliography. high-end 1999 technology, which required 0.58n3 nanoseconds to run. At small n the cubic algorithm dominates, and at large n the linear algorithm dominates: Bentley found that even with this max imal difference in computing environments, the crossover point was low, near n = 5, 800, where both implementations required around 2 minutes of running time. At n = 105 the O(n3) implemen tation with small coefficients required 7 days,
MARCH 2001 NOTICES OF 1HE AMS 311 Presidential Views: Interview with Hyman Bass
Every other year, when a new AMS president takes office, the Notices publishes interviews with the current president and with the president elect. What follows is an edited version of an interview withAMS president Hyman Bass, whose term began on February 1, 2001. The interview was conducted in October 2000 by Notices senior writer and deputy editor Allyn Jackson. An interview with past president Felix E. Browder appeared in the February 2001 issue of the Notices, pages 187-189.
Notices: First, I'd like to ask you about the AMS' Bass: Well, I don't yet know much in detail about role in advocating support for basic research. Where it, but I know from earlier statements that [NSF di do you think the AMS is right now in this area, and rector] Rita Colwell had, within the NSF agenda, how do you see its activities evolving? given a high priority to redressing what were per Bass: I think right now we are probably in a bet ceived as imbalances in the funding for mathe ter position than we have been in many years. A matics vis a vis the other sciences. What was note great deal of the credit goes to [past presidents] worthy is that this case, which had long been made Arthur Jaffe and Felix Browder. Of course, this by mathematicians, was now being made by a life had a lot to do with changing conditions in the gen scientist who is directing the foundation. eral environment for sdence. But I think that Arthur When things get sufficiently out of balance, and Felix deserve credit for mathematics having even the users of mathematics are threatened. played a significant role in improving the resource And because mathematics is an enabling science, situation. The basic principle behind that, as I see there are lots of "users". The life sciences have al it, was that mathematics moved from a stance of ways been better treated by Congress and the pub pursuing the narrow interests of the discipline to lic, because their applications are in some sense one of reaching out to the public with broad ad closer to personal use and recognized social needs. vocacy of support for basic science in general and When the life scientists see ample documentation doing this in partnership with the other sciences. that the long-term health of basic science is under In this mode, whatever gains mathematics enjoyed threat, then I'd like to think that they are per would be achieved in tandem with growth in sup suaded by the logic of the facts and wise policy. port for science in general. The recent legislative But it takes people with a very broad vision of sci initiative for doubling the NSF [National Science ence as a global enterprise to reach conclusions like Foundation] budget over five years is confirmation that. Most people operate in a much more narrow of the wisdom of that strategy. environment, and they seek to maximize resources From my point of view, I don't see the need for only on a more limited scale. major new initiatives by the AMS, but rather the So the bottom line on this-and I would say need for sustaining the progress that's been this as an overarching statement-is that I am achieved. I should add that major credit goes to lucky to come into a situation in which the AMS the AMS Washington Office, especially the con on every front is in a healthy condition, both as an tacts that Sam Rankin has made with congres organization and as a community. The most trou sional staff there. Thanks to Sam's work, the Wash bled part is probably the one where I am spend ington Office has become an extremely effective ing much of my time, which is education. But even operation and a very productive investment on there I think we're making significant progress. the part of the AMS. Notices: Let's talk a little bit about public aware Notices: Have you heard about the new math ini ness of mathematics. tiative at the NSF?1 What do you think of it? Bass: For one thing, there's now a lot more writ ing on mathematics for the general public than 1 See "NSF Mathematical Sciences Initiative" by Philippe there has been in recent years. This is an area Tondeur in this issue of the Notices, and "NSF Launches where a lot of interesting progress has been made, Major Initiative in Mathematics", Allyn jackson, February and of course public urgings to the math com 2001, pages 190-192. munity to give more attention to public outreach
312 NOTICES OF THE AMS VOLUME 48, NUMBER 3 and to writing good expository material helps nur dynamics of expansion of the ture that. We have journals like the Notices which field. For one thing, the size of I think have been very successful at producing a the field is just much, much lot of excellent material, not least your own writ larger. The student enrollment ing, which I have long admired. The appointment in our universities, the number of the new [public awareness] staff at the AMS is of people who need technical another step in that direction.2 training, and then in turn the Notices: Is there a particular emphasis you think number of professional math the AMS should have in its public awareness efforts? ematicians are vastly larger. Bass: I don't have any specific emphasis in mind, Mathematics as a profession but it should be pursued on a fairly broad front, has a social fabric that is quite and I think that our efforts should be treated in different from what it ever had part as a learning experience. I think we should try before. No one worries that to assess pretty carefully what works and what mathematics is an endangered doesn't, because in these arenas we are starting to species; the ideas have gone function essentially as amateurs. For example, in on for many millennia, and it's Washington there is a professional community unthinkable that mathemati- Hyman Bass that does political work, but we are not asking for cal ideas and culture would the work to be done in that culture. We want the not continue to grow in time. But as a large pro work to be done in a way that represents the cul fessional community we inherit questions that are ture of mathematics. In Washington we have de not about the continued growth and development veloped a small cadre of professionally able peo of ideas, but about the sustained capacity to meet ple who have those skills. The same is true with all the human needs and dependencies that are cre public outreach. We should think that the product ated by the large community that is trained to do of our investment is twofold. One is to produce suc this work. We have to think about the difficulties cessful expository material and promotional ma that are created by inadequate resources or dislo terial that reaches a public audience, and the other cations caused by people having to leave the field is to build a professional capacity to do this sort and do something for which their mathematical of work and to train others to do it. We have to training has not equipped them. think of the Washington outreach and public aware So a lot of issues that we face, even though they ness outreach as partly educational and partly de grew organically out of the growth of the field of veloping the professional culture for this kind of mathematics, are not inherently mathematical work. Once we know better how to do this, we can problems. They are really problems of the pro create internship opportunities for young people fessional community. By the same token, one can so that they can engage in this without abandon understand the growth and evolution of the AMS ing their mathematical careers. as responsive to these social dynamics in the pro Notices: You mentioned earlier your involve fession. I can remember my early experiences on ment in education. What is the AMS role in K -12 AMS committees, where it was hotly debated education? whether the AMS should become a large publish Bass: This is an area in which the AMS did not ing organization. The idea of publishing was viewed organizationally decide to move but in some sense as alien to its scholarly purposes and too much was gradually moved into it by external develop tinged with commercialism. ments, developments that reflect the broader Notices: Nobody worries about that anymore! growth of our professional community. Historically, mathematicians' involvement in Bass: Not only do they not worry about it, but K-12 education was usually seen as episodic. in fact that very enterprise enables the AMS to Certain mathematicians chose to turn their inter function as a kind of enlightened patron of the very aspects ests and reflections in those directions, just as of the culture that are somewhat fragile and that would not mathematicians might become interested in phi easily be supported by external organizations losophy or poetry or music. Interest in education . But these things are always trade offs. was not treated as a movement in the field, but as We are a much larger and more structured or ganization. The first AMS meeting something congenial with it. Those efforts were I went to was at Columbia, hospitably received in the mathematical commu and there were on the order of 100 or 120 people, nity and were treated as a wholesome part of the and one attended every lecture. There general culture, but not as central to it. was more the feeling of a philosophical in quiry rather than a major enterprise. The situation is quite different now, but not In the post-Sputnik era what the country needed because of change of individual int erest or concern. was a cadre of highly trained technical profes A lot of it has to do with the whole interlocking sionals, and our system developed a very high ca 2 See "AMS Establishes Public Awareness Office" in the "In pacity to produce that. Many people failed and side the AMS" section of this issue. many were alienated or driven away from
MARcH 2001 NOTICES OF THE AMS 313 mathematics and science in the process, but that teachers we have are not there and that some ideal was considered okay, because the number of peo community of teachers is suddenly going to appear. ple that got through the filter was enough to meet The teachers in the schools are not dumb or stu national needs. pid and stubborn. They are actually very dedicated What we used to accomplish for a limited num people who love what they do. In most cases they ber of students we now must accomplish for nearly wouldn't be there otherwise, because there are all students, without sacrificing quality levels. We very few incentives. Most of them are actually need to be attentive to the ways in which the dis quite smart and able to learn things. But they have cipline has changed, to the presence of technology, had long experience with subject matter and with to appropriate ways of presenting mathematical kids that is very different from mathematicians' ideas in the classroom, and to contemporary un experience. Teachers are very realistic and have a derstanding of instruction and student learning. real sense of survival and pragmatism, and if they This places great new demands on teachers. The feel that mathematicians are people who are going country has undertaken to solve a problem it never to scorn them or humiliate them, they become de has faced before-that is, to help all students at fensive and will not view mathematicians as a tain high levels of mathematical proficiency. source of help. That kind of thing has happened. One of the first things you have to do when you The mathematicians see themselves as kind of in think about education is to decide what are the tellectual philanthropists and believe the teachers goals, what do you want people to learn? In the U.S. do not want to receive the wisdom they're ready this is a matter for states and districts, sometimes to offer. So there is a lot of that kind of alienation. even individual schools. Never in our nation's his I think that that's much of what the "math wars" tory have goals been articulated and shared at the are about. national level. So the NCTM [National Council of I personally think the NCTM has achieved a Teachers of Mathematics] stepped into this policy great deal, and I think that the new PSSM docu vacuum. The standards NCTM created [in 1989] ment3 is an extraordinary achievement that has were based in part on a combination of educa been well informed by the advice that was sought tional research and the views of some disciplinary from other professional communities. The NCTM mathematicians, but largely also on the wisdom of has made serious and bona fide efforts to ground practice and the knowledge base of professional its policy documents in whatever research is avail practitioners. In my view, it was a positive event able and in solicited advice from other profes that the standards were developed by the profes sional communities. I think that a sensible and sional organization of practicing teachers. constructive way to make improvements is to im Creating standards is the first and the easiest prove the way the NCTM functions. We can't invent step in this business. The next step is curriculum solutions to these educational problems that ignore development, which is complex design work. The the professional community of teachers. The NSF funded many projects to develop curricula rhetoric of mathematicians who publicly protest based on the NCTM standards. Starting in the mid- every single fault and detail in everything the NCTM does is simply not doing the work that's 1990s, these curricula began entering schools. That was the first time this whole movement began going to move us forward. The NCTM has demon strated that it can productively accommodate con to touch people's lives on a significant scale. This structively rendered criticism. precipitated pockets of adverse reaction from par So finally let me answer your question. The ents, whose kids returned with homework that question was, What does K-12 education have to the parents sometimes did not know how to do or do with the AMS? What I've described so far are even recognize. And mathematicians are among ways in which individual mathematicians have parents. It was this concern with their childrens' been drawn into this. On the national level-and schooling that first turned the attention of certain this is now public policy and part of legislation mathematicians toward school mathematics edu it has been recognized that this is a national prob cation. lem and that, in particular, mathematicians and sci When mathematicians first got vivid exposure entists have a special responsibility that extends to what was happening in the schools, many of their traditional roles in research and education at them were outraged. For some it was a perceived neglect of "basic skills", generally understood to 3 Principles and Standards for School Mathematics, pub be the teaching of standard algorithms. This was lished in April 2000, is the updated version of the NCTM often attributed to the early introduction of tech standards. The Notices carried four articles on PSSM: "Re nology into the classrooms. As they looked closer vising the NCTM Standards", january 2000, page 5; they were often alarmed by the seemingly fragile "Updated NCTM Standards Released", june/july 2000, pages 683-684; "Principles and Standards for School mathematical understanding of the teachers. It's Mathematics: A Guide for Mathematicians", September not as if these concerns were without cause. But 2000, 868-876; and "Four Reactions to Principles and the question is, What do you do with what you see? Standards for School Mathematics", October 2000, pages We can't invent solutions that pretend that the 1072-1079.
314 NOTICES OF THE AMS VOLUME 48, NUMBER 3 You the university level to concerns for K-12. This responsibility has taken concrete form in many will like the books. funding programs. There is also the growing recog nition of the fact that the teachers who teach in the schools and whose knowledge of mathemat You.. students ics we deride so much learned their mathematics will like the p ..ices. primarily in mathematics departments. Therefore there is a kind of structural responsibility, even at the university level, to giving more attention to this. So for those various external reasons, the profes sional community of mathematicians and therefore Calculus Lite, 2nd Edition the AMS-because it is the organization of that Frank Morgan community-has an inherent interest in K-12 ed Paperback; 320 pages; $28.00 ucation issues. It is appropriate for the AMS to create a legiti Calculus Lite is an elementary text that mate and respected space for attention to that makes the ideas and practice of calculus easily accessible. The book is used at kind of concern, more or less on the order that it numerous colleges and universities exists right now, with the Committee on Education, including Princeton University, Penn State with programs at the national annual meetings, University, and Mount Holyoke College. with articles in the Notices and elsewhere, and with a few other things. There's no recipe for what order of engagement is appropriate. That's largely up to the opinions and the practices of the com munity. But there are enough active mathemati cians who are interested in educational issues that Discrete Algorithmic the level of attention given to them now seems ap Mathematics, 2nc1 Edition propriate. Mathematicians have important things Stephen B. Maurer and Anthony Ralston to learn-about schools, about teacher education Hardcover; 910 pages; $59.00 and teacher learning, about making change in schools, etc. The AMS can provide opportunities Arguably the best and most comprehen for such learning. sive text, this book is a bargain at 6 cents In addition, there is a huge number of things per page. It will accompany the students beyond the course. about math education now happening in the pub lic domain. So at a very basic level, the AMS sim ply wants to remain aware of these currents. If mathematicians want to voice any concerns or opinions or contribute to educational work and pol icy, we need mechanisms by which connections can The Geometry Toolbox for be made. The Committee on Education affords op ~~~bo-~ Graphics & Modeling portunities for this. Gerald Farin and Dianne Hansford One motif that runs through everything we've Hardcover; 288 pages; $38.00 talked about is outreach-outreach to the public, outreach to other disciplines, and outreach to ed The ideal introduction to linear algebra ucation and to various policy arenas. The AMS, in and analytic geometry for students who want to connect solid mathematics to a measured way and without undermining its cen !.::~~~~~~ current applications, such as computer tral mission and its core commitments to mathe graphics and animation. matical research, is adapting flexibly to these ex panding roles. As long as it performs those roles well and doesn't let them imbalance its central mission, I think that this is a healthy mode of op To Submit a Request for an Examination Copy: Examination copies for text adoption purposes are available to qualified eration. faculty. Please mail or fax your request on department letterhead, including course title, semester offered, enrollment, current text(s), and your adoption decision date.
A K Peters, Ltd. 63 South Ave., Nati ck, MA 01 760-4626 t Tel: (508) 655-9933 Fax : (508) 655-5847 www.akpeters.com • service@ akpeters.com Publishers of Science & Technology
MARCH 2001 NOTICES OF THE AMS 315 Thompson and Uhlenbeck Receive National Medals of Science
considered a founder of geometry based on analytical methods. She is also a leader in encouraging young women to study math ematics. The National Medal of Science was es tablished by Congress in 1959. It was in tended to be bestowed annually by the president of the United States on a select group of individuals deserving of special recognition by reason of their outstanding contributions to knowledge in the physical, biological, mathematical, or engineering John G. Thompson Karen K. Uhlenbeck sciences. Congress expanded this defini tion in 1980 to recognize outstanding work On November 13, 2000, President Clinton in the social and behavioral sciences. In 1962 Pres announced the names of twelve individuals to ident John F. Kennedy awarded the first Medal of receive the National Medal of Science, the nation's Science to the late Theodore Von Karman, profes highest honor for achievements in research in sor emeritus, California Institute of Technology. In twelve 2000 winners, 386 have been science and engineering. Among those honored cluding the awarded the Medal of Science. Since 1995 the fol were two mathematicians, JoHN G. THOMPSON and lowing mathematical scientists have received the KAREN K. UHI.ENBECK. medal: Felix E. Browder, Ronald Coifman, Leo P. is the Graduate Research Professor Thompson Kadanoff, Richard Karp, Cathleen S. Morawetz, of Mathematics at the University of Florida, Louis Nirenberg, Stephen Smale, and S.-T. Yau. Gainesville. He is considered one of the foremost A committee of twelve scientists and engineers group theorists of all time, and his name is asso is appointed by the president to evaluate the ciated with one of the monumental achievements nominees for this award. The National Science of the twentieth century, the classification of all Foundation (NSF) administers the National Medals finite simple groups. He was awarded the Fields of Science for the White House. Medal in 1970. - Compiled from NSF press releases Uhlenbeck holds the Sid W. Richardson Foun dation Chair in Mathematics at the University of Texas at Austin. She made pioneering contributions to global analysis and gauge theory that r esulted in advances in mathematical physics and the theory of partial differential equations. She is
316 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Book Review
The Parrot's Theorem Reviewed by QueenaN. Lee-Chua
The Parrot's Theorem proofs do not fall DenisGuedj into the wrong Translated from the French by Frank Wynne hands? As befits a Weidenfeld & Nicolson, London, 2000 good thriller, this ISBN0-297-64578-1, Hardcover, £12.99 book also has a (First published as Le Theoreme du Perroquet, satisfying twist at Editions du Seuil, Paris, 1998) the end. Yet it is the and Scheduled U.S. Publication: September 2001 mathematics St. Martin's Press the mathemati cians that hold the ISBN 0-312-28055-6, Hardcover, $23.95 novel together. In Literature for young adults often has elements order to solve the of adventure and fantasy, as seen in the works of mystery, the char ]. R. R. Tolkien, C. S. Lewis, and Lewis Carroll. The acters must all take Parrot's Theorem is not only an adventure story but a mathematical his a mathematical novel as well, and it takes the tory course. And reader on an odyssey through history (from Thales what an enlighten- to Wiles). This delightful book is reminiscent of ing course it turns out to be! Here the author, a pro Lewis Carroll's Through the Looking Glass and fessor of the history of science at Universite de Curiosa Mathematica, where mathematical Paris VIII and an award-winning film writer, is at tidbits are golden threads interwoven through a his best. The historical accounts, many of which rich tapestry of fantasy. will be familiar to mathematically inclined readers, The main characters are delineated in consid flow seamlessly through these pages and seem in erable depth-the deaf eleven-year-old Max Liard, no way contrived. How Thales used similar trian whom Euclid would have classified as a solid, since gles to measure the Pyramid of Cheops is explained Max has "length, breadth and depth" (p. 9); the by Mr. Ruche through a slide show, and this con reclusive bookseller Monsieur Pierre Ruche, who cept is extended when Max measures the Obelisk is bequeathed a vast library of mathematical tomes in Paris. Irrational numbers and the Pythagorean by Grosrouvre, a long-lost friend in the Amazon; theorem are the topics of a play by Max and and, of course, Sidney the parrot, a voluble store Mr. Ruche, and the proof that J2 is irrational house of mathematical knowledge-theorems, comes alive in a conversation between Max's proofs, and all. Not since the Cheshire Cat has siblings. To introduce a tour of The Elements, such an entertaining figure been found in the pages Mr. Ruche recounts the origin of one of Euclid's of a mathematical fantasy book. famous, and possibly apocryphal, dictums: When The highly appealing plot has all the elements King Ptolemy asked Euclid if there were no easier replied, of a well-crafted mystery. What exactly caused way to understand mathematics, Euclid to geometry." the sudden death of Grosrouvre in Brazil? What "There is no royal road to expound secrets lurk in Mr. Ruche's library, dubbed the A well-crafted chapter is devoted the three great mathematical problems of Rainforest Library by Max and his family? Why are ing on world-the squaring of the circle, the those goons after Sidney? Who kidnapped Max? the ancient duplication of the cube, and the trisection of an And how can Archimedes, Pythagoras, Fibonacci, angle. The author does not hesitate to use terms Cardano, and others help in ensuring that vital such as cissoid, conchoid, or trisectrix, but through conversation he Queena N. Lee-Chua is associate professor of the help of figures and witty mathematics at the Ateneo de Manila University, Philip manages to convey the meaning to the lay reader pines. Her e-mail address is queena@math. admu. edu. ph. without sacrificing mathematical accuracy.
MARCH 2001 NOTICES OF THE AMS 317 Further explanation of these concepts is found in does it ever occur to us that Native Americans a couple of footnotes which, though they may de refer to a bison as "two-horn"? "Iso-" means "same", tract from the flow of prose in an ordinary novel, and "skelos" means "legs", hence the term are strategically placed here. "isosceles"; but are we aware that "scalene" comes The author does not stick only to ancient math from the Greek word for "limping"? Students know ematics but also includes discussion of the work that algebra comes from al-]abr, but they may be of Newton and Leibniz, Cardano and Tartaglia, intrigued to know that its ancient meaning is Galois, Gauss, Fermat, Cauchy, Pascal, Poisson, "bone-setter". Witness this exchange between and Fibonacci. The book also includes some less Mr. Ruche and his chauffeur, Mr. Habibi (p. 157): familiar but no less fascinating historical figures: the Franciscan monk Luca Pacioli, whose Summa "Al-]abrmeans a bone-setter," said Mr. de arithmetica, published at the height of the Habibi. "Where I come from, if you have Renaissance, was a masterpiece that brought a douar, if you break a thing, you go to Arabic algebra to the West; the physician Nicolas al-]abr and he twist this way-Ow!-and Chuquet, who wrote the earliest algebra textbook that way, and set the bone back in place. in French and was the first to use negative expo Yes, ]abr is someone who fixes some nents; and Bernt Holmboe, whose student Niels thing that is broken." Abel would eventually surpass him. Another chap ter touches on the basics of calculus, such as Mr. Ruche jumped in, "In the story of limits, differentiation, and integration. There is Don Quixote there is an 'algebrist'-a not as much detail as is found in David Berlinski's bone-setter. I never understood why he A Tour of the Calculus, but probably enough to was called an algebrist before. Cer spark the curiosity of the lay reader. The me vantes must have learned the word chanics of logarithms and exponentiation are from the Spanish Moors." gently put forth, and the subsequent segue into However, some typographical errors mar an interest rate problems feels totally natural. Mathematical highlights, such as Goldbach's otherwise engaging read. The architect I. M. Pei, Conjecture, Fermat's Last Theorem, and Euler's is "leo," not "Leo" Ming Pei (p. 199). The sine Conjecture, are also woven into the plot. Aside addition formula is the sum, not the difference, from stories about mathematicians' lives, the book of two products (p. 169). In the display box, the also looks into the origins of common symbols: summation sign is missing in the formula for the among them, the number 0, the equal sign, the total surface area of all the rectangles (p. 237). A square root symbol, the number e, fractional mistranslation appears in the first line of the proof notation, and the coordinate system. of the irrationality of .J'l. Two numbers a and b But perhaps the primary reason this book may are co-prime, not "both prime" (p. 95)1. become a classic in young adult literature is its Still, all's well that ends well. Max is rescued, the literary beauty. The author not only has presented fate of Mr. Ruche's friend is finally revealed, the the mathematics in an entertaining way but has also bad guys receive a well-deserved end, and written some lovely passages that are the linguis Sidney? Ah, Sidney. It would be utterly improper tic equivalents of elegant mathematical proofs. He to reveal the twist, wouldn't it? No wonder the explains the structure of a quatrain ("lines one, two novel became an instant bestseller in France when and four linked by rhyme, while line three is free," it first appeared in 1998. The periodical Le Point p. 144) and discusses what is perhaps the best touts The Parrot's Theorem as "a beautiful book exemplar: Omar al-Khayyam's Rubaiyat, with its glorying in the great adventures of the human haunting verses (pp. 144-5). During a discourse on mind" (inside cover), and it is all this and more. It friendly numbers he mentions that Pythagoras is evidence that there is no fundamental difference reportedly invented the word "friendship" by defin between great mathematics and great literature ing "friend" as "someone who is another me, like both possess an elegance that resonates within the numbers 220 and 284. Two numbers are friends the human soul. if each is the sum of everything that measures the other" (p. 68). This becomes a portent, for shortly Acknowledgment before dying Grosrouvre reminds Mr. Ruche, "What The reviewer thanks Allyn Jackson and the about us-are we friends? What is the sum of those anonymous referee for perceptive comments. things that define you or me? I think perhaps the time is coming when we will find out" (p. 68). References Word play abounds throughout the novel. Gros [1] DAVID BERUNSKI, A Tour of the Calculus, Pantheon rouvre writes Mr. Ruche's first name (Pierre) as a Books, New York, 1995. delightful mathematical-linguistic pun, rrr, 1 Editor's Note: The publisher of the U.S. edition, St. Mar pronounced "pee-air" (in French). Most people tin's Press, told the Notices that these errors would be fixed know that the prefix "bi-" means "two," but in the U.S. edition.
318 NOTICES OF 1HE AMS VOLUME 48, NUMBER 3 AMERICAN MATHEMATICAL SOCIETY
NE.W JOURNAL! 2002 subscriptions available now from the American Mathematical Society
MOSCOW MATHEMATICAL JOURNAL Published by the Independent University of Moscow Editorial Board: Editors: Yulij llyashenko and Michael Tsfasman Dmitri Anosov Vladimir Arnold This international quarterly publishes highest quality research and research-expository papers Sergei Artemov from all over the world. Its purpose is to bring together different branches of our science and to Alexander Belavin achieve the broadest possible outlook on mathematics, characteristic of the Moscow mathematical school in general and of the Independent University of Moscow in particular. Victor Buchstaber An important specific trait of the journal is that it especially encourages research-expository Pierre Cartier papers, which must contain new important results and include detailed introductions, placing the Boris Feigin achievements in the context of other studies and explaining the motivation behind the research. Victor Ginzburg The aim is to make the articles-at least the formulation of main results and their significance John GuckEmheimer readable not only to a narrow class of specialists. Sabir Gusein-Zade 2002 Subscription: Price (four issues) US$150.00* Anatol Katok Askold Khovanski FREE SAMPLE AVAILABLE UPON REQUEST* Alexander Kirillov The journal is published in English beginning in 200 I. An electronic version will be available for Igor Krichever viewing by both subscribers and nonsubscribers through December 200 I. Check Robert MacPherson www.ams.org/bookstore/journals for updated details. Gregory Margulis The first issue appears in January-March 200 I. Subscriptions are available from AMS for the 2002 Nikolai Nadirashvili subscription year. All 2002 subscribers to will receive the inaugural Moscow Mathematical journal Yuri Neretin 200 I issues free of charge as soon as they are published. Sergei Novikov Here is a sample of the papers that will be appearing in this prestigious journal: Jacob Palls • J. Guckenheimer and Yu. llyashenko: The duck and the devil: canards on the staircase. Senya Shlosman • A. Kirillov: Some results on the structure of quantum family algebras. Stephen Smale • Yu. Neretin: Matrix balls, radial analysis of Berezin kernels, and hypergeometric determinants. Alexei Sossinski • S. Shlosman and M. T sfasman: Random lattices and random sphere packing: typical properties. Victor Vassiliev • V. Vassiliev: On combinatorical formulas for cohomology of spaces of knots. Serge Vladut • S. Vladut: Jsogeny classes statistics for abelian varieties over finite fields.
