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Mathematical I MRS JAMS A Problems in INTERNATIONAL MATHEMATICS Engineering RESEARCH SURVEYS Journal of Applied Mathematics and stochastic Analysis Theory, Methods. and Applications

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IMR ~~~:. ~~~e1matione l Mathematics Research Surveys INTERNATIONAL MATHE RESEARCH SURVE Elliptic Equations of Yamabe Type Olivier Druet and Emmanuel Hebey

Morris Weisfeld Editor-in-Chief Dedicated to Maximilien and Theophile [email protected]. edu Contents Nicolas Katz I. Introduction , ...... nmk@m ath.princeton.edu 2. PDE background 3. Euclidean background ...... Terence Tao 4. The Yamabe problem 5. The negative case in Yamabe-type ~~-~~~~~~~ [email protected] 16 6. The H?-theory for blow-up ...... 17 7. Uniqueness in the Hi -theory ...... 32 Vladimir Voevodsky 8. Examples of blowing-up s equences of solutions 37 [email protected] 9. TheC0-theotyforblow-up ...... : : : ::::: :· · ·· · · · · ·· 10. Proof of Theorem 9. 1 . 54 ·· ························· ·· 11 . Remark on the coercivity 58 82 12. The 3-dimensional case ...... 85 13. Higher dimensions ...... 90 14. Compactness and noncompactness 103

Eqitor-in-Chief 1 Introduction duke.edu Our ~im in ~his ~a per is to report on a current ly active research a rea in the field of eo­ weisfeld@math. metric partial differential equations (PDEs). We chose to report on the specific thea~ of

Received I June 2004. Revision received 2 November 2004.

General Information [email protected] Author Proposals · [email protected]

We:report on th e specific theory of elliptic equations ofYamabe type. Such equations h ave been the ;:targe(of investigation for decades. Among other topics, we discuss the very complete H ~ -theory and t9-theofy for the blow-up of sequences of solutions of such equations. We also address the question oflhe1ocalizai~on of blow-up points, and discuss compactness and noncompactness issues. . • . • 1

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Modern Differential Geometric Function Theory Geometric Problems on Geometry in Gauge Explorations in Complex Analysis Maxima and Minima Theories STEVEN G. KRANTZ, Washington University, St Louis, MO TITU ANDREESCU, University of Wisconsin, Whitewater, WI,· American Mathematics Competitions, University of Nebraska, Volume 1: Maxwell Fields Presented from the point of view of modern work in the field, this new book addresses advanced topics in Lincoln, NE; OLEG MUSHKAROV, Institute for Mathematics, ANASTASIOS MALLIOS, University of Athens complex analysis that verge on current areas of Bulgarian Academy of Sciences, Sofia; LUCHEZAR Panepistimioupolis, Athens, Greece STOYANOV, University of Western Australia, Crawley, Western research, including invariant geometry, the Bergman Australia Differential geometry, in the classical sense, is devel­ metric, the automorphism groups of domains, extremal oped through the theory of smooth manifolds. In the length, harmonic measure, boundary regularity of Reflecting the authors' experience as teachers and early 1990s, the author of this work initiated a new kind conformal maps, the Poisson kernel, the Hilbert Olympiad coaches, this carefully developed problem of differential geometry in which all the machinery of transform, the boundary behavior of harmonic and book takes a unique, intuitive approach to extreme­ classical differential geometry can be explained without holomorphic functions, the inhomogeneous value problems, treating them within the framework of any notion of smoothness. This was achieved via sheaf Cauchy-Riemann equations, and the corona problem. Euclidean geometry. Included is a comprehensive theory (geometry) and sheaf cohomology (analysis). The author adroitly weaves these varied topics to reveal selection of maxima and minima problems, from a number of delightful interactions. Perhaps more classical Greek constructions to modern open ques­ This two-volume work systematically applies the tions; detailed, step-by-step solutions to many of the author's sheaf-theoretic approach to such physical importantly, the topics are presented with an under­ standing and explanation of their interrelations with exercises are provided. The reader is exposed to theories as gauge theory. Both volumes contain a wealth algebra, analysis, combinatorics, and topology, and, of detailed and rigorous computations, and will appeal other important areas of mathematics such as harmonic analysis, differential geometry, partial differential throughout the text, emphasis is placed on creative to mathematicians and physicists as well as advanced techniques for problem solving. This volume is ideal for undergraduate and graduate students studying applica­ equations, potential theory, abstract algebra, and invariant theory. use at the junior and senior undergraduate level, as well tions of differential geometry to physical theories. as for enrichment programs and Olympiad training for 2005/APPROX. 450 PP./HARDCOVER 2005/APPROX. 350 PP., 20 ILLUS./HARDCOVER advanced high school students. ISBN O-BI76-437B-B/SI29 .00 (TENT.) ISBN O-BI76-4339-7 j$69.95 (TENT.) 2005/APPROX. 320 PP./SOFTCOVER PROGRESS IN MATHEMATICAL PHYSICS CORNERSTONES ISBN O-BI76-3517-3/S79.95 (TENT.) Volume 2: Yang-Mills Fields Differential Geometry and Differential Geometry of Continuing his point of view started in Volume I, the Analysis on CR Manifolds author extends the application of his sheaf-theoretic SORIN DRAGOMIR, Universitii della Basilicata, Romano, Curves and Surfaces approach to Yang-Mills fields in general. Important Potenza, Italy; GIUSEPPE TOMASSINI, Scuola Normale A Concise Guide topics covered include cohomological classification of Superiore, Pisa, Italy VICTOR A. TOPONOGOV, Sobolev Institute of Mathematics, Yang-Mills fields, the geometry of Yang-Mills A-connec­ This monograph is a unified presentation of several Novosibtrsk, Russia; VLADIMIR Y. ROVENSKI, Technion tions and moduli space of a vector sheaf, and Einstein's differential geometric aspects in the theory of CR Israel Institute of Technology, Haifa, Israel equation in a vacuum. manifolds and tangential Cauchy-Riemann equations. It This book presents traditional material of curves and 2005/APPROX. 360 PP./HARDCOVER presents topics from the Tanaka-Webster connection, a surfaces related to differential geometry along with ISBN O-BI76-4379-6/SI20.00 (TENT.) key contributor to the birth of pseudohermitian geome­ PROGRESS IN MATHEMATICAL PHYSICS important ideas of Riemannian geometry. The author try, to the major differential geometric achievements in introduces the reader to curves, then progresses to the theory of CR manifolds, such as Fefferman's metric, surfaces, and finally to more complex topics in the Finite Congruence Lattices pseudo-Einstein structures and the Lee conjecture, CR concluding section. The book weaves together standard of LaHices immersions, subelliptic harmonic maps as a local theoretical material with more difficult theorems and manifestation of pseudoharmonic maps from a CR complex problems while maintaining an easy separation GEORGE GRATZER, University of Manitoba, Winnipeg, manifold, Yang-Mills fields on CR manifolds, to name Canada between the two. One of the striking features of this several. It also aims at explaining how certain results presentation is the large number of nontrivial and In the past half-century the study of lattices has become from analysis are employed in CR geometry. original problems, some with useful hints and solutions, a large and important field with a great number of 2005/APPROX. 530 PP./HARDCOVER which introduce a motivated student into the real world interesting and deep results and many open problems. ISBN O-BI76-43BB-5/SI25.00 (TENT.) of geometry. Another strong point is the nontrivial One of the world's leading experts in lattice theory, PROGRESS IN MATHEMATICS material connected with the Aleksandrov global angle George Gratzer, presents major results on the subject comparison theorem--one of the cornerstones of in a self-contained exposition featuring the author's modern Riemannian geometry. "Proof by Picture" method. Also included are a discus­ Graphs and Networks Transfinite and Nonstandard 2005/APPROX. 200 PP., 40 ILLUS./SOFTCOVER sion of the techniques used to construct "nice" lattices ISBN O-BI76-43B4-2/S59. 95 (TENT.) and "nice" congruence-preserving extensions, an ARMEN H. ZEMANIAN, State University of New York at Stony appendix featuring complete proofs, and a comprehen­ Brook sive bibliography and index. The book is appropriate for This book gives an exposition of the most recently a one-semester graduate course, and will also be of obtained results concerning path-based and walk-based interest to researchers studying lattices. transfinite connectedness, infinite-ordinal distances, 2005/APPROX. 300 PP., 100 ILLUS./HARDCOVER and applications of nonstandard analysis to establish ISBN O-BI76-3224-7/$79.95 (TENT.) hyperreal current-voltage regimes in transfinite electri­ cal networks and along transfinite transmission lines. 2004/202 PP./SOFTCOVER/ISBN O-BI76-4292-7 /SB4.95

CALL: 1-800-777-4643 • FAX: (201) 348-4505 E-MAIL: [email protected] • www.birkhauser.com Please mention promotion #Y8585 when ordering. Prices are valid in the Americas only Birkhiiuser JiB and are subject to change without notice. For price and ordering information outside the Americas, Boston · Basel · Berlin · please contact Birkhauser Verlag AG by E-mail: [email protected] 4/05 Promotion #Y8585 Notices April 2005 of the American Mathematical Society Feature Articles 402 Flowers of Ice-Beauty, Symmetry, and Complexity: A Review of The Snowflake j ohn A. Adam The reviewer examines a recent, nonmathematical, book on the science of snowflakes, and then provides a mathematical discussion of snowflakes for Notices readers.

41 7 Mathematics of the Heavens Robert Osserman April is designated Mathematics Awareness Mon th and the theme for 2005 is "Mathematics and the Cosmos". The author has written three essays­ variations on this theme.

426 Cohomology, Computations, and Commutative Algebra ]on F. Carlson Cohomology rings of finite groups provide an interesting and important class of commutative rings. Examples can be explored by computer calculations, which in turn lead to conjectures about their properties. The author explains what these properties are, what might be conjecturally true of them, and what has been proven so far.

Communications Commentary ~~------397 A Photographic Look at the Joint 399 Opinion Meetings, Atlanta 2005 400 Letters to the Editor 439 2005 Steele Prizes 43 5 Mathematicians under the Nazi~ -A 443 2005B6cherPrtze Book Review Reviewed by ]ochen Bruning 445 2005 ColePrizeinNumberTheory 459 MathSciNetMatters 447 2005 Satter Prize Norman Richert 449 2005 Book Prize 451 2005 Whiteman Prize 454 2005 Conant Prize 457 2004 Morgan Prize Notices Departments Mathematics People ...... 460 Stein Receives Bergman Prize, Bhargava Receives Blumenthal EDITOR: Andy Magid Prize, Harr ison Awarded von Neumann Prize, Prizes of the ASSOCIATE EDITORS: Mathematical Society of japan, AWM Essay Contest Winners Su sanne C. Brenner, Bill Casselman (G raphics Editor), Announced. Robert ]. Daverman, Nathaniel Dean, Rick Durrett, Su san Friedlander, Robion Kirby, Steven G. Krantz, Mathematics Opportunities ...... 463 Elliott H. Lieb, Mark Saul, Karen E. Smith , Audrey Terras, Lisa Traynor NSF Program in Mathematical, Social, and Behavorial Sciences; SENIOR WRITER and DEPUTY EDITOR: Maria Mitchell Wom en in Science Award; AP Calculus Readers Allyn Jackson Sought; AMS Congressional Fellowship. MANAGING EDITOR: Sandra Frost Reference and Book List ...... 464 CONTRIBUTING WRITER: Elaine Kehoe PRODUCTION ASSISTANT: Muriel Toupin Mathematics Calendar ...... 472 PRODUCTION: Kyle Antonevich, Stephen Moye, Erin Murphy, Lori Nero, Karen Ouellette, Donna New Publications Offered by the AMS ...... 479 Salter, Deborah Smith, Peter Sykes ADVERTISING SALES: Anne Newcomb Classified Advertisements ...... 484

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[Notices of the American Mathematical Society is published monthly except bimonthly in June/July by the .American Mathematical Society at 201 Charles Street, Providence, Rl 02904-2294 USA, GST No. 12189 2046 RT* ***. Periodicals postage paid at Providence, RJ , and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Ma thematical Society, P.O. Box 6248, Providence, Rl 02940-6248 USA.) Publication here of the Society's street address and the other information in brackets above is a tech­ nical requirement of the U.S. Postal Service. Tel: 40 l-4 55-4000, email: noti ces@ams . or g. ©Copyright 2005 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines established to. ensure permanence and durability. Opinions expressed in signed Notices articles are those of the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society. Joint Mathematics .--~:------Meetings

ATLANTA, GEORGIA January 5-8, 2005

See photo key, page 495 Joint Mathematics ...... ;::~~Jo.<::---- Meetings See photo key, page 495

ATLANTA, GEORGIA January 5-8, 2005 Opinion

Most states' standards documents have not been reviewed The K-12 Math Test by an academic mathematician and are rife with inaccuracies. Many are so ambitious they cannot have been reviewed for realism by classroom teachers. It is common to describe ma­ Conundrum terial in "bands" of several grades. It is then consistent with This is a story with disasters, well-intentioned efforts gone the standard to promote students who have learned very lit­ wrong, villains, and no quick fix. There are constructive small tle in the first year of the band, guaranteeing a lot of repeti­ steps, however. tion in the last year. Sometimes material is described in topic­ centered "threads" with no due date at all. Tests Most standards are largely concerned with process or ex­ High-stakes K-12 tests seem to be both necessary and an posure not expected to result in testable skills. The word abomination. High school math skills are falling, ranking on "understand" for instance is clearly distinguished from "able international tests is an embarrassment, colleges' math to work problems with". This lets teachers promote students offerings are dominated by remedial courses. None of this will who "understand" material even if they can't work problems. change without high-stakes tests to provide discipline But this ostensibly higher-level knowledge is not recognized and accountability. On the other hand, they distort teaching: by high-stakes tests. they are high stakes for schools as well as students, so ad­ Even perfect standards documents are not the full ministrators apply pressure to teachers. Teachers pass this solution, since tests would still supplant them as de facto stan­ on to students. As a matter of survival, teachers teach to the dards.2 test, so things skipped on tests are left out of courses. And if "understanding" doesn't help scores, then tricks and me- What We Can Do chanical drill are the order of the day. · The state of U.S. math education is a national emergency, and The tests themselves are antiques: one-shot tests academic mathematicians have a crucial role in resolving it. with multiple-choice questions. Questions are mediocre, It is a delicate role, since there is no agreement on the nature ambiguous, or even wrong. Even the answer format has or even the existence of the emergency. There are politicians consequences. Questions are often numerical, in part to avoid who feel that their election to high office after failing algebra giving clues in answer choices: rrr2 is obviously the area of demonstrates that math is not necessary for success. The K-12 a circle, but 16.6 is not obviously the area of a circle of radius community has been shifting toward qualitative and imagi­ 2.3. This and calculators have shifted the focus to numerical native "understanding" for decades, and a good deal of the work and led to a significant decline in abstract and symbolic estrangement from the academic community results from thinking. our reactionary attachment to testable skills and pedantic pre­ The antique-test problem, at least, has an explanation and cision. some prospect of relief. Each edition of a traditional test is Some suggestions: made essentially from scratch, so it is necessary to weight • If you have an opportunity to participate in development scores to keep outcomes consistent. Determining this weight­ of state standards, take it but be prepared to yield on many ing is the main expense in test development, and the testing issues. Remember, this is an incremental process, and if our industry has an enormous investment in the necessary ex­ viewpoints are correct, then the discipline imposed by high­ pertise. However, this is irrelevant to modern computer-based stakes tests give us a long-term advantage. tests. Each student gets a slightly different test, and if these • If you have an opportunity to discuss K-12 pedagogy, vary a bit in difficulty (or the student has a bad day), the rem­ pass. There are exceptions to this of course, but at the edy is to retake it. It seems unlikely the traditional industry moment alienation is more likely than progress. Again, will make this transition, since it would mean discarding their the discipline of high-stakes tests should help as the issue be­ major asset. However, a new industry is developing, and soon comes more a struggle against a common enemy and less a choices will be available to departments of education with the difference of opinion or philosophy. courage to break with their traditional partners. • Think about tests. Tests will be driving standards and cur­ riculum in the near future, since they provide concrete objec­ Standards tives and quantitative measures of success. Bad tests drive the Each U.S. state has a standards document for mathematics ed­ process in bad directions, so the current test-development ucation. In principle this organizes teaching, texts, and tests process is a serious weak point. Large collections of mathe­ and ought to ease problems with tests by providing an envi­ matically "wholesome" sample problems would be a great re­ ronment designed to prepare students for them. The dys­ source. There is a particular need for symbolic word problems. functional educational situation would thus seem to be due -Frank Quinn to dysfunctional standards documents, and consequently Virginia Polytechnic Institute and State University much of the debate is focused on standards. They do leave much to be desired.l lSee http://www.edexcellence.net/foundation/ 2See http://www.math.vt.edu/people/quinn/ publication/publication.cfm?id=338&pubsubid= educati on/Cou rseDefi ni ti on. pdf for an attempt to 1118 for a recent Fordham Foundation report. address this.

APRIL 2005 NOTICES OF THE AMS 399 Letters to the Editor was the only one whose humour my was to be found on at least fifteen beloved mediaeval historian wife was different websites, in all cases with­ able to share, and a little too readily out formal attribution. Mathematical Jokes at that. I remember laughing twice at The Dawson comment reveals a the classical "Abelian grape" joke: lack of understanding of the aims of I enjoyed the article of Renteln and once when I first heard it at my high many social scientists when he rec­ Duildes on mathematical folk humor school math club and then again that ommends that they should not "waste in the January Notices. However, I night after I'd read enough about would like to suggest a couple cor­ time looking for psychological or group theory to get the joke. political explanations." This unthink­ rections: one in interpretation, the I take issue, however, with the lax­ ing slight of social science suggests other in one of the jokes. ness of citation of one joke in partic­ Firstly, I do not think that the use that psychologists and political sci­ ular, appearing on p. 32, described entists, who surely look for psycho­ of "Zorn's Lemma/ Lemming/ Lemon" as "an interesting metajoke." logical or political explanations for in the well-known riddle needs to be I was the original author of that interpreted as betraying antichoice various phenomena, are wasting their joke, which first appeared in the time. Social scientists might equally sentiments or guilt. The usage is moderated USENET newsgroup rec. adequately explained by the fact well suggest that mathematicians who humor. funny in April 1993 (see devote months or years trying to solve that Zorn's Lemma is about the only http://groups-beta.google. "lemma" (it seems only a historical an arcane mathematical problem are com/group/rec.humor.funny/msg/ wasting their time. accident that it did not pass into 7e8a721f677ef749) and can be The goal of our essay was to sug­ general usage as "Zorn's Axiom" or found in Google a modest twenty­ "Zorn's Theorem") to be really well gest that there is such a thing as math­ seven times. ematical mentality or world view known and to have both a familar and that mathematical humor provides a name and a succinct restatement. - John Chew Contrary to the authors' assertions, bona fide and revealing window on that University of Toronto way of thinking. Now it may well be this wordplay would not work on just [email protected] any lemma. A test case: that our delineation of that mentality is in error, but the fact that such a men­ Q: What's yellow and asserts that (Received January 17, 2005) exponents of commuting elements of tality exists is surely not at issue. The jokes we report are in circulation, and a Banach algebra have the multi­ plicative property? all of them were in theory created Dundes and Renteln Reply by some ingenious individual, even if, A: Lemon 2.12 of Douglas' Banach We are generally pleased with the after the fact, we are not always able Algebra Techniques in Operator response to our survey of mathemat­ Theory. to properly recognize that individual's ical humor but somewhat surprised efforts. The proof of the pudding is in Occam's Razor tells us, then, not to by the number of mathematicians the eating, so the jokes we report are waste time looking for psychological who emailed us with considerable pro­ or political explanations: whoever meant to be savored and enjoyed by prietary fervor claiming the those who best understand them. made up the riddle hadn't (pause ... ) authorship of one or more of the any Choice. jokes. These claimants demonstrated -Alan Dundes Secondly, surely it is obvious that a basic lack of knowledge of the a compact city is one which can lay off University of California, Berkeley nature of folklore. Once A tells a joke -Paul Renteln all but a finite number of its police to B and B transmits it to C and D and force, no matter how shortsighted California State University, the geometric progression begins, A San Bernardino they are. loses forever his control over the sub­ sequent dissemination and appear­ (Received January 20, 2005) - Robert Dawson ance of his creation. Presumably there Saint Mary's University was an originator of such contempo­ Correction Halifax, NS B3L 3C3 rary proverbs as "You snooze, you The following website was inadver­ rdawson@cs . stmarys.ca lose" or "Use it or lose it", but we do tently omitted from the list of refer­ not know who these creative spirits ences in our article: http: I /www. (Received January 10, 2005) were. It is not easy to create new math.uchicago.edu/~conrad/ traditions. If one had as an exercise amused. html. We thank Justin Sinz to create an entirely new folk song for bringing this to our attention. or an entirely new curse, it would not Origins of an Anecdote be an easy task. In fact, the authors -Paul Renteln and Alan Dundes I thoroughly enjoyed Paul Renteln of new mathematical jokes ought to and Alan Dundes' paper "Foolproof: feel gratified that their creation has Learned Societies and the War A Sampling of Mathematical Folk enjoyed some acceptance by mem­ on Terror Humor", especially the section "Mak­ bers of their peer group. We found, for "Mathematician Caught in 'War on ing Fun of Mathematicians", which example, that John Chew's metajoke Terror' Dragnet". This is the lead story

400 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Letters to the Editor in the October 2004 issue of the CAUT refusals to comply by various affected Bulletin (CAUT =Canadian Association societies has forced Treasury to put of University Teachers, counterpart this into abeyance. of the American Association of Such behavior on the part of U.S. University Professors). The full story authorities is not new. It was com­ is available free online. Open http: I I mon during the cold war. One exam­ www. caut. o rg. Click on "Bulletin ple: The International Congress of online", then "archives", finally Mathematicians resumed activity "October". The first item to pop up after World War II in 1950 at Harvard. is this story. Well worth reading. Jacques Hadamard, then 85, had Professor James Lewis of the Uni­ been designated Honorary President. versity of Alberta was en route to a Initially the U.S., mindful of his left mathematics meeting in Chicago views, said he would not be admitted, when he was harrassed by U.S. although he had spent the war years authorities, as had been the case, in the U.S. Other French mathemati­ he discovered, with two men each cians threatened a boycott. There named John Lewis. It doesn't pay to was talk of moving the congress to have a common name, as Senator Canada. Under this pressure, the U.S. Ted Kennedy had already learned authorities relented- somewhat. under similar circumstances. Hadamard was given a visa, but valid CAUT learned that such episodes only for Cambridge, Massachusetts. were frequent. This case was distin­ He was not even allowed transit to guished only by our colleague's will­ Mexico after the congress. ingness to go public. It is worthwhile Nonetheless, securing his partici­ reading the CAUT Bulletin's report on pation and leadership in the congress his comments. A letter column does was an important victory for the vig­ not offer sufficient space to do them orous protest which took place and justice. The implications are scary for the success of the congress. The and lasting. current suspension of the Treasury They apply beyond individual Department's ban on processing sci­ human rights and have possible con­ entific publications from the coun­ sequences for the functioning of tries specified is a current notable learned societies and for the content victory, hopefully lasting. If we are of their meetings. Participants from as alert and vigorous and public on anywhere in the world could be ex­ similar issues, we may not need to cluded at the U.S. border, no matter despair. how long their voyages, to the detri­ ment of other participants as well as -Lee Lorch to themselves. York University The U.S. authorities interfere also [email protected] for purely political purposes, having nothing to do with the "war on terror". (Received January 13, 2005) Quite recently, a Latin American Studies Association held an interna­ tional conference in Las Vegas. At the very last moment, the U.S. govern­ ment refused visas to all sixty-one Cuban specialists who had been in­ vited, including several who had made previous scholarly visits to the U.S. This damaged the professional content of the conference for all participants. Also recently, the U.S. Treasury Department forbad translating or abstracting publications emanating from several countries under the Trading with the Enemy Act. This would have criminalized at least Math­ ematical Reviews. Loud protests and

APRIL 2005 NOTICES OF THE AMS 401 Flowers of lee Beauty, Symmetry, and c_0n}If1le¥t~: A Review of The SiJaWflake: Winter's Secret B~Cl ·lflfi;? ..· • ..¥~ ' I Reviewed by john A. Adam

rowing up as a child in southern Eng­ contains no mathematics, so the first section will land, my early memories of snow in­ address the qualitative features of snow and clude trudging home from school with snowflakes discussed by the author. I will draw Gmy father, gazing at the seemingly enor­ on some of the general descriptions of snow and mous snowdrifts that smoothed the its properties both from this book and those by hedgerows, fields and bushes, while listening to the others mentioned below. I have decided to comment soft "scrunch" of the snow under my Wellington on every chapter individually, because each can boots. In the country, snow stretching as far as I be treated to some extent independently of the could see was not a particularly uncommon sight. others. The second section takes the form of a The quietness of the land under a foot of snow mathematical appendix devoted to an outline of seemed eerie. I cannot remember the first time I some of the mathematical aspects of crystal for­ looked at snowflakes per se; my interests as a mation, with particular emphasis on ice crystals. small child were primarily in their spheroidally In some sense, I hope, this will become a parallel shaped aggregates as they flew through the air. "mathematical universe" to the first section. Many years later, as I cycled home from my office However, the growth of crystals is a complex in Coleraine, Northern Ireland, I remember being combination of thermodynamics and statistical intrigued by a colored blotch of light to the west physics, and no review of this nature could do of the Sun at about the same elevation. Little did I justice to the immense theoretical edifice that has know then that these two events of snowfall and been erected on this topic. "sundogs" (of which more anon) were intimately This slim volume is informally yet clearly writ­ connected. Since that time I have learned rather ten, and the photographs (mostly taken by Patricia more about meteorological optics, and this book Rasmussen) are quite stunning. Obviously, it is about the beauty of snowflakes has challenged aimed at a popular audience. In one sense, though, me to learn more of the physics and mathematics it is the latest in a disjoint set of popular books oh behind crystal formation in general and ice crys­ pattern formation in nature: D'Arcy Thompson's tal formation in particular. On Growth and Form [38]; Stevens's Patterns in As in an earlier review [1], I will divide this Nature [32]; Ball's The Self-Made Tapestry [3]; review into two main sections; Libbrecht's book Stewart's Nature's Numbers [33] and What Shape Is a Snowflake? [34]; and finally, from a very differ­ john A. Adam is a professor of mathematics at Old ent perspective, Bejan's constructal theory as ex­ Dominion University. His email address is j adam@odu. edu. pounded in Shape and Structure, from Engineering The Snowflake: Winter's Secret Beauty, by Kenneth Libbrecht, with photographs by Patricia Rasmussen, to Nature [5]. With the exception of [34], none of Voyageur Press, Inc., 2003, hardcover, $20.00, ISBN these address the nature of the snowflake to any 0-89658-630-8. great extent. Some of Libbrecht's low-key yet vivid

402 NOTICES OF THE AMS VOLUME 52, NUMBER 4 descriptions captured my imagination, and I would depending on context. The two words will be used certainly recommend this book to any person synonymously here unless otherwise noted. wishing to "experience" something of the physical The short Chapter 2 is entitled "Snowflake intuition of a scientist. Watching". In it, Libbrecht records a partial history As might be expected, the study of snowflakes of snowflake watchers from Descartes to the is not new; no doubt people have been fascinated present day. In particular, he makes mention of by their beauty and symmetry since time im­ Wilson Bentley, a Vermont farmer who dedicated memorial. According to [4), [16) the Chinese aware­ much of his life to photographing snowflakes. In ness of this was recorded in 135 B.C., while in the late 1920s Bentley worked with physicist W. J. Europe the Dominican scientist, philosopher, and Humphreys to publish a book containing more theologian, Albertus Magnus, studied them around than 2,000 of his snow crystal images. (There is a 1260 A.D. Not surprisingly, the astronomer and 1962 Dover edition of this book, Snow Crystals.) The mathematician Johannes Kepler was intrigued by scientific discussion begins in earnest in the third snow crystals, writing a small treatise entitled On chapter ("Snow Crystal Symmetry"). It starts with the Six-Cornered Snowf1ake. In 1611 he asked the the following quote from Richard P. Feynman (The fundamental question: There must be some definite Feynman Lectures on Physics, 1963): cause why, whenever snow begins to fall, its initial Poets say science takes away from the shape of a six­ formation invariably displays the beauty of the stars-mere globs of gas chance, why cornered starlet. For if it happens by atoms. I too can see the stars on a desert do they not fall just as well with five corners or night, and feel them. But do I see less seven?In his treatise he compared their symmetry or more? ... What is the pattern, or the with that of honeycombs and the seed arrangement meaning, or the why? It does not do inside pomegranates [4). However, nothing was harm to the mystery to know a little known in Kepler's era of the molecular structure about it. For far more marvelous is the of water, which ultimately determines the hexag­ truth than any artists of the past imag- to onal shape of ice crystals, so Kepler was unable ined it. explain their shape in mechanical terms, though he did attempt to do so using entirely reasonable A quote from Kepler in 1611 is included as he packing arguments. (Indeed, finding the densest pondered the sixfold symmetry of snowflakes, and (not necessarily periodic) packing of spheres is although Libbrecht makes no mention of it, a quote now known as the Kepler problem [14).) In 1665 the from D'Arcy Wentworth Thompson [38) is partic­ scientist Robert Hooke published ularly appropriate here: in which he described his observa­ Micrographia, The beauty of a snow-crystal depends using a microscope. More his­ tions of snowflakes on its mathematical regularity and sym­ be found in [4). torical details may metry; but somehow the association ("The Cre­ In the first chapter of The Snowf1ake of many variants of a single type, all author asks the fundamental ative Genius"), the related but no two the same, vastly do complex questions: How do crystals grow? Why increases our pleasure and admira­ physical sys­ patterns arise spontaneously in simple tion ... . The snow-crystal is further of course, tems?These are very profound questions, complicated, and its beauty is notably lit­ and a considerable portion of past and present enhanced, by minute occluded bubbles erature in applied mathematics and theoretical of air or drops of water, whose sym­ to answer them. physics is devoted to attemptmg metrical form and arrangement are very goes well beyond Obviously, the second question curious and not always easy to explain. formation into the the "mineral kingdom" of crystal Lastly, we are apt to see our snow crys­ on) "animal and vegetable" ones of patterns in (and tals after a slight thaw has rounded ·Libbrecht re­ living things. In particular, though, their edges, and has heightened their minds the reader that snowflakes are the product beauty by softening their contours. of a rich synthesis of physics, mathematics, and chemistry and that they are even fun to catch on Returning to the question of complex patterns one's tongue! The range of such comments reflects in nature, we may not be surprised to be reminded the underlying parallel approaches of the book: on by Libbrecht that all crystals demonstrate the or­ the one hand, a good qualitative and nontechnical ganizational ability to self-assemble. Starting with description of the scientific aspects of snowflake a random collection of molecules, this is an example formation and, on the other, the sheer fun of doing of spontaneous pattern formation. The best non­ science. It is worth noting that in Libbrecht's ter­ mathematical book I know on the broad spectrum minology, snow crystal to snowflake is as tulip to of pattern formation is The Self-Made Tapestry [3), flower. In other words, a snowflake can be an and the present book, much narrower in scope, is individual snow crystal or collection of the same, a good introduction to some of the underlying

APRIL 2005 NoTICEs oF THE AMS 403 concepts explained therein. For example: Libbrecht same conditions at each does a pretty good job of explaining to a nontech­ moment of time; their nical readership the physics of crystal facets and symmetry is evidently a their formation. One basic snow crystal shape is the reflection of their shared hexagonal prism, which possesses two basal facets history. and six prism facets, and de­ It should be noted that pending on which of the two while Nakaya's 1954 book types grows faster, the prism Snow Crystals: Natural can become a long column and Artificial is referred or a thin plate. The slowest to on page 44, no men­ moving facets eventually de­ tion is made of the im- fine the shape of the crystal. portant work by Furukawa, who studied under At this point the author asks Kuroda and Kobayashi, both of whom were students a very significant question: of Nakaya. Many valuable details and references on How can molecular forces, the history and science of snow crystal research can operating only at the be found in Furukawa's essay (http: I /www. nanoscale, determine the lowtem.hokudai .ac.jp/ -frkw/english/ shape of large crystals? An­ a 1 etter. html ). swers to this and related In Chapter 5 ("Morphogenesis on Ice") the search questions are hinted at in the for an explanation of snow crystal complexity is following two chapters. taken yet further. Touch­ And so to Chapter 4 ("Hi­ ing on a point made ear­ eroglyphs from the Sky"), in lier in this review, Lib­ which it is stated that the real brecht temporarily puzzle of snowflakes is their broadens his perspective combination of symmetry by noting that while a and complexity and that major progress in solving flower is an example of this puzzle was made by the physicist Nakaya in biological morphogene­ the 1930s. Eventually, he succeeded in growing in­ sis, even simpler physical dividual snow crystals in his laboratory under many systems exhibit this fea­ different humidity levels and temperatures, and it ture. Thus whether it be was only a matter of time before he was able to clas­ waves on the oceans or sify the symmetry and considerable variety dis­ ripples on snowdrifts and sand dunes, they are all played by these crystals. He developed his famous relatively simple pattern-forming systems in which morphology diagram, demonstrating the remark­ complexity arises spontaneously, but for Libbrecht able sensitivity of the snow crystal form to its en­ the snowflake is the poster child of morphogene­ vironmental conditions. This diagram is repro­ sis. Here as in earlier chapters he does a good job duced on page 45 of The Snowflake. Basically, at of introducing the concept of self-organization in low but fixed levels of supersaturation (degree of physical (and, in passing, biological) systems. This humidity), as the temperature decreases below ooc is all to the good, given the preoccupation in some to about -35oc, snow crystals are essentially plates, quarters with the concept of "intelligent design" then solid prisms, and then plates again. At higher (with all due respect to my fellow Christians). In a supersaturation levels, the evolution is from den­ sentence that I find very drites to needles, hollow columns, sectored plates appealing, Libbrecht notes and dendrites, and then columns again. Essentially, that instabilities like those the overall crystal shape, whether it is platelike or discussed here: are the columnar, reveals something about the temperature heart of pattern fprmation, at which the crystal grew, and the complexity of the and nature is one unstable structure indicates something about the humidity. system heaped on top of However, each crystal falling on one's nose is a another. product of the cumulative history it has under­ Scientifically, the real gone as it has been wafted hither and yon by air meat of the book is to be currents through many different atmospheric con­ ditions. In mathematical terms, we might think of The three photographs of snowflakes (above) its shape being defined by a line integral over its were provided by Yoshinori Furukawa of path through space and time. Generally, the length Hokkaido University, who also provided the scale of variations of temperature and humidity will photographs from which the background be much larger than the dimension of the crystal, snowflakes came. The ordinary symmetry of so each vertex or arm of the crystal experiences the snowflake growth is really extraordinary.

404 NOTICES OF THE AMS VOLUME 52, NUMBER 4 found in this chapter. There are several key com­ faceting and branching is what determines the ments made by Libbrecht that bear repeating as form of a given snow crystal in a kind of morpho­ written: Growth is the key ingredient for the gen­ logical balancing act, and this in turn depends on eration of snow-crystal patterns. Left in isolation for the temperature and humidity history of the grow­ a long time, an ice crystal will eventually turn into ing crystal and also its size. Libbrecht speculates a plain hexagonal prism .... Ornate patterns appear that if snowflakes occur in other planetary atmos­ only when a snow crystal is out of equilibrium, pheres, they may well be different from tht! ones while it is growing. Such circumstances are often we know and love. referred to as nonequilibrium conditions. Another Although he does not identify it as such, the important concept introduced is that of diffusion­ branching instability Libbrecht refers to is the limited growth. A snow crystal grows by assimilating Mullins-Sekerka instability (see the second section molecules of water vapor into the existing ice of this article for more details). Nobody could fault lattice, provided the humidity is high enough. How­ him for simplifying physical phenomena in a ever, continuing crystal growth gradually depletes book of this kind, but as he implies elsewhere, the the vapor from layers of air adjacent to the crys­ picture is just not that straightforward. Indeed, I tal, and the remaining water vapor molecules must am indebted to Professor]. S. Wettlaufer for the diffuse over increasingly larger distances. Since following comment (in reference to a statement the travel time for diffusion (other things remain­ made in [2]) that puts into perspective the com­ ing unchanged) is proportional to the square of plexity of the problem: distance traveled, it is clear that the growth rate the The initial question that needs to be crystal will be inhibited; the growth is now of the asked is: How does a hexagon emerge diffusion-limited. from a nucleus of some 1,000 mole­ initial "seed" crystal consisting of several An cules that, at the point of nucleation, do has a molecularly hundred or thousand molecules not necessarily reflect that symmetry? on the crystallographic rough boundary. Depending This has nothing whatever to do with the orientations may grow more orientation, some structure of the background diffusion (under a spatially uniform rapidly than others field, but rather it is solely controlled by thus a surface is created with growth drive), and the statistical mechanics of adsorbates curvature. "Flatter" por­ both positive and negative on the seed. The seed is much smaller and while tions of the boundary are called facets, than the mean free path in the vapor facets, it may a completely rough seed will not have phase. The enhancement of the vapor regions "fill still grow anisotropically. The curved field at a corner, generically understood from which it is in" more readily than the facets, as the "point effect of diffusion" or the facets seen that ultimately the slower-growing "Berg-effect", may or may not lead to an shape in the absence of signifi­ define the crystal instability. The conditions must be discussed below). cant branching (a mechanism carefully determined; for example, is two mechanisms, in diffusion­ Combining the above the size of the seed much smaller than the crystal ver­ limited growth one might expect that the characteristic scale of the diffusion molecules faster tices would "harvest" water vapor field? There is a plethora of outstand­ projection further than facets by virtue of their ing physical and mathematical prob­ free path of the into the medium (if the mean lems related to the transition between of the system). vapor phase is of order the size nucleation, interface controlled growth loop known This can induce a positive feedback and diffusion limited shapes. Therefore, be noted in the as a branching instability. As will if and/ or when a diffusive/ Mullins­ if this instability second section of this review, Sekerka instability occurs, understand­ stabilizing occurs, there is usually a self-limiting ing the habit of the crystal still requires that eventually mechanism, akin to surface tension, an understanding of the initial value balances the destabilizing one. If this diffusion­ problem and thus the evolution of a and limited branching instability is iterated over crystal from its birth. over, a type of snow crystal known as a dendrite will develop. Ultimately, this may give rise to some kind In Chapter 6 ("Snowflake Weather") Libbrecht of self-similar or fractal-like structure over several makes some statements about snow crystal quan­ orders of magnitude of scale, not unlike a fern leaf. tities that might form the basis for several inter­ This branching process is more pronounced in esting estimation problems, but first it is necessary regions of higher vapor pressure, since then the to note some facts about snow. Consider first a diffusive transport of water vapor molecules is small ice cube 1 cm3 in volume. This has a mass more effective. Conversely, faceted growth is more of one gram, and since the mass of a water mole­ likely to 'occur in a lower vapor pressure environ­ cule is about 3 X 10- 23 g, there are about 3 X 1022 ment. Ultimately, the complex interplay between molecules in this ice cube. A typical snow crystal

APRIL2005 NOTICES OF THE AMS 405 of mass 3 x 10- 5 g therefore may contain about snowmen numbers about 3 x 109 , or about one 1018 molecules. Depending on the type of crystal, half of the present world population. A factor of temperature, wind speed, and other factors, in­ two is not a large discrepancy in this context, and cluding the density of packing, snow has consid­ in any case, that's a lot of snowmen. (There is a de­ erable variation in its water content. An eight-foot lightful "Peanuts" cartoon showing a vast army of blanket of fresh snow may contain as little as 1 inch snowmen built on a cloudy day by that precocious of water per unit of surface area or as much as 3 genius, Linus. He marches up and down declaring feet. According to [16], the majority of U.S. snows their military invincibility under any and all cir­ have a water-to-snow ratio in the range of 0.04 to cumstances. Then the Sun comes out. I use this par­ 0.10. I had to dig around for most of this infor­ ticular cartoon in mathematical modeling classes mation; I wish that facts such as these had been to emphasize the dangers of making false (or at incorporated into the book as an appendix. best, weak) assumptions.) In the beginning of this chapter we read: Later, another estimation beckons us. Libbrecht Snowflakes are being manufactured in the atmos­ states that the total global precipitation per day is phere at an astounding rate-around a mil-lion equivalent to about 1015 liters of water, and each billion crystals each second. Every ten minutes that's of us typically exhales about 1liter of water per day enough snow to make an unstoppable army of snow­ into the atmosphere. By simple proportion, there­ men, one for every person in the world.... Let's think fore, and using his figures, we infer that if our dis­ about this: on the basis of these numbers, after ten contribution to the water cycle were uniformly the globe (seemingly yet another minutes there are about 1015 x 6 x 102 = 6 x 1017 tributed around then our average contribu­ crystals, and estimating a typical snowman to be flawed assumption), content of a snow crystal is composed of two identical spheres of diameter 1.5 tion to the total water about 1018 7 1015 = 103 molecules. On this num­ ft. gives a volume of about 3.5 ft. 3 or approximately ber, not surprisingly, we agree. 3.5 x (0.3) 3 ;::, 0.1 m 3 = 105 cm3. If we assume that The chapter closes with a nice little section all the crystals formed make their way to the ground about ice nucleation: how water molecules, like (a rather dubious assumption) and take an average adolescents, eventually learn to settle down and water-to-snow ratio of 0.07 for U.S. snow (!), then become very cool (or freeze, in the case of the · the mass of crystals falling in a ten-minute inter­ former). There is also a rather intricate table 13g. Meanwhile, the mass val is approximately 2 x 1 0 classifying some of the many types of snow. by the product of our typical snowman is given Chapter 7 ("A Field Guide to Falling Snow") is by 5 = 3 our army of 10 x 0.07 7 x 10 g. This means far the longest in the book and is ac­ companied by many beautiful pho­ tographs. We are introduced to a verita­ ble zoo of snow crystals: diamond dust (of which cirrus clouds are made), stellar dendrites, sectored plates, columns and needles, hollow columns, needles and bullets, capped columns, split stars and split plates, twinned crystals, twelve-sided snowflakes and double stars, chandelier crystals, spatial dendrites, triangular crys­ tals, and rimed snowflakes! What type of ice crystal then was re­ sponsible for the sundog I mentioned earlier (such patches of colored ~ight are also known as mock ~uns or parh,elia)? Bil­ lions of slowly falling horizontally ori­ ented hexagonal plate like crystals (pre­ sent in cirrus clouds) 30 microns or larger are the culprits. They behave like tiny prisms and refract light entering one ver­ tical face (say face 1) and exiting face 3. Unlike rainbow formation, there is no re­ flection involved, and so the red portion This photograph of atmospheric halos was taken at the South Pole, where is always closest to the Sun. They occur conditions are frequently favorable. It was provided by Walter Tape, now at least 22 ° away from the at the ,Univer.sity of Alaska in Fairbanks. It is one of many photographs in Sun. There are many other related phe­ his book [35]. nomena one may observe, such as ice

406 NOTICES OF THE AMS VOLUME 52, NUMBER 4 particulate matter. As to the question not asked in the book: sound waves are readily absorbed by a· thick covering of fresh snow, because all the air pockets are prone to trapping the waves to a certain extent. As noted in [16], as snow ages, it changes from being light and fluffy to smooth and hard, and in this state it can become an efficient reflector of sound waves, and sounds may seem clearer and be heard from greater distances. Chapter 8 ("In Search of Identical Snowflakes") addresses what might be termed "the question we This drawing of halos was made in Danzig, have all been waiting for": Is every snowflake 1661 by the astronomer Hevelius. It is one of unique? In a recently used phrase, the answer to several early records described in the book on this question depends on what is meant by the meteorological optics by J. M. Pernter and F. M. word "is". Less flippantly, the answer does depend Exner. on how powerful a "magnifying glass" one wishes to use to provide an answer. It is an axiom of fun­ crystal halos, and some of them very rare. A com­ damental physics that all electrons are identical, mon example is the 22° halo around the Sun (or the so at this scale the components of all snowflakes Moon, but it is much fainter for obvious reasons). are identical. However, since this applies to ele­ However, only small (less than about 20 microns phants and teapots as well, the concept is not par­ in size) columnar crystals are responsible for this ticularly helpful. Moving up in size from electrons, particular optical manifestion. Again, light is re­ it is also true that "ordinary" (as opposed to fracted between alternate faces of the hexagonal "heavy") water molecules are identical. Heavy water columns, but the crystals are small enough to tum­ is water composed of deuterium, a stable isotope ble, and so are randomly oriented as they fall. I see of hydrogen, occurring with a frequency of about these halos, or parts of them, at least once a month one deuterium atom for every 5,000 of the lighter on the average, and sometimes they can last for an hydrogen atoms. Another stable isotope of oxygen hour or two. There are many other types of halos; has a relative frequency of occurrence of about details and further references may be found in [2] one in 500, so while a snow crystal might contain and [3 5]. A general mathematical setting for halo water molecules, about 1 in 500 water molecules theory is available in [36] (see also [18], in which will be different from the rest. The number of the use of symmetry arguments is further explored, con­ figurations containing together with some speculations on possible halo these heavier oxygen atoms forms produced in the atmosphere of Titan!) is combinatorially enormous, so the likelfuood of two snowflakes being identical In passing, Libbrecht briefly addresses the ques­ is equi..J' ai~nt·r, to that of the sustained existence of the proverbial tion: Why (not who) ·• is snow white? To which we may (~ '' "t" "' add: Why does it seem so quiet outside after a snowball in the proverbial vecy,: hot place. But as Lib- ) _,;i. "" _..... "\.. snowfall? In fact, let us generalize the first ques­ brecht reminds us, crystals ~6ri~ impuriJY atOJ:llS, ~'+ tion by asking why it does appear white, while ice stacking faults, and other types of defects. And this (in sufficient quantities, as in a glacier) appears does not help matters, to say the least. . ~- ··- ·· somewhat blue in color. Snow is made of small crys­ If we relax our requirement of mathematical tals of ice, as we well know by now. There are myr­ uniqueness at th~ atomic an~ m~_lE; ciJJ.ru:: ley~l, using iads of tiny surfaces from which light is reflected, a good opt1cal nncroscope WJ.th>'f'esolutwn down to resulting in a very efficient scattering process in about 10- 3 mm, it migb.t;0ell be'possible to find which very little absorption takes place. This high two hexagonal plates that appear indistinguish­ reflectivity, incidentally, is also the main reason why able, but the more complex crystals are;y eh intri­ snow is melted more by warm air than by direct sun­ cate, with all sorts of minor asymmetries,, so the light. By contrast, ice is a continuous medium (on problem gets worse again. Just as with the early the scale of ice crystals, at least), and so sunlight stages of male pattern baldness, it all debends on is more readily absorbed because of the longer how closely one looks. path-length between scattering centers. Blue light The "afterword" by photographer Patricia Ras­ is absorbed rather less efficiently than are longer mussen is beautiful, not least because of her wavelengths, but this is not obvious in something obvious love for poetry. And with mention of as small as an ice cube! Large quantities of ice the word beauty, it is appropriate to quote both allow an accumulation of the effects of scattering Poincare (as stated in Chapter 8 of the book) and and absorption by not only the ice molecules but Kenneth Libbrecht himself as he ends this, his last the plentiful supply of dust, air bubbles, and other chapter.

APRIL 2005 NOTICES OF THE AMS 407 n A last word on the topic of snow crystals may be of interest: Thanks to the sharp eyes of a Minnesota man, it is possible that two identical snowflakes may fi­ nally have been observed. While out snowmobiling, he noticed a snowflake that looked familiar to him. Searching his memory, he realized it was identical 1(n)-plot to a snowflake he had seen as a child in Vermont. Weather experts, while excited, caution that this may be difficult to verify. See more about snow crystals on Kenneth Lib­ brecht's website www. snowcrysta l s. net.

Equilibrium Mathematical Appendix Crystal A snow crystal pattern is an interfacial phenome­ Shape non, occurring at the boundary between the solid phase and vapor phase of water. By contrast, an ice crystal grows at a solid/liquid boundary. There is a plethora of theoretical papers on the subject of crystal formation (for a sample, see the references), only a small proportion of these, understandably, Figure l: The equilibrium crystal shape (for cubic being devoted to snow or ice crystals. I will at­ symmetry) formed from the Wulff construction; the tempt to provide a general overview of the topic boundary is shown in bold. It is the interior envelope of and then address some of the particular models of the set of perpendiculars to radial rays intersecting the snow and ice crystal formation. y(:N) (surface free energy) polar plot. As noted already several times, many nonbio­ (From [43].) logical patterns in nature arise at the moving The scientist does not study nature be­ interface between two domains or phases, such as cause it is useful; he studies it because an ice-water vapor boundary, with competition between forces tending to stabilize and destabilize he delights in it, and he delights in it the boundary. However, in contrast to complex because it is beautiful. (Jules Henri biological systems, crystal growth perhaps repre­ Poincare, 1908) sents a conceptually simpler example of sponta­ neous pattern formation and self-organization, There is great beauty in a large, sym­ based on the existing "laws" of thermodynamics, metrical stellar crystal. The beauty is en­ statistical mechanics, kinetics, and transport the­ hanced by the magnifying lens that ory. Nevertheless, many of the theoretical problems brings out the fine structures in the ice. associated with these phenomena are quite formi­ The beauty is enhanced still further by dable mathematically. an understanding of the processes that A snowflake with its planar hexagonal symme­ created it. (Kenneth G. Libbrecht, 2003) try is a good illustration of some of the questions

Is seeing believing? By searching through both old and recent snowflake photographs with a computer search tool specially de­ signed for the task, candidates for in­ distinguishable flakes have been found. Because of the poorer quality of the picture on the left, however, some difficult work remains to be done. These photographs were taken on the first day of April, which might account for the similarity of the flakes.

408 NOTICES OF THE AMS VOLUME 52, NUMBER 4 y that can be asked in a more general context. They are patterns that have emerged, apparently, from a structureless environment, and as has been noted previously; they are very sensitive to that

/ environment. Depending on the history of each / I / I ice crystal as it moves through regions of differ­ / I / I / ent temperature, supersaturation, and airspeed, I I I I I I there will be formed several different types of ice ,~ r I I I structure; and since each history is (presumably) I I I unique, so in principle is each snow crystal. I I I According to the review by Langer [19], regularly I faceted crystals will form under a wide variety of I e conditions when the molecules are tightly bound ~~------~------~ on crystallographic planes. At the other extreme, X when the surface molecular binding is sufficiently weak (as in many metals and alloys), growth is Figure 2: A portion ofthe crystal boundary S used in dominated by the mechanism of diffusion close to establishing equations (4) for the total free energy :f and the the solidification front (and may be rapid). In such "area" of the two-dimensional crystal. (The notation of [9] is cases, the fluid-solid interfaces are macroscopi­ used in this figure.) cally smooth but microscopically rough. Ice falls this stabilizing effect come about? The surface between these two extremes: facets grow slowly molecules on a bump with positive curvature have parallel to the basal plane, but rapidly in the hexag­ fewer nearest neighbors than do those on a plane onal directions, and surfaces tend to be rounded. surface and are thus more susceptible to being There are two basic types of mechanism that removed, and the bump will tend to move back to contribute to the solidification process: diffusion the plane configuration. This corresponds to a control (involving long-range processes) and in­ terface control (involving local processes). In [42], reduction in the melting temperature of the bump. models based on these mechanisms are referred to Similarly, molecules on the surface of a negatively respectively as nongeometric and geometric growth curved region have more nearest neighbors and are models. In the latter, the interfacial growth veloc­ more tightly bound; the melting temperature has ity is determined by local conditions only, and been raised. hence diffusion-driven morphological instabilities Thus, a moving interfacial boundary is driven by are absent. However, even this is something of an a diffusion field gradient and inhibited by curvature­ oversimplification, because a geometric model may related forces. A common feature of such compe­ be used to examine diffusion-limited growth if tition between stabilizing and destabilizing influ­ the interfacial speed is sufficiently large. (In this ences is the existence of a characteristic length context the appropriate length scale L is essentially scale for the resulting pattern. It is often difficult the diffusivity divided by the interface speed; large to predict the pattern selection principles that op­ gradients can exist, and the boundary layer of erate in these systems. Such patterns, as noted thickness L effectively defines the interfacial re­ above, are sensitive to the system geometry and ex­ gion.) A model is considered geometric if the ternal conditions in general. A valuable review of normal velocity at an interfacial point depends pattern formation models in this context can be only upon the shape and shape-dependent quan­ found in [17). Mathematically, the problem of a tities of the interface. moving and developing interface between two dis­ By contrast, nongeometric models pertain to tinct media is a Stefan problem, wherein a nonlin­ growth on surfaces that are everywhere rough and ear system evolves dynamically in time. As noted are usually formulated as some type of free bound­ in [7], this problem is especially interesting when ary problem. It transpires that such diffusion­ the interface motion is a nonequilibrium problem, controlled models of simple geometric shapes such where the configuration is initially such that the free as planes, cylinders, and spheres are commonly energy of the system is not at an absolute minimum. unstable. For example, if a plane solid interface develops a small bump extending into the vapor A very useful resource for mathematicians in­ phase or liquid phase, the temperature gradient is terested in the subject of crystal formation is the locally larger than in its immediate neighborhood review in [3 7). Therein, an important convex set, (because the isotherms are closer at that point), known as the Wulff shape (or equilibrium crystal latent heat diffuses away more rapidly, and the shape) is introduced. The equilibrium crystal shape bump continues to grow, at least until the stabi­ is the shape which minimizes the total surface free lizing effect of surface tension (via the curvature energy per unit area ;y(N) for the volume it en­ of the interface) becomes comparable. How does closes, and its boundary Yvy is defined as

APRIL 2005 NOTICES OF THE AMS 409 (1) Yvy = {r: r·N 5 y(N)VN}. (4) F = I(p+p)f(8)d8 and n = ~ J(p+p)p(B)d8. y(N) is orientation-dependent, N is the unit nor­ mal vector, and r is the radius vector of any point The total free energy must be minimized subject on the equilibrium crystal surface (see Figure 1). to the constraint n = constant. If the choice of Yvy is determined uniquely by the Wulff Theorem Lagrange multiplier A is made such that we can (or construction), one proof of which is given below. define the quantity Another important boundary is the steady-state growth shape Yvv defined by (5) Q(B, p, p, p) = ~(p + p)p- A(p + p)f, (2) Yvv = {r: r·N 5 V(N)V N}, then the appropriate Euler-Lagrange equation is where V(N) is the growth rate in the direction N. For further references to the Wulff construction, (6) - dde ( + dd;2 ( = 0· see [11]. ~; ~;) ~;) Wulffs Theorem It follows that In a crystal at equilibrium, the distances of the faces from the centre of the crystal are proportional to (7) (p + p)- A(f +f) = 0, their surface free energies per unit area [9]. The proof and notation therein for the two-dimensional with solution case will be followed here. The fundamental idea (8) p(B) = Csin(B- Bo) + Af(B) is to relate the equilibrium shape of the "crystal" to the polar plot of the surface free energy. This for arbitrary constants C and 80 . By choosing the is done via the Wulff construction (again, see origin in such a way that the crystal has some Figure 1), which is the interior envelope of the set rotational symmetry about it, the choice C = 0 is of perpendiculars to radial rays intersecting the permissible. Then latter [43]. The equilibrium shape is determined by the requirement that the total free energy for a (9) given crystal "area" is a minimum. which establishes that the polar diagram of the edge We consider a point P(x, y) on the boundary S free energy is proportional to the pedal of the equi­ of the crystal; the polar coordinates of Pare (r, ). librium shape of the crystal. Here the arc length parameter will be denoted by It is important to note that the steady-state e for consistency with [9], but also to emphasize shape of a crystal is not as g~neral as the Wulff that we are dealing with rather general equilib­ shape for an equilibrium crystal, because it is not rium boundary curves as opposed to the circular obtained from an initial value problem. Not all ini­ ones discussed in the model below. (Note that e will tial shapes will in fact reach this steady-state shape, be used in two other ways in the models below.) and neither does it indicate what class of geomet­ The line segment OM (see Figure 2) is the perpen­ ric models results in a convex form from an initial dicular from the origin to the tangent line to S at shape. Further details on this ;:tnd other aspects of P; it has length p. Denoting the edge free energy the problem may be found in [41']. per unit length of a boundary element by f(B), the A "Toy'' Model for Crystal Growth total free energy F and the area n (related to the As a precursor to the more physically significant number of molecules) are respectively given by models of crystal growth discussed below, let us examine a specific yet simple evolution equation F =I f(B)ds =I f(B) (x 2 + }'2 ) 112 dB for the interface in a two-dimensional geometrical (3) model and comment on its stability. The model that and n =~I (xy- yx)dB, follows is not particularly realistic physically­ only the lowest order curvature term is retained to where x = dx/dB, etc. From the figure and stan­ exhibit the desired behavior of the boundary-but dard algebra, it must be emphasized that it is merely a mathe­ matical simile; it cannot be derived from the kinetics p = xcosB + ysinB. of crystal growth for realistic crystals. In fact, a The locus of M as the point P moves around the model of this type is one in which the boundary boundary curve S will be the pedal curve of S. The layer effectively defines the interface: as noted ear­ expression above and its derivative with respect to lier, the characteristic length scale Lis the ratio of e can be inverted to obtain the diffusion coefficient to the interfacial speed, and X = p COS tJ - p Sin tJ and y = p Sin tJ + p COS tJ. this would be true in practice only if the crystal growth speeds were excessively high. Neverthe­ Hence less, for applied mathematicians, toy models are

410 NOTICES OF THE AMS VOLUME 52, NUMBER 4 the aesthetic equivalent of "back-of-the-envelope" (14) r (t) = [2t + r2 (0)] 112 . calculations for engineers! If x is the position vector of a point on the Performing a linear stability analysis around this interface corresponding to outward unit normal solution, following [8] (see also [42]) we consider N, then we assume that radial perturbations only, because tangential terms dx can be eliminated by a suitable gauge transforma­ (10) N· dt = U(x; s, K), tion (equivalently, they can be shown to drop out of the linear ~tability analysis). Thus where the speed U is a function of the local sur­ face geometry, being dependent on position, and (15) implicitly on arc length s and curvature K. What Then form might U take? One choice is [7]: iJ 2K (16) (11) U = V(K) + }' 052 .

where Ko = r - 1 , and The second term corresponds to the stabilization mechanism for short wavelengths when ;y > 0. Heuristically we may choose the "curvature (17) potential" V to take the form of a cubic for the following reasons. A planar interface cannot move Since at all (except under very special circumstances), so V(O) = 0. The growth rate of a spherical crystal (or circular in two dimensions) behaves like R- 1 for the linear perturbation E(t, e) may be shown to largeR, so we would expect V "" CXK for small K. satisfy the equation Noting further that a solid with large curvature will contract due to forces of surface tension, U DE = _ V' (K0 ) (£ + 1) at r2 aez must become negative for some value of K. Hence (18) there must be a minimum "bubble" size for nu­ }' iJ2 ( 0 ) cleation, and this corresponds to a term - {3K 3 in X E - r 4 aez aez + 1 E. V, where {3 is related to the minimum size for nu­ cleation. A quadratic term, J1 K2 , is also included in Seeking eigenfunctions of the form V the expression for to account for an asymmetry (19) E(t, e) = Eme crmt COS me, in the solidification process between the freezing and the remelting of the dynamic interface, so we we see that the linear growth rate write (12) V(K) = CXK + J1K 2 - {3K 3 • (20) O"m = (m2r; 1) [ V' u) -;y~ 2 J. Stability Criteria The m = 1 mode corresponds to a uniform rota­ In terms of the unit tangent vector to the curve T tion of the circle and is a neutrally stable pertur­ at a point with polar angle (s being arc length e bation. As noted in [8], the first term is stable for along the boundary curve), the curvature vector K allm(* 1)ifV'(1 / r) > 0, andsinceforsufficiently is variously defined as large radii V ~ K, this will occur even for nonzero K = dT/ds = (d'l; de)J(ds ; de) J1 and {3 in the complete expression for V. This sim­ ple result is the analog of tbe Mullins-Sekerka in­ = (d'l ; de); ldx;del, stability as discussed in [29], [30] (see also [10], [31], where x(e) is the position vector. For a circle with [46]). The second term is always stabilizing for time-dependent radius r(t) and the special case of m > 1 (with ;y playing a role analogous to that of U = V(K) = K (with the unit of length chosen to fix surface tension), so this will act to limit the range ex = 1 (7], [8]), the equation of m-values that are in the unstable regime. More Sophisticated Models dx N· dt = U(x; s, K) Based on more detailed theoretical physics, how might the interface evolution of a crystal be char­ reduces to acterized mathematically? Following the approach in [42], we consider the boundary of a two­ (13) dr = V (!) dt r ' dimensional crystal to be represented by the closed curve C [x (u, t), y (u, t)] in the pl~me, with time de­ with solution pendent components parametrized by the variable

APRIL 2005 NOTICES OF THE AMS 411 u. Let the arc length s be related to u via the rela­ (25) tion 2 ( oK(s,t)) =-(o V 2V)-oKis Vd' ut"' s uS"' 2 + K uS"' o K S. (21) s (u, ) t = iu I oC(u''"' t) I d u.' 0 uu' As pointed out in [8), when the curve Cis parame­ trized by e rather than by u or s' then the evolution Further, let W = loC(u, t)/ou l = ds/du. is now e equation for K becomes strictly local. This is a defined to be the angle between the positive x -axis consequence of the fact that its rate of change at vector (cos sin e) = and the unit tangent '1 = e, fixed e is given by w-1 oC(u, t)/ou, and the unit normal vector N is inward pointing. The boundary evolves in terms of (26) the normal velocity function V (e, bJl) according to

(22) (ac ) = -vN. Thus at u (27) 2 The initial-value problem with V defined as above ~ ~ = -K V(e), can be solved exactly by the method of character­ istics, since the normal velocity of the interface de­ where V = V + V" is named the "velocity stiffness" pends explicitly on the surface orientation alone. in [42). These results also can be derived by con­ sidering the curve as a set of complex numbers in They are rays of the form x(t) = xo + d(e0 )t, x0 the plane [8). For finite this equation has solu­ being a point on the initial curve, and V (80 ; Xo) is V in the direction d (e0 ) . The surface normal direc­ tion K (0) tion is preserved along each characteristic, and so (28) K = - the curve C at time t is defined by the set of all 1 + K(O)VT' points x(t). To discuss the curvature evolution, both global in terms of the initial curvature K (0). If V > 0, the and local, we require that the two appropriate curvature will decrease monotonically in time Frenet equations for the unit tangent (T) and nor­ at nonfaceted orientations for which K (0) * 0. , mal (N) vectors at a point on the boundary curve For orientations in which V < 0, the curvature are diverges at finite time, T = - ( K (0) V) - l . In fact, as discussed below, a shock (intersection of characteristics) develops before the minimum blow-up time. In defining a local normal velocity with reference to a weak "growth drive" 811 (the chemical poten­ tial difference between the surface and external phase), two different kinetic processes are merged via a convexity type of argument involving a tran­ sition function £(e) to form an expression for the local normal velocity of a crystal with n-gonal sym­ metry. This is (29) V (e, 8J1) = Vr (8J1) £(e) + Vr (e, bJl) (1 - £(e)) ',

where

(24) (30) Vr(bJi) = crg(8J1) exp ( ~;;;) / a'l _ av N· ae _ av as - --as · -at - --as· and Since W is essentially a measure of the length of (31) Vr (e, bJl) = CrbJl [ 1 + cosP ( ~e ) J, an infinitesimal displacement on the boundary, the differential equation for W represents its rate where pis an even integer. Vr (for facet) is the nor­ of dilation. The corresponding equations in T and mal growth rate at facet orientations under the e describe the rotation rates of those quantities, "growth drive" 811, while Vr (for rough) describes as determined by the anisotropy of V . After some the normal interfacial motion at vicinal and mole­ subtle reformulation (involving gauge invariance ar- cularly rough orientations (due to surface migra­ . gurnents), a nonlocal integrodifferential equation tion of adsorbed molecules away from facets). for the curvature evolution can be derived, namely: Since 0 ~ £ (e ) ~ 1, this function is a measure of

412 NOTICES OF THE AMS VOLUME 52, N UMBER 4 the distribution between facetlike and roughlike vature at rough orientations. Further, if t0 is the time growth. It is periodic in 2rr In, and ~ ( er) = 1, of corner formation, the facet collides with the ~ ( er + TT In) = 0 for a facet orientation er. In [42) shock at t "" l.lt0 almost immediately after shock the choice of ~(e) = cosm (ne 12), m ;:::: p being formation, consistent with observations [40). even, is made. If in particular e = rr I 4 and n = 4 Green's Functions in the definition of V, then for m = p = 2, The literature on this subject abounds with exam­ V = CrD/1 + 8Vr, which implies that the rate of cur­ ples of the application of Green's function tech­ vature decrease increases with 6 iJ. In short, the niques to the growth and development of crystals polygonalization of crystals corresponds to de­ and might well be a useful source of examples for creasing curvature in rough orientations. graduate classes. This particular analysis is based Thus, in general, a crystal has slowly growing on the approach in [40) (but see also [25)). In con­ molecularly smooth faces (facets) and more rapidly nection with shocking facets, the temperature field growing faces that are rough at the molecular level. T around an expanding surface with a constant den­ At this level, both structures are sensitive to tem­ sity of heat sinks is computed using Green's func­ perature, as one might expect. Facet locations are tions for the two-dimensional diffusion problem. associated with minima of the surface free energy, The expression for the field at (x, z; t) is and indeed a crystal shape may be only partially u(x, z; t) = u (x, z) faceted, but a major question for crystal growth is 0 to find what processes are responsible for the evolution of an initial "seed crystal" towards a (34) - s: dt' r oo [" G (x - x', z - z'; t - t') faceted asymptotic growth shape. This is caHed xQ(x' z'; t')dx' dz', global kinetic faceting and has been observed to occur experimentally and numerically [26) and has where u = (T- Too)/Too, Too being the melt tem­ been predicted theoretically [42). In this process, perature far from the interface. The temperature the rough orientations of partially faceted shapes field before the formation of a shocking facet is u0 , grow out of existence with decreasing curvature. and the standard Green's function for the diffusion Such a decrease implies the presence of disconti­ problem nuities in surface slope (or a jump in the normal to the surface), so it is of considerable interest (3 5) [ :t - k\7 2J G(x, z; t) = 6(x)6(z)6(t) to be able to tie the time-dependence of the local curvature to the ultimate shape of the crystal is boundary. Mathematically, discontinuities of this kind are associated with both singularities in (36) G(x, z; t) = 4~kt exp (- x24:tz2) , the interfacial curvature and the intersection of characteristics [39). The latter viewpoint, while k being the thermal diffusivity of the melt. The complementary to the former, corresponds to the source term is existence of shocks in the solution of the surface evolution equation. The dynamics of such shocks, (37) Q(x, z; t) = qH(ut- x)H(x)D(z), when they exist, are of great importance in the in terms of the Heaviside step-function Hand the interpretation of experimental data. Furthermore, Dirac delta function 6 (remember that distribution it has been shown in [41) that if the initial seed theoretic distinctions and subtleties are usually crystal is convex, then convexity is preserved suppressed in much of the scientific literature). throughout the whole growth process. Forms of The term q is a heat flux constant, proportional V that possess cusps at faceted orientations to the surface density of heat sinks. After some also explain the formation of expanding facets rearrangement separated from rough orientations by a corner, q e-z2 / 4k(t- t' l called shocking facets in [40). Specifically, in place lr , (38) u(x, z; t)- u0(x, z) = - 4k dt , I, of expression (29) for V (e, O!J), now 0 t- t where (32) V (e, Dp) = Vr (DJ1) ~(e) + Vr (e, Dp), I = - t') e-n2 dn, where ~4k(t r:+ry (33) Vr (e, O!J) = CrD/1 (a lsin2el- b sin2 2e), ()( = Xl~4k(t- t'), where the coefficients a and b are related to cer­ 17 = utI ~4k(t - t'). tain angular rates of change. It transpires in this case that models with the parameter range Experimentally, the authors found that ut « J4kf. 2b < a < 7 13b correspond to the formation and ex­ Under the experimentally reasonable assumption pansion of facets accompanied by decreasing cur- that 17 « 1,

APRIL 2005 NOTICES OF THE AMS 413 Personally, I have been both pleasantly surprised and pleased to encounter such a broad swath of whence thoughtfully written papers addressing crystal growth. I hope that this review will be a useful (but clearly not exhaustive) resource for readers wish­ u(x, z; t) - u (x, z) fi(x, z; t) 0 = ing to pursue the subject further. However, a caveat (39) qut loo e-,\ in order: a review of this nature must maintain "'='-- -dA. is 4k (x2 +z2) f 4kt A a balance between material of interest to mathe­ maticians while at the same time doing justice to Noting that the exponential integral [6] is defined the physics of the problem. Inevitably, the simple as oo e- ,\ models and oversimplified physical descriptions £ 1 (x) = f AdA presented here will not satisfy crystal theorists, but 0 (40) as in all topics on the interface between two mag­ 00 ( -l)n+l en must be made. = -ln e - y + I n . n! , isterial disciplines, compromises n ~ l This article is no exception, and I will end it with a warning: The mathematical physics of snow crys­ where y is the Euler-Mascheroni constant, it follows tal formation is much more complicated than might that when (x2 + z 2 )/4kt « 1, be inferred from the discussion presented here!

_ qut ( 4kt ) Acknowledgments (41) u(x, z; t) "" - 4k - y + ln x2 + 22 . I am very grateful to Professor JohnS. Wettlaufer, At the other extreme, for short times such that who has not only provided me with several of his 4kt « x2 + z 2 , u is exponentially small. Using ex­ papers and constructively reviewed an early draft perimental data, Tsemekhman and Wettlaufer [40] of this manuscript but has also patiently, thought­ estimate that the geometric model begins to fail at fully, and enthusiastically responded to my ques­ t "" lOs . From this and related observations, the tions and comments during its preparation. authors conclude that geometric models of the en­ References tire interface are very accurate up to about t = 15s. Ultimately the model breaks down when q is no [1] ]. A. ADAM, Like a bridge over colored water: A math­ longer constant, and nonlinear effects become sig­ ematical review of The Rainbow Bridge: Rainbows in nificant. Studies such as this one confirm that geo­ Art, Myth and Science, Notices of the AMS 49 (2002), are very useful in describing the 1360- 71. metric models [2] __ ,Mathematics in Nature: Modeling Patterns in the early stages of the evolution of a crystal surface. Natural World, Princeton University Press, Princeton, Some of these models predict the possible forma­ N], 2003. tion of a shocking facet and are still applicable for [3] P. BALL, The Self-Made Tapestry, Oxford University some time beyond this point. Indeed, according to Press, Oxford, 1999. Wettlaufer [private communication], even the [4] __ , H2 0 : A Biography of Water, Phoenix, London, timescales cited above do a great injustice to the 1999. utility of the geometric approach: for solid helium [5] A. BEJAN, Shape and Structure, from Engineering to growing from a superfluid, the geometric model is Nature, Cambridge University Press, Cambridge, 2000. the only model of significance, and it never breaks (6] N. BLEISTEIN and R. A. HANDELSMAN, Asymptotic Expan­ down! sions of Integrals, Dover Publications, Inc., New York, 1986. Summary (7] R. C. BROWER, D. A. KESSLER,]. KOPUK, and H. LEVINE, Geo­ metrical approach to moving interface dynamics, Phys. There are many other mathematical aspects of Rev. Lett. 51 (1983), 1111-4. crystal formation that have been investigated over [8] __ , Geometrical models of interface evolution, the last several decades. It is a field rich in theory, Phys. Rev. A 29 (1984), 1335- 42. experiment, and applications (particularly inter­ (9] W. K. BURTON, N. CABRERA, and F. C. FRANK, Th e growth esting, for example, but outside the scope of the of crystals and the equilibrium structure of their sur­ present review, is the asymptotic analysis found in faces, Phil. Trans. R. Soc. A 243 (1 951), 299- 358. [44], [45]). Interesting theoretical papers on dendrite [10] A. A. CHERNov, Stability of faceted shapes, ]. Cryst. formation are presented in the set [20]- [24], [27], Growth 24/ 25 (1 974), 11-31. [28]. In addition, many theoretical developments are [11] M. ELBAUM and]. S. WETTLAUFER, Relation of growth and crystal shapes, Phys. Rev. E 48 (1993), lied math­ equilibrium to be found outside the "standard" app 3180- 3. ematical literature: some of the papers cited below (1 2] T. FUJIOKA and R. F. SEKERKA, Morphological stability are published in the journal of Crystal Growth, of disc crystals,]. Cryst. Growth 24/ 25 (1974), 84- 93. and Physical Review E, for example, so it is en­ (13] M. GAGE and R. S. HAMILTON, The heat equation shrink­ cumbent on those of us interested in entering a field . ing convex plane c urves, ]. Differential Geom. 23 such as this sometimes to walk in pastures new. (1986), 69-96.

414 NOTICES OF THE AMS VOLUME 52, NUMBER 4 [14] T. C. HALES, Cannonballs and honeycombs, Notices of [39] V. TSEMEKHMAN and]. S. WETTLAUFER, Shocks preempt the AMS 47 (2000), 440-9. continuous curvature divergence in interface motion, [15] C. HERRING, Some theorems on the free energies of Phys. Rev. Lett. 87 (2001), 205701-1-4. crystal surfaces, Phys. Rev. 82 (1951), 87-93. [40] __ ,Shocking facets in interface growth,]. Cryst. [16] R. HIGHFIELD, The Physics of Christmas, Little, Brown Growth 235 (2002), 589-95. and Co., New York, 1998. [41] __ , Singularities, shocks and instabilities in in­ (17] D. A. KESSLER, ]. KOPLIK, and H. LEVINE, Pattern selec­ terface growth, Stud. Appl. Math. 110 (2003), 221-56. tion in fingered growth phenomena, Adv. in Phys. 3 7 [42]]. S. WETTLAUFER, M. ]ACKSON, and M. ELBAUM, A geometric (1988), 255-339. model for anisotropic crystal growth,]. Phys. A: Math. [18] G. P. KONNEN, Symmetry in halo displays and symmetry Gen. 27 (1994), 5957-67. in halo-making crystals, Appl. Opt. 42 (2003), 318-31. [4 3] ]. S. WETTLAUFER, Crystal growth, surface phase tran­ [19]]. S. LANGER, Instabilities and pattern formation in crys­ sitions and thermomolecular pressure, NATO ASI Se­ tal growth, Rev. Mod. Phys. 52 (1980), 1-28. ries, Vol. I 56, Ice Physics and the Natural Environment (20]]. S. LANGER and H. MDLLER-KRUMBHAAR, Theory of den­ (]. S. Wettlaufer,]. G. Dash, and N. Untersteiner, eds.), dritic growth-I. Elements of a stability analysis, Acta Springer-Verlag, Berlin, 1999. Metal/. 26 (1978), 1681-7. [44] J-J. Xu, Interfacial Wave Theory ofPattern Formation, [21] __ ,Theory of dendritic growth-II. Instabilities in Springer-Verlag, Berlin, 1998. the limit of vanishing surface tension, Acta Metal/. 26 [45] J-J. Xu and]. SHIMIZU, Asymptotic theory for disc-like (1978), 1689-95. crystal growth. I. Basic state solutions, Discrete Con­ [22] __ ,Mode selection in a dendrite-like nonlinear sys­ tin. Dynam. Systems Ser. B 4 (2004), 1091-116. tem, Phys. Rev. A 27 (1983), 499-514. [46] E. YOKOYAMA and T. KURODA, Pattern formation in snow crystals occurring in the surface kinetic process [23]]. S. LANGER, Existence of needle crystals in local mod­ and the diffusion process, Phys. Rev. A 41 (1990), els of solidification, Phys. Rev. A 33 (1986), 435-41. 2038-49. [24]]. S. LANGER and D. C. HoNG, Solvability conditions for dendritic growth in the boundary-layer model with cap­ illary anisotropy, Phys. Rev. A 34 (1986), 1462-71. [25] K. G.l.IBBRECHT, Cylindrically symmetric Green's func­ Simulating tion approach for modeling the crystal growth mor­ phology of ice, Phys. Rev. E60 (1999), 1967-74. (26] M. MARUYAMA, N. KURIBAYASHI, K. KAWABATA and ]. S. Snowflakes WETTLAUFER, Shocks and curvature dynamics: A test of Snowflake formation is very complicated, and one global kinetic faceting in crystals, Phys. Rev. Lett. 85 would not expect a purely theoretical account to (2000), 2545-8. explain all that occurs. There have been many at­ (27] H. MDLLER-KRUMBHAAR and]. S. LANGER, Theory of den­ dritic growth-III. Effects of surface tension, Acta Met­ tempts to simulate the process in software. Even all. 26 (1978), 1697-708. here, the complexities are overwhelming. The prin­ [28] __ , Sidebranching instabilities in a two-dimensional cipal focus of effort has been to understand the ex­ model of dendritic solidification, Acta Metal/. 29 (1981), traordinary six-fold planar symmetry of snowflakes 145-57. coexisting with a complicated dendritic structure. [29] W. W. MULLINS and R. F. SEKERKA, Morphological sta­ All simulation attempts seem to have been in are­ bility of a particle growing by diffusion or heat flow, stricted two-dimensional environment. ]. App/. Phys. 34 (1963), 323-9. The central problem is that the symmetry sug­ [30] __ , Stability of a planar interface during solidifi­ gests deterministic growth, the dendritic structure cation of a dilute binary alloy,]. Appl. Phys. 35 (1963), randomness. Of course some of the randomness 444-51. is contributed by a rapidly changing environment. (31] K. NAGASHIMA andY. FURUKAWA, Time development of But although the environment is uniform on the a solute diffusion field and morphological instability scale of a snowflake, the symmetry of real on a planar interface in the directional growth of ice crystals,]. Cryst. Growth 209 (2000), 167-74. snowflakes is not exact, and this also suggests that [32] P. S. STEVENs, Patterns in Nature, Atlantic-Little, Brown, local randomness on the snowflake is important. Boston, 1974. Other considerations suggest the role of ran­ [33] I. STEWART, Nature's Numbers, Basic Books, New York, domness. It is easy enough to construct dendrites 1995. in software that, at least locally, look somewhat like [34] _ _ , What Shape Is a Snowflake? W. H. Freeman & snowflakes. The simplest process in which this Co., New York, 2001. happens in a physical manner is diffusion-limited [35] W. TAPE, Folds, pleats and halos, Amer. Scientist aggregation or DLA, in which particles wander (1982), 467-74. around randomly and stick irreversibly onto a cen­ [36] W. TAPE and G. P. KONNEN, A general setting for halo tral, growing core whenever they hit it. theory, Appl. Opt. 38 (1999), 1552-625. [37]]. E. TAYLOR,]. W. CAHN, and C. A. HANDWERKER, Geo­ Diffusion-Limited Aggregation metric models of crystal growth, Acta Metal/. Mater. Particles were initially randomly distributed on a 40 (1992), 1443-74. [38] D. W. THOMPSON, On Growth and Form, Dover, New hexagonal lattice with a single particle in the cen­ York, 1992. tral core. The paths show the motion of particles

APRIL 2005 NOTICES OF THE AMS 415 at all scales. In snowflakes surface tension forbids this, and in any realistic simulation must be taken into account. The combined effect of surface ten­ sion and DLA is to establish a certain characteris­ tic length involved in the growth process. Up until about 1985 all attempts to simulate snowflakes built the symmetry in, by constructing one arm and then reflecting and rotating it. But then Nittman and Stanley showed that a truly random process could in fact give rise to a good approxi­ mate symmetry without artificial forcing. Their process was an extension of DLA in which sites next to the growing core freeze only when they en­ counter several diffusing free particles. This re­ quirement is already one that assures a certain stability in the process and leads to quasi-sym­ metry, but gives rise only to a very restricted range of snowflake shapes. In addition they introduced a somewhat artificial parameter that generates Diffusion-Limited Aggregation. Particles were thicker dendritic arms. With this, quasi-symmetry intitially randomly distributed on a hexagonal is maintained, and a wider range of shapes arise. lattice with a single particle in the central core. Unfortunately, their construction seems to have lit­ The paths show the motion of the particles tle to do with physics. wandering in randomly from the boundary of A much more physically realistic account was the region to attach to the core. that of Yokoyama and Kuroda (of Hokkaido Uni­ versity). Their simulations involved a fairly de­ wandering in randomly from the boundary of the tailed account of actual surafce motion of molecules region to attach to the core. in ice formation, as well as a detailed analysis of The characteristic feature of DLA is that bump how diffusion behaves at corners. Although ex­ growth is unstable-high curvature means a high tensive dendritic growth does not occur in their gradient in the distribution of particles, which are models, it does not seem too far away. therefore attracted to bumps. This instability com­ One recent attempt at imitating snowflakes has bined with microscopic randomness presumably been done with a 2D cellular automaton by Clifford does play a role in snowflake growth. Reiter. The only randomness built into his con­ struction is in the initial state, however, and al­ Diffusion Instability though some of his figures do resemble snowflakes, Bumps attract diffusing particles, which follow the it is hard to see what this process has to do with gradient of a certain potential field. As it grows the real life. gradient increases, so a bump grows unstably. References P. BALL, The self-made tapestry, ll0-127. ]. NITTMANN and H. E. STANLEY, ]. Phys. A 20 (1987), Lll85- 9l. C. REITER, Chaos, Solitons &Fractals 23 (2005), llll-lll9. E. Y oKOYAMA and T. K URODA, Phys. Rev. A 41 (1989), 203849. - Bill Casselman, Graphics Editor

Diffusion Instability. Bumps attract diffusing particles, which follow the gradient of a certain potential field. As it grows the gradient increases, so a bump grows unstably.

The results of DLA don't look much like snowflakes. One simple explanation for this is that dendrites formed in a DLA process are fractal, which means that they possess a similar structure

416 NOTICES OF THE AMS VOLUME 49, NUMBER 4 Mathematics of the Heavens Robert Osserman

Sponsored each April by the Joint Policy Board for Mathematics, Mathematics Awareness Month provides an opportunity to celebrate mathematics and its uses. The theme for Mathematics Awareness Month 2005 is "Mathematics and the Cosmos". In this article, the Notices ·reproduces three Mathematics Awareness Month "theme essays" written by Robert Osserman. Two other theme essays, plus a variety of resources including a Mathematics Awareness Month poster, are available on the website http: I jwww. math aware. org.

by the publication of Laplace's Mechanique Celeste in five volumes Mathematics from 1799 to 1825 and Poincare's Les Methodes Nouvelles de Ia and the Cosmos Mechanique Celeste in three volumes from 1892 to 1899. The mid-nineteenth century produced further important contribu­ Introduction and Brief History tions from Jacobi and Liouville, among others, as well as two The mathematical study of the cosmos has its roots groundbreaking new directions that were to provide the basic tools in antiquity with early attempts to describe the leading to the two revolutionary breakthroughs of twentieth-cen­ motions of the Sun, Moon, stars, and planets in pre­ tury physics; Hamilton's original approach to dynamics became cise mathematical terms, allowing predictions of the springboard for quantum mechanics and the general subject future positions. In modern times many of the of dynamical systems, while Riemann's 1854 "Habili­ greatest mathematical scientists turned their at­ tationsschrift" introduced curved spaces of three and more di­ tention to the subject. Building on Kepler's dis­ mensions as well as the general notion of an n-dimensional man­ covery of the three basic laws of planetary motion, ifold, thus ushering in the modern subject of cosmology leading Newton invented the subjects of "celestial me­ to Einstein and beyond. chanics" and dynamics. He studied the "n-body The twentieth century saw a true flowering of the subject, as problem" of describing the motion of a number of new mathematical methods combined with new physics and masses, such as the Sun and the planets and their rapidly advancing technology. One might single out three main moons, under the force of mutual gravitational at­ areas, with many overlaps and tendrils reaching out in multiple traction. He was able to derive improved versions directions. of Kepler's laws, one of whose consequences yielded First, cosmology became ever more intertwined with astro­ dramatic results just in the past decade when it was physics, as discoveries were made about the varieties of stars and used to detect the existence of planets circling their life histories, as well as supernovae and an assortment of other stars. previously unknown celestial objects, such as pulsars, quasars, The two leading mathematicians of the eigh­ dark matter, black holes, and even galaxies themselves, whose ex­ teenth century, Euler and Lagrange, both made istence had been suspected but not confirmed until the twenti­ fundamental contributions to the subject, as did eth century. Most critical was the discovery at the beginning of Gauss at the turn of the century, spurred by the dis­ the century of the expansion of the universe, with its concomi­ covery of the first of the asteroids, Ceres, on Jan­ tant phenomenon of the Big Bang, and then at the end, the recent uary 1, 1801. The nineteenth century was framed discovery of the accelerating universe, with its associated con­ jectural "dark energy". Robert Osserman is the special projects director of the Second, the subject of celestial mechanics evolved into that of Mathematical Sciences Research Institute in Berkeley. He also serves as chair of the Mathematics and the Cosmos dynamical systems, with major advances by mathematicians Advisory Committee for Mathematics Awareness Month G. D. Birkhoff, Kolmogorov, Arnold, and Moser. Many new dis­ 2005. His email address is ro@ms ri . or g. coveries were made about the n-body problem, both general ones, The author gratefully acknowledges the input of Myles such as theorems on stability and instability, and specific ones, such Standish, Martin Lo, Fabrizio Polara, Susan Lavoie, and as new concrete solutions for small values of n. Methods of chaos Charles Avis of the jet Propulsion Laboratory, and jerry theory began to play a role, and the theoretical studies were both Marsden of Cal tech. informed by and applied to the profusion of new discoveries of

APRIL 2005 NOTICES OF THE AMS 417 planets, their moons, asteroids, comets, and other rockets, artificial satellites, and space probes. On objects composing the increasingly complex struc­ the other hand, almost all of those space vehicles ture of the solar system. were equipped with scientific instruments for gath­ Third, the advent of actual space exploration, ering data about the Earth and other objects in our sending artificial satellites and space probes to solar system, as well as distant stars and galaxies the furthest reaches of the solar system, as well as going back to the cosmic microwave background the astronaut and cosmonaut programs for nearby radiation. Furthermore, the deviations in the paths study, transformed our understanding of the ob­ of satellites and probes provide direct feedback on jects in our solar system and of cosmology as a the gravitational field around the Earth and whole. The Hubble space telescope was just one of throughout the solar system. many viewing devices, operating at all wave lengths, Beyond these direct effects, there are many other that provided stunning images of celestial objects, areas of interaction between the space program and near and far. The 250-year-old theoretical discov­ mathematics. We list just a few: eries by Euler and Lagrange of critical points known • GPS: the global positioning system, as "Lagrange points" saw their practical application • data compression techniques for transmitting in the stationing of satellites. The 100-year-old messages, introduction by Poincare of stable and unstable • digitizing and coding of images, manifolds formed the basis of the rescue of • error-correcting codes for accurate transmis­ otherwise abandoned satellites, as well as the plan­ sions, ning of remarkably fuel-efficient trajectories. • "slingshot" or "gravitational boosting" for Finally, the biggest twentieth-century innova­ optimal trajectories, tion of all, the modern computer, played an ever­ • exploitation of Lagrange points for strategic increasing and more critical role in all of these ad­ placement of satellites, vances. Numerical methods were applied to all • dynamical systems methods for energy­ three of the above areas, while simulations and com­ efficient orbit placing, puter graphics grew into a major tool in deepen­ • finite element modeling for structures such as ing our understanding. The ever-increasing speed spacecraft and antennas. and power of computers went hand-in-hand with Some of the satellites and space probes that the increasingly sophisticated mathematical meth­ have contributed to cosmology and astrophysics are ods used to code, compress, and transmit messages • the Hubble space telescope, and images from satellites and space probes span­ • the Hipparcos mission to catalog the posi­ ning the entire breadth of our solar system. tions of a million stars to new levels of accuracy, • the COBE and WMAP satellites for studying the General References cosmic microwave background radiation, (1) DENNIS RICHARD DANIELSON, The Book of the Cosmos: • the Genesis mission and SOHO satellite for Imagining the Universe from Heraclitus to Hawking, studying Perseus, Cambridge, 2000. the Sun and solar radiation, [2) BRIAN GREENE, The Fabric of the Cosmos. Space, Time, • the ISEE3/ICE space probe to study solar flares and the Texture of Reality, Alfred A. Knopf, Inc., New and cosmic gamma rays before going on to visit the York, 2004. MR2053394. Giacobini-Zimmer comet and Halley's comet, [3) THOMAS S. KuHN, The Copernican Revolution: Planetary • the LAGEOS satellites to test Einstein's pre­ Astronomy in the Development of Western Thought, diction of "frame dragging" around a rotating body. Harvard University Press, Cambridge, 1957. Rather than trying to cover all or even most of [4) ROBERT OsSERMAN, Poetry of the Universe: A Mathemat­ the mathematical links, we focus on two that ical Exploration of the Cosmos, Anchor/ Doubleday, are absolutely essential and central to the whole New York, 1995. endeavor: first, navigation and the planning of [5) GEORGE POLY A, Mathematical Methods in Science, rev. ed. (Leon Bowden, ed.), New Mathematical Library, Vol. 26. trajectories; and second, communication and the Mathematical Association of America, Washington, transmission of images. DC, 1977. MR0439511 (55:12401) [6) STEVEN WEINBERG, The First Three Minutes, A Modern View Navigation, Trajectories, and Orbits of the Origin of the Universe, 2nd ed., Basic Books, 1993. When the U.S. space program was set up in earnest-a process described in detail in the recent History Channel documentary Race to the Moon­ Space Exploration a notable feature was the introduction of the Starting in the twentieth century, the mathemati­ Mission Control Center. The first row of seats in cal exploration of the cosmos became inextricably mission control was known as "the trench", and it entwined with the physical exploration of space. On is from there that the mathematicians whose spe­ one hand, virtually all the methods of celestial me­ cialty is orbital mechanics kept track of trajecto­ chanics that had been developed over the centuries ries and fed in the information needed for were transformed into tools for the navigation of navigation. Their role is particularly important for

418 NOTICES OF THE AMS VOLUME 52, NUMBER 4 operations involving rendezvous between two vehicles, in delicate operations such as landings on the Moon and in emergencies that call on all their skills, the most notable of which was bringing back alive the crew of Apollo 13 after they had to aban­ don the command module and were forced to use the lunar landing module-never designed for that purpose- to navigate back to Earth. The first thing that an astronaut or former astronaut will tell you about navigating in a space­ ship is that no amount of experience piloting a plane will be of any help. On the contrary, previ­ ous experience may be a hindrance, since it reinforces one's natural intuition that if you want to catch up with an object ahead, you go faster, and conversely. But if you are orbiting at a certain speed and have to rendezvous with something ahead, then "stepping on the accelerator" (translate as "applying a forward thrust") will lift you into a higher orbit where first of all, the vertical distance between you and the object orbiting ahead will increase, and second, your average angular veloc­ ity will decrease, by Kepler's third law, and you will An all-sky image of the infant universe, 380,000 find yourself getting further and further behind. years after the Big Bang. In 1992, NASA's COBE mission In fact, the only way in practice to effect a ren­ first detected tiny temperature fluctuations (shown as dezvous and docking maneuver is to feed the data color variations) in the infant universe, a landmark discovery. on both vehicles into a computer and apply the The WMAP image brings the COBE picture into methods of orbital mechanics to plan a trajectory sharp focus. The new, detailed image provides firm answers that brings both vehicles to the same place at the to age-old questions. same time at essentially the same velocities. Math­ ematically, such maneuvers are best described by working in phase space, where each point has six gradual reduction of weight as fuel is used up, or coordinates: three describing its position and three in cases of relativistic speeds, the force is given by describing components of its velocity vector. One' the first derivative of momentum, but the princi­ must find paths for the two vehicles that come ple is the same. together in phase space. In the case of the 2-body problem, where the only The ability to navigate started with Isaac New­ force involved is the gravitational attraction be­ ton. Not only did he formulate his laws of motion tween the two bodies, it is frequently said that and of gravity, but he also developed the calculus Newton was able to give a complete solution. That which allowed him to put those laws into the is not, strictly speaking, the case, if one means by language of mathematical equations. Today, our "a solution" of a differential equation, an expres­ knowledge of the physics involved has been sion for the unknown function whose derivatives improved with the addition of relativity and other appear in the equation. In this case, it would mean factors that may play a role, and calculus has been finding an expression for the position as a func­ further developed into many different branches of tion of time. However, what Newton showed was mathematics. that the orbit of each of the bodies lies on a conic What kind of mathematics? Newton's equations section (in a fixed inertial frame of reference), and involve the gravitational forces acting upon one of in the case considered by Kepler, where the orbit the participating bodies, arising from all of the is an ellipse, there is an explicit expression for other bodies. Since force is mass times acceleration the time as a function of the position. Commonly and since acceleration is simply the second deriv­ known as "Kepler's equation", it is of the form ative of position with respect to time, it is the t = x - e sin x, in suitable units of time t, where x differential calculus that describes the accelera­ is the polar angle from the center of the ellipse and tions. Then, once the accelerations are given, it is e is the eccentricity. What one wants, of course, is necessary to use integral calculus in order to get x as a function oft, and much effort and ingenu­ from the second derivatives to the positions. ity have gone into finding effective means of solv­ In a more general context, where the mass may ing Kepler's equation for x in tenns oft. Lagrange be c hanging with time, such as happens with an did extensive work on the problein, in the course extended application of thrust to a vehicle, with the of which he developed both Fourier series and

APRIL2005 NOTICES OF THE AMS 419 Bessel functions, named after later mathemati­ will require more fuel to correct the trajectory once cians who investigated these concepts in greater the spacecraft starts approaching its final target. detail. Both Laplace and Gauss made major con­ One of the mathematical tools used to optimize tributions, and succeeding generations continued some feature of a flight trajectory, such as fuel con­ to work on the subject. sumption or flight time, is a maximum principle When there are more than two bodies involved, introduced by Pontryagin in 1962. Pontryagin's the problem cannot be solved analytically; instead, theorem characterizes the optimum values of the integration (positions from accelerations) must certain parameters, called the controllers, that be done numerically: now, with high-speed com­ determine a trajectory. puters. So, numerical integral calculus is a major In recent decades, ingenious new methods have factor of spacecraft navigation. been developed to extract the maximum effect One may picture navigation as being the mod­ from the least amount of fuel. One such method eling of mother nature on a computer. At some is known as the "slingshot" or "gravity-assisted time, with the planets in their orbits, a spacecraft trajectory". By aiming a space vehicle in a way that is given a push outward into the solar system. Its crosses the orbit of another planet or moon just subsequent orbit is then determined by the gravi­ behind that body, the path of the vehicle will be tational forces upon it due to the Sun and planets. deflected, sending it on its way to the next target We compute these, step-by-step in time, seeing with minimum expenditure of fuel. A number of how the (changing) forces determine the motion of space probes, such as Cassini-Huygens, have ben­ the spacecraft. This is very similar to what one efited from carefully calculated trajectories that may picture being done in nature. make multiple use of the slingshot effect. Gravity­ How does one get an accurate orbit in the com­ assist methods are equally important for sending puter? The spacecraft's orbit is measured as it pro­ a probe toward the inner planets: Venus and Mer­ gresses on its journey, and the computer model is cury. In that case, one sends the probe to a point adjusted in order to best fit the actual measure­ on the orbit just in advance of the body it is pass­ ments. Here one uses another type of calculus: ing. In both cases, the effect of the fly-by is to alter estimation theory. It involves changing the initial the velocity, changing the direction of flight and "input parameters" (starting positions and veloci­ leaving the end speed relative to the body it is pass­ ties) in the computer in order to make the "output ing unchanged. However, that body will be moving parameters" (positions and velocities at subse­ with considerable quent times) match what is being measured: momentum relative to the Sun, and there will be adjusting the computer model to better fit reality. an exchange of momentum in Also in navigation, one must "reduce" the which the body will be slowed down or speeded up measurements. Usually, the measurements do by an infinitesimal amount, while the probe will be not correspond exactly with the positions in the speeded up or slowed down by a considerable computer; one must apply a few formulae before amount relative to the Sun. a comparison can be made. For instance, the More modern, twentieth-century mathematical positions in the computer represent the centers of methods of dynamical systems have proved mass of the different planets; a radar echo, invaluable in designing complicated fuel-efficient however, measures the path from the radio an­ orbits. These methods include the theory of stable tenna to the spot on a planet's surfaces from which and unstable manifolds, pioneered by Poincare, the signal bounces back to Earth. This processing leading to the subject now known as chaotic involves the use of trigonometry, geometry, and dynamics, and the KAM theory, due to Kolmogorov, physics. Arnold, and Moser, of invariant tori and stability. Finally, there is error analysis, or "covariance" One of the first achievements of the new methods calculus. In the initial planning stages of a mission, was the 1991 rescue of a Japanese spacecraft Hi ten one is more interested in how accurately we will that was stranded without enough fuel to com­ know the positions of the spacecraft and its tar­ plete a planned mission when a second satellite, get, not in the exact positions themselves. With low intended to work in tandem with the first, failed accuracy, greater amounts of fuel are required, to operate. The mathematician Edward Belbruno and it could be that some precise navigation would had designed highly fuel-efficient orbits using not even be possible. Covariance analysis takes methods derived from chaos theory, and that into account (1) what measurements we will have turned out to be just what was needed for this of the spacecraft: how many and how good, (2) rescue operation. Belbruno's methods were incor­ how accurately we will be able to compute the porated into the design used by Giuseppe Racca and forces, and (3) how accurately we will know the the European Space Agency in sending "SMART-1", position of the target. These criteria are then used their first satellite to the Moon, in 2004. Newspa­ in order to determine how closely we can deliver per headlines trumpeted "spacecraft reaches moon the spacecraft to the target. Again, poor accuracy on 5 million miles a gallon" as a dramatic way of

420 NOTICES OF THE AMS VOLUME 52, NUMBER 4 underlining the astonishing fuel efficiency of the be added to all the others to restore the original method. content. For a large data stream, techniques such In the past few years, other applications of the as this can save hours of transmission time and theory of stable and unstable manifolds have been much storage capacity. invoked in trajectory planning. Martin Lo and his "Entropy coding" is a technique that takes into colleagues at the Jet Propulsion Laboratory devel­ account the probability distribution of different sets oped ways to apply the theory in order to place satel­ of data in order to encode more probable data with lites such as Genesis in an orbit around the Lagrange shorter sequences, just as in Morse code where point between Earth and the Sun and then return the letter "E" is represented by a single dot. Image it to Earth. More recently, the same group, together data compression techniques rely on mathemati­ with Jerry Marsden at Caltech, have expanded the cal image probability models that exploit the method for use in interplanetary travel, along what similarities between neighboring small picture they call the "interplanetary superhighway", a route elements to minimize the number of bytes needed derived from the ever-changing configurations of to describe the image. stable and unstable manifolds in the phase space The possibility of detecting and even correcting of our solar system or selected parts of it. A beau­ errors in transmission was first pointed out in a tifully illustrated article by Douglas Smith de­ groundbreaking paper of Richard Hamming in scribing this work can be found in the journal 1950. Since then, an entire field has grown up in Engineering and Science. which, on one hand, ever more refined methods Communication and Image Transmission have been devised for practical applications, and on the other hand, the theory of error-correcting For space exploration and interplanetary probes, codes has turned out to have fascinating links with navigational techniques and orbital mechanics may sphere packing and simple groups, beautifully get you where you want to go, but it is not worth described in the book of Thomas Thompson. much if the data collected cannot be successfully More recent mathematical innovations that have transmitted back to Earth. In the case of the Cassini spacecraft at Saturn, signals have to travel dis­ proved to be of both theoretical interest and great tances on the order of a billion miles or more. Data practical use in this connection are the subjects transmitted from across the planetary system with of fractals and wavelets. An excellent survey of all very limited power are received on Earth as a very aspects of image analysis, transmission, and · faint signal (as low as a billionth of a billionth of reconstruction is the theme essay on Mathematics a watt) embedded in noise. Only through miracles and Imaging on the 1998 Mathematics Awareness of modern technology operating in tandem with Week website. ever-improving mathematical methods is one able to receive the striking and detailed images that References are now on display. Trajectories, Orbits, and Space Navigation Two critical processes come into play for trans­ [1) RicHARD A. BATTIN, An Introduction to the Mathematics mitting messages of all sorts. The first is com­ and Methods ofAstrodynamics, American Institute of Aeronautics and Astronautics, New York, 1987. pression, to be able to transmit the maximum [2) V. A. BRUMBERG, Essential Relativistic Celestial Mechan­ amount of information with the least number of ics, Section 5.1: Equations of motion of Earth's artifi­ bits, and the second is the use of error-correcting cial satellites, Adam Hilger, Bristol, 1991. codes, to overcome problems of noise and distor­ [3) PETER CoLWELL, Solving Kepler's Equation over Three tion. Basically, one wants to eliminate redundancy Centuries, Whitman-Bell, Richmond, VA, 1993. to obtain compression of the data, and then one (4] ]AMES CASE, Celestial mechanics theory meets the has to introduce redundancy in order to catch and nitty-gritty of trajectory design, SIAM News 3 7 (2004), correct errors of transmission. The two operations 1-3 (book review). may at first seem to cancel each other out, but in (5) EDWARD BELBRUN O, Capture Dynamics and Chaotic fact the types of redundancies involved in the two Motion in Celestial Mechanics: With Applications to cases are quite different. the Construction of Low Energy Transfers, Princeton A variety of mathematical techniques are used University Press, Princeton, NJ, 2004. to compress the spacecraft data into fewer bits prior [6) GIUSEPPE D. RAccA, New challenges to trajectory design to transmission to the ground. A simple one avoids by the use of electric propulsion and other new means of wandering in the solar system, Celestial Mech. transmitting all sixteen bits of every data element Dynam. Astron. 85 (2003), 1-24. of a data stream. The value of the first data element [7) V. SzEBEHELY, Theory of Orbits, Academic Press, New is sent, but for the rest of the elements, only the York, 1967. difference from the first is sent. The value of the [8) DouGLAS L. SMITH, Next exit 0.5 million kilometers, first element might require sixteen bits, but the dif­ Engr. and Sci., No. 4 (2002), 6-14. http: I I ferences are so small they might only need two or pr.caltech.edulperiodicalsiEandSI three bits. Once on the ground the first value can articlesiLXV41LoMarsden%20Feature.pdf.