Papers are welcome by email sent to [email protected]. Subscriptions for 2002 can be placed through your usual subscription agents, or can be ordered directly from theAMS*: Moscow Mathematical Journal 2002 American Mathematical Society, P. 0 . Box 6248, Providence, Rl 02940-6248, USA Tel: 401-455-4081, Fax: 401-331-3842, Email: [email protected]
*Institutional subscriptions and sample copies are available from the AMS in the following world regions: North America, United Kingdom, European Union countries, Scandinavia, Switzerland, Israel, Japan, Australasia. For individuals worldwide and institutions in other regions, special prices are applicable (still lower prices for Russian residents and libraries); please direct these subscription inquiries and sample requests to: Moscow Mathematical Journal; Independent University of Moscow, II, B.VIasievsky per., Moscow 121002, Russia; Tel: 7-095-241-0500, Fax: 7-095-291-650 I, Email: [email protected]. AMS AM:eiuCAN MATIIIlMAT!CAL SOCIETY Mathematics People
SARAH S. JOHNSON of Lexington, Kentucky, is a senior at Rhodes Scholarships Awarded Washington University in St. Louis, where she majors in Six mathematics students are among the thirty-two Amer mathematics, environmental studies, and earth and ican men and women chosen as Rhodes Scholars by the planetary sciences. The recipient of both Truman and Rhodes Scholarship Trust. The Rhodes Scholars were Goldwater Scholarships, she has worked on the NASA delegate to the United chosen from 950 applicants who were endorsed by 327 Discovery Mission Team and was a Nations Climate Summit. Johnson is also active as a colleges and universities in a nationwide competition. peer advisor, as a math tutor, and with the Special The names and brief biographical descriptions of the Olympics. At Oxford she will read philosophy, politics, mathematics scholars follow. and economics. of State College, Pennsylvania, is a ZACHARY J BATILES PHilllP N. AssMUs of Madison, South Dakota, is a senior at double major in computer science and mathematics at Luther College in Decorah, Iowa, where he is a triple Pennsylvania State University, where he will also receive major in physics, mathematics, and computer science. A an M.S. in computer science. The winner of many scholar Goldwater Scholar, he has worked at the Fermi National ships and awards in science and engineering, Battles, who Accelerator Laboratory and has won prizes for his work in is blind, has been extremely active in the United States and mathematics. He founded a food drive and is active in internationally in programs for people with disabilities many community and college organizations. Assmus spent and has done community service in Ukraine. At Oxford he a semester at the University of St. Andrews in Scotland and will study for the D.Phil. in numerical analysis. studied women in development in India. He intends to THOMAS M. PALLATHY of Newark, Delaware, graduated study for the M.Phys. in physics at Oxford. from the University of Delaware with a B.A. in mathemat Rhodes Scholarships provide two or three years of study ics and philosophy and an M.A. in linguistics and at the University of Oxford in England. The value of the cognitive sciences. He has published work in neuroscience Rhodes Scholarship varies depending on the academic and is completing a book and documentary on refugee field, the degree (B.A., master's, doctoral), and the Oxford studies. Pallathy has also assisted refugees in Kosovo and college chosen. The Rhodes Trust pays all college and Bosnia, before and after the NATO campaign, and has university fees and provides a stipend to cover necessary advised policymakers on their plight. At Oxford he will expenses while in residence in Oxford, as well as during study for the M.Phil. in development studies. vacations, and transportation to and from England. The BEN M. GooDWIN of Conway, Arkansas, is a senior math total value averages approximately $27,000 per year. ematics major at Hendrix College. A Goldwater Scholar and the winner of many awards for his research and scholar -From a Rhodes Scholarship Trust announcement ship, he has presented mathematical papers at national meetings. Goodwin is also an environmental activist and is involved with Habitat for Humanity and other commu Diaconis Chosen nity and national service organizations and projects. He will read for the B.A. in mathematical sciences at Oxford. von Neumann Lecturer THoMAs S. McCALEB JR. of Tallahassee, Florida, is in his PERsr W. DIACONIS of Stanford University was selected by the final year at the United States Air Force Academy, where Society for Industrial and Applied Mathematics (SIAM) to he majors in mathematics. A Cadet Wing Commander, a deliver the 2000 John von Neumann Lecture. The lecture triathlete, and a member of the Academy parachute team, was titled "Geometry of Markov Chains" and was given at he is ranked second in the Cadet Order of Merit for his the SIAM 2000 Annual Meeting in Puerto Rico. class. He plans to read for the B.A. in modern history and The John von Neumann Lecture prize was established economics at Oxford. in 1959 and is given in the form of an honorarium of
320 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Mathematics People Institute for Mathematical Sciences National University of Singapore $2,500 plus travel expenses to deliver an invited lecture at the SIAM annual meeting. The selected lecturer surveys The National University of Singapore has recently and evaluates a significant and useful contribution to formed the new Institute for Mathematical Sciences, mathematics and its applications. whose mission is to provide an international center of excellence for mathematical research. The institute's -From a SIAM announcement programs will focus on fundamental issues in and appli cations of the mathematical sciences and will also promote interest in those fields and in multidisciplinary 2000 Maria Mitchell Women in research in Singapore and the region. Science Award Each year, the institute will organize two programs, The Maria Mitchell Association has announced the recipi each lasting up to six months, in accordance with devel ents of its Women in Science Award for 2000. They are oping trends in the mathematical sciences and with the CINDA-SUE C. DAVIS, director of the Women in Science and interests of scientists in Singapore and the region. Engineering (WISE) Program, University of Michigan, and Mathematical scientists at junior and senior levels and CATHERINE BANKS, director of the Science and Mathematics graduate students are expected to visit the institute for Center for Women (SMC), Texas Women's University. periods of varying lengths, ranging from one month to Davis has been the director of the WISE program since six months, and to interact with each other through 1984, working to provide enriched learning, research, and workshops, seminars, and informal discussions. mentoring opportunities for girls and women from the precollege years through graduate study. She has been From July to December 2001, the inaugural program of instrumental in developing and hosting a number of the institute will focus on the following areas: important conferences on gender-equity issues in science education. Banks is the founding director of SMC, which Coding Theory and Data Integrity: opened in 1986 and provides resources to enable girls The program will be divided into three parts, each and women to pursue education in science, mathematics, lasting six to eight weeks: computer science, and engineering. The winners each receive a cash award of $5,000. 1. Mathematical foundations (computational number The Maria Mitchell Association makes the annual theory, algebraic curves, and related topics); award to recognize an individual, program, or organization 2. Coding and cryptography (constructions of codes that encourages the advancement of girls and women in and cryptosystems, and related topics); studies and careers in science and technology. Maria 3. Applied cryptology (implementations, commercial Mitchell (1818-1889) was the first woman astronomer applications, and related topics). and first woman astronomy professor in the United States. The award may be given in the natural and physical Each part of the program will include a one-week tuto sciences, mathematics, engineering, computer science, or rial and a one-week workshop. technology. Organizing Committee: Shih-Ping Chan, Robert Deng, -From a Maria Mitchell Association announcement San Ling, Harald Niederreiter (chair), Eiji Okamoto, Igor E. Shparlinski, Neil J.A. Sloane, and Chaoping Xing. The institute invites applications for membership for Deaths participation in the above program. A limited ERNST AUGUST (PETER) BEHRENS, professor emeritus, McMaster number of fellowships, covering travel and living University, died on December 1, 2000. Born on May 11, 1915, expenses, are available to young mathematical scien he was a member of the Society for 44 years. tists. Applications should be received at least three (3) JOSEPH L. CARMEN, IBM, Raleigh, NC, died on October 28, months before the commencement of membership. 2000. Born on April 17, 1966, he was a member of the Society for 1 year. More information and application forms RADHA G. LAHA, professor emeritus, Bowling Green State are available from: University, died on July 14, 1999. Born on October 1, 1930, http://www.ims.nus.edu.s2 he was a member of the Society for 36 years. or by writing to: RoBERT BRUCE McLAUGHLIN, of Herndon, VA, died on Secretary, Institute for Mathematical Sciences July 12, 2000. Born on August 30, 1959, he was a member National University of Singapore of the Society for 2 years. 2 Science Drive 2 KENNAN T. SMITH, professor emeritus, Oregon State Uni versity, died on November 18, 2000. Born on July 17, 1926, Singapore 117543, Republic of Singapore he was a member of the Society for 51 years.
MARCH 2001 NOTICES OF THE AMS 321 Mathematics Opportunities
and independence, and to promote success in graduate Project NExT: New Experiences school. in Teaching Sixteen women will be selected. Each will receive a travel allowance, campus room and board, and a stipend of Project NExT (New Experiences in Teaching) is a program $1,250. for new or recent Ph.D.'s in the mathematical sciences The application deadline is March 1, 2001. For further who are interested in improving the teaching and learning information see the university's Web site, http: I /www. of undergraduate mathematics. It addresses the full range gwu.edu/~math/spwm.html, or contact the codirectors, of faculty responsibilities in teaching, research, and ser Murli M. Gupta (mmg@gwu. edu) or E. Arthur Robinson Jr. vice, and it provides professional support for new faculty (robi nson@gwu. edu), Department of Mathematics, George as they undertake these activities. Each year about sixty Washington University, Washington, DC 20052; telephone faculty members from colleges and universities through 202-994-4857; fax 202-994-6760. out the country are selected to participate in a workshop preceding the MAA summer meeting, activities during -George Washington University announcement MAA meetings, and an electronic discussion network. Fac ulty for whom the 2001-02 academic year will be the first or second year of full-time employment with significant Maria Mitchell Women in teaching responsibilities at the college/university level are Science Award invited to apply to become Project NExT Fellows. The application deadline is April 13, 2001. For more The Maria Mitchell Association offers an annual award to information see the Project NExT Web page, http: I I recognize an individual, program, or organization that en archives.math.utk.edu/projnext/. courages the advancement of girls and women in studies and careers in science and technology. Maria Mitchell -From a Project NExT announcement (1818-1889) was the first woman astronomer and first woman astronomy professor in the United States. The award may be given in the natural and physical 2001 Summer Program for sciences, mathematics, engineering, computer science, or technology. The winner(s) will be chosen by a national Women in Mathematics jury of distinguished educators and scientists. The total cash award is $10,000. Funding for the award has been The George Washington University has announced the provided by an anonymous donor. Guidelines and nomi 2001 Summer Program for Women in Mathematics (SPWM nation forms are available from the association's Web site 2001) to be held June 30-August 4, 2001. SPWM 2001 is at http: I /www. mmo. org/, or contact the Maria Mitchell an intensive five-week program for mathematically tal Women in Science Award Committee at the Maria Mitchell ented undergraduate women who are completing their Association, 2 Vestal Street, Nantucket, MA 025 54; junior years and may be contemplating graduate study in telephone 508-228-9198. Deadline for nominations is the mathematical sciences. The goals of this program are April 30, 2001. to communicate an enthusiasm for mathematics, to develop research skills, to cultivate mathematical self-confidence - From a Maria Mitchell Association announcement
322 NOTICES OF THE AMS VOLUME 48, NUMBER 3 For Your Information
and probability. U.S. students scored below average in Tll\1SS Repeat Study Released measurement and geometry and above average in the the U.S. place among In early December 2000 the National Center for Education other areas. In none of the areas did Statistics (NCES) released a report containing initial the high-scoring nations. questions as findings from the Third International Mathematics and The TIMSS-R test included multiple-choice had Science Study-Repeat (TIMSS-R). Entitled Pursuing well as "free-response" questions in which students one free-response Excellence: Comparisons ofInternational Eighth-Grade Math to present their reasoning. For example, a club of 86 members ematics and Science Achievement from a U.S. Perspective, mathematics question stated that in students to 1995 and 1999, the report builds on data collected during there were 14 more girls than boys and asked the first round of TIMSS, which was held in 1995. The U.S. calculate the number of boys and girls in the club. Twenty has participated in such international studies for around nine percent of U.S. students got this problem right, forty years, but TIMSS-R is the only one designed specifi compared to 33 percent of students internationally. The cally to allow comparison with a prior study. free-response sample problems in mathematics boiled Unlike TIMSS, which collected data on mathematics and down to a single numerical answer, but one of the science science achievement at the fourth-, eighth-, and twelfth problems was open to multiple interpretations. The grade levels, TIMSS-R focused primarily on eighth-grade problem presented a "food web", which is a diagram show achievement. Forty-one countries participated in the ing relationships among dependencies on food sources original TIMSS, and thirty-eight participated in TIMSS-R. for a small group of animals, and asked students to use The latter group included twenty-six nations that had the diagram to figure out what would happen to the participated in TIMSS in 1995, plus twelve others that were robin population if the corn crop failed. The report states participating for the first time. Among those countries that, provided logical explanations were given, three that decided against participating in the repeat study are possible answers were acceptable for full credit: the Austria, France, Germany, and Switzerland. In the U.S. the robin population d ecreases, the robin population TIMSS-R sample consisted of about 9,000 eighth-graders increases, and the robin population stays the same. in 220 schools. Nevertheless, only 3 5 percent of U.S. students got The report separates the participating nations into three this problem right, and the international average was groups whose scores were similar. The U.S. was in the 26 percent. middle group, where scores were above the international The report includes comparative data about the average, along with the Czech Republic, Malaysia, Bulgaria, performance of different U.S. student populations. For Latvia, England, and New Zealand. The top group consisted example, the report states that there was no difference of fourteen nations, including Canada and five Asian between U.S. boys' and girls' performance on the mathe nations. Seventeen nations were in the group scoring at or matics part of TIMSS-R. White students outperformed black below the international average. The report also provides and Hispanic students by a considerable margin. Students data about the highest-scoring students, those whose scores whose parents completed college had higher scores than put them in the top 10 percent internationally. Only 9 per those whose parents did not. cent of U.S. eighth-graders were among these top scorers. Part of the report is devoted to comparing data from The five nations with the largest percentages of top TIMSS and TIMSS-R. No difference was found between the scoring students were all in Asia; these percentages ranged 1995 achievement of eighth-graders on TIMSS and the 1999 from 46 percent in Singapore to 33 percent in Japan. performance of eighth-graders on TIMSS-R. Most nations The report discusses achievement in five mathematics that participated in both studies also found little change. content areas: fractions and number sense; measurement; The only ones registering a significant rise in average scores geometry; algebra; and data representation, analysis, were Latvia, Canada, and Cyprus, and the only one with a
MARcH 2001 NOTICES OF THE AMS 323 For Your Information
significant drop in average score was the Czech Republic. The study found that U.S. black eighth-graders performed significantly better in 1999 than in 1995. A group of seventeen nations participated in the fourth grade TIMSS testing and in the eighth-grade TIMSS-R testing. Comparing data from the two rounds of testing can provide some insight into how students fare as they move through the educational system. Thus one can compare where U.S. fourth-graders placed in this group in 1995 to where U.S. eighth-graders placed in this group in 1999. Such a comparison indicates that the performance of U.S. students deteriorates between the fourth and the eighth grade, going from around average to below average in this group. By contrast, Canadian fourth-graders performed well below average in 1995, but Canadian eighth-graders performed slightly above average in 1999. The period 1995-99 coincides with a period of reform of mathemat ics teaching in many U.S. school districts. However, the report cautions: "Findings from comparisons between the results of TIMSS and TIMSS-R cannot be interpreted to indicate the success or failure of mathematics and science reform efforts in the United States." One part of TIMSS-R collected information about rt?""···· .. ~b'% ~~ teachers' background and preparation. Forty-one percent I of U.S. eighth-grade mathematics teachers held degrees in ~·.·- ~f. mathematics, compared to 71 percent internationally. Still, /" ""V'' . > I .... U.S. teachers, more frequently than their international f counterparts, reported feeling very well prepared to teach l mathematics. On average, 90 percent of U.S. students are being taught by teachers who reported feeling very well ~· prepared to teach mathematics; this percentage was { ~ ;N'' H' ?"i&~~ exceeded only in Macedonia. By contrast, in Japan, where TIMSS-R scores were quite high, just 2 3 percent of students are taught by teachers who reported feeling very well prepared to teach mathematics. The report also notes that, according to information collected from U.S. mathematics teachers, professional development programs tend to emphasize topics like curriculum, teaching methods, and the use of technology and tend not to emphasize :~ %'i'
-Allyn Jackson
Correction: The Notices has recently undergone a change in the manner in which the publication is sub mitted to the printer. As a result of this transition, an erroneous font substitution occurred in the mathemat ical equation in the cover graphic for the February issue. The mathematics should have appeared as:
V(2,l,Ol ® V(2,l,Ol = V(4, 2,0l e V(~~2 .1> e V(4,I,Il e V(3 ,3,0l e V(2,2,2l The Notices regrets this error.
324 NOTICES OF lHE AMS VOLUME 48, NUMBER 3 Inside the AMS
Math Camp (Southwest Texas State University), and the AMS Epsilon Fund University of Michigan Math Scholars. For many years now, summer programs in mathematics The AMS will make a similar number of awards to have provided mathematically talented youngsters with support activities in summer 2001, for a total of around their first serious mathematical experiences. Participation $80,000, again from the Program Development Fund. Once in such "Young Scholars Programs" was a major factor the Epsilon Fund endowment has reached the targeted in the career choice of many mathematicians active in the amount, the AMS intends to award a total of $100,000 in field today. Program participants who eventually chose Epsilon grants each year. career paths outside of mathematics retained a deeper un For further information about the Epsilon Fund for derstanding and appreciation of the field than they would Young Scholars, visit the Web site http: I /www. ams. o rg/ likely have gained through mathematics courses alone. gi vi ng-to-ams/, or contact development@ams. org, Despite their strong track record, many of these pro telephone 800-321-4267, extension 4111, or 401-455-4111. grams experienced financial difficulties during the 1990s, Information about how to apply for Epsilon grants in part because of changes in funding from the National is available at http://www.ams.org/employment/ Science Foundation. As the mathematical community be epsilon.html. A fairly comprehensive listing of came aware of the precarious financial state of the pro summer programs for mathematically talented high grams, leaders in the AMS decided to take action. In 1999 school students (including those with and without the Society started the Epsilon Fund for Young Scholars Epsilon grants) is available at http: I /www. ams . o rg/ to help support summer programs for mathematically employment/mathcamps.html. talented high school students. The name for the fund was chosen in remembrance of the late Paul Erdos, who was fond -Allyn Jackson of calling children "epsilons". At its meeting in November 2000 the AMS Board of Trustees approved the Society's engagement in a sustained AMS Establishes Public effort to raise an endowment for the Epsilon Fund. In addition, a Board-designated fund of $500,000 has been Awareness Office created as a start for the endowment. The AMS hopes to build the endowment to around $2 million through indi In recent years AMS committees and officials have vidual donations and grants from foundations. highlighted the need to raise public appreciation and As a start for the program, the AMS used $75,000 from understanding of mathematics. They believe that mathe its Program Development Fund to award Epsilon grants to maticians must communicate to the general public about seven programs for activities during summer 2000. The the essential role mathematics plays in human culture and modest size of the grants means that they cannot cover society. Such communication could help bolster efforts to the whole cost of the programs. However, the grants can improve mathematics education and to increase funding have a large effect on attracting additional funding. The support for mathematics research. In addition, this programs that received Epsilon grants for summer 2000 kind of outreach naturally feeds into efforts to increase were: All Girls/ All Math (University of Nebraska, Uncoln), collaborations between mathematicians and researchers Hampshire College Summer Studies in Mathematics, Math in other areas of science and engineering. camp, PROMYS (Boston University), Ross Young Scholars To address this need, the AMS has established a Program (Ohio State University), SWT Honors Summer Public Awareness Office at the Society's headquarters in
MARCH 2001 NOTICES OF THE AMS 325 Inside the AMS
Providence. Increasing the coverage of mathematics in the popular press is one goal, but the office will also work on a variety of other fronts. For example, efforts are under way to establish relations with other scientific societies, such as the American Physical Society and the American Chem ical Society, which already have well-established public awareness efforts. The possibility of joint projects to produce radio and TV spots highlighting the impor tance of mathematics and science is being explored. The AMS Public Awareness Office will also provide assistance to the AMS Washington Office in developing materials for outreach efforts in the nation's capital. Other activities include organizing public events and developing Web materials about mathematics. In addition to its main purpose of promoting appreciation and understanding of mathematics, the AMS Public Awareness Office has as a secondary purpose the promotion of AMS activities and achievements. Michael Breen and Annette Emerson have been appointed as AMS public awareness officers. A mathematician cur rently on leave from Tennessee Technological University, Breen has experience in radio and television broadcasting and newspaper writing. Emerson was previously the man ager of the AMS Promotions Department, where among other duties she oversaw the AMS Bookstore on the Web. For further information on the activities of the AMS Public Awareness Office, visit the Web site http: I jwww. ams. org/publ i c-awareness/, or contact . Life. -in the fast lane • . !1: )lsuilly . . . [email protected]. involves ·· a few sacrif~ces. • Your. insu~c:e· ·· coverage doesn't have to be· one of theli!. Whether ·you•re moving on or even ,out on , -Allyn Jackson your own,_ insuranee.. offe?:ed through your , AMS me~rship wc;m' t end j 'uSt beCause · · you •v.e clianged j(Jbs-. ·It travels right in . your back pock~t. .. · ·- , .. .· Take advantage: of one of·· your be$t. mem-. PE!rship benefits. Affordable coverage~ .Reliable providers. Portable ._. be~efits • . call &DO ~2~,&&3 to_speak to ·a customer service representative• .Because an estab- . lished benefits-package fits your changing 'lifestyle. . .' · ' · · '
This program is administered by Seabury & Smith, a Marsh & McLennan Company. Some plans may not be available in all states. The comprehensive healthcare insurance plan is undenvritten by New York Life Insurance Company, 51 Madison Avenue, New York, NY 10010. The member assistance and disability income protection insurance plans are underwritten by Unum Life Insurance Company of America, 15 Corporate Place South, P.O. Box I 387, Piscataway, NJ 08855. The catastrophe major medical and high limit accident insurance plans are undenvritlen by The United States Life Insurance Company, itz the city of New York, (M~mbu Am f'ricm • Gf'nf'rol Fimmcial Gro11p), 3600Roure66, P.O. Box 1580, Neptune, NJ07754-1580. The tenn life insurance plan is undenvritlen by Connecticut General Life Insurance Company, a Cigna Company, Hartford, CT06152.
326 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Reference and Book List
The Reference section of the Notices Technology Research Program. See May 1, October 1, 2001: Applica is intended to provide the reader with http: I /www. itr. nsf. gov/. tions for NSF I AWM Travel Grants for frequently sought information in April 13, 2001: Applications for Women. See http: I lwww. awm-math. an easily accessible manner. New Project NExT. See "Mathematics org/travel grants. html; telephone information is printed as it becomes Opportunities" in this issue. 301-405-7892; e-mail: awm@math. available and is referenced after the April15, 2001: Applications for umd.edu. first printing. As soon as information the second competition for NRC Re May 1, 2001: Burroughs Wellcome updated or otherwise changed, it is search Associateships. See http: I I Fund Career Awards at the Scientific will be noted in this section. www4.nationalacademies.org/ Interface. See http: I lwww. bwfund. org/interfaces_in_science.htm, Contacting the Notices osep/ rap/, or contact the National Research Council, Associateship or call Debi Linkous, program associ The preferred method for contacting ate, telephone 919-991-5116. Programs(TJ 2114), 2101 Constitution the Notices is electronic mail. The August 15, 2001: Applications for editor is the person to whom to send Avenue, NW, Washington, DC 20418; the third competition for NRC Re articles and letters for consideration. telephone 202-334-2760; fax202-334- search Associateships. See http: I I Articles include feature articles, 2759; e-mail: rap@nas. edu. www4.nationalacademies.org/ memorial articles, communications, April 23-25, 2001: Full proposals osep/rap/, or contact the National opinion pieces, and book reviews. The for large projects for NSF Information Research Council, Associateship Pro editor is also the person to whom to Technology Research Program. See grams (TJ 2114), 2101 Constitution send news of unusual interest about http:llwww.itr.nsf.govl. Avenue, NW, Washington, DC 20418; other people's mathematics research. April 30, 2001: Nominations for The managing editor is the person telephone 202-334-2760; fax 202-334- the Maria Mitchell Women in Science to whom to send items for "Mathe 2759; e-mail: rap@nas. edu. matics People", "Mathematics Oppor Award. See "Mathematics Opportuni October 1, 2001: Nominations for tunities", "For Your Information", ties" in this issue. the Emanuel and Carol Parzen Prize. "Reference and Book list", and "Math ematics Calendar". Requests for Where To Find It permissions, as well as all other A brief index to information that appears in this and previous issues of the inquiries, go to the managing editor. Notices. The electronic-mail addresses are AMS Bylaws-November 1999, p. 1252 [email protected] of the editor and noti ces@ams. org in AMS e-Mail Addresses-November 2000, p. 1288 the case of the managing editor. The AMS Ethical Guidelines- June 1995, p. 694 fax numbers are 979-845-6028 for the AMS Officers 1999 and 2000 (Council, Executive Committee, editor and 401-331-3842 for the man Publications Committees, Board of Trustees)-May 2000, p. 591 aging editor. Postal addresses may be AMS Officers and Committee Members- October 2000, p. 1127 found in the masthead. Conference Board of the Mathematical Sciences- September 2000, p. 913 Upcoming Deadlines Information for Notices Authors-January 2001, p. 39 March 1, 2001: Applications for EDGE Mathematics Research Institutes Contact Information-August 2000, p. 786 Summer Program. See http: I /www. brynmawr.edu/Acads/Math/edge/ National Science Board- February 2001, p. 216 edge. html. New journals for 1999- ]une/]uly 2000, p. 688 March 1, 2001: Applications for NRC Board on Mathematical Sciences and Staff- April 2000, p. 494 the George Washington University NRC Mathematical Sciences Education Board and Staff- April2000, p. 494 2001 Summer Program for Women in NSF Mathematical and Physical Sciences Advisory Committee-March 2001, Mathematics (SPWM 2001). See "Math p. 328 ematics Opportunities" in this issue. Program Officers for Federal Funding Agencies- October 2000, p. 1100 April 9-11, 2001: Full proposals (DoD, DoE); November 2000, p. 129 1 (NSF) for group projects for NSF Information
MARCH 2001 NOTICES OF THE AMS 327 Reference and Book List
Submit nominations to J. H. Matis, Massachusetts Institute of *The Brain: Unraveling the Mystery Department of Statistics, Texas A&M Technology of How It Works (The Neural Network University, College Station, TX 77873- Process), by Thomas L. Saaty. RWS Pub 3143. Bernard V. Khoury (10101) lications, 2000, ISBN 1-888603-02-X. American Association of Physics The Bride of Science, by Benjamin MPS Advisory Committee Teachers Woolley. MacMillan, August 1999. ISBN Following are the names and affilia 0-333-72436-4. tions of the members of the Advisory Thomas B. W. Kirk (10101) Chance Rules: An Informal Guide to Committee for Mathematical and Brookhaven National Laboratory Probability, Risk, and Statistics, by Physical Sciences (MPS) of the National Brian S. Everitt. Springer, August 1999. Science Foundation. The date of the Lon Mathias (10103) ISBN 0-387-98768-1. expiration of each member's term is Department of Polymer Science The Crest of the Peacock: The Non given after his or her name. The Web University of Southern Mississippi European Roots of Mathematics, by site for the MPS directorate may be George Gheverghese Joseph. Princeton found at http: I lwww. nsf. gov I julia Phillips (10103) University Press, October 2000 (new homelmpsl. The postal address is Materials Science & Technologies edition). ISBN 0-691-00659-8. Directorate for the Mathematical and Sandia National Laboratories Divine Harmony: The Life and Physical Sciences, National Science Teachings of Pythagoras, by John Foundation, 4201 Wilson Boulevard, David Siegmund (10103) Strohmeier and Peter Westbrook. Arlington, VA 22230. I• Department of Statistics Berkeley Hills Books, November 1999. Stanford University ISBN 0-965-37745-8. Ronald Brisbois (10102) The Dots and Boxes Game, by Elwyn Chemistry Department j. Anthony Tyson (10101) Berlekamp. A K Peters, July 2000. ISBN Macalester College Lucent Technologies 1-568-81129-2. Duelling Idiots and Other Probabil Arturo Bronson (ex officio) Neil deGrasse Tyson (10103) ity Puzzlers, by Paul ]. Nahin. Prince Materials Center for Synthesis and Hayden Planetarium ton University Press, October 2000. Processing American Museum of Natural ISBN 0-691-009 79-1. University of Texas, El Paso History Education of a Mathematician, by Philip ]. Davis. A K Peters, August Tony Chan (10102) Book List 2000. ISBN 1-568-81116-0. (Reviewed Department of Mathematics January 2001.) Book List books that University of California, Los Angeles The highlights Einstein in Love: A Scientific have mathematical themes and hold Romance, by Dennis Overbye. Viking including Alexandre]. Chorin (10101) appeal for a wide audience, Press, October 2000. ISBN 0-670- Department of Mathematics mathematicians, students, and a sig 89430-3. University of California, Berkeley nificant portion of the general public. Excursions into Mathematics: Mil 'When a book has been reviewed in the lennium Edition, by Anatole Beck, Billy joe Evans (10102) Notices, a reference is given to the Michael N. Cleicher, and Donald W. Department of Chemistry review. Generally the list will contain Crowe. A K Peters, 2000. ISBN 1-56881- University of Michigan, Ann Arbor only books published within the last 115-2. two years, though exceptions may be The Fermat Diary, by C.]. Mozzochi. S. james Gates jr. (10103) made in cases where current events AMS, 2000. ISBN 0-8218-2670-0. Physics Department (e.g., the death of a prominent math The Game's Afoot! Game Theory in University of Maryland ematician, coverage of a certain piece Myth and Paradox, by Alexander of mathematics in the news) warrant Mehlmann. AMS, 2000. ISBN 0-8218- Lila M. Gierasch (10101) drawing readers' attention to older 2121-0. Department of Biochemistry and books. Suggestions for books to include Geometry from Africa: Mathemat Molecular Biology on the list may be sent to the manag ical and Educational Explorations, by University of Massachusetts, ing editor, e-mail: noti ces@ams. org. Paulus Gerdes. Mathematical Associ Amherst ation of America, April1999. ISBN 0- The Advent of the Algorithm: The 88385-715-4. Fiona Goodchild (10103) Idea That Rules the World, by David G6del: A Life of Logic, by John L. Materials Research Laboratory Berlinski. Harcourt, March 2000. ISBN Casti and Werner DePauli. Perseus, University of California, 0-151-00338-6. August 2000. ISBN 0-738-20274-6. Santa Barbara Angles of Reflection: Logic and a G6del Meets Einstein: Time Travel in Mother's Love, by Joan L. Richards. the G6del Universe, by Palle Yourgrau. jacqueline N. Hewitt (10101) W. H. Freeman, May 2000. ISBN 0- Open Court, November 1999. ISBN 0- Department of Physics 7167-3831-7. 812-69408-2.