APRIL 2005 NOTICES OF THE AMS 421 This sequence of images shows one of the first 3D simulations, carried out by members of the European Union Training Network "Sources of Gravitational Waves". The sequence shows two spinning black holes merging in the final stages of a rapidly decaying orbit. The leftmost image shows the two individual black holes about to merge. The individual horizons are shown in the center of the image (the larger black hole is just above a smaller one). The developing burst of radiation is shown shooting out towards the upper left direction and to the lower right. The final image on the right shows the final black hole, with the two original black hole horizon surfaces still seen inside, and it also shows the developing and intensifying burst of gravitational waves. Images by Ed Seidel and Werner Benger of the Albert-Einstein-lnstitut, the Zuse-lnstitut-Berlin, and the Center for Computation & Technology at Louisiana State University. Reproduced with permission of the European Union Training Network "Sources of Gravitational Waves".

Websites the universe was demonstrated to the satisfaction jet Propulsion Laboratory: http: I /www. nasa. gov I of nearly all. Einstein was able to use general rela­ centers/jpl/home/index.html. tivity on the one hand to explain earlier observa­ Martin Lo main page: http: I /www. gg. cal tech. edu/ tions, such as the amount of precession of the -mwl/. planet Mercury, and on the other hand to make new Gravity-assisted trajectories: http: I /saturn. jpl . nasa. predictions for the observers to confirm or refute. gov/mission/gravity-assist-primer2.cfm. The first and most widely heralded of those was DVDs and Videos the prediction of the bending of light as it passed· The Race to the Moon, The History Channel, 2004. close to a large mass such as the Sun. Others, such Ring World: Cassini-Huygens Mission to Saturn and Its as gravitational red-shift, gravitational lensing, and Moons, NASA, ]PL 400-1114, 11/ 03. "frame-dragging" around a rotating body, were Mathematics of Space-rendezvous, NASA, ]SC 1801. confirmed one by one over the course of the cen­ Space Flight: Application of Orbital Mechanics, NASA, tury. Still others, like the existence of gravity waves, CMP 277. remain a high priority for twenty-first-century Communication and Image Transmission experimentalists. THOMAS M. THOMPSON, From error-correcting codes through The Nobel Prize-winning work of Russell Hulse sphere packings to simple groups, Mathematical Associ­ and Joseph Taylor stemmed from their discovery ation of America, Washington, DC, 1983. in 19 7 4 of a pulsar whose "pulses" varied in a them to conclude that it Mathematics Awareness 1998 Theme Essay, Mathematics regular fashion, leading andirnaging:http://www.mathaware.org/mam/98/ had an invisible companion, the pair forming a articles/theme.essay.html. familiar binary system each one circling the other (or actually, their common center of gravity) in an approximately elliptical orbit. In this case, the p·air From Black Holes to Dark consisted of two bodies each as massive as the Sun, but compressed into a tight ball whose diameter Energy: Cosmology in the was the size of a small town, and each c0mpleted its orbit around the other in about eight hours. Twenty-first Century Under such extreme conditions, the relativistic Cosmology in the twentieth century was almost in effects would be considerable. One of those its entirety the outgrowth of Einstein's founda­ effects would be the production of gravity waves, tional paper in 1915 on general relativity. Two and Einstein's equations predicted that those waves years later he presented his first model of the would radiate energy in a way that causes the two universe based on general relativity together with bodies to gradually get closer, which would in turn Riemann's notion of the three-sphere. speed up the rate that they completed each orbit Side-by-side with the theoretical advances, ob­ by a very precise amount. After observing the servational astronomy led to great leaps in astro­ variations in the pulse rate over a period of four physics, as the life cycles of stars were discovered years, Hulse and Taylor were able to show that . and elaborated, the existence of other galaxies out­ the speed-up was indeed taking place at the rate side our own was confirmed, and the expansion of predicted, to within less than 1 percent deviation.

422 NOTICES OF THE AMS VOLUME 52, NUMBER 4 That provided the first experimental evidence for launched in 1978, was an X-ray telescope that was the existence of gravity waves. able to make X-ray images as sharp as an optical That evidence, however, was indirect. In fact, the telescope. Gradually, the scientific community strength of the predicted waves in the case of the became convinced that Cyg X-1 was indeed a real­ binary pulsars was far too weak for any hope of di­ life black hole whose physical characteristics rect detection on Earth. However, the same general corresponded closely to those predicted by the principles would apply to a binary pair of black theory. Evidence has accumulated for other X-ray holes, and there the calculations indicated that the sources arising from the vicinity of black holes, as strength of the waves could be just within the lim­ well as black holes in the center of quasars and its of possible detectability with suitably crafted ap­ large galaxies, such as our own. paratus. Attempts at detection had already begun One tool that has become increasingly more im­ in the late 1950s with Joseph Weber. At that time, portant in the study of black holes as in the rest not only was the reality of gravity waves in doubt, of astronomy and cosmology has been computer but the existence of black holes was generally simulations. By 2001, such simulations were able greeted with skepticism. to predict the nature of the gravitational waves The idea of black holes (although not the name) that we might be able to detect from the collision arose very soon after Einstein formulated general and merging of two black holes and to display the relativity. Karl Schwarzschild, despite the fact that results in dramatic images. The reality of black he was in the German army stationed in Russia holes and, in particular, their role in the produc­ and that it was in the midst of World War I, read tion of gravity waves are now widely enough Einstein's paper and almost immediately was able accepted that large amounts of money are being to solve Einstein's equations for the case of the invested in experimental devices, such as the UGO gravitational field surrounding a (nonrotating) project, to detect associated gravity waves. spherically symmetric body. A few weeks later he Given the extent to which theoretical predic­ was able to solve the equations and describe the tions about black holes appear to be confirmed by space-time curvature in the interior of the body. One observations, why the continued hesitancy about of the consequences of the Schwarzschild solu­ their wholehearted acceptance? One answer is that tion seemed to be that a sufficiently massive body each of the predictions is based on certain simpli­ compressed within a sufficiently small radius fying assumptions and continuing unknowns. For (where "sufficiently" was made precise by the example, the early models were for a spherically Schwarzschild equations) would have the property symmetric nonrotating body, whereas physical that no radiation or matter could ever escape. Oddly, a very similar conclusion was reached by reality almost certainly corresponds to rotating purely Newtonian methods in 1783 by John Michell bodies with concomitant bulges at the equator. in England and became widely known through But most importantly, what was missing in the Laplace's famous five-volume Le Systeme du Monde. early studies and what is conspicuously absent in In both cases, however, the question remained the above discussion is the central role of quantum whether it was possible for a real-world physical effects. That, however, would lead us far afield body to exist within those parameters. The first and can be found in many of the references given theoretical evidence was adduced in a 1939 paper below. Instead, we indicate briefly two further sub­ by Robert Oppenheimer and Hartland Snyder, who jects of particular mathematical interest. calculated the space-time geometry around an In 1953, a young differential geometer, Eugenio imploding massive star, under certain simplifying Calabi, made a study of complex manifolds and was assumptions, and concluded that the star would led to conjecture that under very general conditions eventually become invisible. there should be a metric on each manifold of a par­ As for the reality of black holes, it was hard for ticularly symmetric nature. This Calabi conjecture the experts, much less the general public, to decide was a subject of great interest and was finally whether they represented science or science fiction. proved in 1977 by Shing-Tung Yau. Although of con­ Many leaders in the field, from Einstein to John siderable mathematical interest, the Calabi-Yau Wheeler, had serious doubts. It was not until the manifolds, as they came to be known, had no ob­ advent of X-ray astronomy that the balance was vious connection to cosmology until the advent of tilted in favor of science. Since X-rays from outer string theory introduced a whole new dimension­ space do not penetrate our protective atmosphere, or, more precisely, set of dimensions-into play. this research developed hand-in-hand with rocket What the theory required was that, in addition to science. The big discovery was the existence of a our familiar four-dimensional space-time, there powerful X-ray source in the constellation Cygnus, would be six additional "curled-up" space dimen­ designated Cyg X-1. This discovery was made in a sions. Furthermore, the equations of string theory rocket flight in 1964. The first X-ray satellite, Uhuru, imply that this six-dimensional component must was launched in 1970, while its successor, Einstein, have a very particular structure, and in 1984 it

APRIL 2005 NOTICES OF THE AMS 423 was proved that the Calabi-Yau manifolds have suggesting observations that might be made precisely the structure needed. and what to look for in those observations. In The gift of mathematics to physics provided by others, the results of the observations have forced Calabi-Yaumanifolds was amply repaid when physi­ theorists to rethink some of their fundamental cists discovered what was termed "mirror sym­ assumptions. One of the most. striking examples metry" between pairs of geometrically distinct but along those lines was the discovery in 1998, in the physically linked pairs of Calabi-Yau manifolds. course of examining a certain class of supernovae, Using this link, Philip Candelas and his collabora­ that the expansion of the universe, rather than tors were able to suggest precise numerical an­ slowing down under the restraining force of grav­ swers to a problem in algebraic geometry that had ity, appeared.to be speeding up as a result of some seemed far beyond the capabilities of any known mysterious, hitherto undreamed of force, dubbed method to provide: the number of rational curves "dark energy". One immediate thought was that this of given degree on a fifth-degree algebraic hyper­ was due to Einstein's notorious "cosmological con­ surface in projective four-space. Those numbers stant". But even if it worked mathematically, that that algebraic geometers were able to calculate di­ would be no more of a physical explanation than rectly confirmed the predictions arising from when Einstein originally inserted it into his equa­ physics. tion for what turned out to be the wrong reason: The other circle of ideas involves what is known his equations seemed to imply that the universe was as "curvature flow". The simplest example con­ expanding or contracting, rather than static in sists of starting with a smooth closed curve in the time, and this was just before the realization that plane and defining a "flow" by moving the curve the universe actually was expanding. The search in a direction orthogonal to itself at each point for a satisfactory explanation of dark energy is and at a speed proportional to the curvature at the sure to occupy a central place in the mathematics point. Intuition suggests that the curve should be­ of cosmology for some time to come. come progressively more circular. In 1986, Michael Gage and Richard Hamilton were able to prove the General References result, starting from an arbitrary convex curve and [1] MARCIA BARTUSIAK, Einstein's Unfinished Symphony: normalizing the flow to fix the area enclosed. In a Listening to the Sounds of Space-Time, Berkley Pub­ rather different situation, Hamilton was led to lishing, 2003. define and study a "Ricci flow" on an arbitrary [2] BRIAN GREENE , The Elegant Universe: Superstrings, Riemannian manifold, in which the rate of change Hidden Dimensions, and the Quest for the Ultimate of the metric tensor is proportional to the Ricci Theory, W. W. Norton, New York, 1999. tensor. After some rather spectacular successes [3] RoGER PENROSE, The Road to Reality: A Complete Guide in which Hamilton was able to use his method to to the Laws of the Universe, Knopf, New York, 2005. prove that under certain assumptions such a [4] SAUL PERLMUTTER, Supernovae, dark energy, and the ac­ flow tended toward a constant curvature metric, celerating universe, Physics Today, April2003, 53-60. [5] JosEPH SILK, The Big Bang, revised and updated ed., Grigori Perelman announced in 2003 that he Freeman, New York, 1989. used extensions of the method to give complete [6] GEORGE SMOOT and KEAY DAVIDSON, Wrinkles in Time, proofs of Poincare's conjecture and the Thurston William Morrow, New York, 1993. "Geometrization Conjecture". Perelman's proof is [7] K1P S. THORNE, Black Holes and Time Warps: Einstein's still under review by the mathematical community Outrageous Legacy, W. W. Norton, New York, 1994. before being fully endorsed by the experts. Possi­ [8] EDWARD WITTEN, Reflections on the fate of spacetime, ble cosmological implications relate to character­ Physics Today, April1996, 24-30. izing shapes of three-dimensional manifolds that may constitute the universe as it evolves in time. Technical References In another direction, the curvature flow for [1] HUBERT L. BRAY, Black holes, geometric flows, and the curves was generalized to higher-dimensional Penrose inequality in general relativity, Notices A mer. hypersurfaces, leading to a proof of the "Rie­ Math. Soc. 49 (2002), 1372-81. mannian Pemose inequality", first by Huisken and [2] !GNAZIO CIUFOLINI and ]OHN A. WHEELER, Gravitation and Ilmanen in 1997 for a single black hole and then Inertia, Princeton University Press, Princeton, NJ, 1995. for multiple black holes in 1999 by Hubert Bray. [3] DAVID A. Cox and SHELDON KATZ, Mirror Symmetry and Roger Pemose was led to the inequality in 1973 by Algebraic Geometry, Amer. Math. Soc., Providence, RI, a physical argument about the nature of black 2000. [4] CHARLES W. MISNER, K1P S. THORNE, and ]OHN A. WHEELER, holes. Gravitation, W. H. Freeman, New York, 1973. It need hardly be said that the number of [5] BARRETT O'NEILL, Semi-Riemannian Geometry, with topics touched upon here represents a minuscule Applications to Relativity, Academic Press, New York, portion of the activity in recent decades in astro­ 1983. physics, cosmology, and related parts of mathe­ [6] P. ]. E. PEEBLES, Principles ofPhysical Cosmology, Prince­ matics. In some cases, theory has led the way, ton University Press, Princeton, NJ, 1993.

424 NOTICES OF THE AMS VOLUME 52, NUMBER 4 The Open Computer Algebra System

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roup cohomology has roots that go back Another example is the cohomology ring of a to the early part of the last century. The semidihedral 2-group. The semidihedral group of Gtopic has played a significant role in order 16 has the form several mathematical areas such as num­ G = (g,hlg8 = h2 = 1,hgh = g 3 ) . ber theory, algebraic topology, and rep­ resentation theory. Yet only in the last decade, with Again, if k is a field of characteristic 2, then its co­ the aid of modern computers, have many examples homology ring has the form been calculated. The results of the calculations H*(G,k) = k[z ,y,x, w]/(z3, zy, zx, x2 + y 2w). have provided insight into theoretical develop­ ments, and they have stimulated interest in a whole It is interesting to notice in this case that the struc­ new collection of issues involving the structure of ture of the cohomology ring is independent of the cohomology rings. order of the group. All semidihedral2-groups have isomorphic mod 2 cohomology rings. The cohomology ring H*(G, k) of a finite group Other examples are more complicated. There is G is a finitely generated algebra over its coeffi­ a group of order 128 whose cohomology ring (with cients. We usually write it using generators and coefficients in a field of characteristic 2) has the relations, as in form H*(G, k) = k[z, y, x]/(zy). H*(G, k) = k[z, y, x, w, v, u, t, s] /1 This example is the cohomology ring of a dihedral where 1 is the ideal generated by the polynomials 2-group in the case that the coefficient ring k is a zy, y 2 , z 3 + yx, zx + yx, yw, z 2 v + zt, field of characteristic 2. It is the quotient of the poly­ nomial ring k[z .y .x] by the ideal generated by zy. Hence it looks like the coordinate ring of the union of two planes (the z -x plane and the y-x plane) that intersect in a line. and ]on F. Carlson is professor of mathematics at the zwu + zvu + zvt + z 2s + u2 + ut + t 2 • University of Georgia, Athens. His email address is You have probably guessed that this last exam­ jfc@math. uga .edu. ple is a computer calculation. The wonders of mod­ This research was supported in part by a grant from ern technology have permitted us to generate data the NSF. on a scale that we could only have dreamed about The author thanks Lisa Townsley for suggestions on thirty years ago. The output of other computer­ this paper. generated examples can be found in the the book

426 NOTICES OF THE AMS VOLUME 52, NUMBER 4 [10] or on the web page http: I /www. math. uga. Properties of Cohomology Rings edu/-1 val ero/cohoi ntro. html. Experimental To begin, we outline some of the basic structure evidence has aided the development of the subject, of cohomology rings. This structure is measured and it has also provoked some new problems. In with notions such as dimension, depth, and prime this paper we discuss a few of the problems, show ideal spectra. We review some of these ideas in the why they are important in a more general mathe­ discussion. matical setting, and explore the connections to Graded Commutativity and Elementary Abelian other areas of algebra and mathematics. In partic­ Groups ular, the cohomology rings seem to have some very A cohomology ring H*(G, k) is a k-algebra and is special properties which illustrate some basic graded in the sense that it is a direct sum concepts of commutative algebra. The motivation for considering the cohomol­ H*(G, k) = L Hn(G, k) ogy rings of finite groups comes from two very n 2, then the definition almost exactly. the graded commutativity requires that the poly­ In this exposition we delay the actual definition nomial generators be in degree 2. Hence, H*(G, k) of the cohomology ring until the discussion of the has the form computations in the section on Computing Group Cohomology. Instead, we concentrate on the prop­ H*(G,k) ~ k[yl,····Ynl ® A(zl, ... ,Zn) erties of the cohomology rings and how these prop­ where i\ is an exterior algebra. The generators zi erties reflect the properties of the group. The dis­ are in degree 1, while the Yi occur in degree 2. The cussion ends up being something of a primer in elements zi are in every prime or maximal ideal of certain aspects of modern commutative algebra. H*(G, k), and therefore, they are contained in the Both the problems that we encounter and the ques­ Jacobson radical RadH*(G, k) which is the inter­ tions that we want to answer are expressed in the section of all maximal ideals. It follows that language of commutative algebra and algebraic H*(G, k)/ RadH*(G, k) ~ k[y1 , ••• , Ynl is a polyno­ geometry. In addition, some of the tools we employ mial ring even in the case where p is odd. to analyze the experimental data are developments Functorial Properties of modern commutative and noncommutative al­ Group cohomology is a functor. This is a compli­ gebra. cated way of saying that homomorphisms of groups Throughout the paper, G denotes a finite group induce homomorphisms of group cohomology. and k a field. By a ''cohomology ring" we mean a In particular, suppose that H <;::; G is a subgroup. mod p cohomology ring in which the field k has Then there is a homomorphism induced from the finite characteristic p. Moreover, if G is a p-group inclusion called the restriction, for some prime p, then it should be assumed that the field k has characteristic p. resc,H : H*(G, k) ~ H*(H, k).

APRIL 2005 NOTICES OF THE AMS 427 Moreover, if His a normal subgroup of G, then the cyclic center generated by the element e = (g h)2 and quotient map G ~ G 1H induces a homomorphism two maximal elementary abelian subgroups, called the inflation, E = (g, e) and F = (h, e) . The cohomology rings of these two groups can be given as infg,H: H* (GIH,k) ~ H * (G,k). H*(£, k) :: k[u, v] :: H*(F, k). These maps have many nice features. For one, each is transitive. That is, for example, if H c:; K c:; G, The restriction maps have the following form (with then resK.H o resc,K = resc,H· In addition, both the suitable choice of the variables). The restriction restriction and inflation maps are homomorphisms resc,E sends z, y, x to 0, u, v(u + v), respectively, of graded commutative k -algebras. That is, they are while resc.F sends z,y,x to u,O, v(u + v). The in­ linear in k and preserve both the gradings and tersection of the kernels of the restrictions is thus product structures. the ideal (zy) = 1 c:; P = k[z , y, x]. Hence there is no In general, the inflation map can have a large element in H*(G, k) =P 11 that is in the kernel of kernel and may be very far from being surjective. both of these restrictions. The variety of H*(G, k) Likewise, a restriction map may have a large is union of the z-x and y-x planes in k3 . The maps kernel. However, restriction maps must be nearly on varieties have the form surjective. The way we say this is that H*(H, k) must be finitely generated as a module over the res ~ .E(a, b) = (0, a, b(a +b)), image resc,H(H* (G, k)) . res~ .F(a, b) = (a, 0, b(a + b)), Varieties and Quillen's Theorem and hence we can calculate that Vc(k) is the union For the purposes of this section we assume that k images is algebraically closed. A very important theorem of the in group cohomology says that the cohomology Vc(k) = res ~. E(VE (k)) u res ~.F(VF(k)) . rings are finitely generated. The theorem was proved independently by Evens [14] and Venkov In fact, this is the situation in general. The follow­ [21] more than forty years ago. The theorem tells ing theorem was proved by Quillen more than us in the case p = 2 that H*(G, k) =P 11 where Pis thirty years ago. a polynomial ring in a finite number of variables Theorem. Let G be any finite group. Then and 1 is an ideal in P. When p is odd, the theorem implies that H* (G, k) I Rad H*( G, k) (which is a com­ Vc(k) = U res ~.E(VE(k) ) mutative ring) can be written asP 11. This means EE:M'E.Jt that cohomology rings are noetherian and the max­ where the union is over the set Jvl'E .Jl.. of all max­ imal ideals can be classified by varieties. If P is a polynomial ring in n generators, then the variety imal elementary abelian p-subgroups of G. of P 11 is simply the set of simultaneous zeros of Dimension 1. that the polynomials that generated In the case In general, the dimension of a ring or a variety is a G is an elementary abelian subgroup of order pn fairly complicated notion. However, with Quillen's as above, the variety of H* (G, k) is the affine space theorem in view, calculating the dimension of a kn. We denote the variety of H*( G, k) by V c (k). This cohomology ring is an easy exercise. Perhaps it is is the maximal ideal spectrum of the ring. The naive, but still it seems obvious that kn should have points are the maximal ideals in H*(G, k) and a dimension n. Thus it would seem that dimension, subset of Vc(k) is closed if it is the set of all max­ polynomial ring imal ideals that contain a particular ideal. called "Krull dimension", of the The functoriality of the preceding section k[x1 , ... , Xn ] should ben. All of this is correct, and extends naturally to the varieties. For example, if moreover, Quillen's Theorem implies that the Krull H is a subgroup of G, then the restriction homo­ dimension of H*(G, k) is the largest integer r such morphism induces a map on varieties that G has an elementary abelian subgroup of order p'. This number is called the p-rank of G. res~ .H: VH(k) ~ Vc(k). Usually the Krull dimension of a commutative The point is that ifm is a maximal ideal in H*(H, k), ring R is defined as the length f of the longest thenitsinverseimage {u E H*(G, k)l resc, H(u) E m} strictly increasing chain is a maximal ideal in H*( G, k). It should be noted that :P1 c :P2 c · · · c :Pc c R because H*(H, k) is finitely generated over the image resc,H(H*(G, k)), the map res ~. H on varieties is fi­ of proper prime ideals in R . Under the circum­ nite-to-one. Now consider the dihedral group, men­ stances that we encounter, the Krull dimension is tioned at the beginning of the paper. The dihedral also the length of the longest sequence of alge­ group of order 8 has a presentation in the form braically independent elements in R. Such sequences G = (g, h I g 2 = h2 = 1 = (gh)4 ). The group has a are of particular concern to us in later s ections.

428 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Depth and Detection H*(G, k)/(u1) will be annihilated by all elements of We saw in the last section that the Krull dimension positive degree. Consequently, there is no regular of the cohomology ring H*(G, k) is the p-rank of G. sequence oflength 2. This is one of the consequences of Quillen's The­ It is not difficult to prove that this situation orem. In this section we consider another ring­ happens in general. The depth of a commutative theoretic invariant that seems to be only partly graded ring R is never larger than the Krull discernible from the group structure. We are dimension of R /p where pis any associated prime speaking here of the depth of the cohomology ideal. ring. To define the depth of a ring, we first need On the other hand, the depth of a ring need not primes. the notion of a regular sequence. be controlled by the existence of associated As an example, consider the subrin g R of the poly­ and Regular Sequences Depth nomial ring k[z , y] that is generated by the mono­ R generated graded Suppose that is a finitely mials y , z 2 ,z3 ,zy. That is, R is the subring of k). commutative k-algebra such as H*(G, A regular k[z, y ] consisting of all polynomials in which the sequence for R is a sequence u1, u2 , ... , Ur of coefficient on z is zero. This time R is an integral homogeneous elements of positive degree having domain, because k[z, y] is an integral domain. the following properties. First, u1 should be a Hence, every element in R is regular. But it is not regular element, meaning that multiplication by u1 possible to choose a regular sequence of length 2. is an injective map or that the annihilator of u1 in For example, if we choose y as the first element of R is {0}. Then for each 2 ::o; i ::o; r, we must have that the regular sequence, then the element zy (which u; is a regular element on the quotient ring is not in the ideal (y) c;:::; R) is annihilated by every R/(u1 , ••• , U; - d. That is, multiplication by u; on the element of positive degree in R / (y). quotient ring is an injective homomorphism of R­ The depth is one of the ways in which the mod modules. The definition of depth is the following. p cohomology rings of finite groups are special. It Definition. The depth of a finitely generated graded is never the case that H*(G, k) has depth 0. In fact, commutative k-algebra R is the length r of the there is a theorem by Duflot [12] which says that longest regular sequence of elements in R . the depth of a cohomology ring is at least as large as the p-rank of the center. Indeed, it can be shown In fact, a sequence u1, ... , uris regular if and only that if u1, ... , Ur is a sequence of homogeneous if the ring R is a free module over the polynomial elements whose restriction to the center of a Sylow subring k[u 1 , ... , url c;:::; R . Thus, if a sequence of p-subgroup of G is a regular sequence, then elements in R is regular, then any permutation of u1, ... , U r is a regular sequence. the elements in the sequence is also regular. More­ In addition, for all the known computations of over, it can be proved that if u1, ... , Us is a regular cohomology rings of finite groups, there is no sequence, then it can be extended to a regular example of a cohomology ring where the depth is sequence of maximal length. not determined by the existence of an associated For a ring such as a graded polynomial ring, the prime. In every case that we know of, there is a depth is again equal to the Krull dimension. That nonzero element u in H*(G, k) such that the anni­ is, for the ring k[z, y, x] the sequence z , y, x is a hilator p of u is a prime ideal with H*(G, k) /p regular sequence. Such rings are said to be Cohen­ having Krull dimension equal to the depth of Macaulay. Certainly, the depth of any ring is never H*(G, k). So we can ask if this is always the case. greater than the Krull dimension. However, the depth Question. Does every cohomology ring H*(G, k) than the Krull dimension, and this can can be less have an associated prime p such that the depth of An to consider happen for several reasons. example H*(G, k) /p coincides with the Krull dimension of is the cohomology ring of the semidihedral group H*(G, k)? over a field of characteristic 2 as in the introduction. Ithas theformH*(G,k) ~ k[z , y,x,w]j1where1 is There are a few partial results. The answer is theidealgeneratedbyz3, zy , zxandx2 + y 2w.Note affirmative if the p-rank of G is one [8]. The same that the generators z, y, x, ware in degrees 1, 1, 3, happens if the depth is equal to the p-rank of the and 4, so that the relationx2 + y 2w is homogeneous center of a Sylow p-subgroup of G [1 6]. Beyond that, in degree 6. In this case, H*(G, k) has depth 1 while not much is known. Indeed, there does not seem its Krull dimension is 2. The pointis thatthe annihi­ to be much known about the class of rings with this lator of the elementz2 is the ideal A = (z, y, x).Prime property. ideals that are the annihilators of elements in a ring Detecting Cohomology are called associated primes for the ring. In this case The question raised above is not an idle one. It is an element of positive degreeinH*(G, k) is regular if intimately involved with at least one study that andonlyifitis not contained in A. So no matter what is central to many of the e fforts to compute regular element u1 may be chosen for the first ele­ group cohomology. This is the question of detec­ ment in a regular sequence, the class of z 2 in tion of cohomology. We say that a collectionS of

APRIL 2005 NOTICES OF THE AMS 429 subgroups of G detects the cohomology of G, Computing Group Cohomology provided the intersections of the kernels of the In this section, we describe the computer calcula­ restriction maps to the subgroups in S is zero: tions for cohomology. Embedded in the algorithms n Kernel(resc,H: H*(G, k)- H*(H, k)) = {0}. is the definition of group cohomology and the H ES product structure. At the present time, there are at least two different implementations of the We should point out that if Sis a Sylow p-subgroup algorithms. The older implementation [8, 10] by the of G, then the restriction map resc,s is always in­ author and his collaborators relies almost entirely jective. Consequently, any collection of subgroups on linear algebra methods in the early stages of the that contains a Sylow subgroup or any subgroup that computation. The newer implementation by David contains a Sylow subgroup is a detecting family. Green [15] uses noncommutative Grabner bases. We saw above that for G a dihedral2-group, the Green's programs run somewhat faster and (more two elementary abelian subgroups detect the coho­ importantly) are less extravagant with memory mology of G. On the other hand, for some p-groups usage. The basic algorithms are the same for both there is no family of detecting subgroups of G that systems. The first system is built into the computer does not include G itself. An example is the quater­ algebra system MAGMA [6], while Greens's ilion group of order 8, which can be presented as system is a stand-alone package. G = (g,h I g 2 = h2 ,g4 = 1,hgh- 1 = g- 1 ). The computer programs are designed to com­ pute the cohomology rings only of p-groups. This Its cohomology ring has the form is arguably the hardest problem, since the first H*(G, k) ~ k[z, y, x] / (z2 + zy + y 2 , 2 2y + zyz) step of most cohomology calculations is to com­ pute the cohomology of the Sylow p-subgroup where the generators z, y, x have degrees 1, 1, and of the group. The main reasons for considering 4. All of the subgroups of order 4 in G are cyclic p-groups have more to do with the reality of com­ and the restriction of any element in degree 2 or puter capabilities. any element in degree 3 in H*(G, k) to any one of On the one hand, while the co­ homology of p-groups is important, the these subgroups is zero. groups are The existence of detecting families has certainly also small, and the projective modules are also played a role in several complicated cohomology small and easily constructible. The Mathieu group calculations (see, for example, [1]). The point is M12 has 95,040 elements, but its Sylow subgroup that if a detecting family is known, then the ideal has order 64 (3]. of relations among the cohomology generators can The other reason that makes the calculation of be found by simply taking the intersection of the the cohomology of p-groups seem practical is that kernels of the restrictions to the members of the the group rings are local rings. If G is a p-group detecting family. It reduces the computational and k is a field of characteristic p, then in kG we problem to finding the generators of cohomology have that (x - 1)P" = xP" - 1 = 0 for any element x and computing their restriction to the elements in of order pn in G. So the augmentation ideal, which the detecting family. This is often a big improve­ is the ideal spanned by all x - 1 for x E G is a ment over trying to compute the relations among nilpotent ideal of codimension 1 in kG. It is the the generators directly. radical of kG, and kG / Rad(kG) ~ k. It follows The connection between depth and detection from this fact that every projective kG-module is comes from the following theorem [7]. free and is isomorphic to a direct sum of copies of kG. Consequently, the projective modules are easy Theorem. Suppose that the cohomology ring to construct on the computer. H*(G, k) has depth d. Then the cohomology is de­ For the rest of this section we assume that G is tected on the centralizers of the elementary abelian a. p-group and that k is the prime field k = 1Fp. p-subgroups of rank d. Smce only the characteristic of the field really The question that often arises is the converse. matters, for computational purposes we use the smallest field possible. Question. Suppose that the cohomology of G is Projective Resolutions detected on the centralizers of the elementary The first step in the cohomology computation is abelian p-subgroups of rank r. Is it necessary that the construction of a projective resolution for the the depth of H*(G, k) be at least r? trivial kG-module k ~kG / Rad(kG). The projective Adem and Karagueuzian (2] have shown this to resolution is an exact sequence of the form be true in the case that the rank of the center of a P 03 02 OJ E Sylow p-subgroup of G coincides with the p-rank · · · - 3 - Pz - P1 - Po - k - 0 of G. The answer to the question is yes for all of where each Pi is a projective kG-module. In prac­ the cohomology rings that we have computed. tice, we take a minimal resolution of k, meaning

430 NOTICES OF THE AMS VOLUME 52, NUMBER 4 that Po ~kG and that for every i, o(P;) <; Rad(P;_1). linear equations to obtain each (; in succession. This insures that in the induced complex In each degree n, the matrices of the monomials a• of degree n in all previously calculated generators 0 ~ Homkc(Po, k) ~ Homkc(P1, k) are computed so as to avoid computing chain a• a• ~ Homkc(P2, k) ~ ... maps for elements that are not generators for the cohomology. all of the maps are zero maps. The definition of the Once we have computed the chain maps for the gr?up cohomology is precisely the cohomology of generators of cohomology, we want the relations. this ct~mplex. That is, H"(G, k) = H"(Homkc(P*, k)), So suppose the generators are elements of the set the n cohomology group of this complex. Because {z, y, x, ... }. Then for each n we compute the in­ we have that we have used a minimal resolution duced maps P, I Rad(kG)P, ~ k for all monomials H"(G, k) = Homkc(P,, k) ~ (P, / Rad(kG)P,)*, where of degree n in the generators. Each such map is a the ( ) * indicates the k-dual. (1 x n)-matrix since the null space of the matrix The calculation of a typical map P, ~ P,_1 in with these matrices (vectors) as rows is the set of the projective resolution proceeds by reasonably all relations among the monomials of degree n. easy steps. First, in order that the sequence be Of course, we obtain far too many relations exact, the image of o, should be the kernel of o,_1. from the computation. For practical reasons, some So we compute the kernel K, of o,_1. If we have rep­ analysis is required. resented o,_1 as a matrix, then we are only com­ Grobner Bases puting a null space. Next, we find a basis for At this point we are given a polynomial ring and a Rad(kG)K, = IxEc (x - 1)K,. Of course, in practice huge collection of polynomials that generate the we need only to let the sum range over a set of gen­ ideal of relations. The programs described in [8] era~ors for G. Third, we find a complementary generate many relations in each degree. For ex­ bas1s u1, ... , Urn for Rad(kG)K, inK,. Finally, we ample, if x2 - zy is a relation in degree 2, then in let P, be the direct sum of t, copies of kG and degree 3 the system would generate the relation construct o, as the map which takes the identity zx2 - z 2y as well as yx2 - zy2 and x3 - zyx. We element 1 in the ith copy of kG to the element would like to have a minimal set of generators for U; E K, <; Pn- 1· relations, so we need some technique Various time- and space-saving tricks are built the ideal of the data. In addition we need to into the program. For example, the spaces K, and for analyzing calculate ring-theoretic properties such as depth, P, are never constructed as actual kG-modules but regular sequences, and associated primes. The rather are just k -vector spaces on which the action method that does all of this is Grabner bases. of an element x E G is determined by the multi­ obtaining a minimal set plication of x on kG. The algorithm for of relations is very simple. Suppose that s = Products and Relations {s;li = 1, ... , t} are the relations that have been of defining the There are several equivalent ways computed. First we order the set by degree, so that structure on group cohomology. One cup product Degree(s;) ::; Degree(s ) whenever i < j. Note that Hopf algebra structure on kG to 1 method uses the we are assuming that all of our relations are ho­ construct a chain map P * ~ P * 0 k P *. This method mogeneous polynomials. Let 51 = {s1}. Then define the computational setting, be­ is not practical in S; inductively by the following rule: for each i, if s; cause the tensor product resolution becomes too is in the ideal generated by S;- 1, then let S; = S;- 1. too fast. Instead, we compute products by big Otherwise, S; = 5;_1 u {s;}. Then a minimal set of compositions of chain maps. The fact is that every generators for the ideal of relations is the set 51. E Homkc(P,, k) = H"(G, k) lifts homomorphism ( Computationally, the hard part in the algorithm to a chain map{(*} : P* ~ P* of degree -n as in is the line "if S; is in the ideal ... ". For this we need commuting diagram the the membership test for an ideal that is provided by Grabner bases. Suppose that P = k[x1 , ... , x,] is a polynomial ring and let 'I be an ideal in P. To define a Grabner basis for '1, we first choose an ordering on the monomials in P. Such an ordering might be lexicographic where the monomials are ordered as words in a dictionary. Or it might be any chain map determines a cohomology likewise, graded lexicographic, where the longer words are Then the definition of the product of two class. higher in the ordering and ties are broken by a lex­ elements can be taken to be the composition of icographic test. In any case, the ordering should their chain maps. have the property that if u, v, and ware monomi­ The next step in the calculation is the con­ als and u::; v, then uw ::; vw. With the ordering, of a chain map for the generators of struction any polynomial fin P has a leading term, which is cohomology. This is primarily a matter of solving the term whose monomial has the largest order. A

APruL 2005 NOTICES OF THE AMS 431 Grabner basis for an ideal1 is a collection 5 = 51 map. The inflation map from a quotient group of of polynomials that generate 1 and also has the G is constructed similarly. property that the leading term of any f in 1 is Other properties of cohomology rings, such as divisible by the leading term of some element of the depth or the associated primes can be computed 5. It is a theorem that every ideal has a Grabner using the Grabner basis machinery. Some of these basis [ll). calculations can be very time consuming when the The membership test goes as follows. Let f be rings are complicated. any element of P. If the leading term of f is not Are We Almost There Yet? divisible by the leading term of any element of 5, The reader must have noticed by this point that we then we are done. Otherwise, there exists an ele­ are computing the cohomology degree by degree. ment s E 51 and a monomial ex E P such that the The problem with this approach should be obvious. leading term of f - cxs is lower in the ordering If we only compute to degree 7, then any genera­ than that of f. So now replace f by f - cxs and tor or any relation in degree 8 will be completely repeat until either f = 0 or it is shown that f is not lost, ignored. On the other hand, the number of in 1. steps that we can compute is finite. Not only do the Computing a Grabner basis for an ideal can be projective resolutions grow to absorb all available difficult and time consuming. For one thing, the memory, but computing time also becomes a fac­ Grabner basis depends heavily on the chosen order tor. A typical calculation of the cohomology ring on the monomials. Consider the example of the of one of the groups of order 64 (assuming the ideal 1 = (x 2, z 2 + yx) c; k[z, y, x]. Suppose that group has rank 3 or more) would use a hundred or the ordering is chosen as graded lexicographic more megabytes of RAM after fifteen to twenty and that z > y > x. Then it can be seen that the steps. In general, the time to compute each step Grabner basis for 1 consists of the two generators would exceed the sum of all the time to compute x 2 and z 2 + yx which have leading terms x 2 and z 2 • the previous steps. The memory requirements also On the other hand, if the ordering is chosen so that increase dramatically with each step. z < y < x, then the leading term of z 2 + yx is yx. As a result, a critical question arises. When are Hence the element z 2x = (z 2 + yx)x- y(x2 ) is in 1, we done? When have we calculated far enough to but its leading term is not divisible by that of get all of the generators and relations? There are either x 2 or z 2 + yx. So in this case the Grabner at least two tests for completion available. Both basis for 1 consists of x 2 , z 2 + yx and z 2x. depend on ring-theoretic properties of the coho­ In the computation [1 0) of the cohomology rings mology ring. We present an outline of some of of the groups of order 64, some of the Grabner these results in the next section. basis calculations took hours, or days. Most of the problems occurred in the actual computation of the Regularity and Tests for Completion Grabner basis for an ideal. The Buchburger algo­ For the calculations in the appendix of [10), we used rithm that computes the Grabner basis can be as a rather complicated test to show that the computer bad as doubly exponential in the number of vari­ program had produced the entire cohomology ring. ables. In practice, the algorithm usually works much Essentially, the same test was used by Green in [15). faster. However, choices such as the ordering of the More recently, Benson [4) has developed a new test variables can be critical in very unpredictable ways. which seems to be more direct and simple. All of In the computation [1 0) of the cohomology ring of these tests involve notions of systems of parame­ group number 187, the first few attempts to con­ ters and regularity in commutative algebra. In this struct the Grabner basis for the ideal of relations section we present some of these ideas and show failed after several days of computing time on how they apply to the computations of cohomol­ each attempt. The particular cohomology ring has ogy rings. We state the definitions for cohomology twenty-six generators, and there was some question rings, although in general they apply to finitely as to whether we would ever succeed. However, generated graded commutative k-algebras. the problem was solved with a simple change in Homogeneous Parameters the ordering of the variables. A homogeneous system of parameters for a coho­ Restrictions, Inflations, and Other Calculations mology ring H*(G, k) is a sequence of homoge­ The computer programs continue by calculating the neous elements x1 , x2, ••• , Xn such that restriction maps to maximal subgroups. The algo­ 1. n is the Krull dimension of H*(G, k), and rithms are similar to those above. If His a subgroup 2. H*(G, k) is finitely generated as a module over of G, then to compute the restriction map H*(G, k) the subalgebra generated by X1, ... , Xn. ~ H*(H, k), we first construct a kH-chain map Note that because of condition (1), it must be that from the minimal projective kH-resolution of k to k[x 1 , ... , Xn] is a polynomial subalgebra ofH*(G, k). the minimal projective kG-resolution. This is the Condition (2) is equivalent to the statement that chain map that lifts the identity on k. Then the re­ the only maximal ideal which contains all of the striction map is the composition with this chain elements x1, ... , Xn is the ideal of all elements of

432 NOTICES OF THE AMS VOLUME 52, NUMBER 4 positive degree in H*(G, k). It is always possible to a ring and as a module over the polynomial sub­ choose a homogeneous set of parameters with the ring k[z, y] are in degrees less than m + n. Likewise, property that the first d elements in the sequence it can be shown that the ideal of relations among is a regular sequence of length equal to the depth the generators is generated in degrees at most d of H*(G, k). 2(m + n). The basic idea behind most of the tests for com­ Quasiregular Sequences and Regularity pletion can be illustrated with the following What the sample test shows is that, while we can example. Here we present only a sketch. More not expect a cohomology ring to have a regular details can be found in Theorem 14.5.2 of [10]. sequence which is as long as its Krull dimension, A Sample Test for Completion we might always be able to find a quasiregular Suppose that the cohomology ring of G has Krull sequence. The notion of a quasiregular sequence dimension 2. Then we can choose a homogeneous was introduced by the author and Benson in [5]. The set of parameters z, y such that the element z is a definition follows. regular element. Let m, n be the degrees of z and y, respectively. Now choose a minimal projective Definition. Suppose that His a graded commutative, finitely generated k -algebra. We say that a sequence resolution (P* , E) of the trivial module k as above. Then the cohomology element z is represented by of homogeneous elements x1, .. . , Xn in degrees n 1 , ... , nn is a quasiregular sequence, provided that a cocycle fz : Pm ~ k. The statement that fz is a cocycle means that the composition for each i we have that the map Pm +l ~ Pm J.:... k Xi: (H/ (Xl, ... , Xi - l))j ~ (H/ (xl, ... , Xi - l))j+ni is the zero map. Hence, fz induces a homomorphism of multiplication by xi is injective for all degrees f' : Pm / o(Pm+d ~ k whose kernel we denote as L z. ('') So we have an exact sequence f' If the cohomology ring has a quasiregular se­ 0 ~ Lz ~ Pm / o(Pm+l ) .2.. k ~ 0, quence of length equal to its Krull dimension, then and there is a corresponding long exact sequence there is a straightforward formula for how far one has to compute in order to obtain a complete set ... ~ Ext~ c (k, k) (til: Ext;;c(.Om(k), k) of generators and relations for the ring. So we must ask the following question. ~ Ext~c(L z , k) ~ Extj;(;1(k, k) ~ ... Question. Does every mod p cohomology ring of Now Ext~ c (k, k) 2: W(G, k) and by degree shifting a finite group have a quasiregular sequence? Extj;c(.Om(k), k) 2: Hm +r(G, k). Moreover the map In the event that the depth H*(G, k) is one less (f; )* is given by multiplication by z. Because z is a regular element, (fJ* is injective, and the con­ than the Krull dimension, then a quasiregular se­ quence always exists. This is proved by an argument necting homomorphisms 8 are all zero maps. Hence we have an exact sequences which is very similar to that in the example. Benson [4] has recently shown that the answer is affirma­ 0 ~ W(G, k) (til: Hr+m(G, k) ~ Extj;c (Lz, k) ~ 0. tive in the case where the depth is no more than two less than the Krull dimension. A partial result in the Consequently, the cokernel of multiplication by z codepth 2 case had also been proved by Okuyama in degrees at least m is the cohomology Ext~c (L z , k). and Sasaki [19]. However, Benson's work goes much Now here is the key. The module Lz must be a further and involves the notion of regularity. periodic module, that is, a module whose coho­ In [4], Benson considers cohomology rings in the mology repeats regularly after some fixed number context of some construction from local cohomol­ of stages. The reason basically is that there is only ogy. In particular, he considers the Castelnuovo­ one parameter left after factoring out z (Proposi­ Mumford regularity of the rings. The definition of tion 9.7.3 of [10]). Moreover, because y is the only this form of regularity is rather complicated, and parameter left, multiplication by y must induce an we refer to Benson's paper or to [13] for a detailed isomorphism on the cohomology of Lz. So we have explanation. All we will say here is that it is related to that the cohomology of a certainKoszul complex associ­ f; : w +m(G, k) JzH'(G, k) ated to a homogeneous set of parameters. Such ~ Hr+m+n(G, k) / zHr+n(G, k) complexes are doubly graded and the regularity is the maximum value of the internal degrees such that is an isomorphism for all r ~ 0, where f; is mul­ the cohomology is nonzero. Benson has conjectured tiplication by y. The result is that every element of an affirmative answer to the following question. Hr+m+n( G, k) can be written in the form z lX + y f3 for some lX E w +n(G, k) and some f3 E Hr+m(G, k). It Question. Is the regularity of H*(G, k) equal to follows that all of the generators of H*( G, k) as both zero?

APRIL 2005 NOTICES OF THE AMS 433 Benson shows that if the answer is yes, then the [5] D.]. BENSON and]. F. CARLSON, Projective resolutions and cohomology ring has a quasiregular sequence. Poincare duality complexes, Trans. Amer. Math. Soc. Moreover, he shows that it has what he calls a 132 (1994), 447-488. strongly quasiregular sequence-a quasiregular [6] W. BosMA and]. CANNON, Handbook ofMagma Functions, sequence as in the definition above except that Magma Computer Algebra, Sydney, 2002. [7] ]. F. CARLSON, Depth and transfer maps in the the condition(*) is replaced j;::: by I;:~(n 1 - 1). cohomology of groups, Math. Z. 218 (1995), 461-468. Moreover, he shows there is an explicit formula in [8] __ , Problems in the computation of group terms of the regularity of the computed cohomol­ cohomology, Progr. Math. 173 (1999), 107-120. ogy ring that determines whether the computation [9] __ , Calculating group cohomology: Tests for is complete. That is, if Rn is the computation of completion,]. Symbolic Comput. 31 (2001), 229-242. H*(G, k) calculated to degree n, and if Rn satisfies (10] ]. CARLSON, L. TOWNSLEY, L. VALERO-ELIZONDO, and Benson's formula, then it must be the case that M. ZHANG, Cohomology Rings ofFinte Groups, Kluwer, Rn := H*(G, k). Dordrecht, 2003. [11] D. A. Cox, ]. LITILE, and D. O'SHEA, Ideals, Varieties So What Is True? and Algorithms. An Introduction to Computational Algebraic Geometry and Commutative Algebra, The investigations of the questions about depth, de­ Springer-Verlag, New York, 1997. tection, and regularity share one curious property. [12]]. DUFLOT, Depth and equivariant cohomology, Com­ The theoretical developments seem to be right at ment. Math. Helv. 56 (1981), 627-637. the edge of the experimental evidence. That is to say, [13] D. EISENBUD and S. GoTO, Linear free resolutions and we can prove most of what we can see. Moreover, minimal multiplicities,]. Algebra 88 (1984), 89-133. seeing the evidence and proving the theorems seem [14] L. EvENS, The cohomology ring of a finite group, to go hand in hand. Trans. Amer. Math. Soc. 101 (1961), 224-239. A case in point is the study of the quasiregular [15] D.]. GREEN, Grabner Bases and the Computation of sequences and regularity. We can prove the results Group Cohomology, Lecture Notes in Math., vol. 1828, Springer-Verlag, Berlin, 2003. we want as long as the codepth of the cohomology [16] _ _ , On Carlson's conjecture in group cohomology, ring is at most 2. This is also approximately what Math. Z. 244 (2003), 711-723. we have computed. There are almost no computed [17] __ , private communication, 2004. examples of groups G such that H*(G, k) has [18] ]. LANNES, Surles espaces fonctionnels dont la source codepth more than 2. Green recently reported that est le classificant d'un p-groupe abelien elementaire, he computed the mod 2 cohomology of codepth 3 Pub/. Math. Inst. Hautes Etudes Sci. 75 (1992), 135-244. of a group of order 128 [17]. In this case he got an [1 9] T. OKUYAMA and H. SASAKI, private communication. affirmative answer to Benson's question (the reg­ [20] D. QUILLEN, The spectrum of an equivariant coho­ ularity of H*(G, k) equal to zero), but one example mology ring, I, II, A nn. of Math. 94 (1971), 549-602. [21] B. B. is hardly strong evidence. VENKOV, Cohomology algebras of some arbitrary classifying spaces, Dokl. Akad. Nauk SSSR 127 (1959), The questions on depth and associated primes 943- 944. (Russian) were developed partly by looking at experimental evidence. The computations have also aided in eliminating some of the earlier speculations. For ex­ ample, it has now been shown that Question 3.3 of [7] has a negative answer. Counterexamples (if they exist) to any of conjectures will likely lie in large and complicated groups, and the computations become more expensive in terms of time and mem­ ory. At the same time both the equipment and the software are improving and perhaps will some day provide us with the insights to answer some of the questions.

References (1] A. ADEM, ]. F. CARLSON, D. KARAGUEUZIAN, and R. ]. MILGRAM, The cohomology of the Sylow 2-subgroup of Higman-Sims, ]. Pure Appl. Algebra 164 (2001), 275- 305. [2] A. ADEM and D. KARAGUEUZIAN, Essential cohomology of finite groups, Comment. Math. H elv. 72 (1997), 101- 109. (3] A. ADEM, ]. MAGINNIS, and R. ]. MILGRAM, The geometry and cohomology of M 12 ,]. Algebra 139 (1 991), 90- 133. [4] D.]. BENSON, Dickson invariants and regularity in group cohomology, preprint.

434 N OTICES OF THE AMS VOLUME 52, NUMBER 4 Book Review

Mathematicians under theN azis Reviewed by ]ochen Bruning

Mathematicians under the Nazis biographical mater­ Sanford L. Segal ial and historical ac­ Princeton University Press, 2003 counts of the period Hardcover, $85.00 in question. Scientists ISBN 0-691-00451-X generally dislike un­ proven assertions Imagine an eminent scientist whose work im­ and unjustified gen­ presses by its depth, by its impact, and by its eralizations, which beauty. It has become a cornerstone of research and are difficult to avoid education alike and provides strong inspiration in historiographic and encouragement for younger scientists to tackle writing. Thus a sci­ difficult problems of similar potential. Now one of entist initiating such the disciples discovers the fact, not usually men­ a study would likely tioned in a scientific context, that this scientist want to examine the supported a totalitarian regime while producing extant original docu­ his celebrated work and, even worse, used his sci­ ments and other entific abilities in a nontrivial, albeit not outright available testimony. Given the right circumstances, criminal, way to promote political causes of ma­ the study could develop into a full-fledged research lignant forces. Such a discovery may well undermine project. the value system that supports total dedication It is tempting to speculate that the author of the to science and lead to disquieting questions: How book under review found himself at some point in is this possible? Will not the clarity of mind so his career in the situation just described, his field essential to major scientific progress prevent such being mathematics, in particular number theory, political blindness? If not, what does this say about and his "fallen hero" being the German mathe­ cian Ludwig Bieberbach. At any rate, Bieberbach the cultural dependence of science and the mean­ mati would satisfy the criteria of scientific brilliance ing of scientific achievements? and active political support of the Nazi movement, Stable scientific communities seem to protect and he is the character most carefully and com­ themselves quite effectively against such poisonous pletely described in this book. As he mentions in thoughts, which may nevertheless affect individual the introduction, the author developed over a pe­ members. A likely first reaction would be to study riod of years an increasing curiosity about and knowledge of German mathematicians under the ]ochen Bruning is professor of mathematics at the Nazis. In the end, he designed a r esearch project Humboldt Universitiit in Berlin. His email address is [email protected]. that met with the support of the Humboldt

APRIL 2005 NOTICES OF THE AMS 435 Foundation, thus allowing him to spend a fruitful that the abstractness of mathematical thinking period in Germany, filled with the searching of creates a state of alienation from practical life archives and the interviewing of surviving wit­ ("Weltfremdheit" in German), which produces a nesses. In this way, the material basis for this book certain naivete or even blindness in judging polit­ was collected. Naturally, his inquiry expanded be­ ical promises and intentions. Taken as a whole, the yond the original motivations to include other mat­ development of the German mathematical com­ ters, such as the origin of the public image of math­ munity in the time span from 1918 to 1945 should, ematicians as "disembodied intellects" and the according to Segal's main thesis, tell us something resulting contempt so often experienced by the significant, independent of our special interest or mathematical community in confronting the po­ qualification in mathematics. litical world. The mathematician-turned-historian To convince his readers, Segal expands his ma­ then faces a dilemma: on the one hand, the moti­ terial over eight chapters totaling 508 pages. After vation to carry out a historical study often has its explaining in Chapter 1 his working principles, as roots in personal experiences asking for explana­ briefly summarized above, he proceeds to discuss tions, which the scholar may hope to find by un­ in the next two chapters the academic crisis in covering hidden mechanisms of history; the need Germany during the Weimar Republic and the for proof, on the other hand, forbids speculation, so-called "Grundlagenkrise" in mathematics, both even when plausible, and asks for ever more facts epiphenomena of the great social and cultural until, eventually, the "laws of history" emerge as self­ restructurings leading into the First World War. evident. Historical methodology was developed ex­ This part of the book builds largely on existing actly to avoid these two impossible extremes. material. With these preliminaries at his disposal, Sanford Segal explains his strategy for avoiding Segal presents in the next chapter three carefully the obvious pitfalls. His basic credo says that "his­ chosen case studies. They are meant to illustrate tory is made by people", by individuals, or, more the strong competition for funds and prestige not often, by groups of individuals connected by suf­ only among individual mathematicians but also ficiently intense communication and equipped with among various agencies of the Nazi administration, a sufficiently large common basis of codes and an at first unintended but then highly welcome values. He views German mathematicians in the consequence of the ominous "Fuhrerprinzip": period between 1918 and 194 5 as such a group, and Hitler saw the Nazi elite arise as survivors from a he proposes to study this group in its entirety as continuous struggle for power. While part of the an interacting body encompassing all professional material in this chapter is new and sharpens the mathematicians, not only those who made it into profile of some characters involved, the structural the textbooks. One expects that all possible ways defects of the administration, as well as the per­ of reacting to and coping with the Nazi adminis­ sonal shortcomings of some administrators, are tration would appear within any such group, but already well known. the probability distribution may vary with the Chapter 5 provides us with a fairly complete view social status of that group and possibly other of the professional life of mathematicians with parameters. academic degrees, again forged from a number of Segal sees his study as being of interest even to interesting case studies, based on original docu­ those with little mathematical knowledge, because ments and a careful evaluation of existing work. The the mathematical method of axiomatic thinking next chapter is entitled "Mathematical Institutions" and deductive reasoning, commonly believed to and presents another twelve case studies that be­ be of universal validity, was severely tested by the long under this heading only if we give the word Nazi "axiom of racial compatibility", which had to "institution" its most general meaning. Neverthe­ be applied to everything in life, in particular to less, we find here a lot of interesting and little­ thinking. This axiom was happily embraced by known material. The events described vary greatly some mathematicians, like Bieberbach and Teich­ in significance, ranging from the rather marginal muller, tolerated by many others, and ignored by "Lambert project" to the efforts to found the Ober­ the rest. However, at that time the issue of whether wolfach Institute and the organization of mathe­ or not there were decisive "racial"-or rather matical "working groups" at two concentration cultural-differences in mathematical theory and camps, a phenomenon whose significance up to proof could hardly be avoided by anyone taking the now has been perhaps undervalued. We get a vivid mathematical profession seriously. There is the impression of the burden the Nazi administration hope, already mentioned above, that intellectual and the war put on the mathematical community: training in mathematics would immunize against the elimination of Jewish (or Jewish-related) col­ demagogy, giving one the means to detect the leagues painfully questioned its solidarity and even hidden interests embedded in flawed argumenta­ its definition, while drafting and diversion of tion and to unmask their innumerable disguises and research power for military purposes emptied the dissimulations. The contrary has also been argued, ranks and considerably weakened scientific output.

436 NOTICES OF THE AMS VOLUME 52, NUMBER 4 The latter point, in particular the total contribution case studies that examine the fate of Nazi victims, of mathematicians to the war effort, is still some­ like Hausdorff and Landau, but also of some Nazi what unclear in its quantitative and qualitative di­ sympathizers or at least ardent German national­ mensions and thus might have been given more at­ ists, like Teichmuller, Witt, and Kahler. Other tention by the author. At the time, it was still accounts in this chapter try to evaluate the extent possible to acquire new funds by emphasizing the to which the better-known protagonists featured importance of concentrated research in wartime. in the preceding chapters helped colleagues Such arguments made possible the establishment endangered by the Nazi regime. The picture is of the Oberwolfach Mathematical Research Insti­ rounded off by presenting some lesser-known tute, which took place officially with the appoint­ cases, like the fate of the mentally ill but ingenious ment of Wilhelm Suss as director on January 3, logician Gentz en and the difficulties the statistician 1945. This achievement required the special skills Riebesell experienced after having positively and connections of Suss, who may well be called reviewed a mathematical book written by a Jewish the leader of German mathematics during the Nazi author. The idea is clearly to document the most period. He was involved in practically every major "characteristic" patterns of behavior that were decision within the mathematical community and experienced or displayed by mathematicians dur­ was directing this community in many ways, e.g., ing the Nazi regime while carefully elaborating the by serving as president of the German Mathemat­ special individual aspects of each case. ical Society (Deutsche Mathematiker Vereinigung Among those who were not victims of the Nazi (DMV)) from 1937 till 1945 and as rector of the terror, we do not discover. real heroes, nor do we Universitiit Freiburg from 1940 till1945. find serious wrongdoers, but we learn a lot about The next chapter deals with Ludwig Bieberbach, the art of dissimulation. One has to keep in mind the suspected fallen hero, whose eccentric and that the community of mathematicians by no means puzzling personality may have been what caught represents the social structure of German society; the author's attention in the first place. In this in fact, we are looking at a rather privileged group chapter we find the most complete account of of people here. Of course, those who held offices Bieberbach's thoughts and deeds to date, again of some importance, notably Suss, were in a more based on many primary and secondary sources. difficult position than the average mathematician: Bieberbach was one of the most brilliant mathe­ they could not achieve anything without a certain maticians of his generation but was never really amount of collaboration with Nazi authorities. One satisfied with his place in life. His philosophical handicap Segal had in writing about Suss and his interests and his mathematical experience led him activities is that the archive of the DMV was not yet to conceive a "racial theory of mathematics" that open to him when he was doing his research. New fit perfectly-and hardly by accident-with the material emerged after this archive was incorpo­ Nazi ideology. Anyone hearing about Bieberbach's rated in 1996 into the Universitatsarchiv at Freiburg, "conversion" from moderate German nationalist to particularly concerning Suss's treatment of the ardent Nazi defender between 1932 and 1934 has Jewish (or Jewish-related) DMV members who were wondered about the reason for this striking change, excluded from membership after January 1939. which certainly was not forced upon him. Segal This material recently caused some serious criti­ suggests "personal self-aggrandizement as a cism among historians of Suss's behavior. Segal, on governing theme in Bieberbach's professional life the other hand, collects a lot of evidence in his favor. and his lack of deeply held beliefs," a plausible but It is notoriously difficult to reach a fair moral judg­ somewhat shallow explanation. More enlightening ment of people like Suss. The more recent past of is Segal's excursion into the development of Germany and of other nations provides us with "typological psychology" and its applications to many such cases to be decided by historians of racial theories, which Bieberbach adapted to his future generations. own mathematical typology, thus forming some As a mathematician, and in particular one of sort of intellectual basis for his attempts at German origin, I found reading this book to be a imposing Nazi ideology on the German mathe­ most rewarding experience. The stunning amount matical community. The outright failure of these of carefully researched details and the unusual attempts attests to the stability of this community, selection of characters, if seen together, provide a but also to its relative unimportance for the goals rich picture of the (academic) mathematical com­ of Nazi ideology. munity in Germany between 1918 and 1945, at The last chapter, entitled "Germans and Jews", least for someone already familiar with the Nazi era could be expected to present a culmination and at and the recent history of mathematics in Germany, the same time some sort of summary after the and one can even sense its traces in the decades reader has absorbed an occasionally overwhelming that followed, since Segal's interviews were all amount of facts. This is not so, at least not in terms conducted after 1980. This is certainly a great of the chosen format: Again we find a sequence of achievement for an author who grew up in a very

APRIL 2005 NOTICES OF THE AMS 437 different cultural environment. Another impressive About the Cover aspect of the book is Segal's extreme care in avoid­ This month's cover was produced by Joerg Colberg of the ing unwarranted or a priori biased judgments; as Virgo Consortium (http: I /www. mpa -garchi ng. mpg. de/ Segal puts it: "When evil is present, it is tempting Virgo/). He writes: to see things only in starkly contrasted black and "The image shows white ... the purpose of this book is neither to wash a slice through a sim­ dirty linen publicly and assign assessments of guilt, ulation of N-body nor to fit the events described into some precon­ structure formation in ceived social structure of mathematics or science. a cosmological volume Rather, it is to describe the historical situation and done by our group. development of the mathematics profession in Matter is represented Nazi Germany and its interactions with the state, by millions of com­ allowing conclusions to emerge therefrom." Such puter particles subject conclusions, however, are never stated. To me at to their mutual grav­ least, the most fascinating aspect of Segal's work ity. Calculating the is that it gives another, very detailed, illustration gravitational forces be­ of what has been called the "banality of evil", tween N particles is an i.e., the apparent impossibility of deducing the O(N2) problem, which unimaginable atrocities of the Nazis merely from becomes computa­ the criminal energy of individuals. It seems that tionally too expensive very common weaknesses and vices may add up to unless sophisticated monstrous deeds if only the "right conditions" are algorithms are used. The total force acting on a particle is given-a rather disquieting thought. split into long-range and short-range components. The for­ From its dedication to detail and the habit of mer can be computed using very fast Fourier transforma­ splitting almost all the important chapters of the tions. The latter remains an O(N2) problem, albeit one of book into sequences of case studies stems what is much reduced scale. Simulations with large particle num­ in my view the weakness of this book. There is no bers are typically quite slow, since cosmologists are inter­ impression-and probably no intention-of com­ ested in resolving smaller and smaller scales while keeping prehensiveness, and the events described do not the size of the simulation box large enough to contain a cos­ connect to convincing larger structures, since there mologically representative volume. In other words, while pro­ is so much focus on individuals. One is reminded grammers try to keep the N 2 direct force calculations small of the geographers described by Borges, who the requirements of cosmologists ensure they're always a~ ultimately set out to create a map of their country close to the computer's limits as possible. of scale one to one. The moving forces of history Despite the vast improvements in efficiency gained from are of course only visible in events, in the histori­ advanced algorithms for the force calculation, the simula­ cal details, but these cannot be understood in tion still took about a month of computer time on a 128- themselves; they need a perspective. Refusing to processor Cray T3E parallel supercomputer at the Max provide perspectives may be honest but renders the Planck Society's Computer Center in Garching, Germany. book much less valuable to the general reader. Simulations of N -body behaviour have become invaluable Thus, I think that Segal does not achieve what he tools of cosmologists since the early 1970s. Every decade, was attempting, which is to write a social history N has increased by about two orders of magnitude, with the of German mathematics in the first half of the latest biggest computational effort, also done by the Virgo twentieth century that would be interesting and Consortium, exceeding ten billion particles. While the ear­ informative enough to teach something to the liest N-body simulations were mainly concerned with test­ general public. Those who are genuinely interested, ing the different models available, cosmology has since though, will be rewarded by the contents, includ­ then become a precision science where a canonical model ing a sizeable (if somewhat selective) bibliography is investigated in as much detail as possible. Computer sim­ and a substantial index. ulations have played an important role in this development and, along with better observations, we hope they will lead us to an understanding of the Holy Grail of cosmology, the formation of galaxies and structure from the uniform ini­ tial distribution of gas in the very early universe." -Bill Casselman, Graphics Editor ([email protected])

438 NOTICES OF THE AMS VOLUME 52, NUMBER 4 2005 Steele Prizes

The 2005 Leroy P. Steele Prizes were awarded at Mathematical Exposition: the 111 th Annual Meeting of the AMS in Atlanta in Branko Grunbaum january 2005. Citation The Steele Prizes were established in 1970 in Branko Gnlnbaum's book, Convex Polytopes, has honor of George David Birkhoff, William Fogg served both as a standard reference and as an in­ Osgood, and William Caspar Graustein. Osgood spiration for three and a half decades of research was president of the AMS during 1905-06, and in the theory of polytopes. That theory is currently Birkhoff served in that capacity during 1925-26. very active and enjoys connections with many other The prizes are endowed under the terms of a areas of mathematics, including optimization, com­ bequest from Leroy P. Steele. Up to three prizes putational algebra, algebraic geometry, and repre­ are awarded each year in the following categories: sentation theory. Much of the development that led (1) Lifetime Achievement: for the cumulative to the present, thriving state of polytope theory influence of the total mathematical work of the owes its existence to this book, which served as a recipient, high level of research over a period of source of information for workers in the field and time, particular influence on the development of as a source of inspiration for them to enter the field. a field, and influence on mathematics through Despite the passage of time, Convex Polytopes re­ Ph.D. students; (2) Mathematical Exposition: for a tains its value both as an exposition of the theory book or substantial survey or expository-research and as a reference work. Springer-Verlag's deci­ paper; (3) Seminal Contribution to Research: for a sion to issue a second edition in 2003, consisting paper, whether recent or not, that has proved to of Grunbaum's original text plus notes by Volker be of fundamental or lasting importance in its field Kaibel, Victor Klee, and Gunter Ziegler to describe or a model of important research. Each Steele Prize newer developments, will extend the book's influ­ carries a cash award of $5,000. ence to future generations of mathematicians. The Steele Prizes are awarded by the AMS Coun­ Biographical Sketch cil acting on the recommendation of a selection Branko Grunbaum was born in 1929 in what was committee. For the 2005 prizes the members of the then Yugoslavia. In 1948 he started studying math­ selection committee were: Andreas R. Blass (chair), ematics at the University of Zagreb and a year later DanielS. Freed, John B. Garnett, Victor W. Guillemin, emigrated to Israel. After receiving his Ph.D. from Craig L. Huneke, Tsit-Yuen Lam, Robert D. MacPher­ the Hebrew University in Jerusalem in 195 7 under son, Linda P. Rothschild, and Lou P. Van den Dries. the guidance of Aryeh Dvoretzky, he was a mem­ The list of previous recipients of the Steele Prize ber of the Institute for Advanced Study in Princeton may be found on the World Wide Web, http: I I for two years. In 1961 he returned to the Hebrew www.ams.orglprizes-awards. University. Following a visiting appointment at The 2005 Steele Prizes were awarded to BRANKO Michigan State University in 1965, in 1966 he GRONBAUM for Mathematical Exposition, to RoBERT P. became professor at the University of Washington; LANGLANDS for a Seminal Contribution to Research he has been in Seattle ever since, from 2000 on as (this year restricted to the field of algebra), and to professor emeritus. At various times he had visit­ IsRAEL M. GELFAND for Lifetime Achievement. The ing appointments at the University of Kansas, the text that follows presents, for each awardee, the University of California at Los Angeles, and Michigan selection committee's citation, a brief biographical State University. His interests cover much of geom­ sketch, and the awardee's response upon receiving etry and combinatorics, with the principal activity the prize. on convex sets and polytopes, and tilings. In recent

APRIL 2005 NOTICES OF THE AMS 439 Branko Gri.inbaum Robert P. langlands I. M. Gelfand

years, most of his efforts were devoted to config­ special cases class field theory, the Artin conjec­ urations of points and lines in the Euclidean plane tures, and Eichler-Shimura theory, which they and to nonconvex polygons and polyhedra. He is extended to higher dimensional varieties. The happy to be able to give courses on these topics conjectures provided a unifying principle for and to see that the material has started to attract the theory of automorphic forms, and in particu­ attention after a long period of quiescence. lar a relatively clear guide to their relation with Response L-functions. As a result of this paper, the system­ The beginning of Convex Polytopes was in notes and atic relation between global and local theory explanations I prepared for students in my semi­ and the systematic use of adele groups became nar at the Hebrew University in 1963. The main fixtures in the subject. topic concerned the material of Klee's seminal The Langlands conjectures had their origin in preprints about the face vectors of convex poly­ Langlands' theory of Eisenstein series, which was topes and Steinitz's characterization of graphs of itself a major mathematical advance. The conjec­ convex 3-polytopes. In time, the notes expanded tures are still unproved, but many difficult cases and formed the core of the book. I was fortunate have been established recently. It's hard to think to have M. A. Perles contribute to the book his of any other instance in the history of mathemat­ path-breaking results dealing with Gale diagrams ics where conjectures gave so accurate a road and to receive the cooperation of Vic Klee and map of what would turn out to be true in so many G. C. Shephard for other parts of the book. After different situations. And few other conjectures the book went out of print, there were several have generated so much research of such high attempts to publish an updated version; they quality. foundered on the sheer quantity of the relevant new Biographical Sketch material. It took the mathematical depth and Robert P. Langlands was born October 6, 1936, in organizational ability of Gunter Ziegler and the New Westminster, British Columbia, Canada. Here­ help of Volker Kaibel and Vic Klee to complete the ceived his A.B. and M.A. degrees at the University of task. I am greatly indebted to all of them. Naturally, British Columbia in 19 57 and 19 58 respectively and I am deeply appreciative of the Steele Prize and his Ph.D. from Yale University in 1960. His principal greatly honored by it. speciality is the theory of automorphic forms. He is best known for the Langlands Program, which Seminal Contribution to Research: proposes deep links between algebra and analysis, Robert P.langlands having significant ramifications for number theory. Citation Langlands held positions at Princeton University The Steele Prize for a Seminal Contribution to Math­ (1960-67) and at Yale University (1967- 72), and ematical Research is awarded to Robert Langlands since 1972 he has been at the Institute for Advanced for the paper "Problems in the theory of auto­ Study. He is the recipient of numerous honorary morphic forms", Springer Lecture Notes in Math., doctorates and awards, including the AMS Cole Prize vol. 170, 1970, pp. 18-86. This is the paper that in Number Theory in 1982, the Commonwealth introduced the Langlands conjectures. Award in 1984, the National Academy of Sciences The Langlands conjectures asserted deep rela­ Award in Mathematics in 1988, the Wolf Prize in tions among modular forms that encompassed as Mathematics (1995- 96), and La Grande Medaille d'or

440 NOTICES OF THE AMS VOLUME 52, NUMBER 4 de l'Academie des Sciences in 2000. He is a fellow interactions with other mathematicians, including of the Royal Society of Canada (1972) and the Royal students. Here we can only touch on a few high­ Society oflondon(1981); he is also a member of the lights. American Academy of Arts and Sciences (1990), the Gelfand's first major achievement is the theory National Academy of Sciences (1993), the American of commutative normed rings, which he developed Philosophical Society (2004), the American Mathe­ in the late 1930s in his thesis. His use of maximal matical Society, and the Canadian Mathematical ideals was crucial not only in harmonic analysis, Society. Langlands is the author of numerous but also in the subsequent development of algebraic research papers. geometry. Next, in collaboration with Naimark, he Response proved that noncommutative normed rings with The pleasure of learning that one is to be awarded involution may be represented as operators in a Steele Prize or perhaps almost any prize in math­ Hilbert space, a cornerstone of the modern theory ematics is, for anyone with a sense of proportion, of C* -algebras. In the 1940s Gelfand turned to soon followed by the uneasy sentiment that there representation theory and the theory of generalized are others more deserving and, at least if the prize functions. There are also foundational papers from is coveted, that they are quite aware of it. There is this period on integral geometry, geodesic flows on little to be done with the unease but to live with it surfaces of negative curvature, and generalized and to be grateful to those unknown members of random processes. the selection committee who appreciated what you Beginning in the mid-1940s Gelfand led many have tried to do and made an effort to persuade investigations on partial differential equations, the other members of the committee of its merits. and in a well-known paper published in 1960 asked The unease is, in any case, soon followed by a for a topological classification of elliptic opera­ more troubling impulse, the desire to supplement tors, based on the observation that the index is a the citation and to explain what one really had in homotopy invariant of the leading symbol. This led mind. I shan't do that now, except to mention that to the Atiyah-Singer index theorem, with its many in the paper "Problems in the theory of automor­ profound implications and applications. We also phic forms", dedicated, I recall, to Salomon Bochner, mention his work with, among others, Levitan and the emphasis was on what I later came to call func­ Dickii on inverse spectral problems and scattering toriality, thus, in particular, on the Artin conjecture theory. and a possible nonabelian class-field theory. Hasse­ Gelfand, in collaboration with Fuks in the late Weil L-functions were mentioned only in passing 1960s, investigated the cohomology of infinite­ as a more or less obvious-once I had learned of dimensional Lie algebras, particularly those asso­ the Taniyama-Shimura-Weil conjecture-after­ ciated with a manifold. Even for the algebra of thought. By the time the paper was published, I vector fields on the circle there is nontrivial and had reflected for two or three years on the "work­ interesting cohomology. This work led to charac­ ing hypotheses", as I called them, contained in it, teristic classes of foliations. and I no longer had any serious doubts. This brief account omits many fundamental re­ In the following years various mathematicians, sults-the Bernstein-Gelfand-Gelfand resolution of myself included, were able to do something with representations, work on integral geometry and them, even some things quite striking, as with, say, the Radon transform, combinatorial characteristic base change and the Artin conjecture in the tetra­ classes, etc.-as well as recent work on such top­ hedral and octahedral case or with various general ics as determinants, noncommutative polynomials, forms of the Ramanujan conjecture. Nevertheless, etc. Gelfand has also had a parallel career working for lack of courage and historical perspective, I on applied problems, ranging from computation to did not, as I now believe, appreciate until quite biology. recently the real import of the suggestions and Gelfand's mathematical influence has spread the depth one would have to attain to solve the not only through his many research papers, but also problems posed. Unfortunately, it may now, at through his books, lectures, and seminars. His least for me, be too late for boldness. On the other series of five books (with various coauthors) on hand, until serious inroads had been made on what Generalized Functions dates from the late 19 50s and the experts call the fundamental lemma, the time has been a classic for 50 years. A recent book with was not ripe for it. Kapranov and Zelevinski entitled Discriminants, Resultants, and Multidimensional Determinants is Lifetime Achievement: Israel M. Gelfand also a major work. In between are monographs Citation on many other topics. Gelfand's seminar, which The broad and lasting impact of I. M. Gelfand on began in Moscow and continues in Piscataway, has mathematics is difficult to convey in a short space. long been a training ground for participants and He has had a profound influence on many fields speakers. His educational activities extend to of research through his own work and through his younger mathematicians as well, including a cor-

APRIL 2005 NOTICES OF THE AMS 441 respondence school in both Russia and the United States as well as many books on elementary Research Notes in Mathematics mathematics. Biographical Sketch Israel M. Gelfand was born on September 2, 1913, in Krasnye Okny, Ukraine; he received his Ph.D. and r ------·-- D.Sc. in mathematics from Moscow State University }} 1. ::, '' I j1 ' ; .:. ! ' '~ .,, .·I·! in 1935 and 1940 respectively. For almost fifty Computational Aspects years (1941-90) Gelfand served as professor of mathematics at Moscow State University; he has -of Polynomial Identities · held visiting professor positions at Harvard Uni­ versity and the Massachusetts Institute of Tech­ Alexei Kanei-Belov & Louis Halle Rowen nology (1989-90). Since 1990 he has 5erved as professor of mathematics at Rutgers University. Gelfand is the author of more than 800 articles and ISBN 1-56881-163-2 30 books in mathematics, applied mathematics, Hardcover and theoretical biology. He has worked chiefly in the area of functional analysis and representation, $69.00 but he has significantly contributed to many other areas of mathematics as welL Gelfand is the recipient of many awards and In this latest volume in the Research Notes honors, including the State Prize of the U.S.S.R. in Mathematics series, the authors present (1953), the Lenin Prize (1956), the Wolf Foundation a comprehensive study of the main research Prize (1978), the Kyoto Prize (1989), and a done in polynomial identities over the last MacArthur Foundation Fellowship (1994). He was 25 years, including Kerner's solution to the elected a member of the American Academy of Specht problem in characteristic 0 and ex­ Arts and Sciences (1964), the Royal Irish Academy amples in the characteristic p situation. (1970), the National Academy of Sciences (1970), the Royal Swedish Academy (1974), the Acactemie des Sciences of France (1976), the Royal Society of The · author;s :, also cover codimen­ Britain (1977), the Academia dei Lincei of Italy sion theory, star:ting1 '! with Regev'sl (1988), the Japan Academy of Sciences (1989), and theorem and c~ntinuing · tnrough tl;le the European Academy of Sciences (2004). In 2000 Giambruno-Zaicev , e~ponential rank. 1 he was made a Lifetime Member of the New York

" " l •_: ' < ~l : :.: ~ Academy of Sciences. A corresponding member The best proofs oL.dassLcai.Lesults_,_s.u_ch, of the U.S.S .R. Academy of Sciences since 1953, as the existence lot' 'c.·enh'al pqlynomials, !l Gelfand was elected to full membership in 1984. the tensor prodyct t~eorem; " the·· nilpb­ He is also the recipient of many honorary degrees. 1 tence of the radical of''a n' affine PI-algebra,• Response Shirshov's l::lieorem, and characterization I am very touched to receive this award from the of group·algebras·: with PI, are presented. American Mathematical Society. For me it is a con­ firmation that everything that I have worked for j ,'i ! i l through my entire life was not in vain. This recog­ Call fpr AutHor~ . . i 1 nition of my work from my peers, colleagues, and The Research Not~S. ,, in Mathemattc1 friends from the American Mathematical Society is series is growin~r:-;- _-A--~~:-P~t~rs is now especially meaningful for me. Mathematics for me accepting relevant manuscripts and is a universal and adequate language of sciences, proposals. Visit bur 1 w~b 'site for mord and it is an example of how people of different cul­ information for I pr9.$pecti.Je ·authors.; tures and backgrounds can communicate and work together. This is extremely important in our times. I' "1 • 1w.,. w w . a k p e t e r s . c o m

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442 NOTICES OF THE AMS VOLUME 52, NUMBER 4 2005 B6cherPrize

The 2005 Maxime B6cher Memorial Prize was committee's citation, a awarded at the 111 th Annual Meeting of the AMS brief biographical in Atlanta in January 2005. sketch, and the Established in 1923, the prize honors the mem­ awardee's response ory of Maxime B6cher (1867-1918), who was the upon receiving the Society's second Colloquium Lecturer in 1896 and prize. who served as AMS president during 1909-10. B6cher was also one of the founding editors of Citation Transactions of the AMS. The original endowment The B6cher Prize is was contributed by members of the Society. The awarded to Frank prize is awarded for a notable paper in analysis pub­ Merle (Cergy-Pontoise, lished during the preceding six years. To be eligi­ France) for his funda­ ble, the author should be a member of the Ameri­ mental work in the can Mathematical Society or the paper should have analysis of nonlinear been published in a recognized North American dispersive equations, journal. Currently, this prize is awarded every three represented most re­ years. The prize carries a cash award of $5,000. cently by his joint work The B6cher Prize is awarded by the AMS Coun­ "Stability of blow-up Frank Merle cil acting on the recommendation of a selection profile and lower committee. For the 2005 prize the members of the bounds for blow-up rate for the critical generalized selection committee were: Charles L. Fefferman, KdVequation" (withY. Martel), Annals ofMath. 155 Leon Simon (chair), and Daniel I. Tataru. (2002), 235-280; "Blow up in finite time and dy­ Previous recipients of the B6cher Prize are: namics of blow up solutions for the L 2-critical gen­ G. D. Birkhoff (1923), E. T. Bell (1924), Solomon eralized KdV equation" (with Y. Martel), ]. Amer. Lefschetz (1924),]. W. Alexander (1928), Marston Math. Soc.15 (2002), 617-664; and "On universal­ Morse (1933), Norbert Wiener (1933), John von ity of blow-up profile for L 2 critical nonlinear Neumann (1938), Jesse Douglas (1943), A. C. Schrodinger equation" (with P. Raphael), Invent. Schaeffer and D. C. Spencer (1948), Norman Levin­ Math. 156 (2004), no. 3, 565-672. son (1953), Louis Nirenberg (1959), Paul]. Cohen Biographical Sketch (1964), I. M. Singer (1969), Donald S. Ornstein (1974), Alberto P. Calderon (1979), Luis A. Caf­ Frank Merle was born November 22, 1962, in Mar­ farelli (1984), Richard B. Melrose (1984), Richard M. seille, France. He received his Ph.D. at the Ecole Nor­ Schoen (1989), Leon Simon (1994), Demetrios male Superieure in 198 7 and held a Centre National Christodoulou (1999), Sergiu Klainerman (1999), de la Recherche Scientifique (CNRS) research position Thomas Wolff (1999), Daniel Tataru (2002), Terence there from 1988 to 1991. From 1989 to 1990 he was Tao (2002), and Fanghua Lin (2002). assistant professor at the Courant Institute. Since The 2005 B6cher Prize was awarded to FRANK 1991 he has been professor of mathematics at the MERLE. The text that follows presents the selection Universite de Cergy-Pontoise. From 1998 to 2003 he

APRIL 2005 NOTICES OF THE AMS 443 was a member of the Institut Universitaire de France particular global existence results) to various and in 1996 and from 2003 to 2004, a member of nonlinear contexts. For nonlinear-type behavior (in the Institute for Advanced Study in Princeton. particular for the qualitative study of breakdown), Over the years he has held various visiting po­ little is known apart from stability results of the sitions at the University of Chicago, Rutgers Uni­ 1980s based on global energy arguments by P.-L. versity, Stanford University, the Courant Institute, Lions and M. Weinstein. the Institute for Advanced Study in Princeton, the The approach we took for these problems was Mathematical Sciences Research Institute in Berke­ not to justify possible formal asymptotics and con­ ley, the University of Tokyo, the CNRS, and Leiden struct one solution with a given behavior; instead, University. we looked for properties of these equations that Merle's awards and honors include the Institut were rigid enough to classify different blow-up Poincare Prize in Theoretical Physics (1997), the Charles-Louis de Saulse de Freycinet Prize of the and dynamical behaviors. Since the 1980s this Academie des Sciences de Paris (2000), and an in­ geometrical approach has had great success in vitation to speak at the International Congress of elliptic theory and in geometry. Earlier research on Mathematicians in 1998. the nonlinear heat equation (Merle and Zaag) sug­ gested that this approach might also be success­ Response ful for evolution equations. Using the Hamiltonian It is a great honor to be awarded the B6cher Memo­ structure, we were able to localize in physical space rial Prize. I am grateful to the prize committee and dispersive effects which occur naturally at infinity. to the American Mathematical Society for their By their local nature, these effects give a new set recognition of this research. I am also deeply grate­ of estimates and provide a dynamical rigidity for ful to Jean Bourgain and Carlos Kenig for their the asymptotic behavior of solutions (by way of a constant support and early recognition of this monotonicity formula, or by local quantities which work, and to George Papanicolaou, who introduced do not oscillate in time or which satisfy a maximum me to these problems and supported me. I would principle). like to thank people who influenced me early For the CGKdV problem, a mechanism of balance in my career and over the years, such as Henri between local dispersive effects and Hamiltonian Berestycki; Haim Brezis; Louis Nirenberg (who was constraints on the solutions allows us to prove a role model); Hiroshi Matano; Robert V. Kohn; Abbas Bahri; Jean Ginibre; my close collaborators and describe blow-up. In the process, we also elim­ Yvan Martel, Pierre Raphael, and Hatem Zaag; and inate the formally expected candidate. Nevertheless, family and friends. getting a sharp lower bound for the blow-up rate The cited work is concerned with the Critical remains an open problem. In the subcritical case, Generalized Korteweg-de Vries (CGKdV) and Crit­ these techniques give asymptotic stability in the en­ ical Schrodinger (CNLS) equations. We considered ergy space of a soliton or finite sum of solitons. the existence and description of solutions which For the CNLS problem, an exact description of break down (or blow up) in finite time and related blow-up is given (at least for solutions with a sin­ qualitative properties of the equations such as gle blow-up point). It confirms that the remarkable long-time behavior of global solutions. Such prob­ conjecture of Papanicolaou (along with M. Landman, lems were proposed as models for understanding C. Sulem, and P.-L. Sulem) is the only generic breakdown in the Hamiltonian context. A number behavior. Additional rigidities for the global be­ of people, including Ya. G. Sinai and V. E. Zakharov, havior of solutions are also exhibited. first investigated these problems in the 19 70s using In the future, I think three directions should be formal asymptotics combined with numerical investigated. The first is to extend this approach methods. Initial work led to less-than-clear results to other dispersive problems. Bearing in mind the for CGKdV and to a controversial blow-up rate qualitative elliptic theory of the 1980s and 1990s, for CNLS. In 1988, for the generic behavior of the second direction is to carry out a similar pro­ the breakdown, Papanicolaou and coauthors sug­ gram in the context of oscillatory integral problems. gested a rate equal to the scaling rate corrected by .,jLoglog(t), but this rate is different from that of In particular, I think questions from the dynami­ the explicit blow-up solution. cal systems viewpoint should be considered, such In the last decade, from Bourgain's seminal work; as classification of connections between critical from the work of Kenig, Gustavo Ponce, Luis Vega; points. The last direction is to develop techniques and now from the work of a large mathematical using localization in both space and frequency to community, a huge breakthrough has arisen out of investigate a new set of questions. analytical methods based on frequency localization Again, I thank the prize committee for honor­ properties of the solution of dispersive equations. ing these lines of research, and I look forward to This approach extends linear-type behavior (in continued work on them.

444 NOTICES OF THE AMS VOLUME 52, NUMBER 4 2005 ColePrizeinNumber Theory

The 2005 Frank Nelson Cole Prize in Number The­ The 2005 Cole Prize in Num­ ory was awarded at the 111 th Annual Meeting of ber Theory was awarded to PETER the AMS in Atlanta in January 2005. SARNAK. The text that follows The Cole Prize in Number Theory is awarded presents the selection commit­ every three years for a notable research memoir in tee's citation, a brief biographical number theory that has appeared during the pre­ sketch, and the awardee's re­ vious five years (until 2001 , the prize was usually sponse upon receiving the prize. awarded every five years). The awarding of this prize alternates with the awarding of the Cole Prize Citation in Algebra, also given every three years. These The Frank Nelson Cole Prize in prizes were established in 1928 to honor Frank Nel­ Number Theory is awarded to son Cole (1861- 1926) on the occasion of his re­ Peter Sarnak of New York Uni­ tirement as secretary of the AMS after twenty-five versity and Princeton University years of service and as editor-in-chief of the Bul­ for his work relating the distrib­ letin for twenty-one years. The endowment was ution of zeros of L-functions in made by Cole, contributions from Society members, certain families to the distribu- Peter Sarnak and his son, Charles A. Cole. The Cole Prize carries tion of eigenvalues in a large com- a cash award of $5,000. pact linear group of a type that The Cole Prize in Number Theory is awarded by depends on the family of L-functions one is con­ the AMS Council acting on the recommendation of sidering. In particular it is awarded for the book a selection committee. For the 2005 prize the mem­ Random Matrices, Frobenius Eigenvalues, and Mon­ bers of the selection committee were: Andrew J. odromy (with N. Katz) in which this Katz -Sarnak phi­ which it is exten­ Granville, Richard L. Taylor (chair), and Marie France losophy is introduced and in verified in the function field case. This Vigneras. sively philosophy has had a major impact on the direc­ Previous recipients of the Cole Prize in Number tion of work in analytic number theory. In addition Theory are: H. S. Vandiver (1931), Claude Chevalley the prize is awarded for the papers "The non­ (1941), H. B. Mann(l946), Paul Erdos (1 951), John T. vanishing of central values of automorphic Tate (1 956), Kenkichi Iwasawa (1 962), Bernard M. L-functions and Landau-Siegel zeros" (with H. Dwork (1962), James B. Ax and Simon B. Kochen Iwaniec) and "Low lying zeros of families of L-func­ (1967), Wolfgang M. Schmidt (1972), Goro Shimura tions" (with H. Iwaniec and W. Luo) in which this (1977), Robert P. Langlands (1 982), Barry Mazur philosophy is tested in the much harder number (1982), Dorian M. Goldfeld (1987), Benedict H. Gross field case. For example, the second paper shows, and Don B. Zagier (1987), Karl Rubin (1992), Paul subject to suitable Riemann hypotheses, that the Vojta(1992),Andrew]. Wiles (1997),Henryklwaniec low lying zeros of the L-functions of modular forms (2002), and Richard Taylor (2002). with root number 1 (resp.- 1) are distributed like

APRIL 2005 NOTICES OF THE AMS 445 the low lying eigenvalues of a random matrix in Of zeros of such L-functions by Mike Rubinstein, S0(2N) (resp. S0(2N + 1)) as N gets large. who was a graduate student at Princeton at that time, gave us valuable evidence for this belief. Biographical Sketch The paper with Henryk Iwaniec and Wenzhi Luo, Peter Sarnak was born on December 18, 1953, in cited above, developed methods to study these Johannesburg, South Africa. He received his Ph.D. questions for L-functions of automorphic forms. from Stanford University in 1980. Sarnak began The paper with Iwaniec does the same for the re­ his academic career at the Courant Institute of lated problem of the quantitative study of nonva­ Mathematical Sciences, advancing from assistant nishing of such L-functions at special points on the professor (1980-83) to associate professor (1983). critical line and its arithmetical applications. This He moved to Stanford University as a professor allowed for the verification of aspects of the con­ of mathematics (1987-91). Sarnak has been a pro­ jectured distribution of zeros as dictated by the fessor of mathematics at Princeton University symmetry. since 1991 and at the Courant Institute since 2001. One of my greatest pleasures in connection with Since 2002, Sarnak has held the position of Eugene these works has been to see how others have picked Higgins Professor of Mathematics at Princeton, up on these ideas and run with them, far beyond having served as the H. Fine Professor (1995-96) what I had anticipated. Let me mention in partic­ and as department chair (1996-99). ular the remarkable conjectures for the moments Sarnak was a Sloan Fellow (1983-85) and a of central values of families of L-functions (Keat­ Presidential Young Investigator (1985-90). In 1991 ing, Snaith, Conrey, Farmer, and Rubinstein) and he was elected to the American Academy of Arts the determination of some of these moments as and Sciences. With P. Deift and X. Zhou, he re­ well as far-reaching quantitative nonvanishing ceived the P6lya Prize of the Society for Industrial results for such central values (Kowalski, Michel, and Applied Mathematics in 1998. Sarnak was Soundararajan, and VanderKam). elected to membership in the National Academy Finally, it was Paul Cohen who many years ago, of Sciences (2002), won the AMS's Levi L. Conant when I was a student at Stanford, pointed me to Prize (jointly with N. Katz) in 2003, and held the Montgomery's work on the pair-correlation of the Rothschild Chair of the Isaac Newton Institute in zeros of zeta and its connection to random matrix Cambridge, UK, and the Aisenstadt Chair of the Cen­ theory and asked, why is it so? tre de Recherches Mathematiques in Montreal in My efforts to try to answer that question began 2004. He has sat on numerous editorial boards, with a paper with Zeev Rudnick on the higher cor­ oversight committees, and advisory committees, relations for zeros of the zeta function and led even­ and he has published extensively in the areas of tually to the works cited above. number theory and automorphic forms. Response It is a great honor for me to receive this prize. I have mostly worked in collaboration with others. Not only has this allowed me to achieve things I could never have done by myself, but it is also more fun (especially when you are stuck, which, of course, is most of the time). This recognition belongs as much to my coworkers as to me. ln my work with Nick Katz cited above, our orig­ inal aim was to determine if there was a function field analogue of the phenomenon (due to Montgomery and Odlyzko) that the local fluctuations of the dis­ tribution of the zeros of the Riemann zeta function are governed by the distributions of the eigenvalues for the Gaussian Unitary Ensemble in random ma­ trix theory. After a lot of false starts and misun­ derstandings, we found such an analogue. Its source lay in the analysis of the large n limit of monodromy groups associated with families of such zeta func­ tions. This led naturally to the possibility that the distribution of low lying zeros of a family of au­ tomorphic L-functions might also be governed in a decisive way by a symmetry type associated with the family. The extensive numerical computations

446 NOTICES OF THE AMS VOLUME 52, NUMBER 4 2005 Satter Prize

The 2005 Ruth Lyttle Satter Prize in Mathematics sketch, and the awardee's re­ was awarded at the 111 th Annual Meeting of the sponse upon receiving the prize. AMS in Atlanta in January 2005. The Satter Prize is awarded every two years to Citation recognize an outstanding contribution to mathe­ TheRuthLyttleSatterPrizeinMath­ matics research by a woman in the previous five ematics is awarded to SvetlanaJit­ years. Established in 1990 with funds donated by omirskayafor her pioneering work Joan S. Birman, the prize honors the memory of Bir­ onnon-perturbative quasiperiodic man's sister, Ruth Lyttle Satter. Satter earned a localization, in particular for re­ bachelor's degree in mathematics and then joined sults in her papers (1) "Metal-in­ the research staff at AT&T Bell Laboratories dur­ sulator transition for the almost ing World War II. After raising a family, she received Mathieu operator", Ann. ofMath. a Ph.D. in botany at the age of forty-three from the (2) 150 (1999), no. 3, 1159-1175, SvetlanaJitomirskaya University of Connecticut at Storrs, where she later and (2) with J. Bourgain, "Ab- became a faculty member. Her research on the bi­ solutely continuous spectrum for 1D quasiperiodic ological clocks in plants earned her recognition in operators",Invent.Math.148 (2002), no. 3,453-463. the U.S. and abroad. Birman requested that the lnher Annals paper, she developed anon-perturba­ prize be established to honor her sister's commit­ tive approach to quasiperiodic localization and solved ment to research and to encourage women in sci­ the long-standing Aubry-Andre conjecture on the al­ ence. The prize carries a cash award of $5,000. most Mathieu operator. Her paperwithBourgain con­ The Satter Prize is awarded by the AMS Council tains the first generalnon-perturbative result on the acting on the recommendation of a selection com­ absolutely continuous spectrum. mittee. For the 2005 prize the members of the selection committee were: Karen E. Smith, Jean E. Biographical Sketch Taylor (chair), and Chuu-Lian Terng. Svetlana Jitomirskaya was born on June 4, 1966, Previous recipients of the Satter Prize are: Dusa and raised in Kharkov, Ukraine, in a family of two McDuff (1991), Lai-Sang Young (1993), Sun-Yung accomplished mathematicians (later three, count­ Alice Chang (1995), Ingrid Daubechies (1997), ing her older brother). She received her under­ Bernadette Perrin-Riou (1999), Karen E. Smith graduate degree (1987) and Ph.D. (1991) from (2001), Sijue Wu (2001), and Abigail Thompson Moscow State University. Since 1990 she has held (2003). a research position at the Institute for Earthquake The 2005 Satter Prize was awarded to SVETLANA Prediction Theory in Moscow. ln 1991 she came with JITOMIRSKAYA. The text that follows presents these­ her family to southern California. She was em­ lection committee's citation, a brief biographical ployed by the University of California, Irvine, as a

APRIL 2005 NOTICES OF THE AMS 447 part-time lecturer (1991-92) and rose through the ranks to become a visiting assistant professor • • (1992-94) and then a regular faculty member (since 1994). She took a leave from UCI to spend nine M months at Caltech (1996). She was a Sloan Fellow (1996-2000) and a speaker at the International Congress of Mathematicians in 2002. She is mar­ cs ried and has three children ranging in age from one to seventeen.