328 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Reference and Book List
Hex Strategy: Making the Right Con Newton's Gift: How Sir Isaac New Uncle Petros and Goldbach's Con nections, by Cameron Browne. A K ton Unlocked the System of the World, jecture, by Apostolos Doxiadis. Blooms Peters, May 2000. ISBN 1-568-81117-9. by David Berlinski. Free Press, October bury USA, February 2000. ISBN 1-582- A History of Algorithms: From the 2000. ISBN 0-684-84392-7. 34067-6. (Reviewed November 2000.) Pebble to the Microchip, edited by Jean Niels Hendrik Abel and His Times: The Universal Computer: The Road Luc Chabert. Springer, September Called Too Soon by Flames Afar, from Leibniz to Turing, by Martin 1999. ISBN 3-540-63369-3. by Arild Stubhaug, translated by Davis. W.W. Norton & Company, Octo Infosense: Turning Data and Infor R. Daly. Springer, May 2000. ISBN 3- ber 2000. ISBN 0-393-04785-7. mation into Knowledge, by Keith 540-66834-9. The Universal History of Computing: Devlin. W. H. Freeman, June 1999. ISBN Number: From Ahmes to Cantor, From the Abacus to the Quantum Com 0-716-73484-2. by Midhat Gazale. Princeton Uni puter, by Georges Ifrah; translated john von Neumann: The Scientific versity Press, March 2000. ISBN 0-691- from the French and with notes by Genius Who Pioneered the Modern 00515-X. E. F. Harding; assisted by Sophie Wood, Computer, Game Theory, Nuclear Philosophy of Mathematics: An Ian Monk, Elizabeth Clegg, and Guido Deterrence, and Much More, by Nor Introduction to a World of Proofs and Waldman. John Wiley & Sons, Novem man Macrae. AMS, October 1999. ISBN Pictures, by James Robert Brown. ber 2000. ISBN 0-471-39671-0. 0-8218-2064-8. Routledge, August 1999. ISBN 0-415- The Universal History of Numbers: The Kingdom of Infinite Number: 12274-0. (Reviewed November 2000.) From Prehistory to the Invention of the A Field Guide, by Bryan Bunch. W. H. The Parrot's Theorem, by Denis Computer, by Georges Ifrah; translated Freeman, January 2000. ISBN 0-716- Guedj. Weidenfeld & Nicolson, June from the French by David Bellos, E. F. and Ian Monk. 73388-9. 2000. ISBN 0-297-64578-1. (Reviewed Harding, Sophie Wood, Mathematical Sorcery: Revealing in this issue.) John Wiley & Sons, December 1999. The Story ISBN 0-471-37568-3. the Secrets of Numbers, by Calvin C. Proofs and Confirmations: The Unknowable, by Gregory Chaitin. Clawson. Plenum Press, May 1999. of the Alternating Sign Matrix Con Springer, August 1999. ISBN 9-814- ISBN 0-306-46003-3. jecture, by David M. Bressoud. MAA Spectrum Series, published jointly 02172-5. The Math Gene: How Mathematical with Cambridge University Press, What Are the Odds? Chance in Thinking Evolved and Why Numbers August 1999. ISBN 0-521-66646-5. Everyday Life, by Michael Orkin. W. H. Are Like Gossip, by Keith Devlin. The Pursuit of Perfect Packing, by Freeman, December 1999. ISBN 0-716- Basic Books, August 2000. ISBN: Tomaso Aste and Denis Weaire. Insti 73560-1. 0465016189. (Reviewed February tute of Physics Publishing, July 2000. Where Mathematics Comes From: 2001.) ISBN 0-750-30648-3. How the Embodied Mind Brings Math Mathematics As Sign: Writing, Imag Riemann, Topology, and Physics, by ematics into Being, by George Lakoff ining, Counting, by Brian Rotman. Michael Monastyrsky; translated by and Rafael Nunez. Basic Books, Octo Stanford University Press, September Roger Cooke, James King, and Victoria ber 2000. ISBN 0-465-03770-4. 2000. ISBN 0-804-73684-7. King. Birkhauser, second edition, May The Wild Numbers, by Philibert Mathematics: Frontiers and Per 1999. ISBN 3-764-33789-3. Schogt. Four Walls Eight Windows, spectives, V. Arnold, M. Atiyah, P. Lax, Squaring the Circle: The War between April 2000. ISBN 1-568-58166-1. and B. Mazur, editors. AMS, December Hobbes and Wallis, by Douglas M. (Reviewed November 2000.) 1999. ISBN 0-8218-2697-2. Jesseph. University of Chicago Press, Women Becoming Mathematicians: Mathematics Success and Fa ilure December 1999. ISBN 0-226-3 9899-4 Creating a Professional Identity in Post among African American Youth: The (hardcover), 0-226-39900-1 (paperback). World War II America, by Margaret A.M. Roles of Sociohistorical Context, Com Stephen Smale: The Mathematician Murray. MIT Press, September 2000. munity Forces, School Influence, and Who Broke the Dimension Barrier, ISBN 0-262-13369-5. Individual Agency, by Danny B. Martin. by Steve Batterson. AMS, February Wonders ofNumbers: Adventures in Lawrence Erlbaum Associates, Decem 2000. ISBN 0-8218-2045-1. (Reviewed Math, Mind, and Meaning, by Clifford A. ber 1999. ISBN 0-805-83042-1. December 2000.) Pickover. Oxford University Press, Mathematics Unlimited: 2001 and Surfing through Hyperspace: Un September 2000. ISBN 0-195-13342-0. Beyond, Bjorn Engquist and Wilfried derstanding Higher Universes in Six Zero: The Biography of a Danger Schmid, editors. Springer, September Easy Lessons, by Clifford A. Pickover. ous Idea, by Charles Seife. Viking Press, 2000. ISBN 3-540-66913-2. Oxford University Press, September February 2000. ISBN 0-670-88457-X. My Numbers, My Friends: Popular 1999. ISBN 0-195-13006-5. (Reviewed October 2000.) Lectures on Number Theory, by Paulo The Symbolic Universe: Geometry Ribenboim. Springer, February 2000. and Physics 1890- 1930, edited by ISBN 0-387-98911-0. Jeremy Gray. Oxford University Press, The Mystery of the A leph: Mathe September 1999. ISBN 0-198-50088-2. matics, the Kabbalah, and the Search Two Millennia ofMath ematics: From for Infinity, by Amir D. Aczel. Four Archimedes to Gauss, by George M. Walls Eight Windows, November 2000. Phillips. Springer, July 2000. ISBN 0- *Added to the "Book List" since the list's ISBN 1-568-58105-X. 387-95022-2. last appearance.
MARCH 2001 NOTICES OF THE AMS 329 AMERICAN MATHEMATICAL SOCIETY
.~~~·-"~~=~--~~~ Tp Ameri~kn Mathe~ftif al SocietY is seeking applications and nominations for can df d~tes for tJle P;osition of Associat¢ \Treasurer of the Society. Jlhe primacy respohsibllitiles of the ~sociate Treasurer are to know and understand the budget of the Sodetj, to moni~or the financial condition of the Society and to f-Uvise tlie Board of Tl.1uste,es conc~rning the financial consequences of its decisions. :f-Ile :ASsociate- reasure-w~cl<.s in dose cooperation with the Treasurer, and serves as ~fieisurer:When:necessaiy.=-:.1 L - . The Associate Treasurer is ,a member of the Board of Trustees, the Agenda and Budg~t Committee, the 'investment Committee, and the Council, and serves on sev- eral other: committees of the Society. As a member of the Council, the Associate Treasurer, with the Treasurer, acts as liaison between the Board of Trustees and the Council and offers advice to the Council on the financial aspects of its deliberations. There are two other areas of major responsibility: ( 1) The Associate Treasurer, along with the Treasurer, serves as a liaison between the Board of Trustees and the Executive Director's office in monitoring staff appointments, promotions, and salaries. (2) The Associate Treasurer serves as the liaison trustee for Mathematical Reviews, and in that capacity, monitors the budget of the Mathematical Reviews. While the term of office is two years, it is anticipated that the person filling this office will be reappointed biennially for a number of terms, to ensure continuity. Applications and nominations should be sent to the chair of the search committee, Linda Keen, or to the Secretary of the Society, Robert J. Daverman. Linda Keen Robert J. Daverman Dept. of Math. and Comp. Science Secretary, AMS Lehman College, CUNY Department of Mathematics Bedford Park Blvd. University of Tennessee Bronx, NY 10468 Knoxville, TN 37996-1330 [email protected] daverman@math. utk.edu Applications or nominations received by 15 September 2001 will be assured full con sideration. The newly appointed Associate Treasurer will take office formally on 01 February 2003 but should be appointed by the Council early in 2002 to permit a smooth transition. All necessary expenses incurred by the Associate Treasurer in the performance of duties for the Society are reimbursed, including travel and communications.
GAMSliNDED \'ti AMERICAN MATHEMATICAL SOCIETY AMERICAN MATHEMATICAL SOCIETY
Leroy P. Steele Prizes·.
tilt The selection committee for this prize requests nominations for consid eration for the 2002 award. Further information about this prize can be c found in the November 1999 Notices, pp. 1258-1269 (also available at 0 http://www.ams.org/ams/prizes.html). Three Leroy P. Steele Prizes are awarded each year in the following cat ·- egories: (I) the Steele Prize for Lifetime Achievement: for the cumulative ~ influence of the total mathematical work of the recipient, high level of c research over a period of time, particular influence on the development of a field, and influence on mathematics through Ph.D. students; (2) the ·- Steele Prize for Mathematical Exposition: for a book or substantial sur E vey or expository-research paper; and (3) the Steele Prize for Seminal Contribution to Research: for a paper, whether recent or not, that has 0 proved to be of fundamental or lasting importance in its field, or a model of important research. In 2002 the prize for Seminal Contribution to z Research will be awarded for a paper in Geometry/Topology. a. Nominations with supporting information should be submitted to the Secretary, Robert J. Daverman, American Mathematical Society, 3 12D 0 Ayres Hall, University of Tennessee, Knoxville, TN 37996-1330. Include a short description of the work that is the basis of the nomination, includ • A curriculum vitae should be included. ing complete biographic citations. - The nominations will be forwarded by the Secretary to the prize selec -te tion committee, which will, as in the past, make final decisions on the u awarding of prizes. Deadline for nominations is March 3 I , 200 I. Edltprs-ln=Chlef Benson s. Farb • Robert l. Zimmer University of Gt/cago, IL, USA
Edltodal Bollrcl Tobias HolckColdlng, New Ycri- U1/velsit); lJ.S4; Kenji Fulcaya, ~ ~ .liJpan, William Mark Goldman, IJniflersity of Maryland, COllege Park, li.S4; RotislaY lvanovich Grigorchuk, Sl1!!ldov Ins/itute of Mathematics, ~ Russia; Gerard B.M. van der Geer, Universily ofAmsterdam, The Nelheriands, Karsten Grove, UniversilyofMaryland, COllege Park, l.54; Joel Hass, IJniversity ofCi1lifomia, .a:Ms; l.54; Toshiyuki Kobayashi, University ofT~ Japan, John W. Lott, University ofMichigan, Ann AltJot; l.54; Alex Lubotzky, The Hebrew lJniversit:}: Jetusalem, Israel, Lee D. Mosher, Rutgers University, Newark, NJ, lJ.S4; Jean-Piene Otal, ENS de L}Of?, France, Alan W. Reid, UniversityofTexas,Austin, lJ.S4; RalfJ.Spatzier, I.JniversityofMichigan,AnnArbot; USA; Shmuel A. Weinberger, University of01iaJgo, 11, l.54
Geometriae Dedicata aims to be a vehicle for excellent publications in geometry and its relationship to topology, group theory and the theory of dynamical systems. Features of the journal include: 1. A fast turn-around time for articles. 2. A "Surveys in Geometry" series: this will be a series of survey articles on active areas in geometry, directed to other geometers. 3. Special issues centered on specific topics.
Forthcoming papers
"An extension theorem for Euler characteristics of groups" ...... J. Stallings "Sure les immeubles hyperboliques" ...... Dam len Gabor.lau. and Frederic Paulin "Actions of semisimple Ue groups on circle bundles" ...... Dave Witte and Robert J. Zimmer "On quasicon\leX subgroups of word hyperbolic groups" ...... G. N. Arzhantseva "The lamplighter group as a group generated by a 2-state automaton, and its spectrum" ...... » » ...... Rostislav I. Grigorchuk and Andrzej Zuk "On Kerckhoff minima and pleating loci for quasi-Fuchsian groups" ...... Caroline Series "Desargues theorem, dynamics, and hyperplane arrangements" ...... Richard Evan Schwartz "Properly discontinuous groups of affine transformations -A survey" ...... Herbert Abels "Simple curves on surfaces" ...... Igor Rlvln "On subgroup separability in hyperbolic Coxeter groups" ...... D. D. long, Alain W. Reid "The proalgebraic completion of rigid groups" ...... » » ...... Hyman Bass, Alexander Lubotzky, Andy R.. Magid, Shahar Mozes "Uniform growth in groups of exponential growth" ...... Pierre de Ia Harpe
ISSN 0046-5755, 2001, Volumes 84-88 (151ssues) Paper Version: NLG 1473.00 I USD 1400.00 Online Version: EUR01473.00 I USD 1400.00 Combined Paper & Online Version: EURO 1767.00 I USO 1680.00 ~ 0 w .._(!) 0 0 ...-LO <(
LL a:w Mathematics Calendar
The most comprehensive and up-to-date Mathematics Calendar information is available on e-MATH at http://www. ams . org/ mathcal/ .
March 2001 submit a bstracts ( maximum 300 words) and full-length papers neatly typed (double-spaced) on one s ide of size four bond paper 1- 3 Fourth New Mexico Analysis Seminar, The University of New to t he organizing secretary according to the schedule. All the Mexico, Albuquerque, New Mexico. (Jan. 2001, p. 51) submitted papers will be reviewed b y the program committee, 2-3 Southeastern Analysis Meeting (SEAM XVII), University of and the recommended reviewed papers will be published in the Georgia, Athens, Georgia. (Nov. 2000, p. 1297) proceedings of the conference. Papers are invited in the following areas and related t opics: combinatorics, structural graph theory, 8- 9 DIMACS Workshop on Protein Structure and Structural Ge extremal graph theory, algebraic graph theory, graph algorithm, nomics: Prediction, Determination, Technology and Algorithms, applications of graph theory. ey. (Jan. DIMACS Center, Rutgers University, Piscataway, New Jers Information: shan@annauni v . edu. 2001, p. 51) 8-1 0 Seminar on Stochastic Processes 2001, The University of 1 6-1 8 AMS Southeastern Sectional Meeting, University of South Florida, Gainesville, Florida. (May 2000, p. 596) Carolina, Columbia, South Carolina. (Jan. 1998, p. 11 3) Information: W. Drady, wsd@ams. org. 8- 11 Workshop on Population Genetics at the Molecular Level, Centre de Recherches Mathematiques, Montreal, Quebec, Canada. 19- 24 IV International Symposium on Hamiltonian Systems (OcL 2000,p. 1145) and Celestial Mechanics, Mathematics Research Center (CIMAT), 9- 11 Around Dynamics. A Conference to Celebrate jack Milnor's Guanajuato, Mexico. (Dec. 2000, p. 1433) 70th Birthday, SUNY at Stony Brook, New York. (Feb. 2001, p. 249) 22- 24 Finite Simple Groups, Geometries, Buildings, and Related 9-11 The Third International Meeting of Origami Science, Math, Topics- Conference in Honor of Ernest Shult, Kansas State and Education, Asilomar, Monterey, California. (SepL 2000, p. 977) University, Manhattan, Kansas. (Dec. 2000, p. 1434) 1 0-1 3 2001 ASL Annual Meeting, Philadelphia, Pennsylvania. (Feb. 22- 24 Richard Brauer (1901-1977): Taking His Ideas into the , p. 1297) 2001, p. 249) 21st Century, Stuttgart, Germany. (Nov. 2000 and 12- 16 DIMACS Workshop on Algorithmic and Quantitative As 25- 30 Sixth International Conference on Approximation in the Caribbean, Guatemala City, Guatemala. (Oct. pects of Real Algebraic Geometry in Mathematics and Computer Optimization Science, DIMACS Center, Rutgers University, Piscataway, New jersey. 2000, p. 1145) (Jan. 2001, p. 51) 26- 2 7 DIMACS Workshop on Resource Management and Schedul ing in Next Generation Networks, DIMACS Center, Rutgers Uni '' 14-1 6 Conference on Graph Theory and Its Applications, Anna versity, Piscataway, New jersey. (Jan. 2001, p. 51) University, Chennai, India. Topics: Interested r esearchers and graph theorists are invited to * 2 6- 2 9Tenth International Conference on Approximation Theory,
This section contains announcements of meetings and conferences should be sent to the Editor of the Notices in care o f the American of interest to some segment of the mathematical public, including ad Mathematical Society in Providence or electronically to notices@ams . org hoc, local, or regional meetings, and m eetings and symposia devoted or mathcal @ams . org. to specialized topics, as well as announcements of regularly scheduled In order to allow participants to arrange their travel plans, organizers of meetings of national or international m athematical organizations. A meetings are urged to submit information for these listings early enough complete list of meetings of the Society can be found on the last page of to allow them to appear in more than one issue of the Notices prior t o each issue. the meeting in question. To achieve this, listings should be received in An announcement will be published in the Notices if it contains a call Providence six months prior to the scheduled date of the meeting. for papers and specifies the place, date, subject (when applicable), and The complete listing of the Mathematics Calendar will be published only the s peakers; a second announcement will be published only if there in the September issue of the Notices. The March, June, and December are changes or necessary additional information. Once an announcement issues will include, along with new announcements, references to any has appeared, the event will be briefly noted in every third issue until previously announced m eetings and conferences o ccurring within the it has been held and a reference will be given in parentheses to the twelve-month p eriod following the month of those issues. New infor month, year, and page of the issue in which the complete information mation about meetings and conferences that will occur later than t he appeared. Asterisks (*) mark those announcements c ontaining new or twelve-month p eriod will be announced once in full and will n ot be revised information. repeated until the date of the conference or meeting falls within the In general, announcements of meetings and conferences held in North twelve-month period. America carry only the date, title of meeting, place of meeting, names of The Mathematics Calendar, as well as Meetings and Conferences of speakers (or sometimes a general statement on the program), deadlines the AMS, is now available electronically through e-MATH on the World for abstracts or contributed p apers, and source of further information. Wide Web. To access e-MA TH, use the URL: http: I I e-math. ams . orgl Meetings held outside the North American area may carry more detailed (or http: I l www. ams . orgl). (For those with VT100-type terminals or for information. In any case, if there is any application deadline with respect those without WWW browsing software, connect t o e-MATH via Telnet to participation in the meeting, this fact should be noted. All commu (t el net e-math. ams . org; login and password e- math) and use the Lynx nications o n m eetings and conferences in the mathematical sciences option from the main menu.)