Response I am very grateful to the AMS for this honor and International Journal of to the members of the Ruth Lyttle Satter Prize Committee for identifying and selecting me. It is Mathematics and Computer humbling to be on the same list with the past Science recipients of this prize. I must say that I have never felt disadvantaged because of being a woman mathematician; in fact, ISSN 1814-0424 (Print), 1814-0432 (Online) the opposite is true to some extent. However, com­ pared to most others, I did have a unique advan­ Managing Editor: tage: a fantastic role model from early on-my Badih Ghusayni mother, Valentina Borok, who would have been much more deserving of such a prize than I am now, Editorial Boar: had it been available in her time. I see my receiv­ Elliot Ward Cheney Farid Mokrane ing this prize as a special tribute to her memory. Samuel L. Cox Frank Pacard It is a pleasure to use this opportunity to say Carlos de Moura Don Redmond some thanks. It was great to be raised by my par­ Saber Elaydi Chris Rodger ents, and I was lucky to be a student of Yakov Philip Feinsilver Ed Sandifer Sinai, who was both my undergraduate (since 1984) and graduate advisor. I am also very grateful to Abel Sylvain Giroux Rene Schott Klein, whose support and encouragement in the Johnny Henderson Samuli Siltanen postdoctoral years were crucial for my career. I had Nicolas Hadjisavvas Gilbert Strang many wonderful collaborators, from each of whom Deborah Hughes Hallett Agnes Szendrei I learned a lot. Three of those particularly stand out, Hajime Ishihara Raafat Talhouk as they have influenced my work in a major way. Michael Johnson Walter D. Wallis They are, in chronological (for me) order: Barry Peter Johnson, Jr. Richard A. Zalik Simon, Yoram Last, and Jean Bourgain. Each of Pierre-Gilles Lemarie AhmedZayed them has not only introduced new techniques to me and had a visible influence on my style and Technical Editor: choice of topics but also provided a special inspi­ Nabil Azzam ration and changed the way I think about mathe­ matics. I am also grateful to Jean for entering, with IJMCS is a refereed journal which publishes his methods and ideas, the area of quasiperiodic original papers in the broad subjects of operators. That certainly brought this field to a mathematics and computer science written in new level and changed how it is perceived by many others. either English or French. The journal's first issue Finally, special thanks go to my family, as I is targeted for January 2006 (special issue by wouldn't have accomplished a fraction of what I did invitation only). An author may submit his/her without patience, support, and a lot of sacrifice on paper for subsequent issues by visiting their part. http://ijmcs.futureintech.net and sending it to the appropriate member of the Editorial Board. Subscription to the print is via: http://ijmcs.futureintech.net/subscibe.htm and to the online version via: http://ijmcs.futureintech.net/subscribo.htm

448 NOTICES OF THE AMS VOLUME 52, NUMBER 4 2005 Book Prize

The AMS Book Prize was awarded at the Joint Math­ contents of these notes cannot ematics Meetings in Atlanta in January 2005. be considered to be a proof of The Book Prize was established in 2003 to recog­ the geometrization conjecture. nize a single, relatively recent, outstanding research They are instead a manifest, book that makes a seminal contribution to the re­ laying out all the key ideas search literature, reflects the highest standards and explaining how things fit of research exposition, and promises to have a deep together. The book, Three-Di­ and long-term impact in its area. The book must have mensional Geometry and Topol­ been published within the six calendar years pre­ ogy, is the first volume of a ceding the year in which it is nominated. Books may multi-volume work projected be nominated by members of the Society, by mem­ to provide all the details of the bers of the selection committee, by members of AMS proof of Thurston's program. It editorial committees, or by publishers. The prize begins at a quite elementary amount is $5,000. This is the firsttime the prize has level, but takes the reader to a been given. rather sophisticated stage of The Book Prize is awarded by the AMS Council classifying the uniformizing acting on the recommendation of a selection com­ geometries of a compact 3- mittee. For the 2005 prize the members of the selec­ manifold. This result is a rna- William P. Thurston tion committee were: Rodrigo Banuelos, Steven G. jor step of the geometrization Krantz (chair), H. W. Lenstra, Dale P. Rolf sen, and program. Even though the geometrization program Bhama Srinivasan. remains unproved, this is exciting and vital mathe­ The 2005 Book Prize was awarded to WrLUAM P. matics. THURSTON. The text that follows presents the selec­ Thurston's book is nearly unique in the intuitive tion committee's citation and a brief biographical grasp of subtle geometric ideas that it provides. It sketch. has been enormously influential, both for gradu­ ate students and seasoned researchers alike. Cer­ Citation tainly the army of people who are working on the Three-Dimensional Geometry and Topology by geometrization program regard this book as "the William P. Thurston, edited by Silvio Levy. touchstone" for their work. A book that has played William P. Thurston's "Geometrization Program" such an important and dynamic role in modern is one of the big events of modern mathematics. mathematics is eminently deserving of the AMS The main thrust of the program is to prove a clas­ Book Prize. sification of all 3-manifolds by showing that each such manifold can be broken up into pieces, each Biographical Sketch of which admits a geometric structure which William P. Thurston was born October 30, 1946, in is hyperbolic, Euclidean, spherical, or one of Washington, D.C. He received his Ph.D. in mathe­ five other model 3-dimensional geometries. A matics from the University of California at Berke­ corollary of the program would be the Poincare ley in 1972. He held positions at the Institute for conjecture. Advanced Study in Princeton (1972-73) and the More than twenty years ago, Thurston wrote an Massachusetts Institute of Technology (1973-74) extensive set of notes explaining the key ideas of his before joining the faculty of Princeton University program. These notes were circulated informally by in 1974. Thurston returned to Berkeley, this time the Princeton Mathematics Department- a copy as a faculty member, in 1991, and became direc­ could be had for the cost of the photocopying-and tor of the Mathematical Sciences Research Institute today the book is in most mathematics libraries. The in 1993. He moved to UC Davis in 1996 and in

APRIL 2005 NOTICES OF THE AMS 449 2003 accepted his current position at Cornell natural to sketch if I were explaining things one­ University, where he holds joint appointments on-one to someone interested in understanding. in the Department of Mathematics and the Faculty This was all much harder than I anticipated. of Computing and Information Science. The process of writing lecture notes flowed rea­ Thurston held an Alfred P. Sloan Foundation sonably smoothly: I just wrote what I thought were Fellowship from 1974 to 1975. In 1976 he was good conceptual explanations, later filling in awarded the AMS Oswald Veblen Geometry Prize pencil sketches in spaces left by the typist. But as for his work on foliations. In 1979 he became the I started to hear of the arduous efforts people second mathematician ever to receive the Alan T. went through to digest the material, I realized that Waterman Award, and in 1982 he was awarded the it wasn't as easy as I had thought to communicate Fields Medal. He is a member of the American mathematics more directly on a conceptual plane. Academy of Arts and Sciences and the National Concepts that sit easily in the brain are often Academy of Sciences. surprisingly hard to build up or to communicate. There is a big difference between proofs that are Response easy to follow but are hard to hold in mind vs. I feel especially honored to receive the AMS Book proofs that may be hard to get straight but that you Prize for Three-Dimensional Geometry and Topol­ see in a glance once you have them. ogy, because I invested so much of myself in it. In writing the book, my goal was to make the This book grew from a portion of lecture notes ideas more accessible by filling in more of the un­ that I distributed to a large mailing list in the olden spoken assumptions. In doing so, I found it hard to avoid switching over to different brain modes days before the Web, now available electronically athttp://www.msri .erg/publications/books/ that are centered more on language and symbols. At the time I started, the hardest problem was to gt3m/. At the time, I had grown dissatisfied with find a workable system for reasonable renditions the usual vehicles for recording and communicat­ of the many figures. None of the artists or graph­ ing mathematics. One issue has been that the ics professionals I tried seemed to be able to get informal drawings and diagrams that mathemati­ the mathematical relationships correct, and I had cians often use when talking with each other are neither the time nor the skill to draw professional­ quite often missing from papers and books. When looking versions myself. Now, of course, there are I was an undergraduate and graduate student, I computer drawing programs that make the job enjoyed learning to study mathematics in a slow, much easier although still laborious. laborious, step-by-step process, but as my study Personally, my mind always used to turn to jelly of mathematics progressed, I went through many when it came time to do homework in school, and experiences of struggling to digest laborious I have similar difficulties when I try to correct and sequential arguments, finally catching on to edit something left over from past efforts. I would whole-brain, instantaneous ways to understand have given up in despair except for the support the concepts and saying to myself, "Oh, is that and hard work of many people, most notably the what they were talking about? Why didn't they say editing by Silvio Levy, who devoted many hours of so?" I started to realize that written mathematics his multifaceted talents and in particular solved the is usually a highly denatured rendition of what sits figure issue by developing good ways to do almost inside mathematicians' heads. It's a hard task that all the figures using Mathematica, in addition to often never happens to translate a detailed step-by­ working out the language and fixing up many step proof or description into a conceptual under­ mathematical issues. The other really remarkable standing, far harder than to translate from concept contribution was through Al Marden, who orga­ to details. We've all also noticed that in seminar nized several bookwriting workshops involving talks and even in informal one-on-one explana­ many colleagues to help with a task much larger tions of mathematics, it's common for one person than I had foreseen. I owe a huge debt to all these to talk completely past another. Why is it so hard people. to communicate mathematics effectively? In the end, I haven't discovered a solution to the I had the ambition to try to communicate on a problem of how to communicate mathematics ef­ more conceptual level, paying attention not only to fectively, but I have a better grasp of the problem the logical aspects of what is correct but also to the than when I started. In any case, I am tremendously psychological aspects of how we can hold it in our gratified to receive this recognition for my efforts heads and understand it. The geometric modules and the efforts of all the people who helped. of our brains are the parts most severely neglected in most mathematical writing. Many papers, even about geometry and topology, lack the figures, or they have figures that are poor or mistaken. I was determined to include all the figures that would be

450 NOTICES OF THE AMS VOLUME 52, NUMBER 4 2005 Whiteman Prize

The 2005 Albert Leon Whiteman Memorial Prize was mathematics and together awarded at the 111 th Annual Meeting of the AMS constitute a major contribu­ in Atlanta in January 2005. tion to our understanding of The Whiteman Prize is awarded every four years the history of mathematics in to recognize notable exposition and exceptional the spirit of the guidelines set scholarship in the history of mathematics. The for the Whiteman Prize. funds donated prize was established in 1998 using The first of Edwards' sev­ of her husband, by Mrs. Sally Whiteman in memory eral major genetic expositions The prize carries a the late Albert Leon Whiteman. was presented in his book Rie­ cash award of $4,000. mann's Zeta Function (1974), The Whiteman Prize is awarded by the AMS which provides the reader Council acting on the recommendation of a selec­ with a deep mathematical un­ tion committee. For the 2005 prize the members of derstanding of Riemann's the selection committee were: Thomas W. Hawkins seminal paper and the many (chair), Victor]. Katz, and Robert Osserman. investigations that were more The first recipient of the Whiteman Prize was or less inspired by it. His sec­ Thomas Hawkins (2001). ond book, Fermat's Last The­ The 2005 Whiteman Prize was awarded to orem: A Genetic Introduction to Harold Edwards HARoLD EDWARDS. The text that follows presents the Algebraic Number Theory selection committee's citation, a brief biographical (1977), was also of this type, its goal being to in­ sketch, and the awardee's response upon receiving troduce the reader to algebraic number theory by the prize. retracing some of the crucial discoveries in their original contexts and with their original motiva­ Citation tions. In particular, the careful177 -page exposition In awarding the Albert Leon Whiteman Prize to of the work of Kummer that it contains provides Harold Edwards, the American Mathematical Soci­ the reader with a solid understanding of the the­ ety pays tribute to his many publications over ory of algebraic numbers as it was perceived by one several decades that have fostered a greater of the principal founders of the theory. In 1984 Ed­ understanding and appreciation of the history of wards published his third book-length genetic ex­ mathematics, especially the theory of algebraic position. Bearing the title Galois Theory, it focused numbers. Edwards' historical work has all been on a clear exposition of the somewhat cryptic work related to the theory of numbers and has been pre­ of Galois, thereby providing the reader with a sented mainly in two forms: mathematical deeper understanding of the mathematical con­ expositions that are organized in the historical siderations that gave birth to present-day Galois order of development so as to convey a genetic un­ theory. Any historian or mathematician interested derstanding of the relevant mathematical theory, in exploring some aspect of the history of the Rie­ and traditional scholarly historical papers. Both mann zeta function, the theory of algebraic num­ forms combine clear and careful historical schol­ bers, or Galois theory would be wise to begin by a arship with an attendant mastery of the underlying careful study of one of Edwards' books.

APRIL 2005 NoTICEs OF THE AMS 451 Edwards' more traditional scholarly historical penetrate. One of Edwards' signal achievements papers have an evident symbiotic relation with his has been to reconstruct and expound Kronecker's genetic expositions. This is especially true of his theory (as well as Dedekind's reaction to it). He book on Fermat's Last Theorem. The masterful began this process in 'The genesis of ideal theory" account of Kununer's mathematics that it contains and completed it in his book Divisor Theory (1990), has its roots in two important, purely historical which provides the sort of systematic and coher­ papers on "The background of Kummer's proof of ent exposition of divisor theory that Kronecker Fermat's Theorem for regular primes" (19 75, 19 77). himself was never able to achieve. Edwards has also Based on a careful reading of the relevant publi­ used the resultant insights into Kronecker's actual cations and the use of unpublished documents, practice of algebraic number theory to provide a these papers present a clear, accurate, and illumi­ more informed interpretation of his scattered­ nating account of an important-and previously and often misrepresented-remarks on the phi­ poorly understood-episode in the history of losophy of mathematics. (His forthcoming paper algebraic number theory. Among the many insights "Kronecker's Fundamental Theorem of General contained in these papers is a critique of the widely Arithmetic" is a good example.) Although Edwards' accepted view that it was Fermat's conjectured personal sympathy for an intuitionist view of math­ theorem that formed the primary motivation for ematics seems to have been the motivation for Kummer's revolutionary theory of ideal factoriza­ much of his historical work relating to Kronecker, tion. A cogent historical case is made for the view the final products of his efforts are characterized that it was actually the loftier quest for higher by their studied objectivity. They have laid to rest reciprocity laws that inspired Kummer. many unfounded anecdotes about Kronecker and Much of Edwards' subsequent historical research his views that had been promulgated by other focuses upon the two men, Kronecker and Dedekind, historians. who in quite different ways sought to develop Edwards' combination of historical insights and Kummer's ideas beyond the special number fields sound mathematical scholarship make him a he had considered. The first fruits of these efforts worthy recipient of the Whiteman Prize. are contained in his paper "The genesis of ideal theory" (1980). In his publications Edwards is frank Biographical Sketch about his preference for Kummer's approach over Harold M. Edwards was born in Champaign, Illinois, the now-familiar approach eventually developed in 1936. He received a B.A. from the University of by Dedekind. His awareness of his own prejudices Wisconsin in 1956, an M.A. from Columbia in 1957, and their potential for misrepresentation has and a Ph.D. from Harvard in 1961. After teaching resulted in remarkably objective and illuminating at Harvard (1961-62) and Columbia (1962-66), he accounts of the work of both mathematicians. went to New York University in 1966, where he has Indeed, it is perhaps because the final set­ remained. He is now an emeritus professor. He theoretic form of Dedekind's theory is neither as has published seven books: Advanced Calculus obvious nor as natural to Edwards as it is to most (1969, 1980, 1993), Riemann's Zeta Function (1974, present-day mathematicians that he has succeeded 2001), Fermat's Last Theorem (1977), Galois The­ so well in delineating the gradual changes Dedekind ory (1984), Divisor Theory (1990), Linear Algebra made to his theory of ideals, which, as he has (1995), and Essays in Constructive Mathematics shown, actually resembled Kummer's in its initial (2005). versions. His paper "Dedekind's invention of ideals" (1982) summarizes cogent historical arguments Response for the radical nature of Dedekind's eventual I am deeply grateful to be awarded the Whiteman approach to ideal theory and for the likely sources Prize, especially so because I am only the second of his inspiration. recipient, the first having been my esteemed That Dedekind's theory of ideals won out over colleague Thomas Hawkins. the rival generalization of Kununer's theory, namely I must echo the pleasure Tom Hawkins expressed Kronecker's theory of divisors, is due at least in part in his response four years ago at this "manifestation to Dedekind's superior expository skill in pre­ of the importance the AMS attaches to the senting his work. Kronecker, on the other hand, historical study of mathematics" as well as his withheld his ideas on divisor theory from publi­ recollection that "when I committed myself to a cation for decades as he sought to work them career in history of mathematics, there was in out in a suitable form. Then in a paper of 1882, as this country no such recognition of historical work a Festschrift in honor of Kummer, Kronecker by professional mathematical societies." finally put something into print, but, much to the Hawkins's phrase, "the historical study of math­ disappointment of his contemporaries, he did no ematics," strikes me as particularly apt. I have more than present a sketch of his ideas that was always felt that my study was mathematics, not the difficult even for experts such as Dedekind to history of mathematics, but the study of the history

452 NOTICES OF THE AMS VOLUME 52, NUMBER 4 has always been for me the easiest-and often the only-point of entry into the study of a given math­ ematical topic. My book on the zeta function began Aworld leading journal thirty-five years ago with a wish to understand and, I admit, a wish to prove the Riemann hypoth­ - now available online and in print esis. For me, the natural approach was to read Riemann's own words, and after I had studied his cryptic eight-page paper in some detail, I thought ASYMPTOTIC that others might profit from an exposition of what ANALYSIS I had learned. Publishing a work of this sort did not appear then to be a very promising career choice, Editor-in-Chief Alain Bensoussan University of Texas at Dallas but it came from a deeply felt attitude toward the study of mathematics and was more an expression u--- Critical insights for the analysis of a need than a career choice. How gratifying it of asymptotic problems is to have the value of the work done for such a • Original mathematical results in the asymptotic theory of prob­ reason confirmed by this prize! lems affected by the presence of small or large parameters I would like to take advantage of this opportu­ • Possible applications to different fields of natural sciences nity to express my gratitude to three individuals who 2005: Volume 41-45; 20 issues have not been mentioned in the acknowledgements €1355 I US$1560 in any of my books, because their influence on (includes electronic access and print) any one book was so indirect, but who were each ISSN 0921-7134 Recommend Asymptotic Analysis lOS immensely important to my career. First, to Raoul Bott, who was my thesis advisor to your librarian Press more than forty years ago. His plain-spoken, com­ Visit our website at www.iospress.nl for additional information mon-sense approach to mathematics has inspired and to download a free sample copy all who have ever heard him lecture, not to men­ tion those of us who had the good luck to start our research careers with him. Second, to Morris Kline, who would certainly be a prime candidate for this prize if he were still alive. He hired me at NYU, and, being a historian himself, he furthered my historical work in many ways. ommunications in And third, to Uta Merzbach, a valued colleague CPartial Differential who took a very helpful interest in my work and Equations publishes whose sharing with me of her expertise and expe­ high quality papers con­ rience in historical research was my only education cerning any theoretical in such work. aspect of partial differential Thank you to the AMS, to the selection commit­ equations, as well as its applications to other areas tee, and to Mrs. Sally Whiteman, who established of mathematics. The journal the prize in memory of her husband, Albert Leon presents significant advances Whiteman. to a wide readership which includes researchers and mathematics graduate students. Mathematical aspects of physics and engineering are also addressed.

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APRIL 2005 NOTICES OF THE AMS 453 2005 Conant Prize

The 2005 Levi L. Conant Prize was awarded at the In eminently readable and unpretentious style, 111 th Annual Meeting of the AMS in Atlanta in the authors give an account of their approach to January 2005. Weyl's problem. After a brief introduction to the The Conant Prize is awarded annually to recog­ 1962 conjecture of Alfred Horn, which recasts the nize an outstanding expository paper published in Weyl problem in terms of a conjectured series of either the Notices of the AMS or the Bulletin of the inequalities for the eigenvalues of the sum matrix AMS in the preceding five years. Established in A+ B, the authors introduce honeycombs, a type 2000, the prize honors the memory of Levi L. of diagram reminiscent of a beehive. Using 15 Conant (1857-1916), who was a mathematician at clearly explained figures that help one to picture Worcester Polytechnic University. The prize carries various combinatorial nuances, the authors ex­ a cash award of $1,000. pertly lead the reader through the intricacies of The Conant Prize is awarded by the AMS Coun­ their work. They gently transport us from Weyl's cil acting on the recommendation of a selection classical problem to a "quantum" analog, involving committee. For the 2005 prize the members of the the Littlewood-Richardson formula for multiplici­ selection committee were: Anthony W. Knapp, Carl ties of representations of unitary groups within Pomerance (chair), and M. B. Ruskai. tensor products. They then explain the key "satu­ Previous recipients of the Conant Prize are: Carl ration conjecture", which connects the classical Pomerance (2001), Elliott Lieb and Jakob Yngvason and quantum problems to each other and implies (2002), Nicholas Katz and Peter Sarnak (2003), and the validity of Horn's conjecture. Having shown Noam D. Elkies (2004). that the saturation conjecture can be reduced to a The 2005 Conant Prize was awarded to ALLEN problem about honeycombs, they sketch its proof, KNuTSON and TERENCE TAO. The text that follows pre­ all the while playing strongly to the reader's intu­ sents the committee's citation, brief biographical ition. The story that is recounted brushes against sketches, and the awardees' responses upon re­ symplectic geometry, invariant theory, combina­ ceiving the prize. torics, and computational complexity, but the Citation authors deftly keep the reader from getting over­ whelmed by technicalities. The Levil. Conant Prize in 2005 is awarded to Allen By skillfully combining honeycomb diagrams Knutson and Terence Tao for their stimulating arti­ with a high level of exposition, Knutson and Tao cle "Honeycombs and Sums of Hermitian Matrices", make this fascinating subject accessible to a wide Notices oftheAMS48 (2001), no. 2, 175-186. In 1912 Hermann Weyl raised the problem of mathematical audience. characterizing the possible sets of eigenvalues of Allen Knutson the sum A+ B of two Hermitian matrices A, B in terms of the sets of eigenvalues of each of them. Biographical Sketch This is a very natural problem with applications to Allen Knutson did his graduate work in symplec­ many areas, particularly to quantum theory. In tic geometry, overlapping with Terence Tao at particular, it allows one to describe the possible Princeton, where their common love of linear algebra results of measurements of the sum of two brought them together to work on Horn's conjec­ observables in terms of those of the individual ture. He finished up at the Massachusetts Institute observables. Yet surprisingly little progress was of Technology, his third graduate school, the made until a full solution was found in 1998. Soon first being the University of California, Santa Cruz, after, Knutson and Tao introduced the concept thus equalling his number of undergraduate of "honeycombs" and used them to simplify the institutions: Caltech, New York University, and the solution and prove some related conjectures. Budapest Semesters in Mathematics Program.

454 NOTICES OF THE AMS VOLUME 52, NUMBER 4 In addition to the Notices article concerning his Wales (1999-2000), and the and Tao's combinatorial work together, Knutson Australian National University has another one solely on "The symplectic and (2001-03). algebraic geometry of Horn's problem", Linear Tao has been supported by Algebra Appl. 319 (2000), nos. 1-3, 61-81. grants from the National Sci­ After a National Science Foundation postdoc at ence Foundation and fellow­ Brandeis with Gerald Schwarz, Knutson moved in ships from the Sloan Founda­ 1999 to the University of California, Berkeley, where tion, Packard Foundation, and he is now associate professor. His awards include the Clay Mathematics Insti­ a Clay research summer fellowship, a Sloan Fel­ tute. He received the Salem lowship, and the International Jugglers' Association Prize in 2000 and the Bacher world record in two-person ball juggling from 1990 Prize in 2002. to 1995. (The record was for 12 balls; nowadays the Tao's research concerns a record is 13.) number of areas, including har­ monic analysis, geometric and Response arithmetic combinatorics, an­ I am extremely honored and gratified to receive the alytic number theory, nonlin- Allen Knutson Conant Prize-almost as much as to receive the ear evolution equations, and al­ initial invitation to write the article! gebraic combinatorics. One of the most mysterious aspects of the origi­ nal conjecture of Horn was a sort of continuous/ Response discrete schizophrenia, in which real eigenvalues I am deeply touched and ho­ were occasionally required to be natural numbers. noured, and perhaps even a This already suggested that there should be other little surprised, to receive this related, naturally discrete, mathematical fields in award. Allen and I were fasci­ which the "eigenvalues" would be automatically nated by this problem of sum­ integral. Three of these have come up: dominant ming Hermitian matrices ever weights of representations of GL(n), Schubert classes since we were graduate stu­ on Grassmannians, and integral honeycombs. dents, and we were always The work of Totaro and Helmke-Rosenthal, and struck by just how much geo­ its more difficult converse by Klyachko, went back metric, algebraic, and combi­ and forth between the Hermitian matrices and the natorial structure underlies Schubert classes. Ours is pretty much entirely in the this innocuous-sounding combinatorial realm, with honeycombs, hives, and problem. This area of mathematics Terence Tao puzzles. Belkale's proof is entirely in the Schubert domain and is being given a very pretty generaliza­ is highly interdisciplinary, ben- efitting from ideas in fields as diverse as algebraic tion by Purbhoo and Sottile, beyond Grassmannians geometry, symplectic geometry, representation to other "minuscule flag varieties". It still seems theory, enumerative combinatorics, linear algebra, amazing that Horn could guess a recursive state­ and the geometry and combinatorics of convex ment completely within the Hermitian framework! polytopes; this topic seems to draw the interest of The saturation problem (as distinct from Horn's mathematicians from many other fields (for in­ conjecture) seems most naturally stated and studied stance, I myself was drawn to this problem de­ purely within representation theory and has received spite being primarily in analysis). I hope our arti­ a solution recently for general groups by Kapovich cle helps popularize this topic further. (An excellent and Millson, "A path model for geodesics in Euclid­ survey of the field can be found in the reference ean buildings and its applications to representation [F2] in the cited article.) theory", math.RT/0411182. Some further progress has been achieved since the publication of the Notices article. For instance, Terence Tao we now understand that while honeycombs (and Biographical Sketch the closely related Littlewood-Richardson rule) Terence Tao was born in Adelaide, Australia, in "solve" the Hermitian matrix and U(n) tensor prod­ 1975. He received his Ph.D. in mathematics from uct problems, they are in fact much more tightly Princeton University in 1996 under the advisor­ connected with the "infinite negative curvature" or ship of Elias Stein. He has been at the University "zero temperature" variants of those problems. of California, Los Angeles, as a Hedrick assistant Indeed, recent work of Speyer has connected professor (1996-98), assistant professor (1999- honeycombs to a variant of the Hermitian matrix 2000), and professor (2000- ). He has also held vis­ problem in which the underlying field Cis replaced iting positions at the Mathematical Sciences Re­ by the field C { t} of Pusieux series, while recent work search Institute (1997), the University of New South of Henriques and Kamnitzer has also connected

APRIL 2005 NOTICES OF THE AMS 455 ~"': ~ "" /~it &'"' ~"'"~' s ~""!!7f ~;'""'ll~w~ k" ~- w ~10-!%' "'fl~'~ "'' *~J, J=t~r AMERICAN'" MATHEMA'I'I CAL S OCIE·'I'"Y , , ,,~"'" ~~,., '"~ >~ ~ ~'"1-1~~£~0 honeycombs to GLn crystal representations. Important New Releases Meanwhile, there have now been several alternative proofs of the Horn and saturation conjectures (in .from the AMS_-·~--" very different settings) which use completely dif­ Find these and thousands of more ferent techniques, such as Derksen and Weyman's tides on the AMS Bookstore: proof of the saturation conjecture via quiver rep­ resentations, Kapovich-Leeb-Millson's proof of www.ams.org/bookstore. Order today! the saturation conjecture via the theory of modules over discrete valuation rings, or Belkale's geomet­ Conformally Invariant ric proof of the Horn and saturation conjectures via Conformally Processes in the Plane the transversality analysis of Schubert varieties. Invariant ,..,. There are clearly some very rich interconnections 'Processes Gregory F. Lawler, Cornell University, in the Plane between very distinct areas of mathematics here, Ithaca, NY Gregory F. Lawler and there is much that is still left to be done; we Mathematical Surveys and are nowhere close to uncovering the underlying Monographs, Volume 114; 2005; 242 theory which explains all of these connections. For •·---- pages; Hardcover; ISBN 0-8218-3677-3; List $59; All AMS members $47; Order code instance, the situation when the underlying group SURY/114 U(n) (or GLn) is replaced by another Lie group is still only partially understood. Another completely Mathematical Modelling: open question is how these honeycombs "con­ verge" in the large dimensional limit (as n goes to A Case Studies Approach infinity), as there should definitely be some con­ Reinhard Illner, C. Sean Bohun, nection with free probability and free convolution. Samantha McCollum, and Thea van Roode, University ofVictoria, BC, Canada Student Mathematical Library,Yolume 27; 2005; 196 pages; Softcover; ISBN 0-8218-3650-1; List $35;AII AMS members $28; Order code STMU27

Tr>IMWions<>f Hilbert Cy-Modules MATHEMATICAL MONOGRAPHS V. M. Manuilov and E. V. Troitsky, Hilbert C'.,·Modules Moscow State University, Russia V. M. Manuilov E. V. Troitsky Translations of Mathematical Monographs, Volume 226; 2005; approxi­ mately 216 pages; Hardcover; ISBN ®--·- 0-8218-381 0-5; List $85;AII AMS members $68; Order code MMON0/226

The Wild World of 4-Manifolds Alexandru Scorpan, University of Florida, Gainesville, FL 2005; approximately 600 pages; Hardcover; ISBN 0-8218-3749-4; List $69;AII AMS members $55; Order code FOURMAN

April is Mathematics Awareness Month! Visit www.mathaware.org for more information. Other resources on the AMS website ------··--·--···----··-··-······--· --·-·-·------Math in the Media, www.ams.org/mathmedia, a monthly magazine from the AMS Feature Column, www.ams.org/featurecolumn, monthly essays on mathematical topics

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456 NOTICES OF THE AMS VOLUME 52, NUMBER 4 2004 Morgan Prize

The 2004 AMS-MAA-SIAM Frank and Brennie Reid W. Barton. The award is based on the research Morgan Prize for Outstanding Research in Mathe­ paper "Packing densities of patterns". matics by an Undergraduate Student was awarded Packing densities were introduced by Herb Wilf at the Joint Mathematics Meetings in Atlanta in in 1992-93. Some of the early questions were January 2005. settled by Alkes Price, Fred Galvin, and Walter The Morgan Prize is awarded annually for out­ Stromquist. Recent contributions were made by standing research in mathematics by an under­ M. H. Albert, M.D. Atkinson, C. C. Handley, D. A. graduate student (or students having submitted Holton, W. Stromquist, A. Burstein, P. Hast6, and joint work). Students in Canada, Mexico, and the T. Mansour. The main goal of Barton's paper is to United· States or its possessions are eligible for extend the theory of packing densities of permu­ consideration for the prize. Established in 199 5, the tations to that of patterns, i.e. words allowing prize was endowed by Mrs. Frank Morgan of repetition of letters. After resolving the basic con­ Allentown, Pennsylvania, and carries the name of ceptual issues elegantly, Barton delves into the her late husband. The prize is given jointly by the study of packing densities for specific families AMS, the Mathematical Association of America of layered patterns. He proves several important (MAA), and the Society for Industrial and Applied results, some generalizing earlier results by the Mathematics (SlAM) and carries a cash award of above-mentioned authors, some opening up new vistas. Barton also outlines a possible program to $1,000. tackle open questions and formulates new conjec­ Recipients of the Morgan Prize are chosen by a tures. This is all in all a remarkable debut paper in joint AMS-MAA-SIAM selection committee. For the the area of pattern research in combinatorics, an 2004 prize the members of the selection commit­ area of considerable current interest. Commentators tee were: Svetlana R. Katok, Herbert A. Medina, consider Barton's paper the best paper so far on Kris Stewart, Philippe M. Tondeur (chair), and Paul packing densities and praise it for its clarity, new Zorn. techniques, and new results. Previous recipients of the Morgan Prize are: Biographical Sketch Kannan Soundararajan (1995), Manjul Bhargava Reid W. Barton is a senior at the Massachusetts In­ (1996), Jade Vinson (1997), Daniel Biss (1998), Sean stitute of Technology majoring in mathematics. A Mclaughlin (1999), Jacob Lurie (2000), Ciprian resident of Arlington, Massachusetts, Reid began his Manolescu (2001), Joshua Greene (2002), and formal studies in mathematics at Tufts University Melanie Wood (2003). while in middle school. As a high school student, he The 2004 Morgan Prize was awarded to REm W. earned four gold medals at the International Math­ BARTO N. Receiving an honorable mention was ematical Olympiad, placing first with a perfect score Po-SHEN LoH. The text that follows presents the in 2001. That year he also placed first at the Inter­ selection committee's citation, a brief biographical national Olympiad in Informatics, earning his second sketch, and the awardee's response upon receiving IOI gold medal. As an undergraduate, he has been the prize. The same information is provided for the designated a Putnam Fellow the past three years and honorable mention. has been a member of MIT's Putnam team, which placed first in 2003 and second in 2001. Reid has Reid W. Barton also competed on MIT's ACM International Colle­ Citation giate Programming Contest team, finishing fifth and The winner of the 2004 Morgan Prize for Outstanding second at the 2003 and 2001 World Finals respec­ Research in Mathematics by an Undergraduate is tively. An accomplished pianist, Reid performs in

APRIL 2005 NoTICES OF THE AMS 457 MIT Chamber Music Society groups. He is an avid Po-Sherr's research interests, combinatorics and bridge player and also enjoys playing intramural its applications, are the product of this varied back­ soccer, hockey, and ultimate. ground. In his spare time at Cambridge, Po-Shen Response explores topics in other fields, tinkers with com­ I am very honored to receive the 2004 Frank and puters, and enjoys the British countryside with his Brennie Morgan Prize. I would like to thank the AMS­ wife, a fellow Caltech graduate. MAA-SIAMMorganPrize Committee for selecting me Response for this award. I would also like to thank Joe Gallian, I feel very honored to be designated Honorable director of the Duluth REU [Research Experiences for Mention for this award, and I am very grateful to Undergraduates], for providing the opportunity to all of the people involved in organizing this prize do research on a challenging problem in a stimulat­ competition. I would like to mention several insti­ ing environment, and all those affiliated with the tutions and individuals who contributed signifi­ Duluth REU who gave me feedback on my research. cantly to this final result. Caltech provided a special close-knit academic and social atmosphere Honorable Mention: Po-Shen Loh that allowed my creativity and imagination to Citation flourish, and its Summer Undergraduate Research The Morgan Prize Committee is pleased to award Fellowship program gave me the opportunity to honorable mention for the 2004 Morgan Prize for explore various fields of research during the sum­ Outstanding Research in Mathematics by an Un­ mers of 2000, 2001, 2002, and 2003. During those dergraduate to Po-Shen Loh. This recognition is summers I worked for three wonderful Caltech based on his senior thesis at Cal tech on "Random advisors: Alain Martin and Leonard Schulman from graphs and the second eigenvalue problem". computer science, and Emmanuel Candes from His result is a probabilistic estimate. It extends applied mathematics. Leonard Schulman super­ the work of Alon and Roichman involving the vised my 2003 project, which evolved into the second-largest eigenvalue of the Cayley graph of senior thesis that won this Honorable Mention. His a sufficiently large group with respect to a subset guidance was essential. I would also like to recog­ of a certain size. The improvement upon the nize the mathematics department at Caltech, in Alon/ Roichman result comes from replacing the whose supportive company I developed the bulk order of the group by the sum of degrees of its of my mathematical knowledge. Finally, thank you irreducible representations. This is considerably to Debbie, my family, and my friends for your smaller for nonabelian groups in general. consistent support and encouragement. The second-largest eigenvalue of a graph is a characterization of the expansion of the graph, which is an important concept in combinatorics and the theory of computation. Graphs with large expansion are used in the derandomization of algorithms, the design of error correcting codes, and other applications. Their investigations have been an active research area for two decades. Po-Shen Loh's contribution is a nice result and the promise of great things to come. Biographical Sketch Po-Shen Loh received his mathematics degree from Caltech in 2004 and is currently studying mathe­ matics at the University of Cambridge on a one-year Winston Churchill Foundation Scholarship. This fall he will start his Ph.D. at Princeton University, aided by fellowships from the Hertz Foundation and the National Science Foundation. As a grade-school student in Madison, Wisconsin, Po-Shen first developed his dual interests in mathematics and computer science through competitions, repre­ senting the United States at the international level in both subjects. At Caltech these interests migrated to research, thanks to many supportive faculty in the mathematics, applied mathematics, and computer science departments and to Cal­ tech's Summer Undergraduate Research Fellow­ ship program.

458 NOTICES OF THE AMS VOLUME 52, NUMBER 4 MathSciNet Matters

default number of headlines, the default language, The "MathSciNet Matters" column appears in and whether or not to display and print reference the Notices several times a year. It includes lists. information on new features of MathSciNet and How Do You View MR? There is a striking strat­ on the underlying Mathematical Reviews ification by age among mathematicians with respect Database, together with tips on how to use to their awareness of MR. Those over a certain age (not MathSciNet to make the most of its richness of to be mentioned here) think of MR in terms o"f over­ structure and content. sized orange volumes that at one time they pored over for hours. Those under that certain age think New Features in MathSciNet. One of the goals of only of MathSciNet. It is worth noting that the "jour­ this column is to have another medium for telling nal of record" is MathSciNet. For example, correc­ the community about new features in MathSciNet. tions to reviews are made only in MathSciNet, where We work on an annual development cycle that cul­ a complete record of original and updated versions minates each year in September with the release is maintained. MR is committed to upholding the of a new version of the Web interface. The new highest standards of electronic publication. features are described under "What's New" on the Factoid. The number of items added to Math­ MathSciNet home page (http: I /www. ams. o rg/ SciNet in 2004 was 13 percent greater than for math sci net). The planning for the new interface 2003. This number has an implication: more re­ begins early in each calendar year. While sugges­ viewers are needed. Have you considered being a tions are appreciated at any time, the best time to reviewer? It is easy to let us know of your interest. send suggestions is probably in the December to Send email to [email protected]. This is a very February range each year. Use the Support Mail important service to the community. link found at the top of all search pages and help Reviewers Corner. The MR Reception is an pages. annual event at the Joint Mathematics Meetings. It is Keeping Authors Straight. In the January col­ always a good umn there was a discussion of the author database, time, and 2005 which is an important component of the MR Data­ in Atlanta was no base. With over 420,000 authors in the database exception. There mistakes of identification can be made. Items ca~ were refresh­ be attributed to two different people with similar ments, many re­ names that really belong to one person. Conversely, viewers in atten­ items can be attributed to one person when in fact dance, a contest they were authored by two different people. These on MR Database mistakes are straightforward to fix if enough in­ trivia with AMS formation is provided. Use Support Mail to de­ gift certificates as prizes, and a scribe the problem, providing as much information slide show with as you know. Data from before 1985 had author pictures from work done in an automatic way. When there was previous recep- ambiguity, items could be attributed to more than one author on the principle that at least one of them Andrew Bucki tions and all as­ pects of MR. One was correct. This leads to the result that an author of the attendees can be his or her own coauthor. We are very ap­ was Andrew Bucki, who for a number of years has preciative of help from the community with author been the most prolific reviewer. In 2004 he submit­ data. ted seventy-one reviews. All reviewers are warmly The New User Profile. The 2005 interface allows invited to stop by the MRReception atthe 2006 Meet­ for a user profile after sign in. Various things can ings in San Antonio. be saved in this profile, among them the contents -Norman Richert of the clipboard, which otherwise vanish after Administrative Editor, Mathematical Reviews twenty-four hours of inactivity. Various settings on search pages can also be saved. For example, the

APRIL 2005 NOTICES OF THE AMS 459 Mathematics People

boundary is the Heisenberg Stein Receives Bergman Prize group) and using it to obtain EUAs M. STEIN of Princeton University has been awarded the estimates for various opera­ 2005 Stefan Bergman Prize. Established in 1988, the prize tors on strongly pseudocon­ recognizes mathematical accomplishments in the areas of vex domains provides one of research in which Stefan Bergman worked. The prize con­ many examples. Furthermore sists of one year's income from the prize fund. Currently Stein's papers and books are this income is about $17,000 per year. renowned for the exceptional The previous Bergman Prize winners are: David W. quality of their exposition. Catlin (1989), Steven R. Bell and Ewa Ligocka (1991), Charles Stein's program includes a systematic analysis of geo­ Fefferman (1992), Yum Tong Siu (1993), John Erik Forn~ss (1994), Harold P. Boas and Emil]. Straube (1995), David E. metric frameworks determined Barrett and Michael Christ (1997), John P. D'Angelo (1999), both by complex structure and Masatake Kuranishi (2000), Laszlo Lempert and Sidney by collections of real vector fields, therebyinfluencingboth Elias Stein Webster (2001), M. Salah Baouendi and Linda Preiss partial differential equations Rothschild (2003), and Joseph ]. Kohn (2004). On the and CR geometry. Stein has studied the Szeg6 and Bergman selection committee for the 2005 prize were Michael Christ, projections as singular integral operators and developed an John P. D'Angelo, and John Erik Forn~ss (chair). appropriate theory for them, revealing deep connections Citation among real, complex, and harmonic analysis. Stein also de­ veloped a flexible theory of anisotropic function spaces, a The Bergman prize is awarded to Elias M. Stein in recogni­ distinguishing and subtle geometric feature of function the­ tion of his work in real, complex, and harmonic analysis. ory in several complex variables. This work connects analy­ Stein has made decisive contributions through his research, sis on nilpotent Lie groups (including the higher step case) his expository efforts, and his training of graduate students. with analysis in the general settings mentioned above. In particular Stein has developed and realized a visionary Stein's fusion of complex analysis, partial differential program for understanding the off-diagonal behavior of equations, analysis on nilpotent Lie groups, and Euclidean the Bergman and Szeg6 kernels associated to wide classes harmonic analysis has deeply influenced countless mathe­ of smoothly bounded pseudoconvex domains and the maticians. His ideas and techniques will continue to impact mapping properties of the associated projection operators. mathematics for years to come. Stein has realized his program through collaborations with many mathematicians, including G. Folland, L. Rothschild, P. Greiner, A. Nagel, S. Wainger, D. Phong, Biographical Sketch N. Kerzman, J.-P. Rosay, D-C. Chang, J. McNeal, and F. Ricci. Elias M. Stein was born in Belgium in 1931 and came to In this vast body of work they have achieved a deep un­ the U.S. at the age of ten. He received his Ph.D. from the derstanding of the kernels and their mapping properties University of Chicago in 1955. Since 1963 he has taught in many situations: strictly pseudoconvex domains, domains at Princeton University, where he has served twice as chair of finite type in dimension two, convex domains of finite of the mathematics department (1968-71 and 1985-87). type, and special classes of domains in higher dimensions. Stein held a National Science Foundation Postdoctoral Stein's research is noted for the beautiful interplay be­ Fellowship (1955-56), an Alfred P. Sloan Foundation tween detailed analysis of model cases and the insights Fellowship (1961-63), and Guggenheim Fellowships linking the models to general theory. His work analyzing (1976-77 and 1984-85). He was elected to membership the Cauchy-Riemann equations on the Siegel domain (whose in the National Academy of Sciences (1974) and the

460 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Mathematics People

American Academy of Arts and Sciences (1982). He received A native of Canada, the von Humboldt Award (1989-90), the Schock Prize from Bhargava is a professor of the Swedish Academy of Sciences (1993), and the Wolf Prize mathematics at Princeton (1999). In 1984 he was awarded the AMS Steele Prize for University and is also the his book Singular Integrals and the Differentiability Prop­ first Clay Mathematics In­ erties of Functions, published in 19 70 by Princeton stitute Five-Year Long­ University Press. In 2002 he received the Steele Prize for Term Prize Fellow. Among Lifetime Achievement and the U.S. National Medal of his awards are the AMS­ Science. MAA-SIAM Frank and Bren­ nie Morgan Prize for Out­ About the Prize standing Undergraduate The Bergman Prize honors the memory of Stefan Bergman, Research in Mathematics best known for his research in several complex variables, (1997), the MAAMertenM. as well as the Bergman projection and the Bergman ker­ Hasse Prize for Exposition nel function that bear his name. A native of Poland, he (2003), and a Packard Manjul Bhargava taught at Stanford University for many years and died in Foundation Fellowship in 1977 at the age of eighty-two. He was an AMS member for Science and Engineering (2004). thirty-five years. When his wife died, the terms of her will The Leonard M. and Eleanor B. Blumenthal Trust for the stipulated that funds should go toward a special prize in Advancement of Mathematics was created for the purpose her husband's honor. of assisting the Department of Mathematics of the Univer­ The AMS was asked by Wells Fargo Bank of California, sity of Missouri at Columbia, where Leonard Blumenthal the managers of the Bergman Trust, to assemble a com­ served as professor for many years. Its second purpose is to recognize distinguished achievements in the field mittee to select recipients of the prize. In addition, the of math­ ematics through the Leonard M. and Eleanor B. Blumenthal Society assisted Wells Fargo in interpreting the terms of Award for the Advancement of Research in Pure Mathe­ the will to assure sufficient breadth in the mathematical matics, which was originally funded from the Eleanor B. areas in which the prize may be given. Awards are made Blumenthal Trust upon Mrs. Blumenthal's death on July 12, every one or two years in the following areas: (1) the 1987. theory of the kernel function and its applications in real The trust, which is administered by the Financial Man­ and complex analysis, and (2) function-theoretic methods agement and Trust Services Division of Boone County in the theory of partial differential equations of elliptic type National Bank in Columbia, Missouri, pays its net income with attention to Bergman's operator method. to the recipient of the award each year for four years. An independent committee selects the winner(s), restricting its -Allyn jackson attention to work published between eight years and one year before the date the award is presented. The recipient(s) accepts the award in person and presents an address on the Bhargava Receives Blumenthal research for which he or she received the award. Prize -Allyn jackson The Leonard M. and Eleanor B. Blumenthal Award for the Advancement of Research in Pure Mathematics has been presented to MANJUL BHARGAVA of Princeton University. The Harrison Awarded award was presented at the Joint Mathematics Meetings in Atlanta in January 2005. von Neumann Prize Bhargava was given the Blumenthal Award for his doc­ The 2004 John von Neumann Theory Prize, the highest toral dissertation, Higher composition laws. He received his prize given in the field of operations research and man­ Ph.D. from Princeton University in 2001. The prize citation agement science, has been awarded to ]. MICHAEL HARRisoN states: "[In his dissertation] Bhargava found a remarkable of Stanford University for his profound contributions to generalization of Gauss's law of composition on binary qua­ two major areas of operations research and management dratic forms to other prehomogeneous vector spaces. science: stochastic networks and mathematical finance. The Using his new understanding of some of these prehomo­ award, which is presented by the Institute for Operations geneous vector spaces, Bhargava is able to count asymp­ Research and the Management Sciences (INFORMS), carries totically the number of quartic number fields of absolute a cash award of $5,000. discriminant at most x, as x goes to infinity. This prob­ The prize citation reads in part: "Over the past 30 years, lem had been open since Davenport and Heilbronn settled Harrison has spearheaded the formulation, development the corresponding problem for cubic number fields in and application of the theory of Brownian networks for 1971. In his thesis Bhargava also established, for the first performance analysis and control of stochastic processing time, a case of the Cohen-Lenstra-Martinet heuristics on networks. He has defined a framework with elegance and the class groups of cubic number fields." depth.... Under his intellectual leadership, heavy traffic

APRIL 2005 NOTICES OF THE AMS 461 Mathematics People theory has gone from being an esoteric pursuit practiced by a small band of devotees to being a powerful and widely AWM Essay Contest Winners accepted technique used by many researchers in the Announced applied probability/queueing community." The Association for Women in Mathematics (AWM) -From an INFORMS announcement has announced the winners of its 2004 essay contest, "Biographies of Contemporary Women in Mathematics". The grand prize went to SAMANTHA VAN ANH TRAN, a student at Presentation High School, San Jose, California, for her Prizes of the Mathematical essay "I Apply Computational Mathematics to Understand the Natural World-Dr. Margot Gerritsen". This essay won Sodety of Japan first place in the grade 9-12 category and, as grand prize The Mathematical Society of Japan (MSJ) has awarded a winner, will be published in the A HIM Newsletter. The first­ number of prizes for the year 2004. place winner in the college category was STEFANI£ COFORJO ARAI TosHIYASU of Kobe University has been awarded the of Hartwick College, Oneonta, New York, for her essay Autumn Prize for his distinguished contributions to "Delving into Bioinformatics: Dr. Susan B. Davidson, Hilbert's second problem. The Autumn Prize is awarded Professor of Computer and Information Science". First to an individual who has made outstanding contributions place in the middle school category (grades 6-8) went to MAllORY BROWN of St. Gregory College Preparatory School, to mathematics within the preceding five years. Tucson, Arizona, for an essay titled "From Neural Networks The Geometry Prizes have been awarded to SETICHI KAMADA to Mentor Networks: Dr. Mary Poulton Teaches of Hiroshima University and SHIN NAYATANI of Nagoya Uni­ Connections". A complete list of the winners, as well as versity. Kamada was recognized for his fundamental copies of their essays, can be found on the AWM website, research on the foundation of two-dimensional braid and http://www.awm-math.org/biographies/ knot theory. Nayatani was honored for research on the con­ contest/2004.html. struction of invariant metrics (often called Nayatani metrics) on the ideal boundaries of real or complex hyperbolic spaces -From an A HIM announcement and its application to vanishing theorems of cohomology groups. The Analysis Prizes were awarded to MAsAFUMI AKAHIRA of Tsukuba University for work on the higher order asymp­ totic theory of statistical estimation, to KATSUNORJ IwAsAKI of Kyushu University for work on the finite dimensionality of the space of polyhedral harmonic functions and on Painleve equations, and to TAKAAKI NISHIDA of Kyoto University for mathematical analysis on the global structure of solutions of nonlinear partial differential equations. The Takebe Senior Prizes have been awarded to MASASHI ISHIDA of Sophia University for the applications of stable Seiberg-Witten invariants to the geometry of 4-dimensional manifolds, to YAsusm TANIUCHI of Shinshu University for the study of hydrodynamic equations, and to ToMoYUKI ARAKAwA of Nagoya University for his proof of the Frenkel­ Kac-Wakimoto conjecture. The Takebe Junior Prizes were awarded to: KURODA SHIGERU of Kyoto University for the study of invariant rings by combinatorial methods, HIDEKAZU FURUSHO of Nagoya University for the study of p-adic multi-zeta values, HIDEAKI SuNAGAWA of Tsukuba University for the study of asymptotic behavior of solutions of nonlinear Klein-Gordon equations, TETSUYA HosAKA of Utsunomiya University for the study of infinite Coxeter groups and CAT(O) spaces, RYo TAKAHASHI of Okayama University for the study of homological properties of Cohen-Macaulay local rings, and TAKUJI NAKAMURA of Osaka City University for the study of positive knots and canon­ ical Seifert surfaces for knots.

-From a Mathematical Society of Japan announcement

462 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Mathematics Opportunities

NSF Program in Mathematical, AP Calculus Readers Sought Sodal, and Behavioral Sdences The Educational Testing Service and the College Board invite interested college faculty to apply to be Readers for The National Science Foundation's (NSF) Directorate for the Advanced Placement Calculus Exam. The AP Calculus Social, Behavioral, and Economic Sciences (SEE) and the exams (AB and BC) were taken by approximately 225,000 Division of Mathematical Sciences (DMS) invite submis­ high school students last year. For the last few years the sion of research proposals for projects that advance the six free-response problems on the exam have been graded mathematical and statistical foundations of research in during seven days in june by more than 650 high school the social, behavioral, or economic sciences. Proposals and college mathematics teachers at Colorado State for workshops or symposia that foster the interaction of University in Ft. Collins, Colorado. This is an excellent social, behavioral, and economic scientists with mathe­ opportunity for teachers, especially those just starting maticians and statisticians also are welcome. their professional careers, to enhance their knowledge It is estimated that nine to fifteen awards will be made, of the AP Calculus Program and of teaching and to meet ranging in duration from one to four years and carrying with other faculty from around the country. To learn award amounts of $150,000 to $650,000. The deadline date more about this opportunity or to apply for a position for full proposals is April 5, 2005. For more information, as a reader, see the website http:llapcentral. see http:llwww.nsf.govlpubsyslodslgetpub. coll egeboard. com and click on the link to "Faculty cfm?nsfOS 542. Involvement" under the dropdown menu for "Colleges & Universities", or send email to ap reade r@ets. o rg. -From an NSF announcement Questions about the reading may be sent to Caren Diefenderfer, Chief Reader for the AP Calculus Progra:m, Maria l\1itchell Women in [email protected]. Sdence Award -Caren L. Diefenderfer, Hollins UniversityUniversity The Maria Mitchell Association offers an annual award to recognize an individual, program, or organization that en­ AMS Congressional Fellowship courages the advancement of girls and women in studies The AMS, in conjunction with the American Association and careers in science and technology. Maria Mitchell for the Advancement of Science (AAAS), will sponsor a Con­ (1818-1889) was the first woman astronomer and first gressional Fellow from September 2005 through August woman astronomy professor in the United States. 2006. The Fellow will spend the year working on The award may be given in the natural and physical sci­ the staff of a member of Congress or a congressional committee, ences, mathematics, engineering, computer science, or as a special legislative assistant in legislative and policy areas technology. The winner will be chosen by a national jury requiring scientific and technical input. Deadline for of distinguished educators and scientists and will receive applications is March 31, 2005. For further information, a cash award of $5,000. Funding for the award has been consult the webpage http: I lwww. ams. o rgl gove rnmentl provided by an anonymous donor. Guidelines and nomi­ congressfell owann. html orcontacttheAMSWashington nation forms are available from the website http: I I office, email: amsdc@ams. o rg; telephone 202-588-1100. 209.68.19 .123lmuseumslwmninsc. php, or contact the Maria Mitchell Women in Science Award Committee at - Allyn jackson the Maria Mitchell Association, 4 Vestal Street, Nantucket, MA 02554; telephone 508-228-9198. Deadline for nomi­ nations is Aprill5, 2005.

-From a Maria Mitchell Association announcement

APRIL 2005 NOTICES OF THE AMS 463 Reference and Book List

The Reference section of the Notices Upcoming Deadlines Behavioral Sciences. See "Mathemat­ is intended to provide the reader with March 31, 2005: Nominations for In­ ics Opportunities" in this issue. frequently sought information in formation-Based Complexity Prize. April 8, 2005: Proposals for 2005 an easily accessible manner. New Contact Joseph Traub at traub@cs. NSF-CBMS Regional Conferences. See information is printed as it becomes columbia. edu. http: I /www. cbms. o rg or contact: Conference Board of the Mathematical available and is referenced after the April1, 2005: Applications for IMA Sciences, 15 2 9 Eighteenth Street, NW, first printing. As soon as information New Directions Short Course. See is updated or otherwise changed, it Washington, DC 20036; telephone: http://www.ima.umn.edu/new­ 202-293-1170; fax: 202-293-3412; will be noted in this section. directions/2005NDshort-course/ email:[email protected] NDcourse-app.php. Contacting the Notices or rosi er@math. georgetown. edu. April 5, 2005: Proposals for NSF April 15, 2005: Nominations for The preferred method for contacting Program in Mathematical, Social, and Maria Mitchell Women in Science the Notices is electronic mail. The editor is the person to whom to send Where to Find It articles and letters for consideration. A brief index to information that appears in this and previous issues of the Articles include feature articles, Notices. memorial articles, communications, AMS Bylaws-November 2003, p. 1283 opinion pieces, and book reviews. The AMS E-mail Addresses-December 2004, p. 1365 editor is also the person to whom to AMS Ethical Guidelines-June/July 2004, p. 675 send news of unusual interest about AMS Officers 2002 and 2003 (Council, Executive Committee, other people's mathematics research. Publications Committees, Board of Trustees)-May 2004, p. 566 The managing editor is the person AMS Officers and Committee Members-October 2004, p. 1082 to whom to send items for "Mathe­ Conference Board of the Mathematical Sciences-September 2004, matics People", "Mathematics Op­ p. 921 portunities", "For Your Information", Information for Notices Authors-june/July 2004, p. 670 "Reference and Book List", and "Math­ Mathematics Research Institutes Contact Information-August 2004, ematics Calendar". Requests for . p. 810 permissions, as well as all other National Science Board-january 2005, p. 76 inquiries, go to the managing editor. New journals for 2003-june/july 2004, p. 672 The electronic-mail addresses are NRC Board on Mathematical Sciences and Their Applications-March 2005, noti ces@math. ou. edu in the case of p. 361 the editor and noti ces@ams. org in NRC Mathematical Sciences Education Board-April 2005, p. 465 the case of the managing editor. The NSF Mathematical and Physical Sciences Advisory Committee-February fax numbers are 405-325-7484 for 2005, p. 261 the editor and 401-331-3842 for the Program Officers for Federal Funding Agencies-October 2004, managing editor. Postal addresses p. 1078 (DoD, DoE); December 2004, p. 1368 (NSF) may be found in the masthead.

464 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Reference and Book List

Award. See "Mathematics Opportuni­ Bat. B 37, B-4000 Liege 1, Belgium; National Academy of Sciences, 500 ties" in this issue. email: j aghi on@ul g. ac. be. Fifth Street, NW, 11th Floor, Wash­ May 1, 2005: Applications for AWM January 1, 2006: Submissions for ington, DC 20001; telephone 202- Travel Grants. See the AWM website, Competition 2006 of the European 334-3294; fax 202-344-1453; email: http://www.awm-math.org/ Mathematical Society. See the website mseb@nas. edu; World Wide Web travel grants. html; telephone: 301- http://www.mat.dtu.dk/people/ http://www7.nationalacademies. 405-7892; email: awm@math. umd. edu. V.L.Hansen/rpa/seeondarteomp. org/mseb/MSEB_Membership.html. May 31, 2005: Registration for In­ html. ternational Mathematics Competition January 1, 2006: Applications for Book List for University Students. See the web­ ICM 2006 Travel Grants. See http: I I The Book List highlights books that site http: I /www. i me-math. o rg or www. i em2006. org or email: grants@ have mathematical themes and are contact John E. Jayne, Department of i cm2006. org. aimed at a broad audience potentially Mathematics, University College Lon­ including mathematicians, students, don, Gower Street, London WClE 6BT, Mathematical Sciences and the general public. When a book United Kingdom; telephone +44-20- Education Board, National has been reviewed in the Notices, a 7679 7322; fax +44-20-7419 2812; Research Council reference is given to the review. Gen­ email: j . j ayne@i me-math. o rg. Thomas Banchoff, Brown University erally the list will contain only books June 1, 2005: Applications for fall ]an de Lange, Freudenthal Institute, published within the last two years, program of the Christine Mirzayan Sci­ The Netherlands though exceptions may be made in ence and Technology Policy Graduate Louis Gomez, Northwestern Uni­ cases where current events (e.g., the Fellowship Program of the National versity death of a prominent mathematician, Academies. See http: I /www7. Javier Gonzalez, Pioneer High coverage of a certain piece of mathe­ nationalaeademies.org/ School, Whittier, CA matics in the news) warrant drawing pol i cyfe ll ows or contact The Na­ Sharon Anne Griffin, Clark Uni­ readers' attention to older books. Sug­ tional Academies Christine Mirzayan versity gestions for books to include on the list Science and Technology Policy Arthur Jaffe, Harvard University may be sent to noti ees-bookl i st@ Graduate Fellowship Program, 500 Eric Jolly, Science Museum of ams .org. Fifth Street, NW, Room 508, Washing­ Minnesota *Added to "Book List" since the ton, DC 20001; telephone: 202-334- jeremy Kilpatrick, University of list's last appearance. 2455; fax: 202-334-1667. Georgia June 2, 2005: Applications for NSF joan Leitzel (chair), University of 13: The Story of the World's Most University-Industry Cooperative Re­ New Hampshire Popular Superstition, by Nathaniel search Programs in the Mathematical Jim Lewis, University of Nebraska, Lachenmeyer. Thunder's Mouth Press, Sciences (UICRP). See http: I /www. Lincoln October 2004. ISBN 1-568-58306-0. nsf.gov/pubsys/ods/getpub. Karen Michalowicz, The Langley 1089 and All That. A journey into efm?nsfOS 504. School, McLean, VA Mathematics, by David Acheson. Ox­ June 30, 2005: Nominations for Kevin F. Miller, University of Michi­ the 2005 Fermat Prize. See http: I I gan, Ann Arbor ford University Press, July 2002. ISBN www.ups-tlse.fr/ACTUALITES/ Marge Petit (vice chair), Consultant, 0-19-851623-1. (Reviewed February Seienees/Prix_Fermat_2004/ North Fayston, VT 2005.) Aregl ement. html. Donald Saari, University of Across the Board: The Mathematics July 31, 2005: Nominations and California, Irvine of Chessboard Problems, by John ]. applications for the Monroe H. Martin Nancy]. Sattler, Terra State Watkins. Princeton University Press, Prize. Contact R. Roy, Director, Insti­ Community College April2004. ISBN 0-691-11503-6. tute for Physical Science and Technol­ Richard ]. Schaar, Texas Instru­ Adam Spencer's Book of Numbers, ogy, University of Maryland, College ments by Adam Spencer. Four Walls Eight Park, MD 20742-2431. Richard Scheaffer, University of Windows, January 2004. ISBN 1-568- August 1, 2005: Submissions for Florida 58289-7. Competition 2005 of the European Frank Wang, Oklahoma School of Alan Turing: Life and Legacy of a Mathematical Society. See the website Science and Mathematics Great Thinker, edited by Christof http://www.mat.dtu.dk/people/ Teuscher. Springer, 2004. ISBN 3-540- V.L.Hansen/rpa/secondartcomp. MSEB Staff 20020-7. html. David R. Mandel, Director Alfred Tarski: Life and Logic, by October 1, 2005: Nominations for Vicki Stohl, Research Associate Anita Burdman Feferman and Lucien Godeaux Prize. Contact Dianna Williams, Senior Program Solomon Feferman. Cambridge Uni­ ]. Aghion, c/o Secretariat of the Royal Assistant versity Press, October 2004. ISBN Society of Sciences of Liege, Institute 0-521-80240-7. of Mathematics of the University of The contact information is: Math­ Alpha and Omega: The Search for Liege, 12 Grande Traverse, Sart Tilman ematical Sciences Education Board, the Beginning and End of the Universe,

APRIL 2005 NOTICES OF THE AMS 465 Reference and Book list by Charles Seife. Viking, July 2003. Kevin C. Knox and Richard Noakes. Gouvea. Oxton House, 2002. ISBN ISBN 0-6 70-03179-8. Cambridge University Press, Novem­ 1-881929-21-3. (Reviewed October Automated Reasoning and the Dis­ ber 2003. ISBN 0-521-66310-5. 2004.) covery of Missing and Elegant Proofs, Gamma: Exploring Euler's Constant, The Mathematical Century: The 30 by Larry Wos and Gail Pieper. Rinton by Julian Havil. Princeton University Greatest Problems of the Last 1 00 Press, December 2003. ISBN 1-58949- Press, May 2003. ISBN 0-691-09983-9. Years, by Piergiorgio Odifreddi, trans­ 023-1. (Reviewed August 2004.) lated by Arturo Sangalli. Princeton Beyond Coincidence, by Martin Geometry: Our Cultural Heritage, University Press, May 2004. ISBN Plimmer and Brian King. Icon Books, by Audun Holme. Springer, April 0-691-09294-X. March 2004. ISBN 1-840-46 5 34-4. 2002. ISBN 3-540-41949-7. (Reviewed Mathematical Journeys, by Peter D. 1' Beyond Reason: Eight Great Prob­ May 2004.) Schumer. Wiley lnterScience, February lems That Reveal the Limits of Science, The Golden Ratio: The Story of Phi, 2004. ISBN 0-471-22066-3. by A. K. Dewdney. Wiley, April 2004. the World's Most Astonishing Number, A Mathematician at the Ballpark: ISBN 0-471-01398-6. by Mario Livio. Broadway Books, Odds and Probabilities for Baseball Cogwheels of the Mind: The Story of October 2002. ISBN 0-767-90815-5. Fans, by Ken Ross. Pi Press, July 2004. Venn Diagrams, by A. W. F. Edwards. (Reviewed March 2005.) ISBN 0-131-4 7990-3. Johns Hopkins University Press, April A Handbook of Mathematical Dis­ Mathematicians as Enquirers: Learn­ 2004. ISBN 0-801-87434-3. course, by Charles Wells. Infinity ing about Learning Mathematics, edited Constantin Caratheodory: Mathe­ Publishing Company, 2003. ISBN by Leone Burton. Kluwer, April 2004. matics and Politics in Turbulent Times, 0-7 414-168 5-9. (Reviewed September Hardbound, ISBN 1-4020-7853-6; pa­ by M. Georgiadou. Springer, Septem­ 2004.) perback, ISBN 1-4020-7859-5; eBook, ber 2004. ISBN 3-540-44258-8. The Heart of Mathematics: An ISBN 1-4020-7908-7. The Constants of Nature: From Invitation to Effective Thinking, by Mathematicians under the Nazis, by Alpha to Omega-The Numbers That Edward B. Burger and Michael Sanford L. Segal. Princeton University Encode the Deepest Secrets of the Uni­ Starbird. Key College Publishing Press, July 2003. ISBN 0-691-00451-X. verse, by John D. Barrow. Jonathan (Springer-Verlag), April 2000. ISBN (Reviewed in this issue.) Cape, September 2002. Pantheon 0-555953-407-9. (Reviewed February Mathematics: A Very Short Intro­ Books, January 2003. ISBN 0-375- 2005.) duction, by Timothy Gowers. Oxford 42221-8. (Reviewed November 2004.) Karl Pearson: The Scientific Life in University Press, October 2002. ISBN Count Down: Six Kids Vie for Glory a Statistical Age, by Theodore M. 0-192-85361-9. (Reviewed February at the World's Toughest Math Compe­ Porter. Princeton University Press, 2005.) tition, by Steve Olson. Houghton Mif­ February 2004. ISBN 0-691-11445-5. Mathematics and War, edited by flin, April2004. ISBN 0-618-25141-3. Kepler's Conjecture: How Some of Bernhelm Booss-Bavnbek and ]ens (Reviewed August 2004.) the Greatest Minds in History Helped H0yrup. Birkhauser, December 2003. The Curious Life of Robert Hooke, Solve One of the Oldest Math Prob­ ISBN 3-764-31634-9. the Man Who Measured London, by lems in the World, by George G. Szpiro. Mathematics in Nature: Modeling Lisa Jardine. HarperCollins, February Wiley, January 2003. ISBN 0-471- Patterns in the Natural World, by John 2004. ISBN 0-060-53897-X. 08601-0. (Reviewed January 2005.) Adam. Princeton University Press, No­ Everything and More: A Compact The Knot Book: An Elementary In­ vember 2003. ISBN 0-691-11429-3. History of Infinity, by David Foster troduction to the Mathematical Theory Meta Math! The Quest for Omega, Wallace. W. W. Norton, October 2003. of Knots, Colin C. Adams. AMS, Sep­ by Gregory ]. Chaitin. April 2004. ISBN 0-393-00338-8. (Reviewed tember 2004. ISBN 0-8218-3678-1. Available at http: I /www. cs. June/July 2004.) The Liar Paradox and the Towers umaine.edu/~chaitin/omega.html. The Fabric of the Cosmos, by Brian of Hanoi: The Ten Greatest Math Puz­ The (Mis)Behavior of Markets: A Greene. Knopf, February 2004. ISBN zles of All Time, by Marcel Danesi. Fractal View ofRisk, Ruin and Reward, 0-375-41288-3. Wiley, August 2004. ISBN 0-471- by Benoit Mandelbrot and Richard Fields Medalists' Lectures, edited 64816-7. Hudson. Basic Books, August 2004. by Sir Michael Atiyah and Daniel Masters of Theory: Cambridge and ISBN 0-465-04355-0. Iagolnitzer. World Scientific, 2nd the Rise of Mathematical Physics, by More Damned Lies and Statistics: edition, December 2003. ISBN 9-812- Andrew Warwick. University of How Numbers Confuse Public Issues, 38259-3. Chicago Press, July 2003. ISBN 0-226- by Joel Best. University of California From Eudoxus to Einstein: A His­ 87375-7. Press, August 2004. ISBN 0-520- tory of Mathematical Astronomy, by Math Magic: How to Master Every­ 23830-3. C. M. Linton. Cambridge University day Math Problems, by Scott Flansburg. More Mathematical Astronomy Press, August 2004. ISBN 0-521- Perennial Currents, revised edition, Morsels, by Jean Meeus. Willmann-Bell 82750-7. August 2004. ISBN 0-060-72635-0. Inc., 2002. ISBN 0-943396-743. From Newton to Hawking: A History Math through the Ages: A Gentle The Music of the Primes: Searching of Cambridge University's Lucasian History for Teachers and Others, by to Solve the Greatest Mystery in Professors of Mathematics, edited by WilliamP. Berlinghoff and Fernando Q. Mathematics, by Marcus du Sautoy.

466 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Reference and Book List

HarperCollins, April 2003. ISBN The Universal Book ofMathematics: 0-066-21070-4. From Abracadabra to Zeno's Para­ The Number rr, by Pierre Eymard doxes, by David Darling. Wiley, July and Jean-Pierre Lafon. AMS, 2004. 2004. ISBN 0-471-27047-4. ISBN 0-8218-3246-8. A World without Time: The Forgot­ Number Theory from an Analytic ten Legacy of Godel and Einstein, by Point of View, by Badih Ghusayni. Palle Yourgrau. Basic Books, January Komati, December 2003. ISBN 9953- 2005. ISBN 0-465-09293-4. 0-0282-7. You Can Do the Math: Overcome Phase Change: The Computer Rev­ Your Math Phobia and Make Better olution in Science and Mathematics, Financial Decisions, by Ron Lipsman. by Douglas S. Robertson. Oxford Praeger Publishers, November 2004. University Press, March 2003. ISBN ISBN 0-2 75-98341-2. 0-195-15748-6. Portraits of the Earth: A Mathe­ matician Looks at Maps, by Timothy G. Feeman. AMS, September 2002. ISBN 0-8218-3255-7. Prime Obsession: Bernhard Rie­ mann and the Greatest Unsolved Problem, by John Derbyshire. Joseph Henry Press, March 2003. ISBN 0-309-08549-7. Probability Theory: The Logic of Science, by E. T. Jaynes, edited by G. Larry Bretthorst. Cambridge Uni­ versity Press, April2003. ISBN 0-521- 59271-2. The Reader of Gentlemen's Mail: , / Herbert 0. Yardley and the Birth of American Codebreaking, by David Kahn. Yale University Press, March 2004. ISBN 0-300-09846-4. The Riemann Hypothesis: The Greatest Unsolved Problem in Mathematics, by Karl Sabbagh. Farrar Straus & Giroux, April 2003. ISBN 0-374-25007-3. The Saga of Mathematics: A Brief History, by Marty Lewinter and William Widulski. Prentice Hall, January 2002. ISBN 0-130-34079-0. Shooting the Sun, by Max Byrd. Bantam, December 2003. ISBN 0-553- 80208-9. Signs of the Inka Khipu: Binary Cod­ ing in the Andean Knotted-String Records, by Gary Urton. University of Texas Press, August 2003. ISBN 0-292- 78540-2. Strange Curves, Counting Rabbits, and Other Mathematical Explorations, by Keith Ball. Princeton University Press, November 2003. ISBN 0-691- 11321-1. (Reviewed in December 2004.) Towards a Philosophy ofReal Math­ ematics, by David Corfield. Oxford University Press, April 2003. ISBN 0-521-81722-6.

APRIL 2005 NOTICES OF THE AMS 467 AMERICAN MATHEMATICAL SOCIETY

The selection committees for these prizes request nominations for consideration for the 2006 awards, which will be presented at the Joint Mathematics Meetings in San Antonio, TX, in January 2006. Information about most of these prizes may be found in the November 2003 Notices, pp. 1289-1299. (Also available at 7 http: I I www.ams.orgl prizes-awards.) PRIZE The George David Birkhoff Prize is awarded jointly by the AMS and SIAM for an outstanding contribution to applied mathematics in its highest and broadest sense. The award was first made in 1968 and now is presented every third year.

The Frank Nelson Cole Prizes are now presented at three year inter­ vals for outstanding contributions in algebra and number theory. The PRIZES award in January 2006 will be the Frank Nelson Cole Prize in Algebra.

The Levi L. Conant Prize, first awarded in January 2001, is presented annually for an outstanding expository paper published in either the Notices or the Bulletin of the American Mathematical Society during the ~t_X._~a_anL preceding five years. PRIZE The Award for Distinguished Public Service is presented every two years to a research mathematician who has made a distinguished contribution to the mathematics profession during the preceding five years.

Nominations should be submitted to the secretary, Robert J. Daverman, DISTINGUISHED American Mathematical Society, 312D Ayres Hall, University of Tennessee, Knoxville, TN 37996-1330. Include a short description of 9a~lic 9e~otce the work that is the basis of the nomination, with complete biblio­ AWARD graphic citations when appropriate. A brief curriculum vitae should be included for the nominee. The nominations will be forwarded by the secretary to the appropriate prize selection committee, which, as in the past, will make final decisions on the awarding of these prizes.

Deadline for nominations is June 30, 2005. he prize is awarded each year to an Tundergraduate student (or students having submitted joint work) for out­ standing research .in mathematics. Any stu­ dent who is an undergraduate in a college or ' university in the United States or its posses­ sions, or Canada or Mexico, is eligible to be considered for this prize. he recipients of the prize are to be selected by a standing joint committee he prize recipient's research need not T of the AMS, MAA, and SIAM. The deci­ be confined to a single paper; it may T sions of this committee are final. The 2005 be contained in several papers. prize will be awarded for papers submitted However, the paper (or papers) to be consid­ for consideration no later than June 30, ered for the prize must be submitted while 2005, by (or on behalf of) students who were the student is an undergraduate; they can­ undergraduates in December 2004. not be submitted after the student's gradua­ tion. The research paper (or papers) may be submitted for consideration by the student or a nominator. All submissions for the prize must include at least one letter of sup­ sent to: · port from a person, usually a faculty mem­ Morgan Prize Committee ber, familiar with the student's research. c/o RobertJ. Daverman, Secretary Publication of research is not required. American Matf:Jematical Society 312D Ayres Hall University of Tennessee • • • • Krioxvil'le, TN 37996-1330 Questions may be directed: Dr. 1\11arthaJ. Siege'l, MAA Secretary Mathematics Department Stephens H~ll302 .. Towson U~iversity 8000 York Road Towson, MD 21252-0001 telephone: 410-704-3091 e-mail: msi egel@towson. edu CAMBRIDGE

NEW AND NOTEWORTHY FROM CAMBRIDGE

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Richard Arratia, Simon Tavare (both University of Southern California, USA) and Andrew Barbour (University of Zurich, Switzerland) Logarithmic Combinatorial Structures: A Probabilistic Approach (EMS Monographs in Mathematics) ISBN 3-03719-000-0, 2003,352 pages, hardcover, 16.5 em x 23.5 em, 69.00 Euro The elements of many classical combinatorial structures can be naturally decomposed into components. For example, permutations can be decomposed into cycles, polynomials over a finite fi eld into irreducible factors, mappings into connected components. This book explains the similarities in asymp­ totic behaviour as the result of two basic properties shared by the structures: the conditioning relation and the logarithmic condition. The discussion is conducted in the lang uage of probability; the book is therfore of particular interest to graduate students and researchers in both combinatorics and probability theory. Jmn Justesen and Tom H0holdt (both Technical University of Denmark) A Course In Error-Correcting Codes (EMS Textbooks in Mathematics) ISBN 3-03719-001-9, 2004, 160 pages, hardcover, 16.5 em x 23.5 em, 39.50 Euro Th is book is written as a text for a course aimed at 3rd or 4th year students. Only some familiarity with elementary linear algebra and probability is directly assumed, but some maturity is required. The students may special ize in discrete mathematics, computer science, or communication engineer­ ing. The book is also a suitable introduction to coding theory for researchers from related fields or for professionals who want to supplement their the­ oretical basis. The book gives the coding basics for working on projects in any of the above areas, but material specific to one of these fields has not been included. The chapters cover the codes and decoding methods that are currently of most interest in research, development, and application.

Katrin Weh rheim (Princeton University, USA) Uhlenbeck Compactness (EMS Series of Lectures in Mathematics) ISBN 3-03719-004-3, 2004, 250 pages, softcover, 17.0 em x 24.0 em, 39.50 Euro This book gives a detailed account of the analytic foundations of gauge theory- Uhlenbeck's compactness theorems for general connections and for Yang-Mills connections. It intends to guide graduate students into the analysis of Yang-Mills theory as wel l as to serve as a reference for researchers in the field. The book is largely self-contained. It contains a number of appendices (e.g. on Sobolev spaces of maps between manifolds) and an introductory part covering the Lp-regularity theory for the inhomogenous Neumann problem .

Arkady Onishchik (Yaroslavl State University, Russia) Lectures on Real Semisimple Lie Algebras and Their Representations (ESI Lectures in Mathematics and Physics) ISBN 3-03719-002-7, 2003, 100 pages, softcover, 17.0 em x 24.0 em, 24.00 Euro In 1914, E. Cartan posed the problem to find all irreducible real linear Lie algebras. A general solution of this problem is contained in a paper of Karpelevich (1955) (written in Russian and not wide ly known), where inclusions between real forms induced by a complex representat ion were stud­ ied. We begin with an exposition of the main part of this paper and relate it to the theory of Cartan-lwahori. We conclude with some tables, where an involution of the Dynkin diagram which allows us to find self-conjugate representations is described and explicit formulas for the index are given. The book is aimed at students and research ersin Lie groups, Lie algebras and their representations.

Athanase Papadopoulos (IRMA, Strasbourg, France) Metric Spaces, Convexity and Nonpositive Curvature (IRMA Lectures on Mathematics and Theoretical Physics) ISBN 3-03719-010-8; 2004,300 pages, softcover, 17.0 em x 24.0 em; 48.00 Euro This book is about metric spaces of non positive curvature in the sense of Busemann, that is, metric spaces whose distance function satisfies a convexi­ ty condition . The book also contains a systematic introduction to the theory of geodesics, as wel l as a detailed presentation of some facets of convexity theory that are useful in the study of nonpositive curvature. The concepts and the techniques are illustrated by many examples from classical hyperbolic geometry and from the theory ofTeichmuller spaces. The book is useful for students and researchers in geometry, topology and analysis.

Yakov Pesin (Pennsylvania State University, USA) Lectures on Partial Hyperbolicity and Stable Ergodicity (Zurich Lectures in Advanced Mathematics) ISBN 3-03719-003-5, 2004, 144 pages, softcover, 17.0 em x 24.0 em, 28.00 Euro This book is an introduction to the modern theory of partial hyperbolicity with applications to stable ergodicity theory of smooth dynamical systems. It provides a systematic treatment of the theory and describes all the basic concepts and major results that have been obtained in the area since its cre­ ation around the early 1970s. It can be used as a textbook for a graduate student course and is also of interest to professional mathematicians work­ ing in the field of dynamical systems and their applications.

Sun-Yung Alice Chang (Princeton University, USA) Non-linear Elliptic Equations in Conformal Geometry (Zurich Lectures in Advanced Mathematics) ISBN 3-03719-006-X; 2004, 100 pages, softcover, 17.0 em x 24.0 em; 24.00 Euro In these lectures, starting from the background material, the author reviews the problem of prescribing Gaussian curvature on compact surfaces. She then develops the analytic tools (e.g. higher order conformal invariant operators, Sobolev inequalities, blow-up ana lysis) in order to solve a fully non ­ linear equation in prescribing the Chern-Gauss-Bonnet integrand on compact manifolds of dimension four.

European Mathematical Society Publishing House Fli ederstrasse 23 [email protected] Seminar for Applied Mathematics, ETH-Zentrum FLI C4 CH-8092 ZOrich, Switzerland www.ems-ph.org Mathematics Calendar

The most comprehensive and up-to-date Mathematics Calendar information is available one-MATH at ht tp://www.ams.org/mathcal/.

April 2005 May 2005 * 8- 9 Workshop on Sampling, Spectral Theory and Their Applica­ '' 4-5 L atent Variable Models and Survey Data for Social and tions, University of West Georgia, Carrollton, Georgia. Health Sciences, Centre de recherches mathematiques, Universite Organizers : Amin Boumenir (UWG), email: boumeni r@westga. edu; de Montreal, Montreal, Quebec, Canada. Vu Kim Tuan (UWG), email: vu@westga . edu. Organizer: Mary E. Thompson (University of Waterloo). Invited Speakers: L. Littlejohn (Univ. of Utah), H. Volkmer (Milwau­ kee Univ.), A. Zayed (DePaul Univ.). '' 9-July 3 Uncertainty and Information in Economics, Institute for Description: This workshop is intended for researchers and grad­ Mathematical Sciences, National University of Singapore, Singapore. uate students working on Sampling and Spectral Theory and Their Description: The program will consist of tutorials and a workshop Applications. The two-day workshop consists of three mini series with ample opportunities for collaborative research among local of lectures delivered by experts in the area and includes the latest and international participants. In addition, the Tenth Conference trends, updates and methods. Each speaker will deliver 3 one-hour on Theoretical Aspects of Rationality and Knowledge will also be lectures; no other contributions are planned. Due to the restricted held on June 10-12, 2005 in conjunction with the workshop in the number of seats, participants should register by April 1st, 2005 by program. e-mailing one of the organizers. The organizers have no funding Progra m: Introducing probabilistic analysis to manage uncer­ to support the attendance of participants. tainty and limited information in basic microeconomic models Information: Contact Micky Weldon, email: mwe l don©westga. edu, has enriched our understanding of economic behavior and made tel: 678-839-6489. fundamental advances in the mathematical theory of economics. This program will have three sub-themes: game theory, information '' 29-May l 2005 Midwest Geometry Conference, Ohio State Uni­ economics, and finance, with uncertainty and information as the versity, Columbus, Ohio. underlying thread connecting these sub-themes. Specific topics Workshop Topics: Positively curved manifolds, Ricci flow and include basic game theory, coalition formation, auctions, incen­ solitons, geometric group theory. tive compatibility, automated and algorithmic mechanism design, Local Organizers: Andrzej Derdzinski, Tadeusz }anuszkiewicz, equilibrium and asset pricing. In addition to participating in the Fangyang Zheng. workshop and tutorials, program visitors will give seminars related Plenary Speakers: Huai-Dong Cao (Lehigh), Ben Chow (UC San to the core themes and engage in research interactions and new Diego), Claude LeBrun (Stony Brook), Igor Mineyev (Urbana­ collaborations. Champaign), Xiaochun Rong (Rutgers), Ralf Spatzier (Ann Arbor). Confirmed participators: Robert Anderson (University of Califor­ Information: Status: Conference is tentative pending approval of nia at Berkeley), Bernard Cornet (Universite Paris 1 and University of support from NSF. http: I /www .math. ohio- stat e. edu;-andrzej/ Kansas), Pradeep Dubey (State University of New York, Stony Brook), mgc05.html Joseph Y. Halpern (Cornell University), Peter Hammond (Stanford

This section contains announcements of meetings and conferences respect to participation in the meeting, this fact should be noted. of interest to some segment of the mathematical public, including ad All communications on meetings and conferences in the mathematical hoc, local, or regional meetings, and meetings and symposia devoted sciences should be sent to the Editor of the Notices in care of the American to specialized topics, as well as announcements of regularly scheduled Mathematical Society in Providence or electronically to notices©ams. org meetings of national or international mathematical organizations. A or mathcal©ams. or g. complete list of meetings of the Society can be found on the last page of In order to allow participants to arrange their travel plans, organizers of each issue. meetings are urged to submit information for these listings early enough An announcement will be published in the Notices if it contains a call to allow them to appear in more than one issue of the Notices prior to for papers and specifies the place, date, subject (when applicable), and the meeting in question. To achieve this, listings should be received in the speakers; a second announcement will be published only if there Providence eight months prior to the scheduled date of the meeting. are changes or necessary additional information. Once an announcement The complete listing of the Mathematics Calendar will be published has appeared, the event will be briefly noted in every third issue until only in the September issue of the Notices. The March, June/July, and it has been held and a reference will be given in parentheses to the December issues will include, along with new announcements, references month, year, and page of the issue in which the complete information to any previously announced meetings and conferences occurring within the twelve-month period appeared. Asterisks (*) mark those announcements containing new or following the month of those issues. New revised information. information about meetings and conferences that will occur later than In general, announcements of meetings and conferences held in North the twelve-month period will be announced once in full and will not be America carry only the date, title of meeting, place of meeting, names of repeated until the date of the conference or meeting falls within the speakers (or sometimes a general statement on the program), deadlines twelve-month period. for abstracts or contributed papers, and source of further information. The Mathematics Calendar, as well as Meetings and Conferences of Meetings held outside the North American area may carry more detailed the AMS, is now available electronically through the AMS website on the information. In any case, if there is any application deadline with World Wide Web . To access the AMS website, use the URL: http: I /www. ams. org/.

4 72 N OTICES OF THE AMS VOLUME 52, N UMBER 4 Mathematics Calendar

University), Sergiu Hart (Hebrew University of Jerusalem), Tom Plenary Speakers: Frederic Campana (Nancy, France), Yves Benoist Benzinger (Ecole Polytechnique Federate de Lausanne), Ali Khan (Paris, France), Jan Willem Klop (Amsterdam, The Netherlands), (Johns Hopkins University), Felix Kubler (Universitaet Mannheim), Francoise Point (Mons, Belgium), and the Brouwer Medal Award Isaac Levi (Columbia University), Eric Maskin (Institute for Ad­ Winner. Evening Leisure Lectures by Jean Doyen (Brussels, Belgium) vanced Study, Princeton), Rohit Parikh (City University of New and Burkard Polster (Monash, Australia). York), John Quah (University of Oxford), Roberto Raimondo (Uni­ Sessions and Organizers: la. Differential Geometry, N. Poncin versity of Melbourne), Kali Rath (University 'of Notre Dame), llya (Luxembourg) and M. Schlichemaier (Luxembourg), 1b. Algebraic Ge­ Segal (Stanford University), Sudhir Shah (University of Delhi), Hemy ometry, 0. Debarre (France), 2a. Mathematical Statistics, R. Gill (The Tulkens (Universite Catholique de Lou vain, Belgium), Anne Villamil Netherlands), M. Hallin (Belgium) and P. Massart (France), 2b. Cod­ (University of Illinois at Urbana-Champaign), Myrna Wooders (Uni­ ing/Crypto Theory, H. van Tilborg(The Netherlands) and A. Canteaut versity of Warwick), Nicholas Yannelis (University of Illinois at (France), 3a. Harmonic Analysis, A. Valette (Belgium/ Switserland) Urbana-Champaign) and Sang-Seung Yi (Seoul National University) and C. Molitor-Braun (Luxembourg), 3b. Partial Differential Equa­ Registration: Registration forms for participation in the tutori­ tions, G. Lebeau (France) and ]. Hulshof (The Netherlands), 4a. als or workshop are available at http: I lwww, ims, nus, edu, sgl Computer Science, B. Hoogewijs (Belgium), 4b. History of Math­ Programsluielindex , htm, Completed forms should be received ematics, G. Alberts (The Netherlands) and M. Bullinck (Belgium), by the Institute at least one month before commencement of each Sa. Non commutative Algebra, L. De Bruyn (Belgium) and ]. Alev activity, Registration is free of charge, Institute membership is not (France), 5b. Model Theory, F. Point (Belgium) and Z. Chatzidakis required for participation, (France). Information: Contacts: For general enquiries, please email email: ims@nus , edu. sg. For enquiries on scientific aspects of the program, June 2005 please email Yen eng Sun at email: mat suny@nus. edu . sg. More information at website: http: I lwww. ims. nus. edu. sgiProgramsl '' 1- 5 1Oth International Conference Mathematical Modelling uie/index. htm. and Analysis and 2nd International Conference Computational Methods in Applied Mathematics, Trakai, Lithuania. '' 14-20 Workshop on Symplectic Field Theory, University of Aim: The Conference focuses on various aspects of mathematical Leipzig, Germany. modelling and usage of finite difference and finite element meth­ Topic: Lecture series by Helmut Hofer on "Fredholm theory in ods for numerical solution of modern problems of science and Polyfolds with Applications". engineering. Organizers: Kai Cieliebak, Helmut Hofer, Dusa McDuff, Klaus Conference organizers: The International Association for Math­ Mahnke, Matthias Schwarz. ematics and Computers in Simulation (IMACS). The European Deadline: January 17, 2005. Consortium for Mathematics in Industry (ECMI). Vilnius Gedim­ Information: http: I lwww .math. uni -leipzig. delwsl. inas Technical University (VGTU). Institute of Mathematics and Informatics, Vilnius (IMI). Vilnius University (VU). Computational '' 1 5- 1 9 SIAM Conference on Optimization, Norra Latin, City methods in applied mathematics (CMAM). Conference Centre, Stockholm, Sweden. Topics: Analysis of numerical methods for solving problems of Information: http: I lwww. siam. orglmeetingsiOP051. For addi­ mathematical physics; Parallel algorithms and parallel computing; tional information, contact SIAM Conference Department at email: Applications of numerical methods; Analysis of ODE and PDE [email protected]. problems and applications; Navier-Stokes equations and Compu­ tational Fluid Dynamics; Image processing; Financial mathematics '' 16-20 Fifth IMACS Seminar on Monte Carlo Methods, Florida and mathematics in economics; Scientific computation. The sci­ State University, Tallahassee, Florida. entific program includes Invited Plenary Talks (40 min), Invited Plenary Speakers: Financial Applications: Ashvin Chhabra, Merrill Semi-plenary lectures (30 min) and Contributed Talks (20 min). The Lynch, USA. Random Number Generation: Pierre L'Ecuyer, Univ. program also includes Poster Sessions. de Montreal, Canada. Biological and Medical Applications: Alex Deadlines: Abstracts: March 30, 2005. Notification of participation Bielajew, University of Michigan, USA. Nuclear Engineering Ap­ and reservation of accommodation: May 10, 2005. plications: Forrest Brown, Los Alamos National Laboratory, USA. Information: Dr. A. Tikonas (MMA2005 & CMAM2), Inst. of Math. Statistical Physics Applications: David Landau, University of Geor­ and Inform., Akademijos 4, LT-08663,Vilnius, Lithuania; tel: (+370 gia, USA. Monte Carlo in Education: Robert Panoff, Shodor Education 5) 210 97 34, (+370) 5 269 89 28; fax: (+370 5) 272 92 09; email: Foundation, USA. Statistical Applications: Jun Liu, Harvard Uni­ [email protected] .lt; and [email protected]; http://www.vtu.lt/ versity, USA. Applied Mathematics and Monte Carlo: Alexandre rclmma2005l. Chorin, University of California at Berkeley, USA. Monte Carlo in Radiation Transport: Eugene Brooks, Lawrence Livermore National '' 5-1 0 31st International Conference "Applications of Mathemat­ Laboratory, USA. ics in Engineering and Economics", Sozopol, Bulgaria. Important dates: February 15, 2005: Deadline for submission Topics: Potential Theory and Partial Differential Equations, Mathe­ of special session proposals. Such proposals should include the matical Analysis and Appl., Operation Research, Numerical Methods session name, the organizer, the list of speakers, and a list of talk and Mathematical Modeling, Computer Science. titles with abstracts. February 21, 2005: Deadline for submission Organizer: Faculty of Applied Mathematics and Informatics by the of extended abstracts for contributed talks. Technical University of Sofia, Bulgaria. Information: email: mcm2005@fsu. edu; http: I lmcm2005 . csi t . Information: http: I lwww. tu-sofia. bglfpmilamee05l, fsu.edulintro .html. * 6-24 Summer School and Conference on Geometry and Topology ,., 20-22 First Joint Meeting of the Mathematical Societies of of 3-Manifolds, ICTP, Trieste, Italy. Belgium, the Netherlands, Luxembourg and France, University Organizers: Michel Boileau, Carlo Petronio, Bruno Zimmermann. of Ghent, Galglaan 2, 9000 Gent, Belgium. Deadline: January 20, 2005. Description: This conference is a joint organization of the Math­ lnformation:http: I /agenda. ictp. trieste. i t/agendalcurrentl ematical societies of France, Luxembourg, The Netherlands and fullAgenda.php?email=O&ida=a04195. Belgium. It is the fourth in the series for Belgium, and counts as the 41st KWG meeting. It focuses on some main current themes in pure '' 8-11 Second Advanced Course in Operator Theory and Complex and applied mathematics. On the meeting, the KWG awards the Analysis, Seville, Spain, Brouwer Medal, and the Award Winner will give a plenary lecture. Information: http: I lwww. us. esl ceacyto2.

APRIL 2005 NOTICES OF THE AMS 473 Mathematics Calendar

* 1 3-1 7 Recent Developments and Open Questions in lwasawa Information: http : I /www .rna. hw. ac. uk/icms/meetings/2005/ theory (in honor of Ralph Greenberg's 60th birthday), Boston klig. University, Boston, Massachusetts. Topics: Main conjectures for modular forms (cyclotomic, anticyclo­ * 27-July 5 Probability and Mathematical Physics, Centre de re­ tomic), non-abelian Iwasawa theory, supersingular Iwasawa theory, cherches mathematiques, Montreal, Quebec, Canada. p-adic variation and p-adic families. Description: Conference celebrating the 65th birthday of Stanislav Funding: Funding may be available for graduate students and Molchanov. recent postdocs. Organizers: DonDawson(Carleton&McGill), VojkanJaksic (McGill), lnformation:http: I /math. bu. edu/people/rpollack/greenberg. Boris Vainberg (UNCC). html. Invited Speakers: G. Ben Arous (Courant), L. Bogachev (England), R. Carmona (Princeton), K. Chen (N. Carolina), M. Cranston (Rochester), * 1 4-1 8 Random Media and Stochastic Partial Differential Equa­ G. Derfel (Ben Gurion), E. Dynkin (Cornel), A. Figotin (Irvine), M. tions, University of Southern California, Los Angeles, California. Freidlin (Maryland), ]. Gaertner (Berlin), I. Goldsheid (England), A. Organizer: Sergey Lototsky (lototsky@math. usc . edu). Gordon (Rochester), R. Grigorchuk (Texas A&M), D. Hundertmark Speakers: M. I. Freidlin (University of Maryland, College Park); (Illinois), S. Jitornirskaya (Irvine), R. Kashrninski (Detroit),* K. Khanin R. Z. Khasminskii (Wayne State University); N. V. Krylov (Uni­ (England), W. Kirsch (Ruhr), A. Kiselev (Wisconsin), A. Klein (Irvine), versity of Minnesota); G. Papanicolaou (Stanford University); A. N. L. Koralov (Princeton), S. Kotani (Japan),'' P. Kuchment (Texas Shiryaev (Steklov Mathematics Institute, Moscow, Russia); R. Sowers A&M), A. Laptev (Stockholm), Y. Last (Hebrew), V. Malyshev (Paris), (Univesity of Illinois, Urbana-Champaign). M. Menshykov (England), N. Minami (Japan), L. Pastur (Ukraine), Information: Program to be posted at http: I /math. usc. edu. A. Ramirez (Chile), B. Simon (Caltech), A. Soshnikov (Davies), T. Spencer (lAS). (")To be confirmed. '' 1 5-1 8 Algebraic and Topological Methods in Non-classical Logics II , University of Barcelona, Barcelona, Spain. '' 30-July 2 Primeras Jornadas de Teoria de Numeros, EPSVG-UPC, Description: The semantic study of non-classical logics is a field Vilanova i la Geltru (Barcelona), Spain. where no single overarching paradigm has been established, and Description: The "Jornadas de Teoria de Nfuneros" (Number Theory where a variety of techniques are currently being explored. An Conference) are born to become a periodic meeting point for the important goal of this meeting is to promote the cross-fertilization arithmetical community working in Spanish language, so that it between the fundamental ideas connected with these approaches. can show the state of the art of research in Number Theory in Spain Information: The names of invited speakers, and further details and promote relationships between different research groups. about travel grants, hotels, etc., will be posted at the congress' web Invited Speakers: P. Bayer, ]. Cilleruelo, E. Friedman, ]. Guardia,]. page http: I /www .mat. ub . esrlogica/meeting2005/. Please visit C. Peral, F. Rodriguez-Villegas. it for any other information on the meeting. Scientific Committee:]. Gonza.lez-Rovira,]. Muiioz-Porras, E. Nart, A. Quiros, N. Vila. '' 21-24 International Workshop on Function Theory in Seoul, Organizing Committee: G. Cardona, ]. Fernandez, ]. Gonzcilez­ Korea University, Seoul, Korea. Rovira, ]. Guardia, V. Rotger. Description: Operator-related function theory and related topics Information: http: I /anduril . epsevg. upc. esrjtn05. will be the main themes of the workshop. Information: http : I /math. korea. ac. kr ;-iwft2005/. July 2005 * 22-24 Logic, Game Theory and Social Choice 4, University of '' 9-1 5 Nonlinear dispersive wave phenomena, Anogia Academic Caen, Caen, France. Village, Crete, Greece. Description: Following LGS1 (Tilburg 1999), LGS2 (St.Petersburg Organizers:]. L. Bona, V. A. Dougalis, ].-C. Saut. 2001) and LGS3 (Siena 2003), LGS4 will bring into focus the develop­ Main speakers: ]. L. Bona, U., A. S. Fokas, F. Merle, J-C. Saut, C. ing theoretical connections between logic and game theory, game Sulem. theory and social choice, logic and social choice. The conference Support: For young scientists (pre- and post-Ph.D.). program will consist of invited lectures and contributed papers. Deadline: April 9, 2005. Submissions of contributed papers are invited. Papers focussing Information: Financial applications: email: euroconf@math . uoc. on connections between the relevant disciplines are especially gr. encouraged by the Program Committee. Keynote Speakers: Eric Maskin (Institute of Advanced Studies, '' 1 0-1 5 Counting Complexity: An International Workshop on Princeton, USA); Philippe Mongin (CNRS and Ecole Polytechnique, Statistical Mechanics and Combinatorics: In Celebration of the Paris, France); Harrie de Swart (TilburgUniversity, The Netherlands). 60th Birthday of Tony Guttmann, Dunk Island, Queensland, Abstract deadline: Complete papers or two-page abstracts should Australia. be sent to Vincent Merlin (merlin@econ. unicaen. fr) or Maurice Organizers:RichardBrak,JandeGier,AleksOwczarek,OleWarnaar. Salles (salles@econ. unicaen. fr) as PDF files before March 1st, Information: http : I /www . complex . org. au/ conferences/t60. 2005. Acceptance will be communicated to the authors by the 15th of April 2005. '' 11-1 5 Novi Sad Algebraic Conference 2005, University of Novi Information: http: I /www. unicaen. fr I crem. Sad, Novi Sad, Serbia/ Serbia and Montenegro. Program: In order to attract younger researchers and others who * 2 7-July 1 Algebraic K- and L-theory of Infinite Groups, ICMS, are not experts in the area, there will be longer plenary talks, Edinburgh, Scotland. 60 minutes in duration. Many of them should be of expository, Organizers: Andrew Ranicki, Ian Hambleton. survey-like nature, which should make them understandable to Main speakers: M. Bridson,]. F. Davis, F. T. Farrell, W. Lueck, E. K. a wider audience. The afternoon sessions will be reserved for Pedersen, F. Quinn. contributed talks in several special sessions, the number of which Invitation only: Space is limited. Anyone interested in an invitation will depend on the number of speakers. The duration of these will should contact Andrew Ranicki (a. ranicki@ed. ac. uk). be 20 minutes, and it is expected that the speakers will present Financial support: For certain categories of participant-see the their most interesting recent research. website for details. Topics: Universal algebra, Lattices and ordered structures, Model Support: By the London Mathematical Society and by the Edinburgh theory and set theory, Clone theory, Algebraic methods in computer node of the EU RTN Network HPRN-CT-2002-00287. science.