MARCH 2001 NOTICES OF THE AMS 333 Mathematics Calendar
St. Louis, Missouri. Organizers: M. Safonov (chair; safonov 334 NOTICES OF TilE AMS VOLUME 48, NUMBER 3 Mathematics Calendar 28-June 1 The Tarski Centenary Conference (including the Univ. Autonoma de Madrid; A. Volcic, Univ. di Triest; S. Wainger*, workshop: Decidability and Complexity, May 27-29), Warszawa, Univ. of Wisconsin-Madison. Poland. (Jan. 2001, p. 52) Information: http : I lwww . matapp . unimib . i tl convex2001l or e 30-June 2 Thirteenth Biennial Conference of the Association of mail:convex2001@matapp .unimid. it. Christians in the Mathematical Sciences, Calvin College, Grand 2 Rapids, Michigan. (Dec. 2000, p. 1434) * 12-14 M SABI'01-The IMACS/IFAC Fourth International Sympo sium on Mathematical Modelling and Simulation in Agricultural 31-June 3 IMACS-ACA'2001, The Seventh International IMACS and Bio-lndustries, Haifa, Israel. Conference on Applications of Computer Algebra, Technical Scope: The symposium will include both contributed papers and Vocational Institute (TVI), Albuquerque, New Mexico. (Jan. 2001, special sessions in the following areas: Modeling strategies, Object p. 52) oriented modeling, Methods and tools for system identification and parameter estimation, Distributed models, Model based control, june 2001 Biophysical modeling, Multi-modal models, Design of "model 3-8 The 6th International Conference on Mathematical Popula biological system" experiments. tion Dynamics, Marrakech, Morocco. (Jan. 2001, p. 52) Special Activities: Special activities for the participants and their companions will include pre- and postsymposium tours to Galilee, 4- 8 International Conference of Computational Harmonic Anal Jerusalem, Eilat, and the Red Sea, the Dead Sea, and the Hashemite ysis, City University of Hong Kong. (Dec. 2000, p. 1435) Kingdom of Jordan. A professional tour to one Israel Ministry of 4-9 Fractals in Graz 2001: Stochastics-Analysis-Dynamics Agriculture Research Station is also planned. Information about the Geometry, Technical University of Graz, Graz, Austria. (Oct. special activities is available from the Symposium Secretariat at C. 2000, p. 1146) Mayer, 59 Ha'atzmaut Road, Haifa 33033, Israel; tel: 972-4-8524254; fax: 972-4-8522491; e-mail: carel@palex. co . il. 4-1 0 Second Goteborg Conference in Harmonic Analysis and Information: Prospective authors and participants are invited to Partial Differential Equations, Chalmers University of Technol 2 ogyi Goteborg University, Goteborg, Sweden. (Jan. 2001, p. 52) visit the M SABI'01 home page located at http: I lwww . technion. ac . il/technion/agr/m2sabi01 .html or contact the general chair: 6-1 0 The 3rd International Conference on Mathematical Biology, P. 0. Gutman, Faculty of Agricultural Engineering, Technion-Israel Guilin, China. (Sept. 2000, p. 978) Institute of Technology, Haifa 32000, Israel; tel: 972-4-8292811; fax: 972-4-8221529; e-mail: [email protected]. 8-1 0 joint Meeting of the Belgian and German Mathematical Societies 2001 BMS·DMV Meeting, University of Uege, Belgium. 14-21 Graphs and Patterns in Mathematics and Theoretical (Sept. 2000, p. 978) Physics, A Conference to Celebrate Dennis Sullivan's 60th * 8-1 0 Nonlinear Dynamics and Chaos: Where Should We Go From Birthday, SUNY at Stony Brook, New York. (Feb. 2001, p. 250) Here?, Burwalls Conference Centre, University of Bristol, UK. 14-2 3 Third International Conference on Geometry, Integrability Workshop Topics: The aim of the meeting is to allow the next and Quantization, Sts. Constantine and Elena Resort (near Varna), generation of nonlinear scientists to find an agenda for research di Bulgaria. (Jan. 2001, p. 53) rections ofthe future, focussed on three key themes: neural systems; spatially extended systems and pattern formation; applications to 16-1 9 Sixteenth Annual IEEE Symposium on Logic in Computer biology, physics, and engineering. Science (LICS 2001 ), Boston, Massachusetts. (Dec. 2000, p. 1435) Invited Speakers: P. Bressloff (Utah), B. Ermentrout (Pittsburgh), J. 18-22 Fourth European Conference on Elliptic and Parabolic Cowan (Chicago), J. Guckenheimer (Cornell), E. Knobloch (Leeds, UK), Problems-Part I, Rolduc, Netherlands. (Jan. 2001, p. 53) Y. Kuramoto (Kyoto, Japan), U. Ander Heiden (Witten, Germany), R. McKay (Warwick, UK), C. Jones (Brown), T. Mullin (Manchester, UK), 18-23 The Fourth St. Petersburg Workshop on Simulation, St. W. Ditto (Georgia Tech), R. Roy (Maryland), A. Winfree (Arizona), M. Petersburg State Univ., St. Petersburg, Russia. (Oct. 2000, p. 1146) Fink (ESPCI, France), J. Casti (Santa Fe Institute). 18-23 Tools for Mathematical Modelling, St. Petersburg State Information: http: I lwww. enm . bris. ac . ukl anml colston. html. Technical University, St. Petersburg, Russia. (Dec. 2000, p. 1435) 1 0-1 3 7th International Meeting on DNA Based Computers, 21 -2 2 (NEW DATE)I nteg ration of Diverse Biological Data, DIMACS University of South Florida, Tampa, Florida. (Jan. 2001, p. 53) Center, Rutgers University, Piscataway, New Jersey. (Sept. 2000, p. 975) 11-1 5 IMA Workshop: Haptics, Virtual Reality and Human Com puter Interaction, IMA, University of Minnesota, Minneapolis, 23-29 International Conference on Operator Theory and Its Minnesota. (Sept. 2000, p. 978) Applications, Dedicated to the Memory of Abraham V. Strauss, * 11-22 Workshop on Fourier Analysis and Convexity, Universita Ulyanovsk, Russia. (Sept. 2000, p. 978) di Milano-Bicocca, Milano, Italy. 23-30 Intermediate Problems of Model Theory and Universal Organizing Committee: L. Brandolini (Bergamo), L. Colzani (Milano Algebra, ERLOGOL-200 1, Technical University, Novosibirsk, Russia. Bicocca), A. Iosevich (Columbia-Missouri), G. Travaglini (Milano (Oct. 2000, p. 1146) Bicocca). 25-29 Cmft2001, Computational Methods and Function Theory, Topics: Many problems related to convex bodies are studied using Aveiro, Portugal. (Jan. 2001, p. 53) Fourier analysis methods. The aim of this conference is to bring together leading researchers willing to learn and share problems 25-291nternational Linear Algebra Conference (9th Conference and techniques related to the interaction between Fourier analysis of the International Linear Algebra Society), Technion, Israel and convexity. Institute of Technology, Haifa, Israel. (Dec. 2000, p. 1436) Speakers: The following speakers will deliver minicourses(*) or one 25-29 Workshop on Groups and 3-Manifolds, Centre de Re hour talks: ]. Beck*, Rutgers Univ.; H. Groemer*, Univ. of Arizona; cherches Mathematiques, Universite de Montreal, Montreal, Quebec, N. Katz, Washington Univ. in St. Louis; A. Koldobsky*, Univ. of Canada. (Feb. 2001, p. 250) Missouri-Columbia; M. Kolountzakis*, Univ. of Crete; A. Magyar, Univ. of Wisconsin-Madison; H. Montgomery*, Univ. of Michigan; 25-July 6 NATO Advanced Study Institute "Symmetric Functions A. Podkorytov, St. Petersburg Univ.; F. Ricci, Scuola Normale 2001: Surveys of Developments and Perspectives", Isaac Newton Superiore diPisa; M. Rudelson, Univ. of Missouri-Columbia; E. Sawyer, Institute for Mathematical Sciences, Cambridge, UK. (Jan. 2001, Me Master Univ.; T. Tao*, Univ. of California Los Angeles; A. Vargas, p. 53) MARCH 2001 NOTICES OF THE AMS 335 Mathematics Calendar 25-July 6 Workshop "Random Walks and Geometry", Erwin achieved in solving questions tackled during the workshop or a SchrOdinger Institute, Vienna, Austria. (Jan. 2001, p. 53) group of people would like to have a session on a specific topic. Bertram , 27- 29 MAA Summer Short Course, Ashland University, Ashland, Organizers: The main organizers of the workshop are: E. awaii . edu), and P. Horak, Kuwait Univ. Ohio. (Feb. 2001, p. 250) Univ. of Hawaii (ed@mat h.h (hor ak@kuc01. kuni v. edu. kw). July 2001 Invited Speakers: R. Fau dree, Memphis S tate Univ.; C. Lindner, Auburn Univ.; V. Sos, Hungarian Acad. of Sci.; R. Thomas, Georgia Warthog Delta'01 Conference on Undergraduate Teaching 1-5 Tech.; C. Thomassen, The Tech. Univ. of Denmark. of Mathematics-Third Southern Hemisphere Symposium on Information: More information, including social events, accommo Undergraduate Mathematics Teaching, Kruger Park, South Africa. dations, p reregistration form, and abstracts of the invited t alks, (Sept. 2000, p. 979) may be found on our Web site, http: I /www. math. hawaii. edu/ '' 1- 7 International Symposium on Computational and Applied ber t r am/. PDEs, Zhangjiajie, Hunan, China. Description: An international symposium on computational and ap 1 6- 20 Algorithms for Approximation IV, University of Hudders plied partial differential equations, this symposium is coorganized field, Huddersfield, UK. (Sept. 2000, p. 979) by the Hunan Institute for Computational and Applied Mathemat 16- 20 Probability on Geometric Structures, CIRM Marseille, ics, Xiangtan Univ. in China; Department of Scientific Computing, Luminy, France. (Nov. 2000, p. 1298) Peking Univ. in China; and the Center for Computational Mathe matics and Applications, The Pennsylvania State University. The 22- 25 International Symposium on Symbolic and Algebraic goal of the meeting is to gather t ogether r esearchers working in Computation (ISSAC 2001 ), University of Western Ontario, London, the field of theory and practical computational methods for PDEs Ontario, Canada. (Sept. 2000, p. 979) and to discuss recent r esearch developments in these areas, open 22-28 International Nearring and Nearfield Conference, James problems, and future directions. Madison University, Harrisonburg, Virginia. (Oct. 2000, p. 1146) Information: More information can be found at http: I / wWTN. math. psu . edu/ccma/ pde2001/ . 23-August 3 Combinatorics and Matrix Theory, University of Wyoming, Laramie, Wyoming. (Jan. 2001, p. 53) 2-6 Singapore International Symposium on Topology and Ge * 27-29 Conference and Reunion in Honor of Arnold Ross, The ometry (SISTAG), Singapore. (Feb. 2001, p. 250) Ohio State University, Columbus, Ohio. 2- 7 4th Operator Algebras International Conference: Operator Organizing Committee: G. Woods, D. Shapiro, K. Rubin, A. Silver Algebras and Mathematical Physics, Constanta, Romania. (Feb. berg, D. Pollack, H. Chu. 2001, p. 251) Program: This conference is a celebration of the 45th year of ematics Program for talented high school 3- 5 Mathematics and Design 2001, Deakin University, Geelong, the Ross Summer Math students. There will be lectures on a variety of mathematical topics Australia. (Feb. 2001, p. 25 1) by alumni and friends of the Ross Program. 3-5 Mathematics & Design 2001: Mind/Ear/Eye/Hand/Digital, Information: http: I /www .math. ohio-state. edu/ r oss/ School of Architecture, Australia School of Computing and Mathe r ossconf2001.ht ml. matics, Deakin University, Melbourne, Australia. (Nov. 2000, p. 1297) 8- 1 3 Second ICMS Workshop on Algebraic Graph Theory, In 30-August 1 7 School on Dynamical Systems, Trieste, Italy. (Jan. ternational Centre for Mathematical Sciences, Edinburgh, Scotland. 2001, p. 54) (Feb. 2001, p. 251) August 2001 9-1 3 Workshop on Geometric Group Theory, Centre de Recherches 5-18 BALTICON 2001, Banach Algebra Theory in Context, Mathematiques, Universite de Montreal, Montreal, Quebec, Canada. Krogerup Hojskole, Denmark. (Dec. 2000, p. 1436) (Feb. 2001, p. 251) 5- 1 8 Groups St. Andrews 2001 , University of Oxford, Oxford, in Scientific Comput 9- 20 SMS-NATO ASI: Modern Methods England. (Apr. 2000, p. 503) ing and Applications, Universite de Montreal, Montreal, Quebec, Canada. (Dec. 2000, p. 1436) 6- 11 2001 ASL European Summer Meeting (Logic Colloquium '01), Vienna, Austria. (Feb. 2001, p. 251) 9-22 European Summer School: Asymptotic Combinatorics with Application to Mathematical Physics, Euler International Mathe 7- 1 0 The 4th Conference on Information Fusion, Montreal, matical Institute, St. Petersburg, Russia. (Nov. 2000, p. 1297) Quebec, Canada. (Feb. 2001, p. 251) 1 2- 14 Fifth SIAM Conference on Control and Its Applications 8- 12 The 9th International Conference on Finite or Infinite Di (CTOl ), Town & Country Hotel, San Diego, California. (Sept. 2000, mensional Complex Analysis and Applications, Hanoi University p. 979) of Technology, Hanoi, Vietnam. (Jan. 2001, p. 54) * 13- 27 Summer 2001 Workshop on Graphs and Combinatorial 1 3- 16 Fourth SIAM Conference on Linear Algebra in Signals, Des igns, University of Hawaii at Manoa, Honolulu, Hawaii. Systems and Control (LASSCO 1) , Boston Park Plaza Hotel, Boston, Aim: The aim of the workshop is to gather people interested in Massachusetts. (Sept. 2000, p. 979) discussing recent advances and open problems in the following 1 5-20 SRTL-2: The Second International Research Forum on areas of graph theory and combinatorial designs: cycles, colorings, Statistical Reasoning, Thinking, and Literacy, University of New and extremal problems. There will be 50-minute invited talks and 2 5- England, Armidale, Australia. (Sept. 2000, p. 979) minute contributed talks. The invited talks will be problem-oriented with the aim of stimulating the active research of participants during '' 1 9- 2 5 9th Prague Topologica l Symposium (General Topology the workshop, and these will be scheduled at the very beginning a nd Its Relations to Modern Algebra and Analysis), Prague, Czech of the program. To keep the character of the meeting as that of Republic. an actively research-based workshop, there will be talks for three Topics: Topology (mainly set-theoretical topology, continua, d e hours per day and slots for 40 contributed talks. In case of wider scriptive topology, categorical topology), Topological Groups and interest, contributions in the areas of the invited talks will be given Sernigroups, Topology of Banach Spaces, Topology and Computer preference. The last four days of the workshop will be e ntirely Science, Topological Dynamics. devoted to active research, and talks are planned on these days Information: e-mail: [email protected]. cuni . cz; http: I /www . only in case some participants would like to report on progress karlin.mff . cuni . cz/-toposym/ . 336 N OTICES OF THE AMS VOLUME 48, NUMBER 3 Mathematics Calendar 20-22 Seventh Annual International Computing and Combi powerful results. The main areas of research to be touched upon are: natorics Conference (COCOON'Ol), Guilin, China. (Nov. 2000, classification and minimal models of varieties, rationally connected p. 1298) varieties, rational and integral points, fundamental groups, and Galois '' 20-24 Ellipticity and Parabolicity in Analysis and Geometry groups. Program: (1) During the first (EPAG 2001 ), University of Potsdam, Potsdam, Germany. week (3-7 September) an instructional Description: This workshop is an activity of the Research Training conference will take place featuring 6 speakers. Each of them will Network "Geometric Analysis" of the European Commission (see give a minicourse of 3 one-hour lectures. Roughly half of the courses will focus on geometric http : I /dipmat. unian. i t/rtn/). It particularly addresses young topics and the other half on arithmetic mathematicians with an interest in analysis and geometry for partial topics. They are intended to be accessible to graduate students differential equations. and the "other half" to specialists. (2) The other two weeks (10-21 September) will be mainly devoted Scope: The aim of this workshop is to bring together young to research work of participants and informal discussions. mathematicians interested in diverse research directions in analysis There will also be a regular seminar featuring lectures on recent results and geometry for partial differential equations. This meeting will to by participants. The program will be elaborated in situ some extent have the character of a summer school, with reserved in order to be as up-to-date as possible. Tutorial sessions for graduate time for discussions. We plan to have 3 or 4 series of overview students are also planned. Speakers: Colliot-Thelene talks and a limited number of contributions describing connections J.-L. (Univ. de Paris-Sud), 0. Debarre between the main topics. (IRMA, Strasbourg), B. Hassett (Rice Univ., Houston), J. Kollar (Princeton Univ.), S. Kovacs (Univ. of Washington, Seattle), E. Topics: Spectral theory, index theory, heat equation and other Peyre (Institut Fourier, Grenoble). parabolic equation methods, existence and regularity of solutions to elliptic/parabolic equations, geometric and spectral invariants. Contact: ari thgeo@renyi. hu. Information: http: I /www. renyi . hu. ;-arithgeo/homepage. Organizers: J. B. Gil (Philadelphia, USA); A. Parmeggiani (Bologna, html. Italy); T. Krainer (Potsdam, Germany); T. Schick (Munster, Germany); 4-7 ECCOMAS 2001, Swansea, UK. (Sept. 2000, p. 980) B. Monthubert (Toulouse, France); I. Witt (Potsdam, Germany). Information: e-mail: epag2001@math. uni -potsdam. de, or http : I I * 9-1 3 4th Dublin Differential Equations Conference, Dublin City epag2001.math.uni-potsdam.de/. University, Dublin, Ireland. Topics: The conference covers the theory, applications and numer 22-26 International Conference on Functional Analysis, Kyiv, ical solution of Differential Equations, and the interplay between Ukraine. (Jan. 2001, p. 54) these. Principal 24-29 5th International Conference on Geometry and Applica Speakers: C. Budd (Bath), E. Delabaere (Angers), 0. Diek mann (Utrecht), P. tions, Varna, Bulgaria. (Dec. 2000, p. 1436) Glendinning (Manchester), S. Howison (Oxford), A. Il'in (Ekaterinburg), M. Lavrentiev (Novosibirsk), J. Ockendon 2 7-31 8th International Conference on Differential Geometry (Oxford), R. O'Malley (Washington), G. Shishkin (Ekaterinburg). and Its Applications, Silesian University in Opava, Opava, Czech Program: In addition to lectures from the invited speakers, other Republic. (Oct. 2000, p. 1146) participants may give shorter contributed talks. 27-31 Equadiff 10, Czechoslovak International Conference on lnformation:http : I /www . deconf . dcu . ie/ or Jurgen. Burzlaff@ Differential Equations and Their Applications, Prague, Czech dcu. ie. Republic. (Apr. 2000, p. 502) 1 0-14 3rd IMACS Seminar on Monte Carlo Methods MCM2001, September 2001 Salzburg University, Austria. (Nov. 2000, p. 1298) 1-6 2001 WSES International Conference on Simulation (SIM'Ol ), 1 8-22 The Fifth International Workshop on Differential Geometry Malta. (Feb. 2001, p. 251) and Its Applications, Timisoara, Romania. (Feb. 2001, p. 251) 1-May 31 lnstitut Mittag-Leffler Call for Proposals, Djursholm, 22-26 Application of Discrete Mathematics, Australian National Sweden. (Sept. 2000, p. 979) University, Canberra, Australia. (Feb. 2001, p. 251) '' 3-8 The Sixth International Conference on Function Spaces, In 24-26 ECCOMAS 2001, Keble College, Oxford, UK. (Sept. 2000, stitute of Mathematics, Wroclaw University of Technology, Wroclaw, p. 980) Poland. 24-28 Fourth European Conference on Elliptic and Parabolic Organizer: Institute of Mathematics, WroclawUniv. of Technology. Problems-Part II, Gaeta, Italy. (Jan. 2001, p. 54) Organizing Committee: M. Burnecki (secretary), R. Grzaslewicz (chair), H. Hudzik, J. Musielak, Cz. Ryll-Nardzewski. 24-30 11th International Symposium on Classical Analysis, Program: The topics are connected with functional analysis, e.g., Kazimierz Dolny, Poland. (Dec. 2000, p. 1437) operator theory, interpolation, geometry, topology, approximation 26-28 ASCM 2001 (The Fifth Asian Symposium on Computer in function spaces. There will be plenary lectures (about 45 min.) Mathematics), Ehime University, Matsuyama, Japan. (Jan. 2001, and short communications (not more than 20 min.). p. 54) Information: http: I /www. im. pwr . wroc. pl;-fsp/. 26-28 First SIAM Conference on Imaging Science, Boston Park 3-21 School on Control Theory, Trieste, Italy. (Jan. 2001, p. 54) Plaza Hotel, Boston, Massachusetts. (Feb. 2001, p. 251) '' 3-21 Higher-Dimensional Varieties and Rational Points, Hungar 26-28 Fluid Structure Interaction 2001, Halkidiki, Greece. ian Academy of Sciences, Budapest, Hungary. October 2001 Cochairs: J.-L. Colliot-Thelene (Orsay) and J. Kollar (Princeton). Local Organizers: K. Boroczky Jr. and T. Szamuely (Budapest). 1-5 International Conference on Numerical Algorithms, Dedi Aim and Scope: The program is organized in order to encourage cated to Claude Brezinski on the occasion of his 60th birthday, collaboration between specialists in higher-dimensional complex Marrakesh, Morocco. (Oct. 2000, p. 1146) geometry and those studying arithmetic/ diophantine questions. In 1 3-14 AMS Eastern Sectional Meeting, Williams College, Williams recent years it has become apparent that the powerful geometric town, Massachusetts. (Sept. 1997, p. 1031) tools elaborated in connection with Mori's Minimal Model Program Information: W. Drady, wsd@ams. org. have applications over arithmetic ground fields as well. We hope that bringing together experts and graduate students specializing 24-26 DIMACS Workshop on Analysis of Gene Expression Data, in higher-dimensional geometry or arithmetic will induce further DIMACS Center, Rutgers University, Piscataway, New Jersey. (Feb. cross-fertilization between the two fields and give rise to new 2001, p. 252) MARCH 2001 NOTICES OF THE AMS 337 Mathematics Calendar November 2001 5-8 Seventh SIAM Conference on Geometric Design (SlAG/GO) (GDOl ), Holiday Inn Capitol Plaza Hotel, Sacramento, California. (Sept. 2000, p. 980) 26-30 AAECC-14, The 14th International Symposium on Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes, RMIT ...... I( '""" p; ji University, Melbourne, Australia. (Jan. 2001, p. 55) December 2001 1 0-14 ICMI Study Conference on the Future of the Teaching and Learning of Algebra, University of Melbourne, Australia. (Jan. 2001, p. 55) .Iliff"-· 1 7-19 Eighth Cryptography and Coding, Royal Agricultural Col - lege, Cirencester, UK. (Sept. 2000, p. 980) (.....,...... ,_..., ...... ,~....,.....,.,~ * 1 9-21 International Conference on Statistics,Combinatorics and Related Areas and The Eighth International Conference of the Fo rum for Interdisciplinary Mathematics, University ofWollongong, Wollongong, NSW, Australia. Conveners: S. N. Mishra, President of the Forum, Department of ~ w; 4 .. , Mathematics and Statistics, Univ. of South Alabama, Mobile, AL 36688-0002; tel: +1-334-461-1642; fax:+1-334-460-7969; mishra@ j J mathstat. usouthal. edu; C. M. Gulati, School of Mathematics and Applied Statistics, Univ. of Wollongong, Wollongong, NSW 2522, Australia; tel: +61-2-4221-3836; fax:+61-2-4221-4845; chandra_ gulati@uow. edu. au. Information: http : I /www . uow . edu. au/informatics/maths/ 1\1- 1\IIM.*lf :X$1 I"'' statconference/. January 2002 1 ~· l-.,.. l • 6-9 Joint Mathematics Meetings, San Diego Convention Center, San Diego, California. (Nov. 1998, p. 1378) l i..'.:.o.!' • ~1 ::- June 2002 ~-...... ,_ - 1 0-16 Aarhus Topology 2002, University of Aarhus, Aarhus, ~ Denmark. 23-28 Fourteenth U.S. National Congress of Theoretical and 1 '""<. Applied Mathematics, Virginia Polytechnic Institute and State ~,·~"'r:&wz--- University, Blacksburg, Virginia. j ~--- August 2002 --~ 1 2-16 MTNS 2002, Fifteenth International Symposium on Math ~ ematical Theory of Networks and Systems, University of Notre ~ Dame, Notre Dame, Indiana. l,_,_,..,..,.,...,...,.,...,.,.....,,...,.,.....,...,."" 20-28 ICM-2002, Beijing, China. (Oct. 2000, p. 1146) "'*"'~ 338 NOTICES OF THE AMS VOLUME 48, NUMBER 3 New Publications Offered by the AMS K. C. O'Meara and C. Vinsonhaler, Generalizations of isomor Algebra and Algebraic phism in torsion-free abelian groups; P. Schultz, Automorphism groups of abelian groups; Contributed papers: Geometry D. M. Arnold, Direct sum decompositions of torsion-free abelian groups of finite rank; M.A. Aviii6 and P. Schultz, The endomorphism ring of a bounded abelian p-group; Abelian Groups, E. Blagoveshchenskaya, G. Ivanov, and P. Schultz, The Baer Kaplansky theorem for almost completely decomposable Rings and Modules groups; A. Blass and j. Irwin, Maximal pure independent sets; Abelian Groups, A. V. Kelarev, University of D. Dikranjan and M. Tkachenko, Characterization of the tori Rings and Modules Tasmania, Hobart, Tasmania, via density of the solution set of linear equations; A. A. Fomin, V. Kelore¥ Quotient divisible mixed groups; L. Fuchs and S. B. Lee, A...... IC. M. Raftgoswamy Australia, R. Gobel, University P. SelwOI> Stacked bases over h-local Pliifer domains; A. J. Giovannini, c._,...... of Essen, Germany, Groups with locally defined heights and products of !R* K. M. Rangaswamy, University groups; R. Gobel and S. Shelah, Reflexive subgroups of the of Colorado, Colorado Springs, Baer-Specker group and Martin's axiom; P. Hill, C. Megibben, P. Schultz, The University of and W. Ullery, L-isotype subgroups of local k-groups; G. Ivanov, Character modules and endomorphism rings of Western Australia, Nedlands, modules over Artinian serial rings; P. Loth, Topologically pure and C. Vinsonhaler, University of Connecticut, extensions; N. R. McConnell and T. Stokes, Rings having Storrs, Editors simple adjoint semigroup; A. Mader, L. G. Nongxa, and M. A. Ould·Beddi, Invariants of global crq-groups; This volume presents the proceedings from the conference on V. H. Mikaelian, On varieties of groups generated by wreath Abelian Groups, Rings, and Modules (AGRAM) held at the products of abelian groups; 0. Mutzbauer, Existence of rigid University of Western Australia (Perth). Included are articles indecomposable almost completely decomposable groups; based on talks given at the conference, as well as a few W. K. Nicholson and M. F. Yousif, C2-rings and the FGF-conjec specially invited papers. The proceedings are dedicated to ture; B. L. Osofsky, Ufting direct sum decompositions of Professor Laszlo Fuchs. The book includes a tribute and a bounded abelian p-groups; K. M. Rangaswamy, On modules review of his work by his long-time collaborator, Professor and submodules with finite projective dimension; L. Strung Luigi Salce. mann and S. L. Wallutis, On the torsion groups in cotorsion Four surveys from leading experts follow Professor Salce's classes; j. Trlifaj, Cotorsion theories induced by tilting and article. They present recent results from active research areas: cotilting modules; J. Zemlicka, Steadiness is tested by a single · Error correcting codes as ideals in group rings, module. · Duality in module categories, Contemporary Mathematics, Volume 273 · Automorphism groups of abelian groups, and · Generalizations of isomorphism in torsion-free abelian March 2001, 308 pages, Softcover, ISBN 0-8218-2751-0, 2000 groups. Mathematics Subject Classification: 05Axx, 06Bxx, 06Fxx, 13Cxx, 15-XX, 16-XX, 20Kxx, 20Exx, Individual member $47, In addition to these surveys, the volume contains 22 research Ust $79, Institutional member $63, Order code CONM/ 273N articles in diverse areas connected with the themes of the conference. The areas discussed include abelian groups and their endomorphism rings, modules over various rings, commutative and non-commutative ring theory, varieties of groups, and topological aspects of algebra. The book offers a comprehensive source for recent research in this active area of study. Contents: Introduction: L. Salce, Laszlo Fuchs and his "moddom" work; Survey articles: A. V. Kelarev and P. Sole, Error-correcting codes as ideals in group rings; B. Olberding, Homomorphisms and duality for torsion-free modules; MARCH 2001 NOTICES OF THE AMS 339 New Publications Offered by the AMS Analysis Function Theory of Several Complex Laguerre Calculus Variables and Its Applications Second Edition on the Heisenberg Steven G. Krantz, Washington Group University, St. Louis, MO Carlos Berenstein, University This work departs from earlier treat ments of the subject by emphasizing of Maryland, College Park, integral formulas, the geometric Der-Chen Chang, Georgetown theory of pseudoconvexity, estimates, partial differential equa University, Washington, DC, tions, approximation theory, the boundary behavior of and Jingzhi Tie, University of holomorphic functions, inner functions, invariant metrics, and Georgia, Athens mapping theory. While due homage is paid to the more tradi tional algebraic theory (sheaves, Cousin problems, etc.), the For nearly two centuries, the relation between analytic func student with a background in real and complex variable theory, tions of one complex variable, their boundary values, harmonic harmonic analysis, and differential equations will be most functions, and the theory of Fourier series has been one of the comfortable with this treatment. central topics of study in mathematics. The topic stands on its It is currently the only book on the subject with exercises and a own, yet also provides very useful mathematical applications. large number of examples. This text provides a self-contained introduction to the corre This item will also be of interest to those working in geometry sponding questions in several complex variables: namely, and topology. analysis on the Heisenberg group and the study of the solu tions of the boundary Cauchy-Riemann equations. In studying Contents: An introduction to the subject; Some integral this material, readers are exposed to analysis in non-commuta formulas; Subharmonicity and its applications; Convexity; tive compact and Lie groups, specifically the rotation group Htirmander's solution of the a equation; Solution of the Levi and the Heisenberg groups-both fundamental in the theory of problem and other applications of a techniques; Cousin prob group representations and physics. lems, cohomology, and sheaves; The zero set of a holomorphic function; Some harmonic analysis; Constructive methods; Inte Introduced in a concrete setting are the main ideas of the gral formulas for solutions to the a problem and norm Calder6n-Zygmund-Stein school of harmonic analysis. Also estimates; Holomorphic mappings and invariant metrics; Mani considered in the book are some less conventional problems of folds; Area measures; Exterior algebra; Vectors, covectors, and harmonic and complex analysis, in particular, the Morera and differential forms; List of notation; Bibliography; Index. Pompeiu problems for the Heisenberg group, which relates to questions in optics, tomography, and engineering. AMS Chelsea Publishing The book was borne of graduate courses and seminars held