474 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Mathematics Calendar

Plenary· speakers: Z. Esik (Univ. of Szeged), R. Freese (Univ. of Description: This biannual event is the premier conference linking Hawaii), P. Idziak (Jagiellonian Univ., Krakow), J. Jezek (Charles algebraists and algebraic geometers from all of Latin America. Univ., Prague), K. Kearnes (Univ. of Colorado at Boulder), R. Topics: Besides the plenary talks and general courses this meeting Mckenzie (Vanderbilt Univ.), R. Poschel (Technical Univ., Dresden), will have seven thematic parallel sessions on the following topics: I. Rosenberg (Univ. of Montreal), A. Szendrei (Univ. of Colorado at Commutative Algebra and Algebraic Geometry, Non-associative Boulder), S. Todoreevice (Univ. of Toronto & Universite Paris 7), F. Algebras and Ring theory, Group Theory, Hopf Algebras and Wehrung (Univ. of Caen), R. Willard (Univ. of Waterloo). Algebraic Coni.binatorics, Homological Methods and Representation Deadlines: The deadline for the registration is April 30, 2005; The Theory, Number Theory, Operator Algebras. A special session on abstract submission deadline is May 31, 2005. Applications of Algebra will also be held. Information: http: llwww. im. ns . ac. yul eventsiNSAC051 default. Speakers: A list of a few of the confirmed speakers is the following: html; email: nsac05@im . ns. ac. yu. Nicolas Andruskiewitsch, Raymundo Bautista, Michel Brion, Ken Brown, Antonio Campillo, Max Karoubi, Jean-Louis Loday, Susan '' 11-22 SMS 200S-NATO Advanced Summer Institute: Equidis­ Montgomery, Adrian Ocneanu, jose Antonio de la Peii.a, Vladimir tribution in Number Theory, Universite de Montreal, Montreal, Popov, Hans-Jurgen Schneider, Aron Simis, Frank Sottile, Richard Canada. Stanley, Boris Tsygan, Mariusz Wodzicki. Speakers: Yuri Bilu (Bordeaux I), William Duke (UCLA), John Organizing and Scientific Committee: Walter Ferrer Santos (Co­ Friedlander (Toronto), Andrew Granville (Montreal), Roger Heath­ ord.), Gerardo Gonzalez-Sprinberg, Alfredo Jones, Alvaro Rittatore, Brown (Oxford), Elon Lindenstrauss (NYU), }ens Marklof (Bristol), Andrea Solotar. ZeevRudnick (Tel Aviv), Wolfgang Schmidt (Colorado at Boulder and Deadline: May 1st, 2005. Vienna), K. Soundararajan (Michigan), Yuri Tschinkel (Gottingen), lnformation:http : llwww . cmat. edu. uyl cmatl eventosl 16clalen; Emmanuel Ullmo (Paris-Sud 11), Akshay Venkatesh (MIT). Walter Ferrer: email: wrferrer@cmat . edu. uy Features: Uniform Distribution and beyond: Applications to Cryp­ tographic protocols, distribution of integers, level statistics. Integer '' 3- 5 30th Sapporo Symposium on Partial Differential Equations, and rational points on varieties: Geometry of numbers, the circle Sapporo, Japan. method, homogeneous varieties via spectral theory and ergodic Description: The Sapporo Symposium on Partial Differential Equa­ theory. The Andre-Oort conjectures: Equidistribution of CM-points tions (PDE) has been held annually to present the latest developments and Heeke points, points of small height. Quantum ergodicity: on PDE with a broard spectrum of interests not limited to the Quantum maps and modular surfaces. methods of a particular school. Note: This summer school is primarily targeted at senior grad Organizers: T. Ozawa, G. Nakamura, S. Jimbo, Y. Giga, K. Tsutaya, students, postdocs and junior faculty. For full consideration, Y. Tonegawa. requests for participation or financial assistance must be received Information: email: cri@math. sci. hokudai. ac. jp; htt p: II coe. before February 28, 2005. Financial support available. math.sci.hokudai.ac.jplsympol sapporol program050803_en. Information: http: llwww .dms. umontreal. calsmsl; email: html. [email protected]. '' 5-11 Logic In Hungary, 2005, Budapest, Hungary. '' 1 6-August 1 The Eighth International Diffiety school, Santo Topics: Set Theory, Foundations of Space-Time, Algebraic Logic, Stefano del Sole (Avellino), Italy. but contributions from all other branches of symbolic logic are The aim of the School is to introduce undergraduate and Aim: welcome. Ph.D. students in Mathematics and Physics as well as post-doctoral Organizing Committee: A. Hajnal, J Suranyi (honorary chair), researchers in a recently emerged area of Mathematics and Theo­ H. Andreka, I. Juhasz, P. Komjath, I. Nemeti (co-chair), G. Sagi retical Physics: Secondary Calculus. A diffiety is a new geometrical (secretary), L. Csirmaz, M. Ferenczi, M. Redei, I. Sain and L. Soukup object that properly formalizes the concept of the solution space (members). of a given system of (nonlinear) PDEs, much as an algebraic variety . renyi. does with respect to solutions of a given system of algebraic Information: Contact: email: lh05@renyi. hu; http: llwww equations. Secondary Calculus is a natural diffiety analogue of the hullh05. standard Calculus on smooth manifolds, and as such leads to a * 7- 1 2 High-dimensional Partial Differential Equations in Science very rich general theory of nonlinear PDEs. Moreover, it appears to and Engineering, Centre de recherches mathematiques, Universite be the unique natural language for quantum physics, just as the de Montreal Montreal, Quebec, Canada. standard Calculus is the natural language for classical physics. Description: High dimensional spatio-temporal partial differential Organizer: Diffiety Institute (Russia). equations are a major challenge to scientific computing of the Deadline: June 15, 2005. future. Up to now deemed prohibitive, they have recently become Contact: Prof. A. M. Vinogradov, Dipartimento di Matematica e manageable by combining recent developments in numerical tech­ Informatica, Universita' di Salerno, Via Ponte del Melillo, 84084 niques, appropriate computer implementations, and the use of Fisciano (SA), Italy; email: ·school OS @ diffiety. org; http: II computers with parallel and even massively parallel architectures. diffiety . ac . ru. This opens new perspectives in many fields of applications. Ki­ * 23- 29 Number fields and curves over finite fields, Anogia netic plasma physics equations, many body Schrodinger equation, Academic Village, Crete, Greece. Dirac and Maxwell equations for molecular electronic structure in Organizer: R. Schoof. and nuclear dynamic computations, options pricing e quations Fokker-Planck and fluid dynamics equa­ Main speakers: G. van der Geer, D. Lorenzini, R. Schoof, M. mathematical finance, and are examples of equations that can now Tsfasman. tions for complex fluids, Support: For young scientists (pre- and post-Ph.D.), especially from be handled. EU Member and Associated States. Scientific Program Committee and Organizers: Andre Bandrauk Deadline: March 23, 2005. (CRC, Chirnie, Universite de Sherbrooke); MichelDelfour (CRMI DMS, Ecole Information: Financial applications: email: euroconf@mat h . uoc. Universite de Montreal, Canada); Claude Le Bris (CERMICS, France). gr. Nationale des Ponts et Chaussees, '' 8- 1 2 NSF-CBMS Regional Conference on Algebraic and Topologi­ August2005 cal Combinatorics of Ordered Sets, San Francisco State University, * 1- 9 XVI Coloquio Latinoamericano de Algebra, Colonia, Uruguay. San Francisco, California.

APRIL 2005 NOTICES OF THE AMS 475 Mathematics Calendar

Speaker: Anders Bjoorner will give ten lectures to introduce September 2005 background material, fundamental results, and recent advances in '' S-8 8th International Conference on Logic Programming and the field of algebraic and topological combinatorics of partially order Non monotonic Reasoning (LPNMR'OS), Diamante, Cosenza, Italy. sets, (oriented) matroids, subspace arrangements, and algebraic shifting etc. Most of the 40-50 participants should expect to obtain Description: LPNMR is a forum for exchanging ideas on declarative funding. Graduate students and postdocs are highly encouraged logic programming, nonmonotonic reasoning and knowledge rep­ to apply. resentation. The aim of the conference is to facilitate interactions Other Speakers: Alexander Barvinok, Winfried Bruns, Gunnar between researchers interested in the design and implementation Carlsson, Persi Diaconis, Isabella Novik, Bernd Sturmfels, Michelle of logic based programming languages and database systems, and Wachs, Neil White. researchers who work in the areas of knowledge representation and nonmonotonic reasoning. Authors are invited to submit papers Organizers: ]. Gubeladze, email: soso@math . sfsu. edu; S. Hosten; presenting original and unpublished research on nonmonotonic email: serkan@math. sfsu. edu. aspects of logic programming and knowledge representation. We Information: http: I /math. sfsu. edu/gubeladze/ cbms. html. particularly encourage papers on application of LPNMR techniques * 14-19 International Conference on Complex Analysis and Re­ to build significant applications. lated Topics: The 1Oth Romanian-Finnish Seminar, "Babes-Bolyai" Topics: Development and mathematical studies of logical systems University, Cluj-Napoca, Romania. with nonmonotonic entailment relations, Implementation of LPNMR Organizers: The Institute of Mathematics "Simion Stoilow" of the systems, and Applications of LPNMR systems. Romanian Academy, the Faculty of Mathematics and Informatics Important dates: Abstract Submission Deadline: March 22, 2005; of the University of Bucharest, the Faculty of Mathematics and GMT; Paper Submission Deadline: March 25, 2005; Notification Informatics of the "Babes-Bolyai" University of Cluj-Napoca, the (Accept/ Reject): May 16, 2005; Conference Schedule: June 6, 2005; Universities of Helsinki, Joensuu and Jyvaskyla from Finland. Final Conference Papers: June 10, 2005; Early Registration Deadline: Topics: Analytic functions of one complex variable; Quasiconformal July 4, 2005. mappings and Teichmuller spaces; Several complex variables; Information: http : I /www .mat . unical . it/lpnmr05/. Potential theory; Functional analytical methods in complex analysis. Registration: A preliminary registration form (including: Name; * 1 2-1 5 Third International Workshop Meshfree Methods for First name; Institution/ Affiliation; Address/ E-mail; Sections' par­ Partial Differential Equations, Rheinische Friedrich-Wilhelms Uni­ ticipation) should be returned (either by e-mail or by standard mail; versitat Bonn, Bonn, Germany. see address below). Description: The numerical treatment of partial differential equa­ Information: Complex Analysis and Related Topics c/ o Institute of tions with meshfree discretization techniques has been a very active Mathematics "Sirnion Stoilow" of the Romanian Academy, P.O. Box research area in recent years. While the fundamental theory of 1-764, 014700, Bucharest, Romania; fax:+40 21 212 51 26; email: meshfree methods has been developed and considerable advances rofinsem@imar . ro. of the various methods have been made, many challenges in the mathematical analysis and practical implementation of meshfree '' 1 7-21 Third Pacific Rim Conference on Mathematics, Fudan methods remain. University, Shanghai, China. Sponsor: Sonderforschungsbereich 611. Topics: All areas of mathematics with focus topics on: Algebra Deadlines: Abstract: About 300 words (preferably in LaTeX format) and Combinatorics; Algebraic Aspects of Lie Theory and Geometry; to email: me shf ree@ins . uni-bonn . de by May 1, 2 00 5. Confirmation Applied Differential Geometry; Asymptotics and Riemann-Hilbert and program: August 1, 2005. Problems; Computational Approach to Complex Dynamical Sys­ Information: http: I /wissrech. ins. uni -bonn. de/meshfree. tems; Kinetic Theory; Low Dimensional Topology and Geometry; Nonlinear Analysis Nonlinear Phenomena, Symmetry and Integrable '' 1 2-16 p-Adic Representations, Centre de recherches mathemati­ Structures; Partial Differential Equations and Applications. ques, Univ. de Montreal, Montreal, Quebec, Canada. Plenary Speakers: Gerard Jennhwa Chang (Taiwan), Shuxing Chen Topics: The main topics are related to a p-adic Langlands corre­ (China), Philippe G. Ciarlet (Hong Kong), Konstantin Mischaikow spondence and its relationship to p-adic families of motives. More (USA), Colin Rogers (Australia), Minoru Wakimoto, (Japan), Shicheng precisely the p-adic Langlands correspondence is a correspondence Wang (China), Roderick Wong (Hong Kong), Shih-Hsien Yu (Hong between p-adic Galois representations of dimension n (of the abso­ Kong). lute Galois group of Qp) and certain representations of GLn(Qp) on Supporter: Fudan University, Mathematical Center of Ministry of p-adic topological vector spaces. This correspondence is supposed Education of China, National Natural Science Foundation of China, to be compatible with p-adic families on both sides. Chinese Mathematical Society, China Society for Industrial and Organizers: Adrian Iovita (Concordia); Henri Darmon (McGill). Applied Mathematics, Liu Bie Ju Centre for Mathematical Sciences Information: http: I /www. crm. umontreal . ca/Number2005/. (City University of Hong Kong), Sino-French Institute of Applied Mathematics '' 12-16 CASC'2005: The 8th International Workshop on Computer Information: Contact: Zhou Chunlian, Sino-French Institute of Algebra in Scientific Computing, CASC'2005, Kalamata, Greece. Applied Mathematics, Fudan University, Shanghai 200433, China; Organizers: CASC General Chairs: V. P. Gerdt (Dubna), E. W. Mayr tel: 81-21-6564 2469; fax: 86-21-6564 8274; email: clzhou@fudan. (Munich) CASC'2005 Conference Chairs: I. Z. Emiris (Athens), I. S. edu.cn;http://PRCM3.fudan. edu.cn. Kotsireas (Waterloo), M. N. Vrahatis (Patras). Deadlines: For submissions: April 1, 2005. Notification of accep­ '' 20-26 Algebraic and Geometric Combinatorics, AnogiaAcademic tance: June 30, 2005. Final version due: July 15, 2005. Village, Crete, Greece. Information: http: I /www. cargo . wlu. ca/ casc2005/; Organizers: V. Batyrev, M. Henk, F. Santos. email: casc2005@in . tum. de. Main speakers: V. Batyrev, L. Billera, A. Bjorner, Sweden, M. Henk, P. McMullen, F. Santos, G. Ziegler. '' 1 3-1 7 5th International Conference on Words, Centre de re­ Support: For young scientists (pre- and post-Ph.D.), especially from cherches mathematiques, Universite de Montreal, Montreal, Quebec, EU Member and Associated States. Canada. Deadline: April 20, 2005. Organizers: Srecko Brlek (Univ. du Quebec a Montreal); Cedric Information: Financial applications: email: euroconf@math. uoc. Chauve (Univ. Bordeaux I, UQAM); Annie Lacasse (Univ. du Quebec gr. a Montreal); Genevieve Paquin (Univ. du Quebec a Montreal).

476 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Mathematics Calendar

· 20-2 ?International Conference Harmonic Analysis and Approx­ Distance Learning and Educational technologies, Mathematics and imations, Ill, Tsahkadzor, Armenia. Science Education. Deadline for Application: March 31, 2005 . Information: http : //sci . tamucc . edu;-iccmm/index .html. Description: The program of the conference will consist of invited 40-minutes plenary lectures and contributed 20-minutes talks. The November 2005 following mathematicians have agreed to give a plenary lecture at '' 4-6 Geometric and Probabilistic Methods in Group Theory and the conference: Borislav Bojanov (Bulgaria), Carl de Boor (USA) , Dynamical Systems, TexasA&MUniversity, College Station, Texas. Ronald DeVore (USA), Nira Dyn (Israel), Hakop Hakopian (Armenia), Organizers: Rostislav Grigorchuk, Gilles Pisier, Zoran Sunik. Kazaros Kazarian (Spain), Gerard Kerkyacharian (France), Sergey Participants: M. Bestvina, M. Bridson, P. Diaconis, A. Eskin, B. Farb, Konyagin (Russia), Michael Lacey (USA), Konstantin Oskolkov (USA), E. Ghys, C. Gordon, E. Guentner, I. Kapovich, A. Katok, S. Katok, A. Allan Pinkus (Israel), Gerald Shrnieder (Germany), Przemyslaw Lubotzky, S. Mazes, A. Olshanskii, S. Popa, A. Reid, L. Saloff-Coste, Wojtaszczyk (Poland). M. Sapir, K. Vogtmann, E. Zelmanov Cont"act Information: Artur Sahakian, Institute of Mathematics, Deadlines: Abstracts and Registration August 31, 2005, Financial Marshal Bagramian ave, 24-B, 375019, Yerevan, Armenia; email: Support July 31, 2005. [email protected]; http://math.sci.am; fax: (3741) 524801. On­ Information: http://www . math. tamu . ectur sunik/05tamu. line registration is available: http: //math. sci. am/ conference/ sept2005/registration.html. December 2005 October 2005 '' 1 2-16 Intersection of Arithmetic Cycles and Automorphic Forms, Centre de recherche mathematiques, Universite de Montreal, '21-26 International Conference of Computational Methods in Montreal, Quebec, Canada. Sciences and Engineering 2005 (ICCMSE 2005), Hotel Poseidon, Purpose: To explore the relationship between intersection numbers Loutraki, Korinthos, Greece. for arithmetic cycles on Shimura varieties, Fourier coefficients of Description: In the past decades many significant insights have automorphic forms, and special values of L-functions. been made in several areas of Computational Methods in Sciences Organizers: Eyal Goren (McGill) and Henri Darmon (McGill). and Engineering. New problems and methologies have appeared. Information: http://www. crm. umontreal. ca/Number2005/. There is permanently a need in these fields for the advancement

of information exchange. 1' 1 4-1 6 CRAMS-OS International Conference on Applied Non har­ Topics: Computational Mathematics, Theoretical and Computa­ monic Fourier Analysis, Business & Computer, University College tional Physics and Theoretical and Computational Chemistry, (BCU), Beirut, Lebanon. Computational Engineering and Mechanics, Computational Biol­ Topics: The conference commemorates the Riemann-Lebesgue ogy and Medicine, Computational Geosciences and Meteorology, Lemma Centennial. Its major themes include, but not limited Computational Economics and Finance, Financial Forecasting, Sci­ to: multidimensional localization principle, generalized Tauberian entific Computation, High Performance Computing, Parallel and theory, time-frequency-scale multiresolution analysis, the Segal­ Distributed Computing, Visualization, Problem Solving Environ­ Bargmann transform, Gabor wavelets and frames, invertibility of ments, Software Tools, Advanced Numerical Algorithms, Modelling Gabor transforms, functional analysis of Gabor frames, almost and Simulation of Complex System, Web-based Simulation and periodic and recurrent functional analysis, reversed filtration and Computing, Grid-based Simulation and Computing, Computational regularization, connections with functional equations and tiling Grids, Fuzzy Logic, Hybrid Computational Methods, Data Mining theory, nonlinear filter theory, analysis with fractal measures, and Information Retrieval, Virtual Reality, Reliable Computing, signal reconstruction in communication theory, and fast numerical Image Processing, Computational Science and Education etc. algorithms. Information: Secretary ICCMSE 2005 (Mrs Eleni Ralli-Simou), email: Deadlines: The deadline for submitting Abstracts is May 15, 2005, iccmse@uop. gr, 26 Menelaou Street, Arnfithea Paleon Faliron, GR- for notification of acceptance is July 15, 2005, and for full-length 175 64, Athens, Greece, fax: +30210 94 20 091 or + 30 2710 paper submission is Sept. 15, 2005. 237397. Information: N.H. S. Haidar, Chairman of the Organizing Committee for CRAMS-05, Business & Computer University College (BCU), ' 24-26 SIAM Conference on Mathematics for Industry: Challenges Commodore Str, Hamra, Beirut, Lebanon; tel: 961 1 752 370-4, ext. and Frontiers, Detroit Marriott Renaissance Center, Detroit, Michi­ 131,961 1 736 511, fax: 9611 340 219, email: nhaidar@suffolk . gan. edu, crams@hnu-crams. org, basicsciences@hu . edu . lb; http: II Information: SIAM's conference on Mathematics for Industry fo­ www.hnu-crams.org/ann.html. cuses attention on the many and varied opportunities to promote applications of mathematics to industrial problems. Since the '' 1 5-January 31, 2006 Semidefinite Programming and its Appli­ SIAM community encompasses enormous talent for integrating cations, Institute for Mathematical Sciences, National University and enriching both industrial work and academic research, this of Singapore, Singapore. conference will stress interactions within the context of mathe­ Program: Will provide a forum for the exchange of ideas among matical models and complex systems, and will encourage other researchers working in theory, applications, algorithms, and soft­ mathematical themes of interest to industry, government, business ware development of SDP. and finance. The program will consist of tutorials and workshops with ample op­ Information: http: //widen@siam. or g. portunities for collaborative research among local and international participants. '' 24-281nternational Conference on Computing and Mathematical Organizing Committee: Chair: Michael Todd (Cornell University). Modeling for Environmental, Social-Economical, and Technical Co-chairs: Jie Sun (National University of Singapore) and Kim-Chuan Systems-2005, East China Normal University, Shanghai, China. Toh (National University of Singapore). Topics: Include, but not limited to: Computing and Modeling of Information: For general enquiries, please email ims@nus. edu . sg. Earth Systems, Coastal and Marine Systems, GIS and Spatial Models, For enquiries on scientific aspects of the program, please email Kim­ Neural Networks, Statistics and Statistical Modeling, Fuzzy Sets and Chuan Toh, mattohkc@nus . edu. sg. Completed forms should be Systems, Numerical Methods and Applications, Optimization and received by the Institute at least one month before commencement Decision Making, Environmental Modeling, Allocation of Resources, of each activity. Registration is free of charge. Institute membership Mathematics and Computing ofRobotics, Sensors and Measurement, is not required for participation. Information about the program

APRIL 2005 NOTICES OF THE AMS 477 Mathematics Calendar

and registration forms are available at the website http : I /www . june 2006 ims.nus.edu.sg/ Programs/semidefinite/index.htm. '' 27-July 3 International Commission on Mathematical Instruc­ tion: Challenging Mathematics In and Beyond the Classroom, january 2006 Trondheim, Norway. for '' 16-27 "Propagation of Waves" CIMPA school and workshop, Scope: The scope of this study will be wide. It will look at, inside and Instituto de Matematicas, UNAM, Cuernavaca, Mexico. instance, the impact of mathematical challenges both challenges in Description: 6 minicourses, 10 invited lectures, contributed talks. outside of the classroom, the role of mathematical for students of all levels of ability, Organizers: Luz de Teresa, Salvador Perez Esteva, Carlos Villegas, supporting the curriculum for propagating mathematical challenges and assessment Arturo Portnoy. vehicles of their effectiveness. We would like to emphasize that we are Deadline: October 30, 2005. interested in students and activities of all type, and want to go far Information: http: I /wwww . mat em. unam. mx/ escuelaCIMPA; http: beyond contests for talented students. -icpam.org//index.html. //www.cimpa Discussion document: Has been prepared by an international committee chaired by Ed Barbeau of the University of Toronto February 2006 (barbeau@math. utoronto. ca) and Peter Taylor of the University of '' 1 3-1 8 L-functions and Related Themes, Centre de recherche Canberra in Australia who is the executive-director of the Australian mathematiques, Univ. de Montreal, Montreal, Quebec, Canada. Mathematical Trust (pjt@olympiad. org). This document defines Focus: The workshop will focus on the analytic theory of L­ terms, describes issues, provides sample situations, and poses functions and how they are used in a variety of questions ranging questions for discussion. Finally, it indicates how to become from arithmetic geometry to classical analytic number theory. involved in the Study Conference. Would-be participants will be Lecturers: Philippe Michel (Montpellier II); Kumar Murty (Toronto); asked to submit a brief curriculum vita and a 6-10 page document K. Soundararajan (Michigan). addressing matters relevant to the study no later than August 31, Organizers: Chantal David (Concordia) and Ram Murty (Queen's). 2005. The committee plans to send out invitations by January 31, Information: http: I /www. crm. umontreal. ca /Number2005/. 2006. The Conference will be followed by a publication. A copy of the discussion document can be obtained by going to the . edu, clicking on "LINKS" and March 2006 website http: I /www. amt . canberra then on "ICMI Study 16". * 1 3-1 7 Anatomy of Integers, Centre de recherche mathematiques Deadline: August 31, 2005. Universite de Montreal, Montreal, Quebec, Canada. Information: http : I /www. amt . canberra. edu/icmis16 .html/. Organizers: Jean-Marie de Koninck (Laval) and Andrew Granville (Montreal). july 2006 Workshop Focus: On multiplicative number theory, divisors, prime '' 7-8 Second International Conference on Nonsmooth/Nonconvex factors, distribution of prime divisors, multiplicative functions, Mechanics with Applications in Engineering, Faculty of Engineer­ smooth/friable numbers etc. ing, Aristotle University, Thessaloniki, Greece. Lecturers: Kevin Ford (Urbana-Champaign); K. Soundararajan Conference Topics: Contact Mechanics-Friction & stick-slip ef­ Gerald Tenenbaum (Institut Elie Cartan Nancy). (Michigan); fects, Elastoplasticity-Shakedown-Limit Analysis, Convex Analysis . ca/Number2005/ . Information: http: I /www. crm . umontreal and Mechanics, Nonsmooth Analysis and Optimization, Nonconvex Mechanics and Duality, Variational, quasivariational and hemivari­ ational inequalities, Energy methods in Mechanics and Structural Dynamics, Structural Optimization, Struc­ The following new announcements will not be repeated until Analysis, Nonsmooth and Identification, Computational Mechanics, Ap­ the criteria in the next to the last paragraph at the bottom of tural Control Mathematical Analysis and Approximation resultsm, the first page of this section are met. plications, Innovative topics (like Chaotic behaviour, Fractal approximation, April 2006 Neural Networks etc.) Deadlines: Submission of Abstract by May 1, 2005. Preliminary ematiques, * 6-12 AdditiveCombinatorics, Centrederecherchemath acceptance by July 30, 2005. ,Submission of full paper by February Universite de Montreal, Montreal, Quebec. 28, 2006. Topics: The topics covered will include: the Freiman-Ruzsa the­ Conference Organizer: c/ o Professor C. C. Baniotopoulos, Institute orem, the structure of set theory addition, Gowers' approach to of Steel Structures, Department of Civil Engineering, Aristotle Szemeredi's theorem and Green and Tao's approach to combinato­ University, GR-54124 Thessaloniki, Greece, tel.: +30 2310 99 5753 rial sets with structure. A mini-school will be organized before this Fax: +30 2310 99 5642, email: nnmae2006@civil . aut h . gr. workshop to introduce more people to this vibrant subject. More Information: http: I /www. c i vil . auth . gr/nnmae200 6/ . information will be made available on this site. Lecturers: Tim Gowers (Cambridge), Ben Green (Cambridge), Irnre * 1 0- 14 1nternational Conference on Analytic Topology, Lake Plaza Ruzsa (Alfred Renyi Institute) and Terence Tao (UCLA). Hotel, Rotorua, New Zealand. Organizers:JozsefSolymosi(UBC)andAndrewGranville(Montreal). Description: The main goal of this conference is to bring together Information: http: I /www . crm. umontreal. ca/Number2005/. a group of researchers from around the world, who are working at the interface between Topology and Analysis, to discuss recent May 2006 developments and future directions of Analytic Topology. Organizers: Warren B. Moors (Auckland University, email: moors@ Diophantine equations, Banff Inter­ * 1 3- 1 8 Analytic methods for math . auckl and . ac . nz); and Jiling Cao (Auckland University, email: national Research Station, Banff, Alberta, Canada. cao@math . auckland. ac. nz ). Description: This meeting brings together the participants of the Information: http: I / www. math. au ckland. ac . nz/- cao/ MSRI and CRM workshops The meeting will be held at the Banff conferen ce06. html. International Research Station. Organi zers: Andrew Granville (Montreal), Yuri Tschinkel (Go ttingen), Michael Bennett (UBC), Chantal David (Concordia) and Bill Duke (UCLA). Information: email: paradis@crm. umontreal. ca.

478 NOTICES OF THE AMS VOLUME 52, NUMBER 4 New Publications Offered by the AMS

Representations of a quiver with automorphism: Generalising Algebra and Algebraic a theorem of Kac; K. Igusa and G. Todorov, On the finitistic global dimension conjecture for Artin algebras; 0. Khomenko, Geometry Some applications of Gelfand-Zetlin modules; D. Kussin, A tubular algebra with three types of separating tubular families; V. Levandovskyy, PEW bases, non-degeneracy conditions and Representations of applications; D. Madsen, Projective dimensions and Nakayama algebras; F. Marko, Borel subalgebras of extension algebras; Algebras and Related F. Marko, Extesion algebras of standard modules; R. Martinez Topics Villa and A. Martsinkovsky, Cohomology of tails and stable cohomology over Koszul quiver algebras; A. Neeman, A survey Representations of Ragnar-Olaf Buchweitz, of well generated triangulated categories; M. Sato, Oneway Algebra and University of Toronto, ON, hereditary rings; M. Schaps, Deformations, tiltings, and Related Topics Canada, decomposition matrices; R. Schiffler, On the multiplication in Ragnar-Olaf Buchweitz and Helmut Lenzing, Helmut Lenzing University of Paderborn, the quantized enveloping algebra of type A; A. Zavadskij, Editors Equipped posets of finite growth. Germany, Editors Fields Institute Communications, Volume 45 Resulting from the Tenth International May 2005, 396 pages, Hardcover, ISBN 0-8218-3415-0, LC Conference on Representations of 2004062747, 2000 Mathematics Subject Classification: 16-06, Algebras and Related Topics held at The Fields Institute 16G20, 16GSO, 16G60, 16G70, 14130, 18F20, 18G20, 18£30, (Toronto, ON, Canada), this collection of research and survey 17B37, All AMS members $87, List $109, Order code FIC/ 45 articles, honoring Vlastimil Dlab's seventieth birthday, reflects state-of-the-art research on the topic. Leading experts contributed papers, demonstrating the An Analogue of a interaction between representation theory of finite dimensional algebras and neighboring subjects. A wide range ME¥,_9IRS Reductive Algebraic of topics are covered, including quantum groups, the theory of Amcrlcnn~latbom"llc o lSoclcty Monoid Whose Unit Lie algebras, the geometry and combinatorics of tilting theory, An Analogue of commutative algebra, algebraic geometry, homology theories, a Reductive Algebraic Group Is a Kac­ and derived and triangulated categories. The book is suitable Monoid Whose Unit Group Is a Moody Group for graduate students and researchers interested in the theory Kac-Moody Group of algebras. Claus Mokler Claus Mokler, University of Contents: E. R. Alvares and F. U. Coelho, On translation Wuppertal, Germany quivers with weak sections; H. Asashiba, Realization of general and special linear algebras via Hall algebras; Contents: Introduction; Preliminaries; R. Bautista and L. Salmeron, On discrete and inductive The monoid G and its structure; An algebras; R. Bautista and R. Zuazua, Exact structures for lift algebraic geometric setting; A categories; G. Benkart and D. Moon, Tensor product generalized Tannaka-Krein reconstruction; The proof of G = e representations of Temperley-Lieb algebras and Chebyshev and some other theorems; The proof of Lie(G) ~ g; polynomials; D. Benson, H. Krause, and S. Schwede, Bibliography. Introduction to realizability of modules over Tate cohomology; Memoirs of the American Mathematical Society, Volume 174, j. Bilodeau, Auslander algebras and simple plane curve Number 823 singularities; I. Burban and Y. Drozd, On derived categories of January 2005, 90 pages, Softcover, ISBN 0-8218-3648-X, LC certain associative algebras; C. GeiB and I. Reiten, Gentle 2004062690, 2000 Mathematics Subject Classification: 17B67, algebras are Gorenstein; j. Y. Guo, On the primeness of an 22£65, Individual member $34, List $56, Institutional Artin-Schelter regular Koszul algebra; D. Happel and L. Unger, member $45, Order code MEM0/ 174/ 823 On the set of tilting objects in hereditary categories; L. Hille, Irreudicible components, nilpotent classes, and the Auslander algebra of k[T] / Tn; M. Hoshino and K. Nishida, A generalization of the Auslander formula; A. Hubery,

APRIL 2005 NOTICES OF THE AMS 479 New Publications Offered by the AMS

AI ~ Malhem aUcal Socley Lie Groups and Analysis TRANSLATIONS Invariant Theory Ernest Vinberg, Moscow State Lie Groups and Maximum Principles Invariant Theory University, Russia, Editor ME¥._9IRS Arn~rlcnn ~lUh~mBhcnl Soctcty Ernest Vinberg on Riemannian Editor Devoted to the 70th birthday of A. L. Onishchik, this volume contains a Maximum Principles on Manifolds and collection of articles by participants in Riemannian Manifolds Applications the Moscow Seminar on Lie Groups and Applications· Stefano Pigola and Invariant Theory headed by E. B. MarcoRigoll Stefano Pigola and Marco Vinberg and A. L. Onishchik. The book Alberto G. Setu Rigoli, University of Milan, is suitable for graduate students and Italy, and Alberto G. Setti, researchers interested in Lie groups and related topics. Universita dell'Insubria, Como, Contents: D. N. Akhiezer, Real forms of complex reductive Italy groups acting on quasiaffine varieties; A. Alexeevski, Component groups of the centralizers of unipotent elements This item will also be of interest to in semisimple algebraic groups; D. V. Alekseevsky and those working in geometry and topology. V. Cortes, Classification of pseudo-Riemannian symmetric Contents: Preliminaries and some geometric motivations; spaces of quaternionic Kahler type; I. V. Arzhantsev and Further typical applications of Yau's technique; Stochastic D. A. Timashev, On the canonical embeddings of certain completeness and the weak maximum principle; The weak homogeneous spaces; A. G. Elashvili and V. G. Kac, maximum principle for the <)?-Laplacian; Q?-parabolicity and Classification of good gradings of simple Lie algebras; some further remarks; Curvature and the maximum principle S. Gindikin, The separation of the continuous spectrum on for the <)?-Laplacian; Bibliography. symmetric spaces of Cayley type; V. Gorbatsevich, Connected sums of compact manifolds and homogeneity; P. I. Katsylo, Memoirs of the American Mathematical Society, Volume 174, On curvatures of sections of tensor bundles; Number 822 Y. Khakimdjanov, Affine structures on filiform Lie algebras; February 2005, 99 pages, Softcover, ISBN 0-8218-3639-0, LC A. Lebedev, D. Leites, and I. Shereshevskii, Lie superalgebra 2004062691, 2000 Mathematics Subject Classification: 58]35; structures in C (n; n) and H' (n; n); A. L. Onishchik and 58]65, Individual member $34, List $57, Institutional member A. A. Serov, On isotropic super-Grassmannians of maximal $46, Order code MEM0/ 174/822 type associated with an odd bilinear form; D. I. Panyushev, Ideals of Heisenberg type and minimax elements of affine Weyl groups; V. L. Popov, Projective duality and principal nilpotent elements of symmetric pairs; V. Serganova, On Geometry and Topology representations of Cartan type Lie superalgebras; E. B. Vinberg, Construction of the exceptional simple Lie algebras; E. B. Vinberg, Short S03-structures on simple Lie On Dynamical algebras and associated quasielliptic planes. ME¥._9IRS Arnerlcnn~lnthcm m t!cnlSoctdy Poisson Groupoids I American Mathematical Society Translations-Series 2 (Advances in the Mathematical Sciences), Volume 213 Luen-Chau li, Pennsylvania On Dynamical State University, University May 2005, approximately 280 pages, Hardcover, ISBN 0-8218- Poisson Groupoids I 3733-8, LC 91-640741, 2000 Mathematics Subject Luen-Chau Ll Park, PA, and Serge Classification: 22-06; 14L24, All AMS members $87, List $109, Serge Parmentier Parmentier, Universite Lyon, Order code TRANS2/213 Villeurbanne, France Contents: Introduction; A class of Arncrlcnn ~lnthom o tloo\ Sooloty biequivariant Poisson groupoids; Duality; An explicit case study of duality; Coboundary dynamical Poisson groupoids - the constant r-matrix case; Appendix; Bibliography. Memoirs of the American Mathematical Society, Volume 174, Number 824 February 2005, 72 pages, Softcover, ISBN 0-8218-3673-0, LC 2004062689, 2000 Mathematics Subject Classification: 53D17; 58H05, Individual member $32, List $53, Institutional member $42, Order code MEM0/174/824

480 NOTICES OF THE AMS VOLUME 52, NUMBER 4 New Publications Offered by the AMS

The Wild World Number Theory of 4-Manifolds Alexandru Scorpan, University Local Zeta Functions of Florida, Gainesville, FL ME¥.9IRS Amer!oanl'>lathematloaLSoclety Attached to the The book gives an excellent overview of 4-manifolds, with many figures and Local Zeta Functions Minimal Spherical Attached to the historical notes. Graduate students, Minimal Spherical Series for a Class of nonexperts, and experts alike will enjoy Series for a Class browsing through it. of Symmetric Spaces Symmetric Spaces -Robion C. Kirby, University of Cali- Nicole Bopp fornia Berkeley Nicole Bopp and Hubert Rubenthaler, University of This gives a panorama of the topology of simply-connected smooth manifolds of dimension four. AmerlcanMnthematlcniScolcty Strasbourg, France Dimension four is unlike any other dimension; it is large Contents: Introduction; A class of real enough to have room for wild things to happen, but too small prehomogeneous spaces; The orbits of G in v+; The to have room to undo them. For example, only manifolds of symmetric spaces G/H; Integral formulas; Functional equation dimension four can exhibit infinitely many distinct smooth of the zeta function for Type I and II; Functional equation of structures. Indeed, their topology remains the least the zeta function for Type III; Zeta function attached to a understood today. representation in the minimal spherical principal series; Appendix: The example of symmetric matrices; Tables of The first part of the book puts things in context with a survey simple regular graded Lie algebras; References; Index. of higher dimensions and of topological 4-manifolds. The second part investigates the main invariant of a 4-manifold­ Memoirs of the American Mathematical Society, Volume 174, the intersection form-and its interaction with the topology of Number 821 the manifold. The third part reviews complex surfaces as an February 2005, 233 pages, Softcover, ISBN 0-8218-3623-4, LC important source of examples. The fourth and final part of the 2004062692, 2000 Mathematics Subject Classification: 17B20, book presents gauge theory. This differential-geometric 17B70, 22E46, 22Dl0, 32M15, 43A85, Individual member $44, method has brought to light the unwieldy nature of smooth 4- List $74, Institutional member $59, Order code manifolds; and although the method brings new insights, it MEM0/ 174/821 has raised more questions than answers. The structure of the book is modular and organized into a Fo' main track of approximately 200 pages, which are augmented Brauer Type (!]Classroom Use with copious notes at the end of each chapter, presenting FIELDS INSf!TUTE many extra details, proofs, and developments. To help the Embedding reader, the text is peppered with over 250 illustrations and Problems has an extensive index. Arne Ledet, Texas Tech Contents: Contents of the notes; Background scenery: Higher dimensions and the h-cobordism theorem; Topological 4- University, Lubbock manifolds and h-cobordisms; Smooth 4-manifolds and This monograph discusses Galois intersection forms: Getting acquainted with intersection forms; theoretical embedding problems of so­ Intersection forms and topology; Classifications and called Brauer type with a focus on 2- counterclassifications; A survey of complex surfaces: Running groups and on finding explicit criteria through complex geometry; The Enriques:Kodaira for solvability and explicit classification; Elliptic surfaces; Gauge theory on 4-manifolds: constructions of the solutions. The Prelude, and the Donaldson invariants; The Seiberg-Witten advantage of considering Brauer type embedding problems is invariants; The minimum genus of embedded surfaces; their comparatively simple condition for solvability in the Wildness unleashed: The Fintushel-Stern surgery; Epilogue; form of an obstruction in the Brauer group of the ground List of figures and tables; Bibliography; Index. field. May 2005, approximately 600 pages, Hardcover, ISBN 0-8218- The book presupposes knowledge of classical Galois theory 3749-4, LC 2004062768, 2000 Mathematics Subject and the attendant algebra. Before considering questions of Classification: 57-02, 57N13; 14J80, 32Q55, 57R17, 57R57, reducing the embedding problems and reformulating the 57R65, All AMS members $55, List $69, Order code solvability criteria, the author provides the necessary theory of FOURMAN Brauer groups, group cohomology and quadratic forms. The book will be suitable for students seeking an introduction to embedding problems and inverse Galois theory. It will also be a useful reference for researchers in the field. This item will also be of interest to those working in algebra and algebraic geometry.

APRIL 2005 NOTICES OF THE AMS 481 New AMS-Distributed Publications

Contents: Galois theory; Inverse Galois theory and embedding theorie elementaire des groupes libres; T. Szamuely, Groupes problems; Brauer groups; Group cohomology; Quadratic de Galois de corps de type fini; Table par norms d'auteurs. forms; Decomposing the obstruction; Quadratic forms and Asterisque, Number 294 embedding problems; Reducing the embedding problem; Pro­ finite Galois theory; Bibliography; Index. November 2004, 470 pages, Softcover, ISBN 2-85629-156-2, 2000 Mathematics Subject Classification: 11-XX, 14-XX, 52-XX, Fields Institute Monographs, Volume 21 35-XX, 03-XX, Individual member $106, List $118, Order March 2005, 171 pages, Hardcover, ISBN 0-8218-3726-5, 2000 code AST/294 Mathematics Subject Classification: 12Fl2, 16K50; 16S35, 12G05, All AMS members $42, List $52, Order code FIM/21 Rational

RATIONAL REPRESENTATIONS, Representations, The THE STEENROD ALGEBRA AND FUNCTOR HOMOLOGY Steenrod Algebra Vincent Franjou Eric M. Friedlander and Functor Teimuraz Pirashvill New AMS-Distributed Lionel Schwartz Homology Publications Panoramasetsyntheses Vincent Franjou, Universite de Nantes, France, Eric M. Friedlander, Northwestern Algebra and Algebraic University, Evanston, IL, Teimuraz Pirashvili, A. M. Geometry Razmadze Mathematical Institute, Tbilisi, Republic of Georgia, and Lionel Schwartz, Universite Paris XIIL Villetaneuse, France 294 Seminaire Bourbaki 2004, The book presents aspects of homological algebra in functor Volume 2002/2003 categories, with emphasis on polynomial functors between Exposes 909/923 vector spaces over a finite field. With these foundations in SEMINAIRE BOURBAKI place, the book presents applications to representation theory, VOLUME 2002-2003 This is a collection of surveys on EXPOSES 909-923 algebraic topology and K-theory. As these applications reveal, topics of current interest. Topics functor categories offer powerful computational techniques Avec table par noms d'auteurs include number theory, partial de 1948/49 a 2002/0S and theoretical insights. differential equations, group theory, polyedras, p-adic cohomology, set T. Pirashvili sets the stage with a discussion of foundations. theory, the Birch and Swinnerton-Dyer E. Friedlander then presents applications to the rational conjecture, L 2 Betti numbers and type representations of general linear groups. L. Schwartz Ih factors, algebraic geometry, and emphasizes the relation of functor categories to the Steenrod Galois groups of fields of finite type. algebra. Finally, V. Franjou and T. Pirashvili present A. Scorichenko's understanding of the stable K-theory of rings as This item will also be of interest to those working in number functor homology. theory, geometry and topology, differential equations, and logic and foundations. The book is suitable for graduate students and researchers interested in algebra and algebraic geometry. A publication of the Societe Mathematique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from A publication of the Societe Mathematique de France, Marseilles (SMF), other countries should be sent to the SMF. Members of the SMF receive distributed by the AMS in the U.S., Canada, and Mexico. Orders from a 30% discount from list. other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Contents: Novembre 2002: Y. F. Bilu, Catalan's conjecture; S. Fischler, Irrationalite de valeurs de zeta; G. Metivier, Contents: Introduction to functor homology; Lectures on the Exemples d'instabilites pour des equations d'ondes non cohomology of finite group schemes; Algebre de Steenrod, lineaires; J.-M. Schlenker, La conjecture des soufflets; modules instables et foncteurs polynomiaux; L'algebre de A. Valette, Nouvelles approches de la propriete (T) de Steenrod en topologie; Stable K-theory is bifunctor homology Kazhdan; Mars 2003: A. Chambert-Loir, Points rationnels et (after A. Scorichenko); Index of notation; Index; Index groupes fondamentaux: applications de la cohomologie p­ terminologique. adique; P. Dehornoy, Progres recents sur !'hypothese du Panoramas et Syntheses, Number 16 continu; E. Ghys, Groupes aleatoires; F. Morain, La primalite November 2004, 132 pages, Softcover, ISBN 2-85629-159-7, en temps polynomial; F. Rousset, Systemes hyperboliques et 2000 Mathematics Subject Classification: 14Ll5, 18G60, 19D55, viscosite evanescente; ]uin 2003: P. Colmez, La conjecture de 55-02, 55S10, Individual member $32, List $36, Order code Birch et Swinnerton-Dyer P-adique; A. Connes, Nombres de PASY/16 Betti L 2 et facteurs de type I h; I. Itenberg, Arnibes de varietes algebriques et denombrement de courbes; F. Paulin, Sur la

482 NOTICES OF THE AMS VOLUME 52, NUMBER 4 New AMS-Distributed Publications ETH Eidgenossische Technische Hochschule Zurich Memoires Integrales orbitales Swiss Federal Institute of Technology Zurich de 1:1 SOCJI':rJ! MATHI!t.lATIQUE 0E FRANCE unipotentes stables The

INTEGRALES ORBITALES et leurs transformees Department of Mathematics of the ETH Zurich UNIPOTENTES STABLES ET LEURS TRANSFORMEES DE invites applications for several SATAKE de Satake

Gia-Vuong NGUYEN-CHU Gia-Vuong Nguyen-Chu, Max­ Planck Institute of Heinz Hopf Lecturerships beginning 1 October 2005 or earlier. The positions are awarded for a 2004 Mathematics, Bonn, Germany period of 3 years, with the possibility of an extension by 1 year. In this volume, the author addresses Duties of Heinz Hopf lecturers include research and teaching in mathe­ some questions arising from harmonic matics. Together with the other members of the department, the new analysis on p-adic groups-more lecturers will be responsible for undergraduate and graduate courses precisely, in Satake transforms of stable unipotent for students of mathematics, natural sciences, and engineering. The distributions in the case of split groups. On one hand, this moderate teaching load leaves ample room for further professional problem is motivated by M. Assem's work on the computation development. Courses at Master level may be taught in English. of unipotent orbital integrals, and on the other hand, by J-L. Applicants should have proven excellence in research in any area of Waldspurgers' work on the determination of the space of mathematics and possess potential for further outstanding achieve­ stable unipotent distributions. This question is easy for ments. Some research and teaching experience after the Ph. D. is usu­ general linear groups but unknown in general. ally expected. Applications with curriculum vitae and a list of publications should be This volume deals with the groups Sp(2n) . For n = 2, it is submitted to the chairman of the Department of Mathematics, ETH shown that these Satake transforms are regular functions over Zentrum, 8og2 Zurich, Switzerland, by 30. April2oos. Later applications the rank-2 unitary real torus. It is then shown that these can be considered for remaining positions. In addition, three letters of functions can be recovered by the Satake transform of some recommendation supporting the application should be sent directly to distributions of a totally different kind: the twisted compact us. ETH Zurich specifically encourages female candidates to apply. traces of an explicit family of representations of GL(5). This phenomenon may be explained by twisted endoscopy between Sp(2n) and GL(2n + 1), as noted by Arthur. For n > 2, the author shows, in some cases, that the Satake transforms of these traces are actually regular functions, of a common form, over the rank-n unitary real torus. In particular, when n s; 4, it is true in general. With these computations, a quite precise Support lor Ph} 'iiC afl( ornpuiJIJOilclf 5CJenll.,h 8/0iou.t conjecture is proposed that describes the Satake transforms of Bnd~mg Ente11n~ stable unipotent distributions on Sp(2n) . 2006 Career Awards at the Scientific Interface This volume is suitable for graduate students and research mathematicians interested in algebra and algebraic geometry. Deadline: May 2, 2005 A publication of the Societe Mathematique de France, Marseilles (SMF), $500,000 award over five years for postdoctoral fellows distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive • These portable awards support up to two years of advanced post­ a 30% discount from list. doctoral training and the first three years of a faculty appointment Contents: Introduction; Une formule pour les traces tordues • Candidates must hold a Ph.D. in mathematics, physics, biophysics, compactes; Les traces tordues compactes sur GL ( 2n + 1); Le chemistry (physical, theoretical, or computational), computer cas GL(5); Integrales orbitales unipotentes stables sur Sp(4); science, statistics, or engineering and must not have accepted, Le cas general, reduction au cas ou {3 = 0; Le cas rr = St2n+1; or be in serious negotiations for, a faculty appointment at the time of application Le cas rr = St2h+l x ~St2k + l x ~St1; Le cas rr = St1 x ~St2h + l x~St2k+l ; Le cas rr = St2h+l x St2k+l x St1; En guise de • Candidates should propose innovative approaches to answer conclusion; Bibliographie. important biological questions Memoires de la Societe Mathematique de France, Number 97 • BWF encourages proposals that include experimental validation of theoretical models November 2004, 110 pages, Softcover, ISBN 2-85629-157-0, 2000 Mathematics Subject Classification: 22£35, 22£50, • Degree-granting institutions in the U .S. and Canada may Individual member $33, List $3 7, Order code SMFMEM/97 nominate up to rwo candidates • Complete program information, eligibility guidelines, and appli­ cation forms are available on BWF's website at www.bwfund.org

t 919.991.5 100 BU RROUGHS f 919.991.5160 www.bwf und.org WELLCOME Post Office Box 13901 21 T. W. Alexander Drive FUND ~ Research Triangle Park, NC 27709-3901 Tbc Burroughs WHkorne Fund is an indtpmdent private Jimnd.aion derlioltal to ,zdvrmcing tht biomediod scimas by supporting nsmrch ami otha scimtijir and educatimltt! tlctivitie)·.

APRIL 2005 NOTICES OF THE AMS 483 Classified Advertisements Positions available, items for sale, services available, and more

of Assistant Professor, starting June or CALIFORNIA September 2005. Earned doctorate or All OREGON But Dissertation (ABD) in Mathematics with CALIFORNIA STATE UNIVERSITY, LOS a strong background in Algebra/ Number OREGON STATE UNIVERSITY ANGELES Theory from an accredited institution of Department of Mathematics Department of Mathematics higher education is required. Preference will be given to individuals with the doc­ The Department of Mathematics invites Inquiries are invited for a possible tenure­ torate from an accredited university, and applications for a tenure-track Assistant track position as Director of Develop­ individuals who will have the doctorate Professor position specializing in applied mental Mathematics at the level of Assis­ from an accredited university at the time probability and stochastic processes. Ap­ tant/ Associate Professor, starting June or of appointment. Doctorate required for plicants should have a Ph.D. in mathe­ September 2005. Earned doctorate or All tenure. Ability to teach a range of under­ matics or a closely related field, significant But Dissertation (ABD) in Mathematics or graduate mathematics classes is essential. active research engagement in applied Mathematics Education from an accred­ Publications in peer-reviewed journals probability or stochastic processes, and ex­ ited institution of higher education is re­ and/ or grant activity is required for cellence in teaching. Interest in partici­ quired. Preference will be given to indi­ tenure/ promotion. CSULA is on the quar­ pating in team-based interdisciplinary re­ viduals with the doctorate from an ter system. Send letter of application, vita, search and graduate education involving accredited university, and individuals who three letters of recommendation and of­ ecosystem processes is preferred. The ap­ will have the doctorate from an accredited ficial transcript from institution awarding pointee will be expected to maintain a vig­ university at the time of appointment. doctorate to Dr. P. K. Subramanian, Chair, orous research program in the field of ap­ Doctorate required for t enure. Must Department of Mathematics, California plied probability and stochastic processes demonstrate interest and experience work­ State University at Los Angeles, 5151 State and participate in teaching, advising and ing with remedial/basic skills mathemat­ University Drive, Los Angeles, CA 90032. mentoring at the graduate and under­ ics program. Ability to teach a range of un­ An Equal Opportunity, Title IX, Disabled, graduate levels. Applicants should send a dergraduate/ graduate mathematics classes Employer. letter of interest and a detailed curriculum is essential. Publications in peer-reviewed 0 0 0 040 vitae including a description of current journals and/ or grant activity is required and future research interests and a list of for tenure/ promotion. CSULA is on the publications to: quarter system. Send letter of application, Search Committee: Probability vita, three letters of recommendation and MICHIGAN Department of Mathematics official transcript from institution award­ Oregon State University ing doctorate to Dr. P. K. Subramanian, MICHIGAN STATE UNIVERSITY Corvallis, OR 97331-4605 Chair, Department of Mathematics, Cali­ East Lansing, Ml48824 Additionally three letters of recommen­ fornia State University at Los Angeles, proMSc Program in dation, one of which addresses teaching, 5151 State University Drive, Los Angeles, Industrial Mathematics are required. They should be sent directly CA 90032. An Equal Opportunity, Title IX, to the above address. For full considera­ Physically Challenged, Employer. Direct your students toward one of the tion, complete application materials must 000041 professional M.Sc. programs. Industry arrive by April15, 2005. Further informa­ needs business-savvy mathematicians. See tion is available at http:// www.math. CALIFORNIA STATE UNIVERSITY, LOS http: I /www. sci encemasters . com/. oregonstate.edu/hiring. ANGELES 000019 OSU is an Affirmative Action/ Equal Op­ Department of Mathematics portunity Employer. 000042 Inquiries are invited for a possible tenure­ track position in Mathematics at the level

Suggested uses for classified advertising are positions available, books or 2005 issue-May 26, 2005; September 2005 issue-June 27, 2005; October lecture notes for sale, books being sought, exchange or rental of houses, issue-July 25, 2005; November 2005 issue-August 26, 2005. and typing services. U.S. laws prohibit discrinlination in employment on the basis of color, age, The 200S rate is $100 per inch or fraction thereof on a single column(one­ sex, race, religion, or national origin. "Positions Available" advertisements inch minimum), calculated from top of headline. Any fractional text of 1/ 2 from institutions outside the U.S. cannot be published unless they are inch or more will be charged at the next inch rate. No discounts for multi­ accompanied by a statement that the institution does not discriminate on ple ads or the same ad in consecutive issues. For an additional $10 charge, these grounds whether or not it is subject to U.S. laws. Details and spe­ announcements can be placed anonymously. Correspondence will be cific wording may be found on page 1373 (vol. 44). forwarded. Situations wanted advertisements from involuntarily unemployed math­ Advertisements in the "Positions Available" classified section will be set ematicians are accepted under certain conditions for free publication. Call with a minimum one-line headline, consisting of the institution name above toll-free 800-321-4AMS (321-42 67) in the U.S. and Canada or 401-455-4084 body copy, unless additional headline copy is specified by the advertiser. worldwide for further information. Headlines will be centered in boldface at no extra charge. Ads will appear Submission: Promotions Department, AMS, P.O. Box 6248, Providence, in the language in which they are submitted. Rhode Island 02940; or via fax: 401-3 31-3842; or send email to There are no member discounts for classified ads. Dictation over the cl assads@ams. or g. AMS location for express delivery packages is telephone will not be accepted for classified ads. 201 Charles Street, Providence, Rhode Island 20904. Advertisers will be Upcoming deadlines for classified advertising are as follows: May 2005 billed upon publication. issue-February 25, 2005; June/July 2005 issue-April 27, 2005; August

484 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Meetings & Conferences oftheAMS

IMPORTANTINFORMATION 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 withlinkstotheabstractforeachtalkcanbefoundon theAMSwebsite.See http: I /www. ams. org/meeti ngs/.Programs and abstracts will continue to be displayed on the AMS website in the Meetings and Conferences section until about three weeks after the meeting is over. Final programs for Sectional Meetings will be archived on the AMS website in an electronic issue of the Notices as noted below for each meeting.

Robert J. McCann, University of Toronto, Optimal Bowling Green, convergence rates for the fastest conservative nonlinear diffusions. Kentucky M. Susan Montgomery, University of Southern California, On some connections between finite groups and Hop{ Western Kentucky University algebras. James J. Zhang, University of Washington, Searching for March 18-,19,2005 quantum projective spaces. Friday - Saturday Special Sessions Meeting #1 004 Advances in the Study of Wavelets and Multiwavelets, Southeastern Section Douglas P. Hardin, Vanderbilt University, and Bruce Associate secretary: Matthew Miller Kessler, Western Kentucky University. Announcement issue of Notices: January 2005 Commutative Ring Theory, Michael C. Axtell, Wabash Col­ Program first available on AMS website: February 3, 2005 lege, and Joe Alyn Stickles Jr., University of Evansville. Program issue of electronic Notices: March 2005 Dynamic Equations on Time Scales and Applications, Issue of Abstracts: Volume 26, Issue 2 Ferhan M. Atici and Daniel C. Biles, Western Kentucky Deadlines University, and Billur Kaymak.;alan, Georgia Southern University. For organizers: Expired For consideration of contributed papers in Special Sessions: Geometric Topology and Group Theory, Jens E. Harlander, Expired Western Kentucky University. For abstracts: Expired Graph Theory, Mustafa Atici, Western Kentucky University. Hop{ Algebras and Related Topics, David E. Radford, The scientific information listed below may be dated. For University of Illinois at Chicago, and Bettina Richmond, the latest information, see www. ams. org/amsmtgs/ Western Kentucky University. sectional . html. Knot Theory and Its Applications, Yuanan Diao, University of North Carolina, Charlotte, and Claus Ernst, Western Ken­ Invited Addresses tucky University. Bennett Chow, University of California San Diego, Title to L-Functions, Heather Russell, Nilabh Sanat, and Dominic be announced. Lanphier, Western Kentucky University.

APRIL 2005 NOTICES OF THE AMS 485 Meetings & Conferences

Nonlinear Analysis and Applied Mathematics, Robert J. Special Sessions McCann, University of Toronto, and Daniel P. Spirn, Uni­ Arithmetic Groups and Related Topics, Alexander versity of Minnesota. Lubotzky, Hebrew University of Jerusalem, and Andrei Numerical Analysis, Approximation, and Computational Rapinchuk, University of Virginia. Complexity: Interdisciplinary Aspects, David Benko, Western Kentucky University, and Steven B. Damelin, Asymptotic Behavior of Evolution Equations, Gaston M. Georgia Southern University. N'Guerekata, Morgan State University, and Nguyen Van Partial Differential Equations and Their Applications, Zhong­ Minh, University of West Georgia. wei Shen and Changyou Wang, University of Kentucky. Designs, Codes, and Geometries, James A. Davis, Univer­ Recent Advances in Noncommutative Algebra, Ellen E. sity of Richmond, Keith E. Mellinger, University of Mary Kirkman, Wake Forest University. Washington, and Qing Xiang, University of Delaware. Representation Theory, Markus Hunziker, Baylor Univer­ Frontiers on Complex Fluid Flows: Analytic and Computa­ sity. tional Methods, L. Pamela Cook and Louis F. Rossi, Uni­ Semigroups of Operators and Applications, Khristo Boy­ versity of Delaware. adzhiev, Ohio Northern University, Lan Nguyen, Western Geometric Analysis, Xiuxiong Chen, University of Wis­ Kentucky University, and Quoc-Phong Vu, Ohio University. consin, Madison, Pengfei Guan, McMaster University, Topology, Convergence, and Order, in Honor of Darrell Zhiqin Lu, University of California Irvine, and Jeff A. Kent, Gary Richardson, University of Central Florida, and Viaclovsky, Massachusetts Institute of Technology. Thomas A. Richmond, Western Kentucky University. High Dimensional Probability, Wenbo Li, University of Delaware, and Joel Zinn, Texas A&M University. Homotopy Theory (in Honor of Donald M. Davis's and Newark, Delaware Martin Bendersky's 60th Birthdays), Kenneth G. Monks, University of Delaware University of Scranton, and W. Stephen Wilson, Johns Hopkins University. April2-3, 2005 Integral and Operator Equations, Charles W. Groetsch, Saturday - Sunday University of Cincinnati, and M. Zuhair Nashed, Univer­ sity of Central Florida. Meeting #1 005 Mathematical Biology, David A. Edwards, University of Eastern Section Associate secretary: Lesley M. Sibner Delaware. Announcement issue of Notices: February 2005 Mathematical Methods for Efficient Simulation of Stochas­ Program first available on AMS website: February 17, 2005 tic Nonlinear Optical Systems, Richard 0. Moore, New Program issue of electronic Notices: April 2005 Jersey Institute of Technology, and Tobin A. Driscoll, Issue of Abstracts: Volume 26, Issue 2 University of Delaware. Deadlines Mathematical Methods in Electromagnetic Wave Propaga­ tion, Fioralba Cakoni and Peter B. Monk, University of For organizers: Expired Delaware. For consideration of contributed papers in Special Sessions: Expired Probabilistic Paradigms in Combinatorics, Joshua N. For abstracts: Expired Cooper, Courant Institute of Mathematics, NYU, and Jozef Skokan, Universidade de Sao Paulo. The scientific information listed below may be dated. For Recent Progress in Thin Fluid Flows, Richard J. Braun, Uni­ the latest information, see www. ams. o rg/amsmtgs/ versity of Delaware. sectional.html. Singular Analysis and Spectral Theory of Partial Differen­ Invited Addresses tial Equations, juan B. Gil, Pennsylvania State University, Altoona, and Gerardo A. Mendoza, Temple University. Xiuxiong Chen, University of Wisconsin, Madison, Folia­ tion by holomorphic discs and its application in Kahler Spectral and High-Order Discretization Methods for Partial geometry. Differential Equations, Tobin A. Driscoll, University of Anna C. Gilbert, AT&T Labs-Research, Analysis, approxi­ Delaware. mations, and algorithms. Symmetry Methods for Partial Differential Equations, Philip Alexander Lubotzky, Hebrew University of Jerusalem, Broadbridge, University of Delaware, and Danny Arrigo, Counting primes, groups, and manifolds. University of Central Arkansas. Lorenz Schwachhoefer, University of Dortmund, Special symplectic connections.

486 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Meetings & Conferences

Graph Theory, John C. George, Eastern New Mexico Uni­ Lubbock, Texas versity, and Walter D. Wallis, Southern Illinois University at Carbondale. Texas Tech University Homological Algebra and Its Applications, Alex Martsinkovsky, Northeastern University, and Mara D. April8-l 0, 2005 Neusel, Texas Tech University. Friday - Sunday Invariants ofLinks and 3-Manifolds, Mieczyslaw Krzysztof Meeting #1 006 Dabkowski, University of Texas at Dallas, Razvan Gelca, Texas Tech University, andJozefHenrykPrzytycki, George Central Section Washington University. Associate secretary: Susan]. Friedlander Announcement issue of Notices: February 2005 Partial Differential Equation and Its Application in Bio­ medical Study, Jay R. Walton, Texas A&M University, and Program first available on AMS website: February 24, 2005 Padmanabhan Seshaiyer and Akif Ibragimov, Texas Tech Program issue of electronic Notices: April 2005 University. Issue of Abstracts: Volume 26, Issue 3 Real Algebraic Geometry, Anatoly Korchagin and David Deadlines Weinberg, Texas Tech University. For organizers: Expired Recent Advances in Complex Function Theory, Brock For consideration of contributed papers in Special Sessions: Williams, Roger W. Barnard, and Kent Pearce, Texas Tech Expired University. For abstracts: Expired Statistical Image Processing and Analysis and Applications, Victor Patrangenaru, Texas Tech University. The scientific information listed below may be dated. For Theory and Application of Stochastic Differential Equa­ the latest information, see www. ams. o rg/ amsmtgs/ tions, Edward J. Allen, Texas Tech University, and Ar­ sectional.html. mando Arciniega, University of Texas at San Antonio. Topology of Continua, Wayne Lewis, Texas Tech Univer: Invited Addresses sity. Nikolai Ivanov, Michigan State University, Title to be Topology of Dynamical Systems, Brian Raines, Baylor Uni­ announced. versity. Mattias Jonsson, University of Michigan, Title to be Undergraduate and Graduate Student Research (and Related announced. Poster Session organized by Ali Khoujmane and Mara D. Nicolas Monod, University of Chicago, Title to be Neusal, Texas Tech), Ali Khoujmane, Edward W. Swim, Edward J, Allen, and Padmanabhan Seshaiyer, Texas Tech announced. University. Hee Oh, California Institute of Technology, Title to be announced.

Special Sessions Santa Barbara, Classical and Differential Galois Theory, Lourdes Juan and California Arne Ledet, Texas Tech University, and Andy R. Magid, University of Oklahoma. University of California Santa Barbara Differential Geometry and Its Applications, Josef F. Dorfmeis­ April16-17, 2005 ter, Munich University of Technology, Magdalena D. Toda, Saturday - Sunday Texas Tech University, and Hongyou Wu, Northern Illinois University. Meeting #1 007 Discrete Groups, Homogeneous Spaces, Rigidity, Alex Gorod­ Western Section nik, University of Michigan, Ann Arbor, Hee Oh, California Associate secretary: Michel L. Lapidus Institute of Technology, and Nicolas Monod, University of Announcement issue of Notices: February 2005 Chicago. Program first available on AMS website: March 3, 2005 Extinction, Periodicity, and Chaos in Population and Epidemic Program issue of electronic Notices: April 2005 Issue of Abstracts: Volume 26, Issue 3 Models, Linda J. S. Allen, Texas Tech University, Sophia Ruey-Jen Jang, University of Louisiana at Lafayette, and Deadlines Lih-Ing W. Roeger, Texas Tech University. For organizers: Expired Future Directions in Mathematical Systems and Control For consideration of contributed papers in Special Sessions: Theory, David Gilliam and W. P. Dayawansa, Texas Tech Expired University. For abstracts: Expired

APRIL 2005 NOTICES OF THE AMS 487 Meetings & Conferences

Huisgen-Zimmermann, University of California Santa The scientific information listed below may be dated. For Barbara, and Edward L. Green, Virginia Polytech Institute the latest information, see www. ams. o rg/amsmtgs/ & State University. sectional . html. Ricci Flow/Riemannian Geometry, Guofang Wei and Invited Addresses Rugang Ye, University of California Santa Barbara. Mei-Chu Chang, University of California Riverside, Set addition and set multiplication. Mainz, Germany Mischa Kapovich, University of California Davis, General­ ized triangle inequalities and their applications. June 16-19,2005 Mihai Putinar, University of California Santa Barbara, Thursday - Sunday Positive polynomials, a hilbertian perspective. Meeting #1 008 James Sethian, University of California Berkeley, Advances in advancing interfaces: New techniques for propagating ]oint International Meeting with the Deutsche Mathematiker­ fronts in wave propagation and materials sciences. Vereinigung (DMV) and the Oesterreichische Mathematis­ che Gesellschaft (OMG) Special Sessions Associate secretary: Susan J. Friedlander Announcement issue of Notices: February 2005 Algebraic Geometry and Combinatorics, Alexander Yong Program first available on AMS website: Not applicable and Allen Knutson, University of California Berkeley. Program issue of electronic Notices: Not applicable Arithmetic Geometry, Adebisi Agboola, University of Issue of Abstracts: Not applicable California Santa Barbara, and Cristian Dumitru Popescu, University of California San Diego. Deadlines Automorphisms of Surfaces, Anthony Weaver, Bronx For organizers: Expired Community College of the City University of New York, For consideration of contributed papers in Special Sessions: and Peter Turbek, Purdue University Calumet. March 15, 2005 Complexity of Computation and Algorithms, Mark Burgin, For abstracts: March 31, 2005 University of California Los Angeles. Curvature in Group Theory and Combinatorics, Laura M. The scientific information listed below may be dated. For Anderson, State University of New York at Binghamton, the latest information, see www. ams. o rg/amsmtgs/ Noel Patrick Brady, University of Oklahoma, Robin i nternmtgs. html. Forman, Rice University, and Jonathan P. McCammond, University of California Santa Barbara. Watch the website maintained by the local organizers athttp://www.mathematik.uni-mainz.de/mainz2005/ Dynamical Systems in Neuroscience, Eugene M. Izhike­ for additional program vich, The Neurosciences Institute. · details and links to sites for hotels, tours, and other local information. Function Theory, Mihai Putinar and Stephan R. Garcia, University of California Santa Barbara. Invited Addresses Geometric Methods in Three Dimensions, Daryl Cooper, Helene Esnault, University of Essen, Deligne's integrality David Darren Long, and Martin G. Scharlemann, Univer- theorem in unequal characteristic and rational points over sity of California Santa Barbara. · finite fields. Geometry and Physics, Xianzhe Dai, University of Califor­ Richard Hamilton, Columbia University, The Ricci flow. nia Santa Barbara, and Zhiqin Lu, University of California Irvine. Michael J. Hopkins, Massachusetts Institute of Technol­ ogy, Title to be announced. History of Mathematics, Shawnee L. McMurran, California State University, San Bernardino, and James J. Tattersall, Christian Krattenthaler, University of Lyon-I, Exact and Providence College. asymptotic enumeration of vicious walkers with a wall Noncommutative Geometry and Algebra, Kenneth R. Good­ interaction. earl, University of California Santa Barbara, J. T. Stafford, Frank Natterer, University of Muenster, Imaging and University of Michigan, and ]. ]. Zhang, University of inverse problems for partial differential equations. Washington. Horng-Tzer Yau, New York University and Stanford Uni­ Recent Advances in Combinatorial Number Theory, Mei­ versity, Dynamics of Bose-Einstein condensate. Chu Chang, University of California Riverside, and Van Ha Vu, University of California San Diego. Special Sessions Representation Theory of Algebras (in Honor of Claus Affine Algebraic Geometry, Shreeram Abhyankar, Pur­ Michael Ringel), Alex Martsinkovsky, Northeastern due University, Hubert Flenner, Ruhr University Bochum, University, Dan Zacharia, Syracuse University, Birge K. and Makar Limanov, Wayne State University.

488 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Meetings & Conferences

Algebraic Combinatorics, Patricia Hersh, Indiana Univer­ Mathematics Education, Gunter Torner, Universitat Duis­ sity-Bloomington, Christian Krattenthaler, University of burg-Essen, and Alan Schoenfeld, School of Education, Lyon-!, and Volkmar Welker, Philipps University Marburg. Berkeley. Algebraic Cryptography, Dorian Goldfeld, Columbia Modules and Comodules, Sergio L6pez-Permouth, Ohio University, Martin Kreuzer and Gerhard Rosenberger, University, and Robert Wisbauer, University of Dusseldorf. Universitat Dortmund, and Vladimir Shpilrain, The City Multiplicative Arithmetic ofIntegral Domains and Mono ids, College of New York. Scott Chapman, Trinity University, San Antonio, Franz Algebraic Cycles, Eric Friedlander and Marc Levine, North­ Halter-Koch, University of Graz, and Ulrich Krause, Uni­ western University, and Fabien Morel, Universite Paris. versitat Bremen. Algebraic Geometry, Yuri Tschinkel, Georg-August­ Nonlinear Elliptic Boundary Value Problems, Thomas Universitat Gottingen, and Brendan E. Hassett, Rice Bartsch, Universitaet Giessen, and Zhi-Qiang Wang, Utah University. State University. Dirac Operators, Clifford Analysis and Applications, Klaus Nonlinear Waves, Herbert Koch, University of Dortmund, Giirlebeck, University of Weimar, Mircea Martin, Baker Uni­ and Daniel I. Tataru, University of California Berkeley. versity, John Ryan, University of Arkansas, and Michael Shapiro, IPN Mexico. Ordinary Differential, Difference, and Dynamic Equations, Werner Balser, Universitat Ulm, Martin Bohner, University Discrete Geometry, Jacob Eli Goodman, The City College of Missouri-Rolla, and Donald Lutz, San Diego State Uni­ of New York (CUNY), Emo Welzl, Eidgen Technische Hochschule, and Gunter M. Ziegler, Technical University versity. of Berlin. Quantum Knot Invariants, Anna Beliakova, Universitat Function Spaces and Their Operators, Ernst Albrecht, Uni­ Zurich, and Uwe Kaiser, Boise State University. versitat des Saarlandes, Raymond Mortini, Universite de Representations and Cohomology of Groups and Algebras, Metz, and William Ross, University of Richmond. Dave Benson, University of Georgia, and Henning Krause, Functional Analytic and Complex Analytic Methods in Universitat Paderborn. Linear Partial Differential Equations, R. Meise, University Set Theory, Joel Hamkins, City University New York, Peter of Dusseldorf, B. A. Taylor, University of Michigan, and Koepke, Universitat Bonn, and Benedikt Lowe, Univer­ Dietmar Vogt, University of Wuppertal. siteit van Amsterdam. Geometric Analysis, Victor Nistor, Pennsylvania State Spectral Analysis of Differential and Difference Operators, University, and Elamr Schrohe, Universitat Hannover. Evgeni Korotyaev, Humboldt-University Berlin, Boris Geometric Topology and Group Theory, Cameron MeA Mityagin, The Ohio State University, and Gerald Teschl, Gordon, The University of Texas at Austin, Cynthia University of Vienna. Hog-Angeloni, Johann Wolfgang Goethe-Universitat, and Stochastic Analysis on Metric Spaces, Laurent Saloff-Coste, Wolfgang Metzler, University of Frankfurt. Cornell University, Karl-Theodor Sturm, University of Group Theory, Luise-Charlotte Kappe, SUNY at Bingham­ Bonn, and Wolfgang Woess, Graz Technical University. ton, Robert Fitzgerald Morse, University of Evansville, Topics in Applied Mathematics: Algebraic Approaches to and Gerhard Rosenberger, University of Dortmund. Preconditioning, Heike Fassbender, Technical University Hilbert Functions and Syzygies, Uwe Nagel, University of of Braunschweig, and Andreas Frommer, University of Kentucky, Irena Peeva, Cornell University, and Tim Romer, Wuppertal. Universitat Osnabnick. Topics in Applied Mathematics: Control Theory, Peter History of Mathematics (including a special workshop on Benner, Technical University of Chemnitz. Mathematics and War), Thomas W. Archibald, Acadia Uni­ Topics in Applied Mathematics: Mathematical Problems of versity, John H. McCleary, Vassar College, Moritz Epple, University of Stuttgart, and Norbert Schappacher, Tech­ Mechanics, Friedrich Pfeiffer and Jurgen K. Scheurle, nische Universitat Darmstadt. Technical University of Munich. Homotopy Theory, Paul G. Goerss, Northwestern University, Topics in Applied Mathematics: Multiscale Problems, Oscil­ Hans-Werner Henn, Institut de Recherche Mathematique lations in Partial Differential Equations, and Homogeniza­ Avancee, Strasbourg, and Stefan Schwede, Universitat tion, Alexander Mielke, University of Hannover. Bonn. Topics in Applied Mathematics: Numerical Partial Differential Hop{ Algebras and Quantum Groups, M. Susan Mont­ Equations/Equations with Inherent Conditions, Rolf Jeltsch, gomery, University of Southern California, and Hans­ Eidgen Technische Hochschule, Maria Lukacova­ Jurgen Schneider, University of Munich. Medvidova, Technical University of Hamburg, and]. Mac Mathematical Physics, Laszlo Erdos, Mathematisches Hyman, Los Alamos National Laboratory. Institut der Albert Ludwigs Universitat, and Michael P. Loss, Topology of Manifolds, Matthias Kreck, University of Georgia Institute of Technology. Heidelberg, and Andrew Ranicki, University of Edinburgh.

APRIL 2005 NOTICES OF THE AMS 489 Meetings & Conferences

Infinite Groups (Code: SS lOA), Anthony M. Gaglione, Annandale-on­ United States Naval Academy, Benjamin Fine, Fairfield University, and Dennis Spellman, Philadelphia University. Hudson, New York Invariants of Graphs and Matroids (Code: SS 8A), Gary Bard Callege Gordon and Lorenzo Traldi, Lafayette College. Measurable, Symbolic, and Tiling Dynamical Systems (Code: October 8-9, 2005 SS 9A), Natalie Preibe Frank, Vassar College, and Samuel J. Saturday - Sunday Lightwood, Western Connecticut State University. Meeting #1 009 Special Functions and Orthogonal Polynomials: Theory and Applications (Code: SS 7A), Diego Dominici, State Univer­ Eastern Section Associate secretary: Lesley M. Sibner sity of New York at New Paltz. Announcement issue of Notices: August 2005 Theory ofInfinite-Dimensional Lie Algebras, Vertex Operator Program first available on AMS website: August 25, 2005 Algebras, and Related Topics (Code: SS SA), Antun Milas, Program issue of electronic Notices: October 2005 SUNY at Albany, Alex J, Feingold, Binghamton University, Issue of Abstracts: Volume 26, Issue 4 and Yi-Zhi Huang, Rutgers University. Deadlines For organizers: Expired For consideration of contributed papers in Special Sessions: Johnson City, June 21, 2005 For abstracts: August 16, 2005 Tennessee

The scientific information listed below may be dated. For East Tennessee State University the latest information, see www. ams. org/amsmtgs/ October 1 5-16, 2005 sectional . html. Saturday - Sunday Invited Addresses Meeting #1 010 Persi Diaconis, Stanford University, Title to be announced (Erdos Memorial Lecture). Southeastern Section Associate secretary: Matthew Miller Harold Rosenberg, University of Paris VII, Title to be announced. Announcement issue of Notices: August 2005 Program first available on AMS website: September 1, 2005 Alice Silverberg, University of California Irvine, Title to be Program issue of electronic Notices: October 2005 announced. Issue of Abstracts: Volume 26, Issue 4 Christopher Sogge, Johns Hopkins University, Title to be announced. Deadlines Benny Sudakov, Princeton University, Title to be For organizers: March 15, 2005 announced. For consideration of contributed papers in Special Sessions: Special Sessions June 28, 2005 For abstracts: August 23, 2005 Geometric Group Theory (Code: SS lA), Sean Cleary, The City College of New York, and Melanie I. Stein, Trinity The scientific information listed below may be dated. For College. the latest information, see www. ams. o rg/amsmtgs/ Geometric Transversal Theory (Code: SS 3A), Richard sectional . html. Pollack, Courant Institute, New York University, and Jacob Eli Goodman, The City College of New York. Invited Addresses Global Theory of Minimal Surfaces (Code: SS 6A), David A. Alberto Bressan, Pennsylvania State University, Title to be Hoffman, Mathematical Sciences Research Institute, and announced. Harold Rosenberg, University of Paris VII. Assaf Naor, Microsoft Research, Title to be announced. The History ofMathematics (Code: SS 2A), Patricia R. Allaire, Queensborough Community College, CUNY, Robert E. Prasad V. Tetali, Georgia Institute of Technology, Title to Bradley, Adelphi University, and Jeff Suzuki, Bard College. be announced. Homological Aspects of Commutative Algebra (Code: SS 4A), Rekha R. Thomas, University of Washington, Title to be Alexandre Tchernev, University of Albany, SUNY, and announced. Janet Vassilev, University of Arkansas.

490 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Meetings & Conferences Lincoln, Nebraska Eugene, Oregon University of Nebraska in Lincoln University of Oregon

October 21-23, 2005 November 12-1 3, 2005 Friday - Sunday Saturday - Sunday

Meeting #1 011 Meeting #1 012 Central Section Western Section Associate secretary: Susan]. Friedlander Associate secretary: Michel L. Lapidus Announcement issue of Notices: September 2005 Announcement issue of Notices: August 2005 Program first available on AMS website: September 29, Program first available on AMS website: September 8, 2005 2005 Program issue of electronic Notices: October 2005 Program issue of electronic Notices: November 2005 Issue of Abstracts: Volume 26, Issue 4 Issue of Abstracts: Volume 26, Issue 4 Deadlines Deadlines For organizers: March 22, 2005 For organizers: Aprill2, 2005 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: July 5, 2005 July 26, 2005 For abstracts: August 30, 2005 For abstracts: September 20, 2005

The scientific information listed below may be dated. For The scientific information listed below may be dated. For the latest information, see www.ams.org/amsmtgs/ the latest information, see www. ams. o rg/amsmtgs/ secti anal. html. sectional . html.

Invited Addresses Invited Addresses Howard Masur, University of Illinois at Chicago, Title to Matthew Foreman, University of California Irvine, Title to be announced. be announced. Mark Haiman, University of California Berkeley, Title to be Alejandro Uribe, University of Michigan, Title to be announced. announced. Wilhelm Schlag, California Institute of Technology, Title Judy Walker, University of Nebraska, Title to be announced. to be announced. Jack Xin, University of Texas, Title to be announced. Hart H. Smith, University of Washington, Title to be announced. Special Sessions Algebraic Geometry (Code: SS lA), Brian Harbourne, Uni­ Special Sessions versity of Nebraska-Lincoln, and Bangere P. Purnaprajna, K- Theory in M- Theory (Code: SS 6A), Gregory D. Landweber, University of Kansas. University of Oregon, and Charles F. Doran, University of Dynamic Equations on Time Scales (Code: SS 5A), Lynn H. Washington. Erbe and Allan C. Peterson, University of Nebraska­ Noncommutative Algebra and Noncommutative Birational Lincoln. Geometry(Code: SS 3A), Arkady Drnitrievich Berenstein, Uni­ versity of Oregon, and Vladimir Retakh, Rutgers University. Geometric Methods in Group Theory and Semigroup Theory (Code: SS 6A), Susan M. Hermiller and John C. Meakin, Partial Differential Equations with Applications (Code: SS University of Nebraska-Lincoln, and Zoran Sunik, Texas 4A), Alexander Panchenko, Washington State University, A&M University. R. E. Showalter, Oregon State University, and Hong-Ming Yin, Washington State University. Large Cardinals in Set Theory (Code: SS 4A), Paul B. Regular Algebras and Noncommutative Projective Geom­ Larson, Miami University, Justin Tatch Moore, Boise State etry (Code: SS 2A), Brad Shelton, University of Oregon, University, and Ernest Schimmerling, Carnegie Mellon Michaela Vancliff, University of Texas at Arlington, and University. James J. Zhang, University of Washington. Mathematical and Engineering Aspects of Coding Theory Representations of Groups and Algebras (Code: SS 5A), (Code: SS 3A), Lance Perez and Judy Walker, University Jonathan W. Brundan, Alexander S. Kleshchev, and Viktor of Nebraska, Lincoln. Ostrik, University of Oregon. Recent Progress in Operator Algebras (Code: SS 2A), Allan P. Resolutions (Code: SS lA), Christopher Alan Francisco, Uni­ Donsig and David R. Pitts, University of Nebraska. versity of Missouri, and Irena Peeva, Cornell University.

APRIL 2005 NOTICES OF THE AMS 491 Meetings & Conferences Taiwan Miami, Florida December 14-18, 2005 Florida International University Wednesday - Sunday April1-2, 2006 Meeting #1 013 Saturday - Sunday First ]oint International Meeting between the AMS and the Meeting #1 015 Taiwanese Mathematical Society. Southeastern Section Associate secretary: John L. Bryant Associate secretary: Matthew Miller Announcement issue of Notices: June 2005 Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Issue of Abstracts: To be announced

Deadlines Deadlines For organizers: To be announced For organizers: September 1, 2005 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: To be announced To be announced For abstracts: To be announced For abstracts: To be announced San Antonio, Texas Notre Dame, Indiana University of Notre Dame Henry B. Gonzalez Convention Center April8-9, 2006 January 1 2-1 5, 2006 Saturday - Sunday Thursday - Sunday Meeting #1 016 Meeting #1 014 Central Section ]oint Mathematics Meetings, including the 112th Annual Associate secretary: Susan]. Friedlander Meeting of the AMS, 89th Annual Meeting of the Mathe­ Announcement issue of Notices: February 2006 matical Association of America, annual meetings of the Program first available on AMS website: To be announced Association for Women in Mathematics (A Tl\fM) and the Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced National Association of Mathematicians (NAM), the winter meeting of the Association for Symbolic Logic (ASL), with Deadlines sessions contributed by the Society for Industrial and For organizers: September 9, 2005 Applied Mathematics (SIAM). For consideration of contributed papers in Special Sessions: Associate secretary: Matthew Miller To be announced Announcement issue of Notices: October 2005 For abstracts: To be announced Program first available on AMS website: November 1, 2005 Program issue of electronic Notices: January 2006 Issue of Abstracts: Volume 27, Issue 1 Durham, Deadlines New Hampshire For organizers: April1, 2005 For consideration of contributed papers in Special Sessions: University of New Hampshire To be announced April22-23, 2006 For abstracts: To be announced Saturday - Sunday For summaries of papers to MAA organizers: To be announced Meeting #1 017 Eastern Section Associate secretary: Lesley M. Sibner Announcement issue of Notices: To be announced Program first available on AMS website: To be announced

492 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Meetings & Conferences

Program issue of electronic Notices: To be announced For consideration of contributed papers in Special Sessions: Issue of Abstracts: To be announced To be announced For abstracts: To be announced Deadlines For organizers: September 22, 2005 For consideration of contributed papers in Special Sessions: Fayetteville, To be announced For abstracts: To be announced Arkansas San Francisco, University of Arkansas November 3-4,2006 California Friday - Saturday Southeastern Section San Francisco State University Associate secretary: Matthew Miller Announcement issue of Notices: To be announced April29-30, 2006 Program first available on AMS website: To be announced Saturday - Sunday Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Meeting #1 018 Western Section Deadlines Associate secretary: Michel L. Lapidus For organizers: April 3, 2006 Announcement issue of Notices: To be announced For consideration of contributed papers in Special Sessions: Program first available on AMS website: To be announced To be announced Program issue of electronic Notices: To be announced For abstracts: To be announced Issue of Abstracts: To be announced

Deadlines For organizers: September 30, 2005 New Orleans, For consideration of contributed papers in Special Sessions: To be announced Louisiana For abstracts: To be announced New Orleans Marriott and Sheraton The scientific information listed below may be dated. For New Orleans Hotel the latest information, see www. ams. org/amsmtgs/ sectional.html. January 4-7,2007 Thursday - Sunday Special Sessions ]oint Mathematics Meetings, including the 113th Annual Meeting of the AMS, 90th Annual Meeting of the Mathe­ History of Mathematics (Code: SS 1A), Shawnee L. McMurran, California State University, San Bernardino, matical Association of America (MAA), annual meetings of and James J. Tattersall, Providence College. the Association for Women in Mathematics (A HIM) and the National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Cincinnati, Ohio Applied Mathematics (SIAM). University of Cincinnati Associate secretary: Susan]. Friedlander Announcement issue of Notices: October 2006 October 21-22, 2006 Program first available on AMS website: To be announced Saturday - Sunday Program issue of electronic Notices: January 2007 Central Section Issue of Abstracts: To be announced Associate secretary: Susan ]. Friedlander Announcement issue of Notices: To be announced Deadlines Program first available on AMS website: To be announced For organizers: April 1, 2006 Program issue of electronic Notices: To be announced For consideration of contributed papers in Special Sessions: Issue of Abstracts: To be announced To be announced For abstracts: To be announced Deadlines For summaries of papers to MAA organizers: To be For organizers: March 21, 2006 announced

APRIL 2005 NOTICES OF THE AMS 493 Meetings & Conferences Oxford, Ohio Washington, District Miami University of Columbia March 16-1 7, 2007 Marriott Wardman Park Hotel and Omni Friday - Saturday Shoreham Hotel Central Section Associate secretary: Susan]. Friedlander January 7-10, 2009 Announcement issue of Notices: To be announced Wednesday - Saturday Program first available on AMS website: To be announced ]oint Mathematics Meetings, including the 115th Annual Meeting of the AMS, 92nd Annual Meeting of the Mathe­ Program issue of electronic Notices: To be announced matical Association ofAmerica (MAA), annual meetings of Issue of Abstracts: To be announced the Association for Women in Mathematics (A VVM) and the National Association of Mathematicians (NAM), and the Deadlines winter meeting of the Association for Symbolic Logic (ASL). For organizers: To be announced Associate secretary: Lesley M. Sibner For consideration of contributed papers in Special Sessions: Announcement issue of Notices: October 2008 To be announced Program first available on AMS website: November 1, 2 008 For abstracts: To be announced Program issue of electronic Notices: January 2009 Issue of Abstracts: Volume 30, Issue 1 Deadlines San Diego, California For organizers: April1, 2008 For consideration of contributed papers in Special Sessions: San Diego Convention Center To be announced January 6-9, 2008 For abstracts: To be announced For summaries of papers to MAA organizers: To be Sunday - Wednesday announced ]oint Mathematics Meetings, including the 114th Annual Meeting of the AMS, 91st Annual Meeting of the Mathe­ matical Association ofAmerica (MAA), annual meetings of San Francisco, the Association for Women in Mathematics (A VVM) and the National Association of Mathematicians (NAM), and the California winter meeting of the Association for Symbolic Logic (ASL). Associate secretary: Michel L. Lapidus Moscone Center West and the Announcement issue of Notices: October 2007 San Francisco Marriott Program first available on AMS website: November 1, 2007 January 6-9, 201 0 Program issue of electronic Notices: January 2008 Wednesday - Saturday Issue of Abstracts: Volume 29, Issue 1 ]oint Mathematics Meetings, including the 116th Annual Meeting of the AMS, 93rd Annual Meeting of the Mathe­ Deadlines matical Association of America (MAA), annual meetings of For organizers: April1, 2007 the Association for Women in Mathematics (A VVM) and the For consideration of contributed papers in Special Sessions: National Association of Mathematicians (NAM), and the To be announced winter meeting of the Association for Symbolic Logic (ASL). For abstracts: To be announced Associate secretary: John L. Bryant For summaries of papers to MAA organizers: To be Announcement issue of Notices: October 2009 Program first available on AMS website: November 1, 2009 announced Program issue of electronic Notices: January 2010 Issue of Abstracts: Volume 31, Issue 1 Deadlines For organizers: April1, 2009 For consideration of contributed papers in Special Sessions: To be announced For abstracts: To be announced For summaries of papers to MAA organizers: To be announced

494 NOTICES OF THE AMS VOLUME 52, NUMBER 4 Meetings & Conferences New Orleans, Photo Index to Pages 397, 398 Louisiana 1 13 New Orleans Marriott and Sheraton New Orleans Hotel 7 8 15 january 5-8,2011 Wednesday - Saturday 9 10 ]oint Mathematics Meetings, including the 117th Annual Meeting of the AMS, 94th Annual Meeting of the Mathe­ 11 matical Association of America, annual meetings of the Association for Women in Mathematics (A lVM) and the 12 National Association of Mathematicians (NAM), and the 23 winter meeting of the Association for Symbolic Logic (ASL). Associate secretary: Susan J. Friedlander Announcement issue of Notices: October 2010 Page 397 Page 398 Program first available on AMS website: November 1, 2010 Program issue of electronic Notices: January 2011 Issue of Abstracts: Volume 32, Issue 1 Page 397

Deadlines 1. Keith Comad (left) and Blumenthal Prize winner Manjul For organizers: April 2, 2011 Bhargava on right, following Joint Prize Session. For consideration of contributed papers in Special Sessions: 2. Exhibits area. To be announced 3. Directory of Registrants board. For abstracts: To be announced 4. Left to right: AMS associate secretary Michel Lapidus, Benoit For summaries of papers to MAA organizers: To be Mandelbrot, Notices editor Andy Magid. announced 5. AMS Colloquium lecturer Robert K. Lazarsfeld. 6. Who Wants to Be a Mathematician game. 7. Mara Neusel (left) and AMS executive director John Ewing in Exhibits area. 8. Mathematician Peter Lax. 9. Origami session. 10. AMS president David Eisenbud, MAA secretary Martha Siegel, and MAA president Ron Graham officially opening the Exhibits. 11. Math in the Arts exhibit. 12. Audience in large lecture hall. 13. Zome Tool exhibit. 14. MAA Invited Address speaker Ravi D. Vakil. 15. Email Center. 16. Chatting between sessions. 17. Exhibits area. 18. Crowd entering Exhibits after opening ceremony. 19. Special Session speaker Martin Davis. 20. Notices deputy editor Allyn Jackson and AMS president elect James Arthur. 21. Studying the program. 22. Richard Guy (left) and ]PBM Communication Award winner Barry Cipra. 23. MAA executive director Tina Straley (left) and MAA associate secretary James Tattersall.

Photographs by Mike Breen, Sandy Frost, Allyn Jackson, Gil Poulin, Michael Reeves Photography, and Diane Saxe.

APRIL 2005 NoTICES OF THE AMS 495 Meetings and Conferences of the AMS

Associate Secretaries ofthe AMS Western Section: MichelL. Lapidus, Department of Math­ Eastern Section: Lesley M. Sibner, Department of Mathe­ ematics, University of California, Sproul Hall, Riverside, CA matics, Polytechnic University, Brooklyn, NY 11201-2990; 92521-0135; e-mail: l api dus@math. ucr. edu; telephone: 951- e-mail: [email protected]; telephone: 718-260-3505. 827-5910. Southeastern Section: Matthew Miller, Department of Math­ Central Section: Susan J. Friedlander, Department of Math­ ematics, University of South Carolina, Columbia, SC 29208- ematics, University of Illinois at Chicago, 851 S. Morgan (M/C 0001, e-mail: mill er@math. sc. edu; telephone: 803-777-3690. 249), Chicago, IL 60607-7045; e-mail: susan@math. nwu. edu; tele­ phone: 312-996-3041.

The Meetings and Conferences section of the Notices 2007 gives information on all AMS meetings and conferences January 4-7 New Orleans, Louisiana p.493 approved by press time for this issue. Please refer to the page Annual Meeting numbers cited in the table of contents on this page for more March 16-17 Oxford, Ohio p.494 detailed information on each event. Invited Speakers and Special Sessions are listed as soon as they are approved by 2008 the cognizant program committee; the codes listed are needed January 6-9 San Diego, California p.494 for electronic abstract submission. For some meetings the list Annual Meeting may be incomplete. Information in this issue may be dated. 2009 Up-to-date meeting and conference information can be January 7-10 Washington, DC p.494 found at www. ams. org/meeti ngsl _ Annual Meeting Meetings: 2010 January 6-9 San Franciso, California p.494 Annual Meeting 2005 2011 March 18-19 Bowling Green, Kentucky p.485 January 5-8 New Orleans, Louisiana p. 495 April2-3 Newark, Delaware p.486 Annual Meeting April8-10 Lubbock, Texas p.487 April16-17 Santa Barbara, California p.487 Important Information regarding AMS Meetings June 16-19 Mainz, Germany p.488 Potential organizers, speakers, and hosts should refer to October 8-9 Annandale-on-Hudson, page 100 in the January 2005 issue of the Notices for general New York p.490 information regarding participation in AMS meetings and October 15-16 Johnson City, Tennessee p.490 conferences. October 21-23 Lincoln, Nebraska p.491 Abstracts November 12-13 Eugene, Oregon p.491 Speakers should submit abstracts on the easy-to-use interactive December 14-18 Taiwan p.492 Web form. No knowledge of LAf£X is necessary to submit an electronic form, although those who use WT£X may submit 2006 abstracts with such coding, and all math displays and simi­ January 12-15 San Antonio, Texas p.492 larity coded material (such as accent marks in text) must Annual Meeting be typeset in WT£X. Visit http:llwww.ams.orglcgi-binl April1-2 Miami, Florida p.492 abstractslabstracts.pl. April8-9 Notre Dame, Indiana p.492 Questions about abstracts and requests for paper forms April 22-23 Durham, New Hampshire p.492 may be sent to abs-i nfo@ams. org. April29-30 San Francisco, California p.493 Paper abstract forms must be sent to Meetings & Confer­ October 21-22 Cincinnati, Ohio p.493 ences Department, AMS, P.O. Box 6887, Providence, RI 02940. November 3-4 Fayetteville, Arkansas p.493 There is a $20 processing fee for each paper abstract. There is no charge for electronic abstracts. Note that all abstract dead­ lines are strictly enforced. Close attention should be paid to specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated.

Conferences: (see http: I lwww. ams. o rglmeeti ngsl for the most up-to-date information on these conferences.) June 5-July 21, 2005: Joint Summer Research Conferences in the Mathematical Sciences, Snowbird, Utah (see November 2004 Notices, page 1294). July 25-August 12, 2005: Summer Research Institute on Algebraic Geometry, Seattle, Washington (see November 2004 Notices, page 1293). Co-sponsored conference: June 2006: Fifth Conference on Poisson Geometry, Tokyo, Japan (watch http: I ltmugs. math. metro-u. ac. jplgeneral . html for future information).

496 NOTICES OF THE AMS VOLUME 52, NUMBER 4 ADVERTISERS' fORUM ~~0~ ASA AMERICAN STATISTICAL ASSOCIATION For more information, email [email protected] or call800-225-6911 The nation s leading professional association for statistics and statisticians since 1839 POWELL'S 1429 Duke Street, Alexandria, VA 22314-3415 Toll-free: 1 (888) 231-3473 Phone: (703) 684-1221 TECHNICAL BOOKS Fax: (703) 684-8037 33 NW Park Avenue, Portland, OR 97209 powells. com [email protected] • www.amstat.org

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