at January 2001, 564 pages, Hardcover, ISBN 0-8218-2724-3, the University of Maryland (College Park), the University of LC 00-059363, 2000 Mathematics Subject Classification: 32-02, Toronto (ON), Georgetown University (Washington, DC), and 32Bxx, 32Dxx, 32Fxx, 32Hxx, 32Txx, 32Uxx, 32Wxx; 35N15, the University of Georgia (Athens). Readers should have an All AMS members $50, List $55, Order code CHEL/340.HN advanced undergraduate understanding of Fourier analysis and complex analysis in one variable. Recommended Text Titles in this series are copublished with International Press, Cambridge, Analysis MA. Contents: The Laguerre calculus; Estimates for powers of the Second Edition sub-Laplacian; Estimates for the spectrum projection operators Elliott H. lieb, Princeton of the sub-Laplacian; The inverse of the operator University, N], and D"' = I.J=l (XJ - XJ+n) - 2iaT; The explicit solution of the a Neumann problem in a non-isotropic Siegel domain; Injectivity Michael Loss, Georgia Institute of the Pompeiu transform in the isotropic Hn; Morera-type of Technology, Atlanta theorems for holomorphic J{P spaces in Hn (I); Morera-type theorems for holomorphic J{P spaces in Hn (II). Praise for the previous edition ... AMS/IP Studies in Advanced Mathematics, Volume 22 I find the selection of the material covered in the book very attractive and May 2001, approximately 328 pages, Hardcover, ISBN 0-8218- I recommend the book to anybody who 2761-8, LC 00-054826, 2000 Mathematics Subject Classification: wants to learn about classical as well as modern mathematical 22E30, 33C20,42C10,43A80,47G30,32W05,32A10,30E99, analysis. All AMS members $51, List $64, Order code AMS1P/ 22N -European Mathematical Society Newsletter The essentials of modern analysis ... are presented in a rigorous and pedagogical way ... readers ... are guided to a level where they can read the current literature with understanding .. . treatment of the subject is as direct as possible. -Zentralblatt fiir Mathematik 340 NOTICES OF THE AMS VOLUME 48, NUMBER 3 New Publications Offered by the AMS Lieb and Loss offer a practical presentation of real and func tional analysis at the beginning graduate level ... could be used Differential Equations as a two-semester introduction to graduate analysis .. . not all of the topics covered are typical. The authors introduce the subject Supplementary Readong with a thorough presentation ... [an] informative exposition. -CHOICE Ordinary Differential Significantly revised and expanded, this new Second Edition Equations provides readers at all levels-from beginning students to practicing analysts-with the basic concepts and standard Second Edition tools necessary to solve problems of analysis, and how to apply these concepts to research in a variety of areas. Stephen Salaff and Shing-Tung Yau Authors Elliott Lieb and Michael Loss take you quickly from basic topics to methods that work successfully in mathematics A publication of the International and its applications. While omitting many usual typical text Press. book topics, Analysis includes all necessary definitions, proofs, These articles arose from original explanations, examples, and exercises to bring the reader to an lectures in differential equations given advanced level of understanding with a minimum of fuss, and, by the authors at Chung Chi College at at the same time, doing so in a rigorous and pedagogical way. the Chinese University of Hong Kong. This second edition Many topics that are useful and important, but usually left to makes this material available to a broader audience. Topics advanced monographs, are presented in Analysis, and these include: existence, uniqueness, and continuous dependence, give the beginner a sense that the subject is alive and growing. continuation of solutions, the linear equation, power series This new Second Edition incorporates numerous changes since solutions, linear systems with constant coefficients, the publication of the original 1997 edition, and includes: autonomous systems, limit cycles, I, and limit cycles, ll. · a new chapter on eigenvalues that covers the min-max prin Distributed worldwide, except in Japan, by the American Mathematical ciple, semi-classical approximation, coherent states, Society. Lieb-Thirring inequalities, and more International Press · extensive additions to chapters covering Sobolev Inequalities, including the Nash and Log Sobolev inequalities December 2000, 72 pages, Softcover, ISBN 1-57146-065-9, 2000 · new material on Measure and Integration Mathematics Subject Classification: 34-01, All AMS members · many new exercises $34, List $42, Order code INPR/ 35N · and much more ... The Second Edition continues its no-nonsense approach to the topic that has made it one of the best selling books on the General and subject. It is an authoritative, straight-forward volume that readers-from the graduate student, to the professional math Interdisciplinary ematician, to the physicist or engineer using analytical methods-will find useful both as a reference and as a guide to real problem solving. Mikio Sato About the authors: Elliott Lieb is Professor of Mathematics A Great Japanese and Physics at Princeton University and is a member of the US, Austrian, and Danish Academies of Science. He is also the Mathematician of the recipient of several prizes including the 1988 AMS/ SlAM Birk Twentieth Century hoff prize. Michael Loss is Professor of Mathematics at the Georgia Institute of Technology. Masaki Kashiwara, Kyoto University, japan, Contents: Measure and integration; LP -spaces; Rearrangement inequalities; Integral inequalities; The Fourier transform; Distri Shing-Tung Yau, Harvard butions; The Sobolev spaces H 1 and H 1i 2; Sobolev inequalities; University, Cambridge, MA, Potential theory and Coulomb energies; Regularity of solutions and Tak.ahiro Kawai, Kyoto of Poisson's equation; Introduction to the calculus of varia University, Japan, Editors tions; More about eigenvalues; References; List of symbols; Index. A publication of the International Press. Graduate Studies in Mathematics, Volume 14 This issue of the Asian Journal of Mathematics was dedicated May 2001, approximately 326 pages, Hardcover, ISBN 0-8218- to Professor Mikio Sato in the year of his seventieth birthday. 2783-9, 2000 Mathematics Subject Classification: 28-01, 42-01, Sato has made renowned mathematical contributions. His 46-01,49-01;26D10,26D15, 31B05,31B15,46E35,46F05, introduction of the sheaf of microfunctions opened a new era in analysis. His powerful collaborations with Kawai and Kashi 46Fl0, 49-XX, 81Q05, All AMS members $31, List $39, Order wara led to the establishment of microlocal analysis, one of the code GSM/ 14.RN most important discoveries in microfunctions and pseudo differential equations. In this volume, leading researchers such as Yau, Schapira, Jacquet, Kashiwara, Melrose, and others, present their work in honor of Professor Sa to. Distributed worldwide, except in Japan, by the American Mathematical Society. MARCH 2001 NOTICES OF 1HE AMS 341 New Publications Offered by the AMS Contents: S. Iyanaga, What I could do for the mathematician gtn_; Y. I. Manin, Three constructions of Frobenius manifolds: A Mikio Sato; H. Komatsu, The early days of the theory of hyper comparative study; B. Mazur, Visualizing elements of order functions and differential equations; M. J. Ablowitz, three in the Shafarevich-Tate group; W. Schmid and S. Chakravarty, and R. Halburd, The generalized Chazy equa K. Vilonen, On the geometry of nilpotent orbits; C. H. Taubes, tion and Schwarzian triangle functions; T. Aoki, T. Kawai, and Moduli spaces and Fredholm theory for pseudoholomorphic Y. Takei, On the exact WKB analysis for the third order ordi subvarieties associated to self-dual, harmonic 2-forms; nary differential equations with a large parameter; A. D'Agnolo R. S. Ward, The integrable systems related to hyperbolic and P. Schapira, Global propagation on causal manifolds; monopoles; Index through Volume 2, Issue 1. E. D'Hoker and D. H. Phong, Order parameters, free fermions, International Press and conservation laws for Calogero-Moser systems; H. Hida, Automorphic induction and Leopoldt type conjectures for December 2000, 332 pages, Hardcover, ISBN 1-57146-080-2, GL(n); J. Hong, S. J. Kang, T. Miwa, and R. Weston, Vertex 2000 Mathematics Subject Classification: OOB15, All AMS models with alternating spins; H. Jacquet, A theorem of members $34, list $42, Order code INPR/ 38N density for Kloosterman integrals; M. Kashiwara and T. Tanisaki, Kazhdan-Lusztig conjecture for symmetrizable Kac-Moody lie algebras. III-Positive rational case; R. Mazzeo Geometry and Topology and R. B. Melrose, Pseudodifferential operators on manifolds with fibred boundaries; H. P. McKean, Breakdown of a shallow water equation; M. Mulase and M. Penkava, Ribbon graphs, quadratic differentials on Riemann surfaces, and algebraic The Third Pacific curves defined over Q; S. Novikov, Discrete Schrodinger oper Rim Geometry ators and topology; T. Oshima, Completely integrable systems with a symmetry in coordinates; S. J. Patterson, The cubic Conference Shimura correspondence; K. Saito, Duality for regular systems jaigyoung Choe, Seoul of weights; K. Takasaki, Spectral curves and Whitham equa tions in isomonodromic problems of Schlesinger type; National University, Korea, S.-T. Yau and S. S.-T. Yau, Existence and decay estimates for Editor time dependent parabolic equation with application to duncan Mortensen-Sakai equation. A publication of the International Press. International Press This volume contains 20 papers from December 2000, 1149 pages, Hardcover, ISBN 1-57146-081-0, the proceedings of the third Pacific 2000 Mathematics Subject Classification: OOB15, All AMS Rim Geometry Conference held at the Korea University (Seoul). members $34, list $42, Order code INPR/ 39N Topics include the existence theorem of harmonic objects via the Green function, strings of Riemannian invariants, inequali ties, ideal immersions, and their applications, eigenvalue Sir Michael Atiyah problems, Grassmann geometry and J-holomorphic curves of a 6-dimensional sphere, the structure of the image of analytic A Great Mathematician disks attached to totally real submanifolds, Mayer-Vietoris of the Twentieth formula for determinants of elliptic operators of Laplace Century Beltrami type, gauge-theoretic equations for harmonic maps into symmetric spaces, canonical flows of Einstein-Weyl mani Shing-Tung Yau, Harvard folds, Jeffrey-Weitsman-Witten invariants of Seifert fibred University, Cambridge, MA, manifolds, complex-analyticity of harmonic maps, and more. and Raymond H. Chan, This item will also be of interest to those working in analysis. Chinese University of Hong Distributed worldwide, except in Japan, by the American Mathematical Kong, Hong Kong, Editors Society. Contents: S. Bando, The existence theorem of harmonic A publication of the International Press. objects via Green function; B.-Y. Chen, Strings of Riemannian This special issue of the Asian Journal of Mathematics was invariants, inequalities, ideal immersions and their applica dedicated to Sir Michael Atiyah in the year of his seventieth tions; R. Chen, On some eigenvalue problems; H. Hashimoto, birthday. Many distinguished researchers contributed articles Grassmann geometry and J-holomorphic curves of a 6-dimen to this volume in celebration of Atiyah's profound contribu sional sphere; H. Kamada, Self-duality of neutral metrics on tions to topology, geometry, representation theory, and partial four-dimensional manifolds; I. Kim, Marked length spectrum differential equations. on the finite set of elements determines the irreducible repre Distributed worldwide, except in Japan, by the American Mathematical sentation in the isometry group of rank one symmetric space Society. of noncompact type; J. Kim, Compactification and construction of asymptotically Euclidean scalar-flat Kiihler surfaces; Contents: S. K. Donaldson, Moment maps and diffeomor K.-T. Kim, The Wu metric and minimum ellipsoids; D. Kwon phisms; S. Gelfand and D. Kazhdan, Conjectural algebraic and Y.-G. Oh, Structure of the image of analytic disks attached formulas for representations of GLn; V. Guillemin and C. Zara, to totally real submanifolds; Y. Lee, Mayer-Vietoris formula for Equivariant de Rham theory and graphs; N. J. Hitchin, The determinants of elliptic operators of Laplace-Beltrami type moduli space of complex Lagrangian submanifolds; (after Burghelea, Friedlander and Kappeler); K. Matsumoto, R. Lawrence and D. Zagier, Modular forms and quantum I. Mihai, and M. H. Shahid, Certain submanifolds of a invariants of 3-manifolds; B. H. lian, K. liu, and S. T. Yau, Kenmotsu manifold; M. Mukai and Y. Ohnita, Gauge-theoretic Mirror principle II; G. Lusztig, Aperiodicity in quantum affine equations for harmonic maps into symmetric spaces; 342 NOTICES OF THE AMS VOLUME 48, NUMBER 3 New Publications Offered by the AMS H. Naitoh, Grassmann geometries on compact symmetric topy equivalences which is invariant on genus; K. Morisugi, spaces; Y. Nakagawa, Combinatorial formulae for Futaki char· Composition structure of the self maps of SU(3) or Sp(2); acters and generalized Killing forms of toric Fano orbifolds; J. Mukai, Self-homotopy of a suspension of the real 4-projec Y.-G. Oh, Naturality of Floer homology of open subsets in tive space; J. Pan and M. H. Woo, Phantom elements and its Lagrangian intersection theory; T. Okayasu, An extension of applications; J. W. Rutter, Homotopy equivalences of lens Chern-Lashof theorem to other space forms; S.-H. Paeng, Topo spaces of one-relator groups; H. Shiga, K. Tsukiyama, and logical entropy for geodesic flows under Ricci curvature and T. Yamaguchi, Principal S1-bundles and forgetful maps; conjugate radius bounded below; H. K. Pak, Canonical flows of S. B. Smith, Rational type of classifying spaces for fibrations; Einstein-Weyl manifolds; J. Park, Jeffrey-Weitsman-Witten M. Arkowitz, Problems on self-homotopy equivalences. invariants of Seifert fibred manifolds; K. Ueno, Complex Contemporary Mathematics, Volume 274 analyticity of harmonic maps. March 2001, 315 pages, Softcover, ISBN 0-8218-2683-2, 2000 International Press Mathematics Subject Classification: 55P10, 55P15, 55P20, December 2000, 333 pages, Hardcover, ISBN 1-57146-064-0, 55P35, 55P45, 55P60, 55P62, 55P91, SSROS, 55R35, Individual 2000 Mathematics Subject Classification: 53-XX, All AMS member $47, List $79, Institutional member $63, Order code members $34, List $42, Order code INPR/ 34N CONM/ 274N Groups of Homotopy Geometry of Self-Equivalences Characteristic and Related Topics Classes Groups of Homotopy Self-Equivalences and Related Topics Ken-ichi Maruyama, Chiba Shigeyuki Morita, Tokyo Ken·lcl'll Maruyomo University, japan, and Institute of Technology, japan John W.Rulte1 Edlton W. University of John Rutter, Characteristic classes are central to Liverpool, England, Editors the modern study of the topology and This volume offers the proceedings geometry of manifolds. They were first from the workshop held at the introduced in topology, where, for Gargnano Institute of the University of instance, they could be used to define Milan (Italy) on groups of homotopy self-equivalences and obstructions to the existence of certain fiber bundles. Charac related topics. The book comprises articles of current research teristic classes were later defined (via the Chern-Weil theory) on the group of homotopy self-equivalences, the homotopy of using connections on vector bundles, thus revealing their function spaces, rational homotopy theory, the classification of geometric side. homotopy types, and equivariant homotopy theory. In the late 1960s new theories arose that described still finer Mathematicians from many areas of the globe attended the structures. Examples of the so-called secondary characteristic workshops to discuss their research and to share ideas. classes came from Chern-Simons invariants, Gelfand-Fuks Included are two specially-written articles, by J. W. Rutter, cohomology, and the characteristic classes of flat bundles. The reviewing the work done in the area of homotopy self-equiva new techniques are particularly useful for the study of fiber lences since 1988. Included also is a bibliography of some 122 bundles whose structure groups are not finite dimensional. articles published since 1988 and a list of problems. This book The theory of characteristic classes of surface bundles is is suitable for both advanced graduate students and perhaps the most developed. Here the special geometry of researchers. surfaces allows one to connect this theory to the theory of Contents: J. W. Rutter, Homotopy self-equivalences moduli space of Riemann surfaces, i.e., Teichmilller theory. In 1988-1999; J. W. Rutter, Bibliography on T(X) 1988-1999; this book Morita presents an introduction to the modern theo M. Arkowitz, G. Lupton, and A. Murillo, Subgroups of the ries of characteristic classes. group of self-homotopy equivalences; S. Bauer, M. Crabb, and This item will also be of interest to those working in algebra M. Spreafico, The space of free loops on a real projective and algebraic geometry. space; H.-J. Baues and Y. Drozd, Indecomposable homotopy Contents: De Rham homotopy theory; Characteristic classes of types with at most two non-trivial homology groups; H.- flat bundles; Characteristic classes of foliations; Characteristic J. Baues and N. Iwase, Square rings associated to elements in classes of surface bundles; Directions and problems for future homotopy groups of spheres; P. I. Booth, Fibrations with research; Bibliography; Index. product of Eilenberg-MacLane space fibres I; D. L. Ferrario, Self homotopy equivalences of equivariant spheres; Y. Felix, Two Translations of Mathematical Monographs (Iwanami Series in examples to illustrate properties of the group of self-equiva Modern Mathematics), Volume 199 lences of a finite CW complex X; A. Garvin, A. Murillo, May 2001 , 180 pages, Softcover, ISBN 0-8218-2139-3, P. PavesiC, and A. Viruei, Nilpotency and localization of LC 00-054312, 2000 Mathematics Subject Classification: 54C40, groups of fibre homotopy equivalences; K. A. Hardie and 14£20; 46£25, 20C20, All AMS members $24, List $30, Order K. H. Kamps, The homotopy groups of the homotopy fibre of code MMON0/ 199N an induced map of function spaces; V. Hauschild, Fibrations, self homotopy equivalences and negative derivations; K. Ishiguro, Classifying spaces and a subgroup of the excep tional Lie group Gz; D. Kahn and C. Schwartz, The structure of the Hurewicz homomorphism; H. j. Marcum, joins, diagonals and Hopf invariants; K.-i. Maruyama, A subgroup of self homo- MARCH 2001 NOTICES OF TilE AMS 343 - ··-·-.·· ···.~ .. - New Publications Offered by the AMS :rrans~auon9 oC Simplicial and Mathematical Physics Operad Methods in Algebraic Topology Bosonic Strings: V. A. Smirnov, Moscow State A Mathematical Pedagogical Institute, Russia Treatment In recent years, for solving problems of algebraic topology and, in partic Jftrgen Jost, Max Planck ular, difficult problems of homotopy Institute for Mathematics in the theory, algebraic structures more Sciences, Leipzig, Germany complicated than just a topological monoid, an algebra, a coalgebra, etc., have been used more and Presented in this book is a mathemat more often. A convenient language for describing various ical treatment of Bosonic string theory structures arising naturally on topological spaces and on their from the point of view of global geom cohomology and homotopy groups is the language of operads etry. As motivation, the author and algebras over an operad. This language was proposed by presents the theory of point particles and Feynman path inte ]. P. May in the 1970s to describe the structures on various grals. He considers the theory of strings as a quantization of loop spaces. the classical Plateau problem for minimal surfaces. The conformal variance of the relevant functional, the Polyakov This book presents a detailed study of the concept of an action or (in mathematical terminology) the Dirichlet integral, operad in the categories of topological spaces and of chain leads to an anomaly in the process of quantization. The mathe complexes. The notions of an algebra and a coalgebra over an matical concepts needed to resolve this anomaly via the operad are introduced, and their properties are investigated. Faddeev-Popov method are introduced, specifically the geom The algebraic structure of the singular chain complex of a etry of the Teichmuilller and moduli spaces of Riemann topological space is explained, and it is shown how the surfaces and the corresponding function spaces, i.e., Hilbert problem of homotopy classification of topological spaces can spaces of Sobolev type and diffeomorphism groups. Other be solved using this structure. For algebras and coalgebras useful tools are the algebraic geometry of Riemann surfaces over operads, standard constructions are defined, particularly and infinite-dimensional determinants. Also discussed are the the bar and cobar constructions. Operad methods are applied boundary regularity questions. The main result is a presenta to computing the homology of iterated loop spaces, investi tion of the string partition function as an integral over a gating the algebraic structure of generalized cohomology moduli space of Riemann surfaces. Some new physical theories, describing cohomology of groups and algebras, concepts, such as D-branes, are also discussed. computing differential in the Adams spectral sequence for the homotopy groups of the spheres, and some other problems. This volume offers a mathematically rigorous treatment of some aspects of string theory, employs a global geometry Contents: Operads in the category of topological spaces; approach, systematically treats strings with boundary, and Simplicial objects and homotopy theory; Algebraic structures carefully explains all mathematical concepts and tools. on chain complexes; Aoo-structures on chain complexes; Operads and algebras over operads; Homoloty of iterated loop Titles in this series are copublished with International Press, Cambridge, spaces; Homotopy theories and Eoo-structures; Operad MA. methods in cobordism theory; Description of the cohomology Contents: Point particles; The Bosonic string; Bibliography; of groups and algebras; Homology operations and differentials Index. in the Adams spectral sequence; Bibliography; Index. AMS/IP Studies in Advanced Mathematics Translations of Mathematical Monographs, Volume 198 March 2001, approximately 112 pages, Hardcover, ISBN 0-8218- March 2001, 235 pages, Hardcover, ISBN 0-8218-2170-9, LC 00- 2644-1, LC 00-067543, 2000 Mathematics Subject Classification: 068957, 2000 Mathematics Subject Classification: 55P15, 55S20, 81T30; 83E30, 81T50, 58D30, 32G15, 53Al0, All AMS Individual member $53, List $89, Institutional member $71, members $31, List $39, Order code AMSIP-JOSTlN Order code MMON0/ 198N 344 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Previously Announced Publications Previously Announced Laminations and Foliations in Dynamics, Geometry and Topology Publications Mikhail Lyubich, John W. Milnor, and Yair N. Minsky, SUNY at Stony Brook, NY, Editors Supplementary Readmg This volume is based on a conference held at SUNY, Stony Partial Differential Equations in Several Brook (NY). The concepts of laminations and foliations appear in a diverse number of fields, such as topology, geometry, Complex Variables analytic differential equations, holomorphic dynamics, and So-Chin Chen, National Tsing-Hua University, Hsinchu, renormalization theory. Although these areas have developed each has developed distinct research fields with Taiwan, and Mei-Chi Shaw, University of Notre Dame, IN deep relations, little interaction among practitioners. In the last few decades, significant progress was made in the the articles by F. Bonahon, "Geodesic study of Cauchy-Riemann and tangential Cauchy-Riemann opera Of particular interest are on Surfaces", and D. Gabai, "Three Lectures on tors; this progress greatly influenced the development of PDEs Laminations Foliations and Laminations on 3-manifolds", which are based and several complex variables. After the background material in that took place during the conference. complex analysis is developed in Chapters 1 through 3, the next on minicourses three chapters are devoted to the solvability and regularity of Contemporary Mathematics, Volume 269 the Cauchy-Riemann equations using Hilbert space techniques. March 2001, 233 pages, Softcover, ISBN 0-8218-1985-2, LC 00- The second part of the book gives a comprehensive study of the 052559, 2000 Mathematics Subject Classification: 53-06, 57-06, tangential Cauchy-Riemann equations, another important class 37-06,32-06, 53Cl2, 57R30, 57Mxx, 57M25, 37Fxx, 37Fl0, by Lewy. of equations in several complex variables first studied 37F75, Individual member $35, List $59, Institutional member 2 An up-to-date account of the L theory for ab operator is given. $47, Order code CONM/269RT103 This fairly self-contained book provides a much-needed intro to several complex variables and PDEs. It also ductory text Recommended Text provides a rich source of information to experts. Algebraic Geometry 2 Titles in this series are copublished with International Press, Cambridge, MA. Sheaves and Cohomology AMS/IP Studies in Advanced Mathematics, Volume 19 Kenji Ueno, Kyoto University, japan December 2000, 380 pages, Hardcover, ISBN 0-8218-1062-6, LC Modern algebraic geometry is built upon two fundamental 00-049584, 2000 Mathematics Subject Classification: 32-XX; notions: schemes and sheaves. The theory of schemes was 35-XX, All AMS members $39, List $49, Order code explained in Algebraic Geometry 1: From Algebraic Varieties to AMSIP/19RT103 Schemes (see Volume 185 in the same series, Translations of Mathematical Monographs). In the present book, Ueno turns to the theory of sheaves and their cohomology. Loosely speaking, Operator Theoretical Methods a sheaf is a way of keeping track of local information defined A. Gheondea, R. N. Gologan, and D. Timotin, Romanian on a topological space, such as the local holomorphic func Academy, Bucharest, Romania, Editors tions on a complex manifold or the local sections of a vector bundle. To study schemes, it is useful to study the sheaves A publication of the Theta Foundation. defined on them, especially the coherent and quasicoherent This volume contains carefully selected contributions by partici sheaves. The primary tool in understanding sheaves is coho pants at the Seventeenth International Conference on Operator mology. For example, in studying ampleness, it is frequently Theory held at the University of Timi$oara (Romania). A large useful to translate a property of sheaves into a statement variety of topics are covered, including single operator theory, about its cohomology. C* -algebras, spectral theory, special classes of concrete opera The text covers the important topics of sheaf theory, including tors, and holomorphic operator functions. types of sheaves and the fundamental operations on them, such Distributed worldwide, except in Romania, by the AMS. as coherent and quasicoherent sheaves, yroper and projective International Book Series of Mathematical Texts morphisms, direct and inverse images, C ech cohomology. August 2000, 415 pages, Hardcover, ISBN 973-99097-2-8, For the mathematician unfamiliar with the language of schemes 2000 Mathematics Subject Classification: 47A20, 44A60, 49199, and sheaves, algebraic geometry can seem distant. However, style 47A48, 16W30, 46L05, 47A55, 35}05, 35}25, 45E10, 47A05, Ueno makes the topic seem natural through his concise are 47A68,47B35,47A57,46L51,43A17,47B37,45P05, 35P05, and his insightful explanations. He explains why things illumi 47A10,47D03,46A32,46B25,46B42,47B38,47C05,46L55, done this way and supplements his explanations with to make algebraic 22D25, 43A20, 43A80, 43A90, 46Ll0, 43A35, 47B25; 46E20, nating examples. As a result, he is able 93B05,93B07,47A10,47A75,47B35,47A20,93B36,46L55, geometry very accessible to a wide audience of non-specialists. 42A50, 47B39, 47A35, 46L05, 20K99, 46L35, 43A07, 43A15, The book contains numerous problems and exercises with 43A22, 28B99, All AMS members $30, List $38, Order code solutions. It would be an excellent text for the second part of a THETA/1RT1 03 course in algebraic geometry. Translations of Mathematical Monographs (Iwanami Series in Modern Mathematics), Volume 197 April 2001, approximately 200 pages, Softcover, ISBN 0-8218- 1357-9, LC 99-22304, 2000 Mathematics Subject Classification: 14-01, 14F99, All AMS members $23, List $29, Order code MMON0/197RT10 MARCH 2001 NOTICES OF THE AMS 345 In Praise of AMS Publications Below are AMS titles with recently received praise from well-known review publications in the mathematical community. The comments continue to attest to the quality and significance of the AMS publications made available through our publishing program. For more about these and other AMS books, visit the AMS Bookstore at www.ams.org/bookstore. Mathematics: Frontiers and Perspectives Domain Decomposition Methods I 0 V. Arnold, University of Paris IX, Paris, France, and Steklov Jan Mandel, University of Colorado, Denver, and Charbel Farhat Mathematical Institute, Moscow, Russia, M. Atiyah, University and Xiao-Chuan Cai, University of Colorado, Boulder, Editors of Edinburgh, Scotland, P. Lax, New York University-Courant Experienced researchers would find these proceedings very useful. Institute of Mathematical Sciences, NY, and B. Mazur, Harvard In particular, various new applications and new algorithms can be University, Cambridge, MA, Editors found here. -SIAM Review This collection demonstrates well that mathematics is alive and vital. Contemporary Mathematics, Volume 218; 1998; ISBN 0·8218·0988-1; -American Scientist 554 pages; Softcover; Individual member $66, List $110, Institutional Individual members of mathematical societies of the IMU member countries can member $88, Order Code CONM/218CT1 03 purchase this volume at the AMS member price when buying directly from the AMS. 2000; ISBN 0·8218·2697·2; 459 pages; Softcover, All AMS members $31, Jacques Hadamard, A Universal List $39, Order Code MFP.SCT103 Mathematician Knotted Surfaces and Their Diagrams Vladimir Maz'ya and Tatyana Shaposhnikova, Linkoping University, Sweden J. Scott Carter, University of South Alabama, Mobile, and Masahico Saito, University of South Florida, Tampa The book is filled with brief and fascinating biographies of many of these colleagues, with amusing anecdotes and incisive quotanons, and with The authors must be congratulated on their heroic endeavors to an illuminating description of the academic milieu in which Hadamard bring known and unknown results into one book. Who should buy was trained. In the second part of the book, Mazya and Shaposhnikova this book? Certainly all topologists with geometric leanings should do discuss the major fields of Hadamard's mathemancs. These chapters, so, and their students too. written with clarity and authority, manage simultaneously to be acces -Bulletin of the London Mathematical Society sible to the novice (with a good undergraduate training in mathemancs Mathematical Surveys and Monographs, Volume 55; 1998; ISBN 0·8218· or science) and to be enlightening to the professional. -siAM Review 0593·2; 258 pages; Hardcover; Individual member $41, List $69, lnst~utional member $55, Order Code SURV/55CT103 Copublished with the London Mathematical Society. Members of the LMS may order directly from the AMS at the AMS member price. The LMS is registered with the Charity Commissioners. History of Mathematics, Volume 14; 1998; ISBN 0·8218·1923-2; 574 pages; Supplementary Readmg Softcover, All AMS members $39, List $49, Order Code HMATH/14.SCT1 03 Pioneers of Representation Theory: Frobenius, Burnside, Schur, and Brauer Recommended Text Charles W. Curtis, University of Oregon, Eugene Riemannian Geometry Parts of the book will appeal to any reader with an interest in the Takashi Sakai, Okayama University, Japan history of mathematics. For instance, there are biographies of each of A good source for teaching a somewhat advanced class in differential the major contributors at the beginnings of the chapters, and geometry and certainly contains enough material for a one-year course. numerous smaller biographical sketches are scattered throughout the {It is] also a good source for the working differennal geometer ... a fine text. There are also revealing glimpses into the mathematical culture book and worthwhile addition to any differential geometer's library. of the early twentieth century found in the numerous citations from the -Bulletin of the American Mathematical Society correspondence between these men. Curtis' accounts of the major upheavals in the lives of German mathematicians caused by the anti· Translations of Mathematical Monographs, Volume 149; 1996; ISBN 0· 8218·0284-4; 358 pages; Softcover, All AMS members $47, List $59, Order semitism of the Nazi regime are also quite engaging. Graduate Code MMON0/149CT103 students just starting out on a research program in algebra or repre sentation theory would benefit immensely from it. Curtis has produced Independent Study a rare but necessary sort of book that fills this need for a more formal introduction to the historical mathematical background of the twen The Prime Numbers and Their Distribution tieth-century mathematics usually seen first in graduate school. Gerald Tenenbaum, Universite Henri Poincare, Nancy I, France, -MAAOnllne and Michel Mendes France, Universite Bordeaux I, France Copublished with the London Mathematical Society. Members of the LMS may order It is a pleasure to read this booklet, written by experts of number directly from the AMS at the AMS member price. The LMS is registered with the theory. Due to the many results, the elegant proofs, and the informal Charity Commissioners. explanation of ideas, it is highly recommended to study this small History of Mathematics, Volume 15; 1999; ISBN 0·8218·2677-8; 292 pages; Softcover; All AMS members $31 , List $39, Order Code HMATH/15.SCT1 03. monograph thoroughly. -Zentralblatt fiir Mathematik. Also available in hardcover. Student Mathematical Library, Volume 6; 2000; ISBN 0-8218·1647-0; 115 pages; Softcover, All AMS members $14, List $17, Order Code STMU6CT103 All prices subject to change. Charges for delivery are $3.00 per order. For optional air delivery outside of the continental U.S., please include $6.50 per nem. Prepayment required. Order from : American Mathematical Society, P.O. Box 845904, Boston, MA 02284·5904, USA. For credit card orders, fax 1-401-455-4046 or call toll free 1-BOQ-321-4AMS (4267) in the U. S. and Canada,1-401·455-4000 worldwide. Or place your order through the AMS Bookstore at www.ams.org/bookstore. Residents of Canada, please include 7% GST. AMERICAN MATHEMATICAL SOCIETY For more publications in your subJect area visit the AMS Bookstore: www.ams.org/bookstore. 346 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Order Online fatbrain.com {*} ~ 8u-JC. rut -L.Js tU .. (.t. WILEY oraer online ana SAVE Save 15% on key Titles in Mathematics For those interested in ordering online, Wiley titles are available at Fatbrain.com Learn more about a special limited offer on key Wiley titles in by visiting http://wwwl.fatbrain.com/fbt/offers/wileymath/ To keep up to date with the latest titles in Mathematics and other professional areas, bookmark http://www.fatbrain.com CLASSIC ALGEBRA Revised Edition P.M. Cohn NEW! Bringing the material of Cohn's three earli NEW! er volumes together into one volume, this Wiley Series in Mathematical Wiley Series in Mathematical text sets a new standard. Proofs have been and Computational Biology and Computational Biology simplified and many new illustrations and exercises included, making this challenging THE MATHEMATICAL THEORY OF COMPUTATIONAL MOLECULAR yet essential subject easier to grasp. "There SELECTION, BIOLOGY: An Introduction is no better text on Algebra than the vol RECOMBINATION Peter Clote & Rolf Backofen umes by Cohn" - Walter Benz, Universitat AND This title provides the background mathematics Hamburg MUTATION required to understand why certain algorithms work. 0-471-87732-8 • 488pp. •10/00. $45.00 pb R. Burger It guides the reader through probability theory, 0-471-87731-X • 442pp. • 01/01 • $99.95 cl This title provides a entropy and combinatorial optimization. unique in-depth study of 0-471-87251-2. 280pp . • 10/00. $120.00 cl Forthcoming appropriate mathemati 0-471-87252-0. 280pp . • 10/00. $55.00 pb MULTI-OBJECTIVE cal methods and models OPTIMIZATION USING for predicting the evolu Forthcoming EVOLUTIONARY tionary change under the combined action of ALGORITHMS GRAPHS selection, mutation and recombination. PERFECT Kalyanmoy Deb 0-471-98653-4. 407pp . • 12/00.$105.00 Jorge Ramirez-Alfonsin, Bruce Reed 0-471-87339-X • 400pp. • 07/01 • 0-471-48970-0. 350pp . • 08/01. $105.00 $85.00 To order, contact: John Wiley & Sons, Inc., Attn: N. Laverde, 9th Floor, 605 Third Avenue, New York, NY 10158. Or call Custome Service at 1-800-225-5945. Fax: 212-850-8888. E-mail: [email protected]. For orders outside of North America, please call: +44 (0) 1243 779777. Or fax: +44 (0) 1243 843296. See our complete listing online at: www.wilev.com/mathematics Prices subject to change without notice and may be higher outside of U.S. Classified Advertisements Positions available, items for sale, services available, and more Department of Mathematical Sciences in ment of 2,700 undergraduates and about ILLINOIS vites applications for two or more antici 1,100 full- and part-time graduate stu pated visiting assistant professorships to dents. Worcester, New England's second SOUTHERN ILLINOIS begin in the fall of 2001. An earned Ph.D. largest city, offers ready access to the di UNIVERSITY CARBONDALE or equivalent degree is required. Success verse economic, cultural, and recreational Department of Mathematics ful candidates must demonstrate strong resources of the region. Temporary Positions research potential and evidence of quality The Mathematical Sciences Department Temporary positions as lecturer for the teaching and will be expected to contribute has 24 tenured/ tenure-track faculty and 2001-02 academic year. Starting date is to the department's research activities and supports B.S., M.S., and Ph.D. programs in August 16, 2001. Master's degree in math to its innovative, project-based educational applied and computational mathematics ematics or admission to candidacy re· programs. and applied statistics. For additional infor quired; Ph.D. preferred. Applicants must At least one position is anticipated in mation, see http: I /www. wpi. edu/ +math/. provide evidence of excellence in teaching each of the following areas: Qualified applicants should send a de and evidence of ability to teach effectively Applied or Computational Mathemat tailed curriculum vitae; a brief statement of in English. Preference given to applicants ics. Preferred research interests are par specific teaching and research objectives; with research interests compatible with tial differential equations with applica and three letters of recommendation, at those of faculty members in the depart tions in fluid and continuum mechanics, least one of which addresses teaching ment. Duties consist of 12 credit hours composite materials, computational mod potential, to either Math or Statistics Vis of undergraduate mathematics instruction eling and simulation, numerical analysis, itor Search Committee, Mathematical Sci each semester. Review of applications will parallel computing, optimization, control ences Department, WPI, 100 Institute Road, begin April 2, 2001, and will continue un theory, applied probability, and discrete Worcester, MA 01609-22 80. til positions are filled. Send applications mathematics. Applicants will be considered on a con (including transcripts) to: Temporary Po Statistics. Applicants are especially en tinuing basis beginning January 1, 2001, sitions, Department of Mathematics, Mail couraged in the areas of time series, until all positions are filled. code 4408, Southern Illinois University experimental design, Bayesian methods, To enrich education through diversity, Carbondale, Carbondale, IL 62901-4408. Monte Carlo methods, data mining meth WPI is an Affirmative Action/ Equal Oppor Southern Illinois University Carbondale is ods, biostatistics, sample survey methods, tunity Employer. an Equal OpportunityI Affirmative Action and survival analysis. Employer. Women and minorities are par Appointments will be made for two WORCESTER POLYTECHNIC INSTITUTE ticularly encouraged to apply. years, contingent on satisfactory teaching, Mathematical Sciences Department with possible renewal for a third year at the discretion of WPI and the appointee. The WPI Department of Mathematical Sci MASSACHUSETTS The teaching load will be either five ?·week ences invites applications and nominations undergraduate courses or four of these for the Harold ]. Gay endowed professor WORCESTER POLYTECHNIC INSTITUTE plus one semester-long graduate course; ship. Candidates are sought in areas of Mathematical Sciences Department the total load is equivalent to ten or eleven applied and computational mathematics Visiting Assistant Professorships semester hours per year. supportive of the strategic development WPI is a private and highly selective of the department and the university. It The Worcester Polytechnic Institute (WPI) technological university with an enroll- is expected that the holder of the profes· Suggested uses for classified advertising are positions available, books issue-April 25, 2001; August 2001 issue-May 24, 2001; September 2001 or lecture notes for sale, books being sought, exchange or rental of issue-June 26, 2001; October 2001 issue-July 26, 2001. houses, and typing services. U.S. Jaws prohibit discrimination in employment on the basis of color, The 2000 rate is $100 per inch or fraction thereof on a single column age, sex, race, religion, or national origin. "Positions Available" advertise· (one-inch minimum), calculated from top of headline. Any fractional text ments from institutions outside the U.S. cannot be published unless they of l/2 inch or more will be charged at the next inch rate. No discounts for are accompanied by a statement that the institution does not discriminate multiple ads or the same ad in consecutive issues. For an additional $10 on these grounds whether or not it is subject to U.S. laws. Details and charge, announcements can be placed anonymously. Correspondence will specific wording may be found on page 1373 (val. 44). be forwarded. Situations wanted advertisements from involuntarily unemployed Advertisements in the "Positions Available" classified section will be set mathematicians are accepted under certain conditions for free publi· with a minimum one·line headline, consisting of the institution name cation. Call toll-free 800·321·4AMS (321-4267) in the U.S. and Canada or above body copy, unless additional headline copy is specified by the 401 ·455-4084 worldwide for further information. advertiser. Headlines will be centered in boldface at no extra charge. Ads Submission: Promotions Department, AMS, P. 0. Box 6248, Providence, will appear in the language in which they are submitted. Rhode Island02940; or via fax: 401-331-3842; or sende·mail to class ads@ There are no member discounts for classified ads. Dictation over the ams. org. AMS location for express delivery packages is 201 Charles telephone will not be accepted for classified advertising. Street, Providence, Rhode Island 02904. Advertisers will be billed upon Upcoming deadlines for classified advertising are as follows: April 2001 publication. issue-January 25, 2001; May 2001 issue-February 23, 2001 ; June/July 348 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Classified Advertisements sorship will have a distinguished research mathematics education. Rank and salary professor and founding chair of a new record, a firm commitment to education, commensurate with experience. Nondis academic department in mathematics. The and the ability to exercise strong leader criminatory Affirmative Action Employer. faculty is seeking a dynamic leader with ship in advancing the academic programs Applicants should send a CV /bibliog proven administrative abilities, significant of the department and the university. raphy, description of experience, a state and current record of research, and ex WPI is a private and highly selective ment on teaching, and three letters of cellent teaching credentials at the under technological university with an enroll recommendation to: Professor B. A. Taylor, graduate and graduate levels. The chair ment of 2,700 undergraduates and about Chair, Department of Mathematics, Univer will direct and facilitate further develop 1,100 full- and part-time graduate stu sity of Michigan, Ann Arbor, MI 48109- ment and growth of this newly organized dents. Worcester, New England's second 1109; e-mail: [email protected]. department, currently consisting of fifteen largest city, offers ready access to the di edu. Further information is available on faculty with a variety of research interests verse economic, cultural, and recreational our home page, http: I lwww . math .lsa. in mathematics. For more information see resources of the region. umich.edul. Applicants considered on a the UNLV World Wide Web site at http: The Mathematical Sciences Department continuing basis. I lwww. un1v. edul. Salary is commensu has 24 tenured/ tenure-track faculty and rate with qualifications and experience. supports B.S., M.S., and Ph.D. programs in Position is contingent upon funding and applied and computational mathematics NEVADA pending institutional and system approval and applied statistics. For additional infor of the new department. Twelve-month-per mation, see http: I lwww . wpi. edul+mathl. UNIVERSITY OF NEVADA year position. A Ph.D. in the mathematical Nominations and applications should be Department Head sciences required. sent to: Search Committee, Mathematical Applicants should submit a detailed cur Sciences Department, WPI, 100 Institute Professor and Founding Chair, Applied riculum vitae; a statement that addresses Road, Worcester, MA 01609-2280. To be Mathematics and Statistics. The University administrative philosophy, research, and complete, applications must include a de of Nevada, Las Vegas, invites applications teaching interests; and the names and ad tailed curriculum vitae, a brief statement of for the position of professor and founding dresses (including e-mail) of five references specific teaching and research objectives, chair of a new academic department in ap to: Professor David G. Costa, Chair of the and the names of four references with plied mathematics and statistics. The fac Search Committee, Department of Math mail/e-mail addresses and telephone/ fax ulty is seeking a dynamic leader, scholar, ematical Sciences, University of Nevada, numbers. teacher, and administrator to direct and Las Vegas, 4505 Maryland Parkway, Box Applications will be considered on a facilitate further development and growth 454020, Las Vegas, NV 89154-4020; phone: continuing basis beginning December 1, of this newly formed department. The 702 -895-0359; e-mail: costa@nevada . edu. 2000, until the position is filled. new department now consists of thir Review of completed applications will be To enrich education through diversity, teen faculty with research interests in gin January 20, 2001, and will continue WPI is an Affirmative Action/Equal Oppor applied/computational mathematics and until the position is filled. tunity Employer. statistics. A Ph.D. in mathematics, statis tics, or a related field in which the methods of applied mathematics and/or statistics NEW JERSEY MICHIGAN are integral to the discipline is required. The successful candidate will have an EDUCATIONAL TESTING SERVICE UNIVERSITY OF MICHIGAN established record of scholarly productiv Positions Available Department of Mathematics ity, demonstrated excellence in teaching, and proven administrative skills. Experi Educational Testing Service is the nation's The department seeks candidates for alec ence in directing doctoral dissertations leading educational assessment organiza turer position beginning September 2001, is desirable. Salary is commensurate with tion and a leader in educational research. involving the operation and direction of its qualifications and experience. Position is We develop and administer achievement, introductory program in precalculus and contingent upon funding and pending in occupational, and admission tests for calculus and the training of instructors for stitutional and system approval of the clients in education, government, and busi these courses. Duties will include the direc new department. The university has an ness. We have the following positions tion of one large multisection precalculus excellent fringe benefits package. For more available in Princeton, NJ: or calculus course per semester; the teach information see: http: I lwww. un1v. edul. Assessment Specialist II ing of one or two sections of the course Applicants should submit a detailed cur Comp. Science/ Mathematics being directed; assistance with our pro riculum vitae and a statement addressing Plan, develop, and evaluate tests and test gram to train and supervise new teachers administrative philosophy, research, and ing programs in computer science and in the introductory program; and general teaching interests. Also, names, addresses, mathematics, including AP computer sci help with the planning, direction, and ad and e-mails of five references should be ence. Use a high level of knowledge in all ministration of the program. Applicants sent to: Dr. George Miel, Chair of the Search phases of test development, including item should have demonstrated excellence in Committee, Department of Mathematical writing, review, and evaluation; test assem the teaching of college mathematics, ex Sciences, University of Nevada, Las Vegas, bly; and scoring. Work independently and perience directing multisection courses in 4505 Maryland Parkway, Box 454020, Las as part of a team; may also work with the first two years of college mathemat Vegas, NV 89154-4020; phone: 702 -895- outside experts in field of specialty. Regu ics, and expertise in modern pedagogical 0360; e-mail: mie1<1lnevada. edu. Review of larly provide guidance and training to less methods. Experience in working with out completed applications will begin January experienced assessment specialist. reach programs is also desirable. Those 20, 2001, and will continue until the posi Master's degree in computer science or who do not precisely fit this description tion is filled. mathematics and six years of increas but who are very strong in several of these ingly responsible professional experience, areas will also be considered. A doctorate UNIVERSITY OF NEVADA including experience in educational mea in mathematics or a closely related area Department Head surement, applied statistics, teaching or is preferred, but all strong candidates will editing, are necessary. Some teaching ex be considered. Preference will be given to Professor and Founding Chair, Mathemat perience is necessary; experience teach candidates who are also involved in mathe ics. The University of Nevada, Las Vegas, ing the contents of AP computer science matical research or scholarship, including invites applications for the position of course is desirable. Knowledge of C++ or MARCH 2001 NOTICES OF THE AMS 349 Classified Advertisements Java is also desirable. An equivalent com HUNTER COLLEGE bination of education and experience may OF THE CITY UNIVERSITY OF be considered. NEW YORK Assessment Specialist I Department of Mathematics and Mathematics Statistics Develop tests in the field of mathematics, including writing, reviewing, and revising Tenure-track assistant professor faculty test questions for the SAT program and position anticipated for September 2001. assembling tests or pools of questions to Salary range: $32,703-$57,049 depending meet specifications. Work independently on experience. Doctoral degree required. and as part of a team; may also work with We are interested in mathematicians (all outside experts in field of specialty. fields). Send CV and three letters of refer ence to: Master's degree in mathematics or a Professor Ada Peluso, Chair related field and three years of increas Department of ingly responsible professional experience, Mathematics and Statistics including experience in educational mea Hunter College surement, applied statistics, teaching, or 695 Park Avenue editing. An equivalent combination of edu New York, NY 10021 cation and experience may be considered. Equal OpportunityI Affirmative Action Em ETS offers competitive salaries, an excel ployer (M/F/D/V). lent benefits package, and an ideal envi ronment for professional growth. Please send 2 copies of your resume along with cover letter and salary require PUBLICATIONS WANTED ments to: Cheryl Wallace, Educational MATHEMATICS BOOKS PURCHASED Testing Service, Rosedale Road, Prince ton, NJ 08541; fax: 609-497-6022; e-mail: Pure & appl. adv. & research level, any cxwallace NEW YORK COLGATE UNIVERSITY Mathematics Department The Mathematics Department of Colgate University is accepting applications for two 1-year leave-replacement positions be ginning August 2001. A Ph.D. is preferred. We invite applications representing all ar eas of mathematics. Colgate University is a highly selective liberal arts college with 2,700 students. Faculty members teach five courses per year and are encouraged to participate in all university programs. An application, including at least a vita, an unofficial graduate transcript, and three letters of recommendation, should be sent to: Search Committee Department of Mathematics Colgate University 13 Oak Drive Hamilton, NY 13346-1398 Screening of applications will begin Jan uary 1, 2001. Colgate is an Equal Opportu nity/Affirmative Action Employer. Appli cations from women and minorities are encouraged. 350 NOTICES OF 1HE AMS VOLUME 48, NUMBER 3 Abstract and Applied Analysis Editor-in-Chief: A G. Kartsatos (University of South Florida) Journal of • Aims & Scope: AAA is devoted exclusively to the publication of original research APPlied MathematiCS papers in the fields of abstract and applied analysis. Emphasis is placed on important developments in classical analysis, linear and nonlinear functional analysis, -··-~:100' ordinary and partial differential equations, optimization theory, and control theory. Subscription Information (ISSN 1085-3375, 2001, volume 6, 8 issues): $295 (print and electronic), $195 (electronic only), $195 (personal print and electronic), $95 (personal electronic only). journal's web site: http://aaa.hindawi.com. International journal of Mathematics and Mathematical Sciences Founding Managing Editor: L. Debnath (University of Central Florida) Aims & Scope: UMMS is devoted to the publication of original research papers, research notes, and research expository and survey articles with emphasis on unsolved problems and open questions in mathematics and mathematical sciences. All areas listed on the cover of Mathematical Reviews are included within the scope of the journal. Subscription Information (ISSN 0161-1712, 2001, volumes 25-28, 48 issues): $495 (print and electronic), $395 (electronic only), $395 (personal print and electronic), $95 (personal electronic only). journal's web site: http://ijmms.hindawi.com. journal of Applied Mathematics Editors-in-Chief: C. Brezinski (Universite des Sciences et Technologies de Lil/e), L. Debnath (University of Central Florida), H. Nijmeijer (Eindhoven University of Technology) Aims & Scope: JAM is devoted to the publication of original research papers and review articles in all areas of applied, computational, and industrial mathematics. Subscription Information (ISSN 111 0-757X, 2001, volume 1, irregular): $295 (print and electronic), $195 (electronic only), $195 (personal print and electronic), $95 (personal electronic only). journal's web site: http://jam.hindawi.com. Bundle Subscription: All three journals for $995 (print and electronic), $695 (electronic only), $695 (personal print and electronic), $195 (personal electronic only). Online availability: All journals' back volumes are available online. Subscription to cur rent year enables online access to all back volumes during the subscription period. Hindawi Alert: Register with "Hindawi Alert" at http://www.hindawi.com/alert.html to receive email notifications with tables of contents as new issues appear online. Backlog Information: Current detailed backlog information of all journals is available at http://www.hindawi.com/backlog.html . Electronic Submission: Submit manuscripts to UMMS at [email protected] and to JAM at submit®jam.hindawi.com. Books online: Symmetry in Nonlinear Mathematical Physics. Proceedings of EWM 8th and 9th general meetings. EWM Workshop on Moduli Spaces in Mathematics and Physics. Visit http://www.hindawi.com for more mathematics publications. ;--~ . Hindawi Publishi ng Corporation, P.O. Box 5873, Akron, O H 44334~8 n , "~: Fax: 1 888 5224561 (USA toll-free), 31 20 5248282 (Amsterclam, H0lland) . ~" ~ HINDAWI URL: http://www.hindawi.com, Email : [email protected] J.:.r3·~ technische universiteit eindhoven for a professorship in statistics rtment of Mathematics and Computing Science has a vacancy for a full ip in Statistics. of Mathematics and Computing Your skills undergraduate and M.Sc. programs • A deep and broad insight into statistics, as is reflected and Applied Mathematics and in Computer in a strong international reputation in the field and The Department has research collaborations a large number of-- also recent-- publications in the Departments at the Technische Universiteit international scientific literature; as well as with a large number of other • Much affinity with, and knowledge of, stochastics; and companies, both at home and abroad. • Interest in application-oriented research; Department has approximately 200 employees • Experience and expertise in acquisition and and more than 6oo students. consultancy; The chair of Statistics is one of the eight chairs • Excellent didactical qualities; in Mathematics, and one of the three chairs in the • Leadership and management qualities. Statistics and Operations Research Group, the other ones being the chair of Combinatorial Optimization What we have to offer (Prof.dr. ).K. Lenstra) and on Stochastic Operations • A prominent leading position in a stimulating Research (Prof.dr. 0 .). Boxma). The other two groups scientific environment, in which you will work with in Mathematics cover various topics in Analysis and enthusiastic students, trainee design engineers, and Discrete Mathematics. PhD students, as well as postdocs from EURANDOM; • A full-time appointment in accordance with the The Statistics group consists of a full professor, Collective Labour Agreement for Dutch Universities; an associate professor and 5 assistant professors. • A maximum salary of DFL 2oo,ooo per annum The group provides undergraduate and graduate depending on your experience; statistics and probability courses for mathematics • An extensive package of fringe benefits. students and students of other Departments. The research in the group is both fundamental and Inquiries applied, with an emphasis on extreme value theory, For more information about the content of this position, non-parametric statistics, design of experiments and please contact the chairman of the appointment industrial statistics. The presence of the Department committee: Prof.dr. 0 .). Boxma, e-mail: [email protected], of Biomedical Engineering provides additional tel. +31 40 2472858, private tel. +31 40 2845800. opportunities for research in biostatistics. You are welcome to contact Prof.dr. 0 .). Boxma if you The Statistics group participates in two research wish to inform the committee of possible candidates schools: The mathematics research school Stieltjes for the position. For information concerning labour and research school BETA in the area of production conditions, please contact W.C.) . Verhoef, personnel and logistics. The group participates very actively in adviser, tel. +31 40 2472321. EURANDOM, the new European institute for research For general information about Department of in Statistics, Probability and Stochastic Operations Mathematics and Computing Science: www.win.tue.nl. Research. The new scientific director of EU RANDOM, Prof.dr. F. den Hollander, holds a part-time professor How to respond ship in Probability in the Statistics Group. Please submit a written letter of application accompa nied by a recent curriculum vitae within four weeks of What are your duties? the publication of this vacancy. Please state the vacancy • You are expected to stimulate and coordinate the code V 32854 in your letter and on the envelope. fundamental and applied research of the group, to initiate new research directions and to establish Technische Universiteit Eindhoven, Attn. Mrs S. Udo, links with other research programs at our university Managing Director of the Department of Mathematics and with EURANDOM; and Computing Science, HG 6.19, P.O. Box 513, • You give and coordinate courses in Statistics and 5600MB Eindhoven, The Netherlands. Probability. and you are responsible for updating these courses; you advise MSc and PhD students; • You fulfill key management functions in the section and the faculty. www.academicpress.com/mathematics first time in print, proofs of the Double Bubble the Hexagonal Honeycomb Conjecture. $69.95/£43.95, ISBN: 0-12-506851-4 Edition and paperback format, this classic and graduate text now includes a supplemental extended coverage of measure theory. Paperback, 448 pp., $59.95/£37.95, ISBN: 0-12-174151 -6 on Lie Algebras and Superalgebras AJ:tto,nlllto Sciarrino, and Paul Sorba Lie Algebras and Superalgebras presents a of the structure of Lie algebras and Lie an extensive study of their finite dimensional Includes CD-ROM. $69.95/£37.95, ISBN: 0-12-265340-8 TlfClite«mf,erg, Cornelis Oosterlee, and Anton Schuller me,thclds are the most efficient solvers for elliptic partial nttlllie,au;!tlc>ns and belong to the fastest methods in scientific are invaluable to engineers and scientists in a wide o[ idis,ciplli nE!S from computational physics to financial of a classic textbook includes material covering stability · radius, and applications to signal analysis and the of partial differential equations. 200 I, c. 304 pp., $69.95/£45.00, ISBN: 0-12-772955-0 ~tlllthrarlate Polyspllnes ications to Numerical and Wavelet Analysis , -,. FranklinW Olin College of Engineering Faculty Position. MatheDJ.atics The Franklin W. Olin College of Engineering is a new institution that strives to provide the best and most innovative engineering education to the world's brightest and most enterprising students. The College is seeking exceptional faculty dedicated to exemplary undergraduate teaching Independent Study and committed to innovation and intellectual vitality through one or more The Prime creative endeavors. Numbers and Their Distribution Faculty are expected to become inspirational teachers of undergraduates, work with other faculty to develop new programs, and obtain national Gerald Tenenbaum, visibility in their field. Preference will be given to experienced candidates Universite Henri Poincare, with a record of demonstrated excellence or with conspicuous ability and Nancy~ France, and motivation. We are especially interested in candidates with backgrounds in Michel Mendes France, applied mathematics or statistics, however exceptional candidates in other Universite Bordeaux I, France areas such as dynamical systems, probability theory/stochastic processes We have been curious about num are encouraged to apply. Familiarity with current issues and approaches bers-and prime numbers-since antiquity. The goal of this book is in teaching college mathematics are important. Experience with the issues to provide insights into the prime in engineering education, and with the different approaches being taken at numbers and to describe how a various institutions are also attractive. sequence so tautly determined can incorporate such a striking amount of randomness. A full The Franklin W. Olin College of Engineering, established in 1997 by a description of the distribution of major commitment from the Franklin W. Olin Foundation, will provide all the prime numbers is closely students a full 4-year scholarship. An entirely new campus is currently connected with the Riemann under construction in Needham, MA, adjacent to Babson College. While Hypothesis, a long-standing Olin College is a completely independent institution, access to Babson's problem which has recently been world-class programs and other colleges near Boston's Route 128 included in the Clay Institute's "Millennium Prize Problems" high-technology corridor will enrich the opportunities available to Olin (www.claymath.org), each worth a faculty and students. million dollars. Student Mathematical Library, Volume 6; 2000; 115 pages; Softcover; To apply, please send an application letter describing your teaching, research ISBN 0-8218-1647-0; List $17; All AMS and other professional goals and accomplishments with a current resume to: members $14; Order code STMU6NA Mathematics Faculty Search, c/o Dr. David V. Kerns, Jr., Provost, Franklin W. Olin College of Engineering, MS-MA, 1735 Great Plain Ave., Needham, MA 02492-1245. Email: [email protected] Applications and nominations will be considered until all open positions are filled. The Franklin W. Olin College of Engineering is an All prices subject to change. Charges for delivery Equal Opportunity Employer. are $3.00 per order. For optional air delivery outside of the continental U. S., please include $6.50 per item. Prepayment required. Order from: American For more information Mathematical Society, P. 0. Box 845904, Boston, MA 02284-5904, USA. For credit card orders, fax 1- 401 -455-4046 or call toll free 1-800-321-4AMS visit: W\IV\IV.olin.edu (4267) in the U.S. and Canada, 1-401-455-4000 worldwide. Or place your order through the AMS bookstore at www.ams.org/bookstore. Residents of Canada, please include 7% GST. APPliCATION FOR 2001 MEMBERSHIP JANUARY--D[C EMBER WWW.AMS.ORG/MEMBERSHIP/ Please read the "Membership Categories" seaion of this form to determine the Fields of Interest membership category for which you are eligible. Then (Ill out this application and If you wish to be on the mailing lists to receive information about publications in return it as soon as possible. fields of mathematics in which you have an interest, please consult the list of major headings below. These categories will be added to your computer record so that Date ______20 you will be informed of new publications or special sales in the fields you have indi- cated . Family Name First Middle 00 General 46 Functional analysis 01 History and biography 47 Operator theory Place of Birth: 03 Mathematical logic and foundations 49 Calculus of variations and optimal City State Country 05 Combinatorics control; optimization 06 Order, lattices, ordered algebraic 51 Geometry Date of Birth: structures 52 Convex and discrete geometry Day Month Year 08 General algebraic systems 53 Differential geometry 11 Number theory 54 General topology If formerly a member of AMS, please indicate dates ------12 Field theory and polynomials 55 Algebraic topology 13 Commutative rings and algebras 57 Manifolds and cell complexes 14 Algebraic geometry 58 Global analysis, analysis on Check here if you are now a member of either 0 MAA or 0 SIAM 15 Linear and multilinear algebra; manifolds matrix theory 60 Probability theory and stochastic Degrees, with institutions and dates ------16 Associative rings and algebras processes 17 Nonassociative rings and algebras 62 Statistics 18 Category theory, homological 65 Numerical analysis algebra 68 Computer science 19 K-theory 70 Mechanics of particles and systems 20 Group theory and generalizations 74 Mechanics of deformable solids 22 Topological groups, Lie groups 76 Fluid mechanics Present position------26 Real functions 78 Optics, electromagnetic theory 28 Measure and integration 80 Classical thermodynamics, heat Firm or institution ------30 Functions of a complex variable transfer 31 Potential theory 81 Quantum theory 32 Several complex variables and 82 Statistical mechanics, structure of analytic spaces matter 33 Special functions 83 Relativity and gravitational theory City State Zip/Country 34 Ordinary differential equations 85 Astronomy and astrophysics 35 Partial differential equations 86 Geophysics Primary Fields of Interest (choose five from the list at right) 37 Dynamical systems and ergodic 90 Operations research, mathematical theory programming, 39 Difference and functional equations 91 Game theory, economics, social 40 Sequences, series, summability and behavioral sciences Secondary Fields of Interest (choose from the list at right) 41 Approximations and expansions 92 Biology and other natural sciences 42 Fourier analysis 93 Systems theory; control 43 Abstract harmonic analysis 94 Information and communication, 44 Integral transforms, operational circuits calculus 97 Mathematics Education 45 Integral equations Address for all mail Prepayment Methods and Mailing Addresses All prices quoted in U.S. dollars . Payment by check must be drawn on U.S. bank if paid in U.S. dollars. Send checks, money orders, UNESCO coupons to American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904. To use credit cards, fill in information requested and mail to American Telephone: home:( __) ______office:( __) ______Mathematical Society, P.O. Box 6248, Providence, Rl 02940-6248 or call Fax: ( __) ______(401) 455-4000 or 1-800-321-4AMS. E-mail:------For Foreign Bank Transfers: Citizens Bank, 1 Citizens Drive, Riverside, Rl 02915, ABA #011-500-120, for American Mathematical Society, Acct #11 07802021 . 0 American Express 0 Discover OVISA 0 MasterCard Signature 0 Check here to receive free quarterly PUBLICATIONS UPDATE by mail. Account number Expiration date r:-1M=-=-o-=-=o N""'"'o=-1 Membership Categories Reciprocating Societies Please read the following to determine what membership category you are eligible for, D Allahabad Mathematical Society D Sociedad Matematica de Ia Republica Dominicana and then indicate below the category for which you are applying. D Australian Mathematical Society D Sociedad Matematica Mexicana Introductory ordinary member rate applies to the first five consecutive years of ordi D Azerbaijan Mathematical Society D Sociedad Uruguaya de Matematica y nary membership. Eligibility begins with the first year of membership in any category D Balkan Society of Geometers other than student and nominee. Dues are $51 . Estadistica D Berliner Mathematische D Sociedade Brasileira Matematica For ordinary members whose annual professional income is below $75,000, the dues Gessellschaft are $1 02; for those whose annual professional income is $75,000 or more, the dues are D Sociedade Brasileira de Matematica D Calcutta Mathematical Society $136. Aplicada e Computacional D Canadian Mathematical Society For a joint family membership, one member pays ordinary dues, based on his or her D Sociedade Paranaense de income; the other pays ordinary dues based on his or her income, less $20. (Only the D Croatian Mathematical Society Matematica member paying full dues will receive the Notices and the Bulletin as a privilege of mem D Cyprus Mathematical Society D Sociedade Portuguesa de bership, but both members will be accorded all other privileges of membership.) D Dansk Matematisk Forening Matematica Minimum dues for contributing members are $204. The amount paid which exceeds D Societal Catalana de Matematiques the higher ordinary dues level and is purely voluntary may be treated as a charitable con D Deutsche Mathematiker-Vereinigung D Edinburgh Mathematical Society D Societatea de $1iinte Matematice din tribution. Romania For either students or unemployed individuals, dues are $34, and annual verification D Egyptian Mathematical Society is required . D Societatea Matematicienilor din D European Mathematical Society Romania The annual dues for reciprocity members who reside outside the U.S. are $68. To be eligible for this classification, members must belong to one of those foreign societies with D Gesellschaft fUr Angewandte D Societe MatMmatique de Belgique Mathematik und Mechanik which the AMS has established a reciprocity agreement, and annual verification is D Societe MatMmatique de France D Glasgow Mathematical Association required. Reciprocity members who reside in the U.S. must pay ordinary member dues D Societe MatMmatique du ($1 02 or $136). D Hellenic Mathematical Society Luxembourg The annual dues for category-S members, those who reside in developing countries, D Icelandic Mathematical Society D Societe Mathematique Suisse are $16. Members can chose only one privilege journal. Please indicate your choice below. D Indian Mathematical Society D Societe Mathematiques Appliquees Members can purchase a multi-year membership by prepaying their current dues rate D Iranian Mathematical Society et lndustrielles for either two, three, four or five years. This option is not available to category-S, unem D Irish Mathematical Society D Society of Associations of ployed, or student members. Mathematicians & Computer D Israel Mathematical Union Science of Macedonia D Janos Bolyai Mathematical Society D Society of Mathematicians, 100 I Dues Schedule (January through December) D The Korean Mathematical Society Physicists, and Astronomers of Slovenia Ordinary member, introductory rate...... D$51 D London Mathematical Society D South African Mathematical Society Ordinary member ...... D$102 D$136 D Malaysian Mathematical Society D Southeast Asian Mathematical Joint family member (full rate) ...... D$102 D$136 D Mathematical Society of Japan Society Joint family member (reduced rate) ...... D$82 D$116 D Mathematical Society of Serbia D Suomen Matemaattinen Yhdistys Contributing member (minimum $204) ...... D D Mathematical Society of the D Svenska Matematikersamfundet Philippines Student member (please verify) 1...... D $34 D Ukrainian Mathematical Society D Mathematical Society of the Unemployed member (please verify) 2 ...... D$34 Republic of China D Union Mathematica Argentina Union of Bulgarian Mathematicians Reciprocity member (please verify) 3 ...... D$68 D$102 D$136 D Mongolian Mathematical Society D D Union of Czech Mathematicians and Category-$ member 4 ...... ,...... D$16 D Nepal Mathematical Society Physicists Multi-year membership ...... $ ..1 ...... for...... years D New Zealand Mathematical Society D Union of Slovak Mathematicians and D Nigerian Mathematical Society Physicists 1 Student Verification (sign below) I am a full-time student at ______D Norsk Matematisk Forening D Unione Matematica ltaliana D bsterreichische Mathematische D Vijnana Parishad of India Gesellschaft ______currently working toward a degree. D Wiskundig Genootschap D Palestine Society for Mathematical Sciences 2 Unemployed Verification (sign below) I am currently unemployed and actively seeking employment. D Polskie Towarzystwo Matematyczne D Punjab Mathematical Society 3 Reciprocity Membership Verification (sign below) I am currently a member of D Ramanujan Mathematical Society the society indicated on the right and am therefore eligible for reciprocity membership. D Real Sociedad Matematica Espanola D Saudi Association for Mathematical Sciences D Singapore Mathematical Society Signature D Sociedad Colombiana de Matematicas 4 D send NOTICES D send BULLETIN D Sociedad Espanola de Matematica Aplicada D Sociedad de Matematica de Chile Name ------ Customer code ______ Change effective as of------ Old mailing address New mailing address Members of the Society When changing their who move or change posi addresses, members are tions are urged to notify the urged to cooperate by sup Providence Office as soon as plying the requested infor New position------possible. mation. The Society's If mailing address is not that of your employer, records are of value only to journal mailing lists must be the extent that they are cur printed four to six weeks please supply the following informations: rent and accurate. before the issue date. New em pi oyer ------Therefore, in order to avoid If your address has changed disruption of service, mem or will change within the Location of employer (city, state, zip code, country) bers are requested to pro next two or three months, vide the required notice well please fill out this form, sup in advance. ply any other information appropriate for the AMS Besides mailing addresses records, and mail it to: for members, the Society's Telephone/home: ------office: ------records contain information Customer Services Fax: ______about members' positions AMS and their employers (for P.O. Box 6248 publication in the Combined Providence, Rl 02940 e-mail ------Membership List). In addi or send the information on Recent honors and awards tion, the AMS maintains the form by e-mail to: records of members' honors, awards, and information on [email protected] or Society service. [email protected] Fold here NO POSTAGE NECESSARY IF MAILED IN THE UNITED STATES BUSINESS REPLY MAIL FIRST-CLASS MAIL PERMIT NO. 5548 PROVIDENCE, Rl POSTAGE WILL BE PAID BY ADDRESSEE Membership and Customer Services AMERICAN MATHEMATICAL SOCIETY P. 0. Box 6248 Providence, RI 02940-9943 111 ••••• 1.11.1 ••• 1•• 111 ••• 1.1 •• 1.1 ••• 1•• 1•• 11.11 ••• 1 Fold here Staple here - - ~ew Titles for Res~arch and-Teaching ----;;,) - - -- - ~~ - PERSPECTIVES IN FLUID DYNAMICS FOURIER ANALYSIS AND PARTIAL DIFFERENTIAL A Collective Introduction to Current Research EQUATIONS George Batchelor, Keith Moffatt, and An Introduction Grae Worster, Editors Rafael Jose Iorio, Jr. and Provides coverage of topics including thin-film flows, Valeria de Magalhaes Iorio Saffman-Taylor fingering, flows in arteries and veins, con Covers everything from the fundamentals to research vective and absolute instabilities, turbulence, natural con topics in the area of nonlinear evolution equations, vection, magnetohydrodynamics, solidification, geological including a fairly complete discussion of local and global fluid mechanics, oceanography and atmospheric dynamics. well-posedness for the nonlinear Schrodinger and the 2001 646 pp. Korteweg-de Vries equations. 0-521-78061-6 Hardback $160.00 Cambridge Studies in Advanced Mathematics 70 2001 416 pp. ORTHOGONAL POLYNOMIALS OF SEVERAL VARIABLES 0-521-62116-X Hardback $69.95 Charles Dunk/ and Yuan Xu Describes the general theory and emphasizes the classical DERIVATION AND INTEGRATION types of orthogonal polynomials whose weight functions Washek F. Pfeffer are supported on standard domains such as the cube, the Considers additive functions defined on the family of all simplex, the sphere and the ball. bounded BV sets that are continuous with respect to a Encyclopedia ofMathematics and its Applications 81 suitable topology. The main applications are related to 2001 400 pp. the Gauss-Green and Stokes theorems. 0-521 -80043-9 Hardback $80.00 Cambridge Tracts in Mathematics 140 2001 288 pp. NONLINEAR AND NONSTATIONARY SIGNAL 0-521-79268-1 Hardback $64.95 PROCESSING W. J. Fitzgerald, R. L. Smith, A. T. Walden, and GEOMETRIC PARTIAL DIFFERENTIAL EQUATIONS AND P. C. Young, Editors IMAGE ANALYSIS Introduces modern methods in the fields of statistics, Guillermo Sapiro engineering, environmental science and finance, to ad Brings a number of new concepts into the field, dress the shortcomings of simplistic assumptions about providing a very fundamental and formal approach the linearity and stationarity of the underlying processes. to image processing. Applications covered include 2001 520 pp. image segmentation, shape analysis, image enhancement, 0-521-80044-7 Hardback $95.00 and tracking. 2001 440 pp. THE GEOMETRY OF PHYSICS 0-521-79075-1 Hardback $64.95 An Introduction Theodore Frankel THE MATHEMAT/CA® BOOK, Discusses analytical and fluid dynamics, electromagnetism, VERSION 4 thermodynamics, the deformation tensors of elasticity, Stephen Wolfram soap films, special and general relativity, the Dirac New to this version: visual tour of key features, practical operator and spinors, and gauge fields, including Yang tutorial introduction, full descriptions of 1100 built-in Mills, the Aharonov-Bohm effect, Berry phase, and functions, a thousand illustrative examples, easy-to-follow instanton winding numbers. descriptive tables, and an award-winning gallery of 1999 678 pp. Mathematica® graphics. 0-521-38753-1 Paperback $36.95 1999 1496 pp. 0-521-64314-7 Hardback $49.95 TURBULENCE STRUCTURE VORTEX DYNAMICS J. C. R. Hunt and J. C. Vassi/icos, Editors Examines a number of key questions that have engaged Availablein ~CAMBRIDGE turbulence researchers for many years. Most involve mathe bookstoresorfrom ~-~ UNIVERSITY PRESS matical analysis, but some describe numerical simulations and experimental results that focus on these questions. 40 West 20th Street, New York, NY 10011-4211 Call toll-free 800-872-7423 2001 300 pp. 0-521-78131-0 Hardback $80.00 Web site: us.cambridge.org/ mathematics AmEx/ MasterCard/VISA accepted. Prices subject to change. Meetings & Conferences oftheAMS IMPORTANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found one-MATH. See http: I /www. ams. org/meeti ngs/. Programs and abstracts will continue to be displayed one-MATH in the Meetings and Conferences section until about three weeks after the meeting is over. Final programs for Sectional Meetings will be archived one-MATH in an electronic issue of the Notices as noted below for each meeting. Carl Pomerance, Bell Laboratories - Lucent Technologies, Columbia, South Babe Ruth, Hank Aaron, Paul Erdos, and me (Erdos Memo rial Lecture). Carolina Special Sessions University of South Carolina Algebraic Structures Associated with Lie Theory, Ben L. March 16-18, 2001 Cox, Elizabeth Jurisich, and Oleg Smirnov, College of Charleston. Meeting #963 Algebras, Lattices, Varieties, George F. McNulty, University Southeastern Section of South Carolina, and Ralph S. Freese and James B. Associate secretary: John L. Bryant Nation, University of Hawaii. Announcement issue of Notices: January 2001 Program first available one-MATH: February 1, 2001 Analytic Number Theory, Michael A. Filaseta and Ognian Program issue of electronic Notices: April 2001 Trifonov, University of South Carolina, Columbia. Issue of Abstracts: Volume 22, Issue 2 Approximation and Wavelets, Konstantin Oskolkov, Pencho Petrushev, and Vladimir Temlyakov, University Deadlines of South Carolina, Columbia. For organizers: Expired For consideration of contributed papers in Special Ses- Banach Spaces, George Androulakis, Stephen Dilworth, and sions: Expired Maria K. Girardi, University of South Carolina, Columbia. For abstracts: Expired Combinatorics and Graph Theory, Jerrold R. Griggs, University of South Carolina, Columbia. Invited Addresses Discrete and Computational Geometry and Graph Drawing, David C. Brydges, University of Virginia, Analysis with Farhad Shahrokhi, University of North Texas, and Laszlo A. functional integrals. Szekely, University of South Carolina, Columbia. Albert A. Cohen, University of Paris VI, Harmonic analy Geometry of Curves and Surfaces, Mohammad Ghomi and sis and image processing. Ralph E. Howard, University of South Carolina, Columbia. Herbert Edelsbrunner, Duke University and Raindrop Geomagic, Mathematics problems in the (re-)construction of Mathematical Biology, Douglas B. Meade, Matthew Miller, shape. and David Wethey, University of South Carolina, Columbia. Daniel J. Kleitman, Massachusetts Institute of Technology, Semigroups and Evolution Equations, Anton R. Schep, Title to be announced. University of South Carolina, Columbia. 360 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Meetings & Conferences Progress in Numerical Linear Algebra, Ralph Byers, Lawrence, Kansas University of Kansas. of Kansas Quantization and Operator Algebras, Albert Sheu, University University of Kansas. March 30-31, 2001 Set Theoretic Topology and Boolean Algebra, William Fleissner, University of Kansas. Meeting #964 Central Section Associate secretary: Susan]. Friedlander Las Vegas, Nevada Announcement issue of Notices: January 2001 University of Nevada Program first available one-MATH: February 15, 2001 Program issue of electronic Notices: May 2001 April21-22, 2001 Issue of Abstracts. Volume 22, Issue 2 Meeting #965 Deadlines Western Section For organizers: Expired Associate secretary: Bernard Russo For consideration of contributed papers in Special Ses Announcement issue of Notices. February 2001 sions: Expired Program first available one-MATH: March 8, 2001 For abstracts: Expired Program issue of electronic Notices: May 2001 Issue of Abstracts. Volume 22, Issue 2 Invited Addresses S. Dale Cutkosky, University of Missouri, Title to be Deadlines announced. For organizers: Expired For consideration of contributed papers in Special Ses- le to be Alexandre Eremenko, Purdue University, Tit sions: Expired announced. For abstracts: February 27, 2001 Ken Ono, University of Wisconsin-Madison, Title to be announced. Invited Addresses Yongbin Ruan, University of Wisconsin-Madison, Title to Panagiota Daskalopoulos, University of California Irvine, be announced. Title to be announced. Randall J. LeVeque, University of Washington, Title to be Special Sessions announced. Algebraic Geometry, Christopher Peterson, Colorado State Vera Serganova, University of California Berkeley, Title to University, and B. P. Purnaprajna, University of Kansas. be announced. C*-Algebras and Crossed Products, Steve Kaliszewski, Lynne Walling, University of Colorado, Title to be Arizona State University, and May Nilsen, Texas A&M announced. University. Special Sessions Commutative Algebra, Craig Huneke and Daniel Katz, University of Kansas. Analysis and Applications ofNonlinear PDEs (Code: AMS SS Tehrani, Complex Variables, Pietro Poggi-Corradini, Kansas State G1), David G. Costa, Zhonghai Ding, and Hossein University. University of Nevada, Las Vegas. Analysis and Applications (Code: AMS SS Deformation Theory, Yan Soibelman and David Yetter, Finite Element Li, Michael Marcozzi, George Miel, and Kansas State University. B1), Jichun Darrell W. Pepper, University of Nevada. Harmonic Analysis and Applications, Rodolfo Torres, Geometric and Computational Group Theory (Code: AMS SS University of Kansas. A1), Eric M. Freden, Southern Utah University, and Eric L. Number Theory, Ken Ono, University of Wisconsin Swenson, Brigham Young University. Popescu, University of Texas at Austin, Madison, Cristian Graphs and Digraphs (Code: AMS SS Cl), Michael Jacobson, and Tonghai Yang, Harvard University. University of Louisville, and K. Brooks Reid, California Optimal Control, Calculus of Variations, and Nonsmooth State University, San Marcos. Analysis, Michael Malisoff, Texas A&M University, Corpus History of Mathematics (Code: AMS SS ]1), Shawnee L. Christi, and Peter R. Wolenski, Louisiana State University. McMurran, California State University, San Bernadino, PDEs and Geometry, Marianne Korten and Lev Kapitanski, Adrian Rice, Randolph-Macon College, and James Tattersall, Kansas State University. Providence College. Polytopes, Margaret Bayer and Carl Lee, University of Number Theory with a Geometric Flavor (Code: AMS SS Kl), Kansas. Arthur Baragar, University of Nevada, Las Vegas. MARCH 2001 NOTICES OF THE AMS 361 Meetings & Conferences PDEs from Fluid Mechanics: Applied Analysis and Numerical Computational Group Theory (Code: AMS SS Cl), Robert Methods (Code: AMS SS H1), L. Steven Hou, York University Gilman, Stevens Institute of Technology, and Alexei and Iowa University, and Xiaoming Wang, Iowa State Myasnikov, Vladimir Shpilrain, and Sean Cleary, City University. College, New York. Physical Knotting and Unknotting (Code: AMS SS D1), Jorge Deformation Quantization and Its Applications (Code: AMS Alberto Calvo, North Dakota State University, Kenneth C. SS K1), Siddhartha Sahi, Rutgers University, and Martin J. Millett, University of California Santa Barbara, and Eric J. Andler, University of Versailles. Rawdon, Chatham College. Graph Theory (Dedicated to Frank Harary on His BOth Set Theory(Code: AMS SS E1), Douglas Burke and Derrick Birthday) (Code: AMS SS M1), Michael L. Gargano and Dubose, University of Nevada, Las Vegas. Louis V. Quintas, Pace University, and Charles Suffel, Stevens Institute of Technology. Topology ofLinks (Code: AMS SS Fl), Jeff Johannes, Univer sity of Nevada, Las Vegas, and Swatee Naik, University of History ofMathematics (Code: AMS SS E1 ), Patricia R. Allaire, Nevada, Reno. Queensborough Community College, CUNY, and Robert E. Bradley, Adelphi University. Waves in Heterogeneous Media (Code: AMS SS Ll), Randall]. LeVeque, University of Washington, and Knut Solna, Uni Matchings in Graphs and Hypergraphs (Code: AMS SS Fl), versity of California Irvine. Alexander Barvinok, University of Michigan, and Alex Samorodnitsky, Institute for Advanced Study. Quantum Error Correction and Related Aspects of Coding Theory (Code: AMS SS ]1), Harriet S. Pollatsek, Mount Hoboken, New Jersey Holyoke College, and M. Beth Ruskai, University of Mass Stevens Institute of Technology achusetts, Lowell. Ricci Curvature and Related Topics (Code: AMS SS G1), April28-29, 2001 George I. Kamberov, Stevens Institute of Technology, Christina Sormani, Lehman College, CUNY, and Megan M. Meeting #966 Kerr, Wellesley College. Eastern Section Singular and Degenerate Nonlinear Elliptic Boundary Value Associate secretary: Lesley M. Sibner Problems (Code: AMS SS D1), Joe McKenna, Changfeng Announcement issue of Notices: February 2001 Gui, and Yung Sze Choi, University of Connecticut. Program first available one-MATH: March 15, 2001 Program issue of electronic Notices: May 2001 Stability of Nonlinear Dispersive Waves (Code: AMS SS H1), Issue of Abstracts: Volume 22, Issue 2 Yi Li, Stevens Institute of Technology, and Keith S. Promislow, Simon Fraser University. Deadlines Surface Geometry and Shape Perception (Code: AMS SS Ll), For organizers: Expired Gary R. Jensen, Washington University, and George I. For consideration of contributed papers in Special Ses Kamberov, Stevens Institute of Technology. sions: Expired Wavelets, Multiscale Analysis, and Applications (Code: AMS For abstracts: March 6, 2001 SS N1), Ivan Selesnick, Polytechnic University, and Gerald Schuller, Bell Laboratories. Invited Addresses Alexander Barvinok, University of Michigan, Ann Arbor, Title to be announced. Morelia, Mexico Robert Calderbank, AT&T Laboratories Research, Title to May 23-26,2001 be announced. Alexei Miasnikov, City College, New York, Title to be Meeting #967 announced. Fifth International ]oint Meeting of the AMS and the Sociedad Frank Sottile, University of Massachusetts, Amherst, A Matematica Mexicana (SMM). Gromov-Witten invariant in the real world. Associate secretary: John L. Bryant Announcement issue of Notices: To be announced Special Sessions Program first available one-MATH: To be announced Program issue of electronic Notices: To be announced Analytic Number Theory (Code: AMS SS A1), Milos A. Dostal, Issue of Abstracts: To be announced Stevens Institute of Technology, and Werner G. Nowak, Vienna, Austria. Deadlines Computational Algebraic Geometry and Its Applications For organizers: To be announced (Code: AMS SS B1), Serkan Hosten, San Francisco State For consideration of contributed papers in Special Ses University, and Frank Sottile, University of Massachusetts, sions: To be announced Amherst. For abstracts: To be announced 362 NOTICES OF 1HE AMS VOLUME 48, NUMBER 3 Meetings & Conferences Invited Addresses Associate secretary: Lesley M. Sibner Announcement issue of Notices: To be announced Victor Perez Abreu, CIMAT, Title to be announced. Program first available one-MATH: To be announced Eric M. Friedlander, Northwestern University, Between Program issue of electronic Notices: To be announced algebraic and topological K -theory. Issue of Abstracts: To be announced Helmut H. W. Hofer, Courant Institute, New York Univer sity, Periodic orbits, holomorphic curves, and algebraic Deadlines invariants. Fororganizers: To be announced Emesto A. Lacomba, UAM-I, Title to be announced. For consideration of contributed papers in Special Ses sions: To be announced Claude R. LeBrun, SUNY at Stony Brook, On the curvature For abstracts: To be announced of 4-manifolds. Antonmaria Minzoni, IIMAS-UNAM, Title to be announced. Invited Addresses Sun-Yung Alice Chang, Princeton University, Title to be Special Sessions announced. Algebraic Geometry, Pedro Luis del Angel, CIMAT, and Jean-Pierre Demailly, Universite de Grenoble, Title to be Javier Elizondo, IMATE-UNAM. announced. Algebraic Topology and K-Theory, Miguel Xicotencatl, Persi Diaconis, Stanford University, Title to be announced. CINVESTAV. Robert Gardner, University of Massachusetts, Amherst, Biomathematics, Jorge Velasco, UAM-I. Title to be announced. Combinatorics and Graph Theory, Emesto Vallejo, IMATE Claude Le Bris, Universite de Paris IX-Dauphine, Title to be UNAM. announced. Complex Analysis, Enrique Ramirez de Arellano, Yves Meyer, Ecole Normale Superieure de Cachan, Title to CINVESTAV. be announced. Differential Geometry, Adolfo Sanchez Valenzuela, CIMAT, Michele Vergne, Ecole Polytechnique, Title to be announced. and RaUl Quiroga, CINVESTAV. Dynamical Systems with Emphasis on Holomorphic Dy Special Sessions namics, Hamiltonian Systems and Variational Systems, Additive Number Theory, Melvyn B. Nathanson, Herbert H. Joaquin Delgado-Fernandez, UAM-I, and Hector Sanchez Lehman College (CUNY), and Jean-Marc Deshouillers, Morgado, IMATE-UNAM. Universite de Bordeaux II. Functional and Harmonic Analysis, Lourdes Palacios, Commutative Algebra and Its Interactions with Algebraic UAM-I, and Salvador Perez-Esteva, IMATE-UNAM. Geometry, Marc F. Chardin, Universite Pierre et Marie General Topology, Alejandro Illanez, IMATE-UNAM. Curie-Paris VI, and Claudia Polini, University of Oregon. Nonlinear Analysis, Gustavo Cruz, IIMAS-UNAM, and Differential Geometric Methods in Mathematical Physics, Monica Clapp, IMATE-UNAM. Johannes Huebschmann, Universite Lille I, Yvette Kosmann Numerical Methods in Differential Equations, Pablo Barrera, Schwarzbach, Ecole Polytechnique, and Richard W. UNAM, and Francisco Solis, IIMAS-UNAM. Montgomery, University of California Santa Cruz. Quasigroups, J. D. Phyllips, and Lev V. Sabinin, University Dynamics of Nonlinear Waves, Christopher K. R. T. Jones, of Quintana Roo. Brown University, and Jean-Michel Roquejoffre, Universite Representation Theory of Algebras and Related Topics, Toulouse III. Raymundo Bautista, IMATE-UNAM. Fractal Geometry, Number Theory, and Dynamical Systems, Ring Theory, Francisco Raggi, IMATE-UNAM, and Sergio R. Michel Lapidus and Machiel van Frankenhuysen, University Lopez-Permouth, Ohio University. of California Riverside, and Michel Mendes-France, Universite de Bordeaux. Stochastical Analysis, Mogens Blatt, IIMAS-UNAM, and Maria E. Caballero, IAMTE-UNAM. Gauge Theory, jean-Claude Sikorav, Ecole Normale Superieure de Lyon, and Ronald Fintushel, Michigan State There will be no sessions for contributed papers. University. Geometric Group Theory, Gilbert Levitt, Universite Toulouse Lyon, France III, and Karen Vogtmann, Cornell University. Geometric Methods in Low Dimensional Topology, Hamish July 17-20,2001 Short, and Daryl Cooper, University of California Santa Barbara. Meeting #968 Geometric Structures in Dynamics, M. Lyubich, SUNY at First ]oint International Meeting between the AMS and the Stony Brook, Etienne Ghys, Ecole Normale Superieure de Societe Mathematique de France. Lyon, and Xavier Buff, Universite Toulouse III. MARCH 2001 NOTICES OF THE AMS 363 Meetings & Conferences Geometry and Representation Theory ofAlgebraic Groups, Algebraic Cycles, Algebraic Geometry (Code: AMS SS A1), Michel Brion, Universite de Grenoble I, and Andrei Roy joshua, Ohio State University. Zelevinsky, Northeastern University. Coding Theory (Code: AMS SS B1), Tom Dowling, Ohio History of Mathematics, Thomas W. Archibald, Acadia State University, and Dijen Ray-Chaudhuri. University, Christian Gilain, Universite Pierre et Marie Commutative Algebra (Code: AMS SS C1), Evan Houston, Curie-Paris VI, and james J. Tattersall, Providence College. University of North Carolina, Charlotte, and Alan Loper, Logic and Interaction: From the Rules ofLogic and the Logic Ohio State University. of Rules, jean-Yves Girard, Universite de Marseille, and Philip Scott, University of Ottawa. Fractals (Code: AMS SS P1), Gerald Edgar, Ohio State Uni versity. Mathematical Fluid Dynamics, Yann Brenier, Universite Pierre et Marie Curie-Paris VI, Susan J. Friedlander, Group Theory (Code: AMS SS Fl), Koichiro Harada, Surinder University of illinois at Chicago, and Emmanuel Grenier, Seghal, and Ronald Solomon, Ohio State University. Ecole Normale Superieure de Lyon. Multivariate Generating Functions and Automatic Compu Mathematical Methods in Financial Modelling, Marco tation (Code: AMS SS H1), Robin Pemantle, Ohio State Uni Avellaneda, Courant Institute, New York University, and versity. Rama Cont, Ecole Polytechnique. Number Theory(Code: AMS SS ]1), David Goss, Ohio State Model Theory, Gregory L. Cherlin, Rutgers University, and University. Frank Wagner, Universite Claude Bernard Lyon I. Proof Theory and the Foundations of Mathematics (Code: Partial Differential Equations and Geometry, Fabrice Bethuel, AMS SS K1), Timothy Carlson, Ohio State University. Universite Pierre et Marie Curie-Paris VI, and Paul C. Yang. Quantum Topology (Code: AMS SS Ll), Thomas Kerler, Probability, Gerard Benarous, Ecole Normale Superieure, Ohio State University. and George C. Papanicolaou, Stanford University. Rings and Modules (Code: AMS SS M1), S. K. Jain, Ohio University, and Tariq Rizvi, Ohio State University. Spectral Theory of Schrodinger Operators (Code: AMS SS Columbus, Ohio N1), Boris Mityagin, Ohio State University, and Sergei Ohio State University Novikov, University of Maryland. September 21-23,2001 Meeting #969 Chattanooga, Central Section Associate secretary: Susan ]. Friedlander Tennessee Announcement issue of Notices: June/July, 2001 University of Tennessee, Chattanooga Program first available one-MATH: August 9, 2001 Program issue of electronic Notices: October 2001 October 5-6,2001 Issue of Abstracts: Volume 22, Issue 3 Meeting #970 Deadlines Southeastern Section For organizers: February 21, 2001 Associate secretary: John L. Bryant For consideration of contributed papers in Special Ses- Announcement issue of Notices: August 2001 sions: June 5, 2001 Program first available one-MATH: August 23, 2001 For abstracts: July 31, 2001 Program issue of electronic Notices: November 2001 Invited Addresses Issue of Abstracts: Volume 22, Issue 3 Alex Eskin, University of Chicago, Title to be announced. Deadlines Dennis Gaitsgory, Harvard University, Title to be announced. For organizers: March 5, 2001 Yakov B. Pesin, Pennsylvania State University, Title to be For consideration of contributed papers in Special Ses- announced. sions: June 19, 2001 Thaleia Zariphopoulou, University of Texas at Austin, Title For abstracts: August 14, 2001 to be announced. Special Sessions Special Sessions Asymptotic Behavior of Solutions of Differential and L 2 Methods in Algebraic and Geometric Topology (Code: AMS Difference Equations (Code: AMS SS B1), john R. Graef, SS G1), Dan Burghelea and Michael Davis, Ohio State University of Tennessee at Chattanooga, and Chuanxi Qian, University. Mississippi State University. 364 NOTICES OF 1HE AMS VOLUME 48, NUMBER 3 Meetings & Conferences Commutative Ring Theory (Code: AMS SS A1), David F. Ergodic Theory (Code: AMS SS H1), Cesar Silva, Williams Anderson and David E. Dobbs, University of Tennessee at College. Knoxville. Geometry and Topology of the Universe (Code: AMS SS El), Mathematical and Numerical Aspects of Wave Propagation Colin C. Adams, Williams College, Glenn Starkmann, Case (Code: AMS SS Fl), Boris P. Belinskiy and Yongzhi Xu, Western Reserve University, and Jeffrey R. Weeks, Canton, University of Tennessee at Chattanooga. New York. New Directions in Combinatorics and Graph Theory(Code: Harmonic Analysis Since the Williamstown Conference of AMS SS Cl), Teresa Haynes and Debra J. Knisley, East 1978 (Code: AMS SS G1), Janine E. Wittwer, Williams College, Tennessee State University. and David Cruz-Uribe, Trinity College. Real Analysis (Code: AMS SS D1), Paul D. Humke, Saint Olaf History of Mathematics (Code: AMS SS A1), Glen R. Van College, and Harry I. Miller, University of Tennessee at Brummelen, Bennington College, Della D. Fenster, Chattanooga. Richmond University, and James J. Tattersall, Providence College. Topics in Geometric Function Theory (Code: AMS SS £1), Lelia Miller-Van Wieren, Penn State Berks Campus, and Bruce P. Number Theory, Holomorphic Dynamics, and Algebraic Palka, University of Texas at Austin. Dynamics (Code: AMS SS B1), Robert L. Benedetto, University of Rochester, John W. Milnor, IMS and SUNY at Stony Brook, and Kevin M. Pilgrim, University of Missouri Williamstown, at Rolla. Massachusetts Irvine, California Williams College University of California Irvine October 13-14,2001 November 10-11,2001 Meeting #971 Meeting #972 Eastern Section Western Section Associate secretary: Lesley M. Sibner Associate secretary: Bernard Russo Announcement issue of Notices. August 2001 Announcement issue of Notices. September 2001 Program first available one-MATH: August 30, 2001 Program first available one-MATH: September 27, 2001 Program issue of electronic Notices. November 2001 Program issue of electronic Notices. December 2001 Issue of Abstracts: Volume 22, Issue 4 Issue of Abstracts: Volume 22, Issue 4 Deadlines Deadlines For organizers: March 13, 2001 For organizers: April10, 2001 For consideration of contributed papers in Special Ses- For consideration of contributed papers in Special Ses- sions: June 26, 2001 sions: July 24, 2001 For abstracts: August 21, 2001 For abstracts: September 18, 2001 Invited Addresses Invited Addresses Hubert Bray, Massachusetts Institute of Technology, Title Gigliola Staffilani, Stanford University, Title to be an to be announced. nounced. Robin Forman, Rice University, Title to be announced. Jonathan Weitsman, University of California Santa Cruz, Emma Previato, Boston University, Title to be announced. Title to be announced. Yisong Yang, Polytechnic University, Title to be announced. Special Sessions San Diego, California Algebraic and Topological Combinatorics (Code: AMS SS D1), Eva Maria Feichtner, ETH, Ziirich, Switzerland, and San Diego Convention Center Dmitry N. Kozlov, KTH, Stockholm, Sweden. January 6-9, 2002 Commutative Algebra (Code: AMS SS C1), Susan R. Loepp, ]oint Mathematics Meetings, including the 1 OBth Annual Williams College, and Graham J. Leuschke, University of Meeting of the AMS, 85th Meeting of the Mathematical Kansas. Association of A merica (MAA), annual meetings of the Diophantine Problems (Code: AMS SS Fl), Edward B. Burger, Association for Women in Mathematics (A WM) and the Williams College, and Jeffrey D. Vaaler, University of Texas National Association of Mathematicians (NAM), and the at Austin. winter meeting of the Association for Symbolic Logic (ASL). · MARCH 2001 NOTICES OF THE AMS 365 Meetings & Conferences Associate secretary: John L. Bryant Announcement issue of Notices: October 2001 Montreal, Quebec, Program first available one-MATH: November 1, 2001 Program issue of electronic Notices: To be announced Canada Issue of Abstracts: Volume 23, Issue 1 Centre de Recherches Mathematiques, Deadlines Universite de Montreal For organizers: April 4, 2001 For consideration of contributed papers in Special Ses May 3-5, 2002 sions: August 7, 2001 Eastern Section For abstracts: October 2, 2001 Associate secretary: Lesley M. Sibner For summaries of papers to MAA organizers: To be an Announcement issue of Notices: To be announced Program first available on e-MATH: To be announced nounced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Ann Arbor, Michigan Deadlines University of Michigan For organizers: October 3, 2001 For consideration of contributed papers in Special Ses March 1-3,2002 sions: To be announced To be announced Central Section For abstracts: Associate secretary: Susan]. Friedlander Announcement issue of Notices: To be announced Program first available one-MATH: To be announced Pisa, Italy Program issue of electronic Notices: To be announced June 12-16,2002 Issue of Abstracts: To be announced First ]oint International Meeting between the AMS and the Unione Matematica Italiana. Deadlines Associate secretary: Lesley M. Sibner For organizers: August 3, 2001 Announcement issue of Notices: To be announced For consideration of contributed papers in Special Ses Program first available one-MATH: To be announced sions: To be announced Program issue of electronic Notices: To be announced For abstracts: To be announced Issue of Abstracts: To be announced Deadlines Georgia For organizers: To be announced Atlanta, For consideration of contributed papers in Special Ses Georgia Institute of Technology sions: To be announced For abstracts: To be announced March 8-1 0, 2002 Southeastern Section Invited Addresses Associate secretary: John L. Bryant Luigi Ambrosio, Scuola Normale Superiore, Title to be Announcement issue of Notices: To be announced announced. Program first available one-MATH: To be announced Luis A. Caffarelli, University of Texas at Austin, Title to be Program issue of electronic Notices: To be announced announced. To be announced Issue of Abstracts: Claudio Canuto, University of Torino, Title to be announced. Deadlines L. Craig Evans, University of California Berkeley, Title to For organizers: October 8, 2001 be announced. For consideration of contributed papers in Special Ses Giovanni Gallavotti, University of Rome I, Title to be sions: To be announced announced. For abstracts: To be announced Sergiu Klainerman, Princeton University, Title to be For summaries of papers to MAA organizers: To be an announced. nounced Claudio Procesi, University of Rome, Title to be announced. Special Sessions The program committee for the first joint meeting of the AMS and the Unione Matematica Italiana (UMI) solicits 366 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Meetings & Conferences proposals for Special Sessions. Each Special Session must Deadlines have at least one organizer from each of the sponsoring For organizers: March 12, 2002 organizations (AMS and UMI). Proposals should include For consideration of contributed papers in Special Ses names of the organizers, their mailing addresses, e-mail sions: To be announced addresses, session title, and a brief description. Proposals For abstracts: To be announced should be sent to Professor Lesley Sibner, Lsi bner@ duke.poly.edu by March 1, 2001. Baltimore, Maryland Portland, Oregon Baltimore Convention Center Portland State University January 1 5-18, 2003 ]oint Mathematics Meetings, including the 1 09th Annual june 20-22, 2002 Meeting of the AMS, 86th Annual Meeting of the Mathe Western Section matical Association ofAmerica (MAA), annual meetings of Associate secretary: Bernard Russo the Association for Women in Mathematics (A WM) and the Announcement issue of Notices: To be announced National Association of Mathematicians (NAM), and the Program first available one-MATH: To be announced winter meeting of the Association for Symbolic Logic (ASL). Program issue of electronic Notices: To be announced Associate secretary: Susan]. Friedlander Issue of Abstracts: To be announced Announcement issue of Notices: To be announced Program first available one-MATH: To be announced Deadlines Program issue of electronic Notices: To be announced For organizers: November 20, 2001 Issue of Abstracts: To be announced For consideration of contributed papers in Special Ses Deadlines sions: To be announced For abstracts: To be announced For organizers: April15, 2002 For consideration of contributed papers in Special Ses sions: To be announced For abstracts: To be announced Boston, For summaries of papers to MAA organizers: To be an Massachusetts nounced Northeastern University Phoenix, Arizona October 5-6, 2002 Phoenix Civic Plaza Eastern Section Associate secretary: Lesley M. Sibner January 7-10,2004 Announcement issue of Notices: To be announced Associate secretary: Bernard Russo Program first available one-MATH: To be announced Announcement issue of Notices: To be announced Program issue of electronic Notices: To be announced Program first available one-MATH: To be announced Issue of Abstracts: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Deadlines For organizers: March 6, 2002 Deadlines For consideration of contributed papers in Special Ses For organizers: April 2, 2003 sions: To be announced For consideration of contributed papers in Special Ses For abstracts: To be announced sions: To be announced For abstracts: To be announced For summaries of papers to MAA organizers: To be an Madison, Wisconsin nounced University of Wisconsin-Madison October 12-1 3, 2002 Central Section Associate secretary: Susan]. Friedlander Announcement issue of Notices: To be announced Program first available one-MATH: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced MARCH 2001 NOTICES OF THE AMS 367 Meetings and Conferences of the AMS Associate Secretaries of the AMS Western Section: Bernard Russo, Department of Mathe Eastern Section: Lesley M. Sibner, Department of Mathe matics, University of California Irvine, CA 92697; e-mail: matics, Polytechnic University, Brooklyn, NY 11201-2990; [email protected]; telephone: 949-824-5505. e-mail: 1si bner@duke. po 1y. edu; telephone: 718-260-3505. Central Section: Susan J. Friedlander, Department of Math Southeastern Section: John L. Bryant, Department of Math ematics, University of illinois at Chicago, 851 S. Morgan (M/ C 249), ematics, Florida State University, Tallahassee, FL 32306-4510; e Chicago, ll. 60607-7045; e-mail: susan@math. nwu. edu; telephone: mail: bryant@math. fsu. edu; telephone: 850-644-5805. 312-996-3041. The Meetings and Conferences section of the Notices gives October 12-13 Madison, Wisconsin p.367 information on all AMS meetings and conferences approved by press time for this issue. Please refer to the page numbers cited in the table of contents on this page for more detailed 2003 information on each event. Invited Speakers and Special Ses January 15-18 Baltimore, Maryland p.367 sions are listed as soon as they are approved by the cognizant Annual Meeting program committee; the codes listed are needed for electronic abstract submission. For some meetings the list may be in 2004 complete. Information in this issue may be dated. Up-to January 7-10 Phoenix, Arizona p.367 date meeting and conference information is available on the Annual Meeting World Wide Web at www. ams. org/meeti ngs/. Important Information regarding AMS Meetings Meetings: Potential organizers, speakers, and hosts should refer to 2001 page 87 in the January 2001 issue of the Notices for general information regarding participation in AMS meetings and March 16- 18 Columbia, South Carolina p. 360 conferences. March 30-31 Lawrence, Kansas p. 361 April21-22 Las Vegas, Nevada p. 361 Abstracts April28-29 Hoboken, New Jersey p. 362 Several options are available for speakers submitting abstracts, May 23-26 Morelia, Mexico p. 362 including an easy-to-use interactive Web form. No knowledge July 17-20 Lyon, France p. 363 of LaTeX is necessary to submit an electronic form, although September 21-23 Columbus, Ohio p. 364 those who use LaTeX or AMS-LaTeX may submit abstracts October 5-6 Chattanooga, Tennessee p. 364 with such coding. To see descriptions of the forms available, October 13-14 Williamstown, MA p. 365 visit http: I /www. ams. org/abstracts/i nstructi ons. html, November 10-11 Irvine, California p. 365 or send mail to abs-submit@ams. org, typing help as the sub ject line; descriptions and instructions on how to get the tem plate of your choice will be e-mailed to you. 2002 Completed abstracts should be sent to abs-submi t@ January 6-9 San Diego, California p. 365 ams. org, typing submission as the subject line. Questions Annual Meeting about abstracts may be sent to abs-i nfo@ams. org. March 1-3 Ann Arbor, Michigan p. 366 Paper abstract forms may be sent to Meetings & Conferences March 8-10 Atlanta, Georgia p.366 Department, AMS, P.O. Box 6887, Providence, RI 02940. There May 3-5 Montreal, Quebec, Canada p.366 is a $20 processing fee for each paper abstract. There is no June 12-16 Pisa, Italy p.366 charge for electronic abstracts. Note that all abstract deadlines June 20-22 Portland, Oregon p. 367 are strictly enforced. Close attention should be paid to spec October 5-6 Boston, Massachusetts p. 367 ified deadlines in this issue. Unfortunately, late abstracts can not be accommodated. Conferences: (See http: I /www. ams . o rg/meeti ngs/ for the most up-to-date information on these conferences.) June 10-August 9, 2001: Joint Summer Research Conferences in the Mathematical Sciences, Mount Holyoke College, South Hadley, MA. See pages 1327-1331, November 2000 issue, for details. Cosponsored Conferences: February 15-20, 2001: AAAS Annual Meeting, San Francisco, CA. See page 1477, December 2000 issue, for details. May 28-30, 2001: 2001lnternational Conference on Computational Science, Hilton San Francisco and Towers Hotel, San Fran cisco, CA 368 NOTICES OF THE AMS VOLUME 48, NUMBER 3 Inspiration - pure and simple. Maple 6 is the most comprehensive computational tool available for engineers. For fac:Jity, Maple 6 will accelerate research and innovate teaching. For students, Maple 6 is a complete analytical platform that takes learning from theory to practice. Never before has a single environment offered such a complete array of functionality. Maple 6 delivers a comprehensive, intelligent, and easy-to-use environment for all your technical computations and analysis -from first principles to advanced theoretical explorations. It's the technical assistant every engineer has been waiting for! Adopt the proven analytical tool that will take your research and teaching to the next level! Contact Waterloo Maple for more information. maple6 © 2001 Waterloo Maple Inc. Maple is a registered trademark of Waterloo Maple Inc. All other trademarks are property of their respective owners SPRINGER FOR MATHEMATICS 5. ABBOTT, Middlebury College, Middlebury, VT VLADIMIR G. BOLTYANSKII, CIMAT, Guanajuato, Announcing the LAtest Releases - Mexico, and V. A. EFREMOVICH, deceased, Translated Springer's Classics in Mathematics UNDERSTANDING ANALYSIS by ABE SHENITZER, York University, North York, This book outlines an elementary, one Ontario, Canada J.L. DOOB, University of Illinois, Urbana, IL semester course which exposes students to INTUITIVE COMBINATORIAL CLASSICAL POTENTIAL both the process of rigor, and the THEORY rewards TOPOLOGY AND ITS PROBABILISTIC inherent in taking an axiomatic approach to the study of functions of a real variable. The This book deals with COUNTERPART philosophy of this book is to focus attention the topology of curves ''''"n•• • ••••~·"'' "This huge book on questions which give analysis its inherent and surfaces as well as written in several fascination. Does the Cantor set contain any with the fundamental years by one of the irrational numbers? Can the set of points concepts of homotopy few mathematicians where a function is discontinuous be arbitrary? and homology, and does -·-Classical Is an this in a lively and Potential Theory able to do ir, appears infinitely differentiable function neces V.G. 8ohyanskif well and liS Probabilistic sarily the limit of its Taylor series? Giving V.A.Efr...-movich motivated way. The Counterpart as a precise and impressive study ... these topics center stage, the motivation for a book is well suited not that replaces, in rigorous approach is justified by the fact that on Iy as preparation for a coherent way, they are inaccessible without it. students who plan to without being 2001/264 PP./HARDCOVER take a course in algebraic topology but also ~ / $39.95 .... encyclopaedic, ISBN 0387-950605 for advanced undergraduates or beginning a large librwy of books and papers UNDERGRADUATE TEXTS IN MATHEMATICS graduates interested in finding out what scattered without a uniform language." topology is all about. The book has more than -METRIKA 1986, REVIEWED BY L. PERKO, Northern Arizona University, Flagstaff 200 problems. many examples, and over 200 MARCEL BRELOT DIFFERENTIAL EQUATIONS illustrations. 2000/846 PP./SOFTCOVER/$34.95 2001/216 PP ., 210 ILLUS./HARDCOVER/$49.95 ISBN 3-54041206-9 AND DYNAMICAL SYSTEMS ISBN 0387-95114-8 Third Edition UNIVERSITEXT R.C. LYNDON and P.E. SCHUPP, University of Reviews from the first edition- FRED BRAUER, University of British Columbia, lllinois, Urbana " The text succeeds admirably ... Examples Vancouver, Canada and CARLOS CASTILLO-CHAVEZ, COMBINATORIAL abound, figures are used to advantage, and Cornell University, Ithaca, NY a reasonable balance is maintained between GROUP THEORY what is proved in derail and what is asserted MATHEMATICAL MODELS '""'"'"""''"'"" "This book .. defines with supporting references... Each section IN POPULATION BIOLOGY the boundaries of closes with a set ofproblems, many of which AND EPIDEMIOLOGY the subject now are quite interesting and round out the text -"-·-.. called cmnbinarorial material... this book is robe highly recom As the world population Combina - torial Group Theory group theory.... it mended both for use as a text, and for exceeds the six billion is a considerable professionals in other fields wanting to gain mark, questions of achieventent to insight into modem aspects of the geometric population explosion, have concentrated theory of continuous (i.e., nor discrete) of how many people the a survey of the dynamical systems." earth can support and +- subject into 339 - MATHEMATICAL REVIEWS under which conditions, pages.... the book 2000/APPROX. 562 PP./HARDCOVER/$49.95 become pressing. Some is a valuable and welcome addition to ISBN 0387-9477&7 of the questions can be the literature, containing many results TEXTS IN APPLIED MATHEMATICS, VOL. 7 addressed through the nor previously available in a book." use of mathematical -MATHEMATICAL REVIEWS J.M. STEELE, University of Pennsylvania, Philadelphia models, but not all. The goal of this book is 2000/340 PP., 18 ILLUS./SOFTCOVER/$34.95 to search for a balance between simple and ISBN 3-5404115&5 STOCHASTIC CALCULUS AND analyzable models and unsolvable models FINANCIAL APPLICATIONS which are capable of addressing important questions such as these. This book will include D. GILBARG, University of Stanford, CA and A discussion of simple random walk and both examples and exercises. N.S. TRUDINGER, The Australian National the analysis of gambling games are used University, Canberra, Australia to motivate the theory of martingales and 2001/408 PP ./HARDCOVER/ $59.95 ISBN 0387-98902·1 continuous time stochastic process ELLIPTIC PARTIAL motion. TEXTS IN APPLIED MATHEMATICS, VOL. 40 DIFFERENTIAL EQUATIONS The book then takes up the Ito integral and enough of the theory of the diffusion equation OF SECOND ORDER to be able to solve the Black-Scholes POE and prove the uniqueness of the solution. " ..., ,, <> R ..,u• ~ • '" "This is a book of interest to anyone The foundations for the martingale theory of ,_.. _.... L _ having ro work arbitrage pricing are then prefaced by a well ORDER TODAY! with differential motivated development of the martingale Elliptic representation theorems and Girsanov theory. Partial Differential equations. either as CALL: 1-800-SPRINGER or !?~:~~on"J Order a reference or as a 2000/ 344 PP./HARDCOVER/ $69.95 book to learn from. ISBN 038"7·95015-8 FAX: (201) 348-4505 The authors have APPLICATIONS OF MATHEMATICS, VOL. 45 WRITE: Springer-Verlag New York, Inc ., raken trouble to Dept. S2534, PO Box 2485, Secaucus, make the treatment NJ 07096-2485 +- self-contained. Jr (is) VISIT: Your suitable required reading for a Ph.D. local technical bookstore student." E·MAIL: orders@springer-ny .com - NEWSLETTER. NEW ZEALAND ~ INSTRUCTORS: Call or write for info on MATHEMATICAL SOCIETY textbook exam copies 2001/518 PP./SOFTCOVER/$34.95 ' Springer ISBN 3-540.411607 Your 30-Day return privilege is always guaranteed! www .springer-ny .com 3/01 Promotion #52534