AMERICAN MATHEMATICAL SOCIETY FORTHCOMING TITLES FROMTHEAMS

Elements of Elements of Combinatorial COURSE Comb~natonal and Differential Topology ADOPTION and D1fferent1al Topology V. V. Prasolov, Independent University of Moscow, Russia Graduate Studies in Mathematics, Volume 74; 2006; approximately 342 pages; Hardcover; ISBN 0-8218-3809-1; List US$59; All AMS members US$47; Order code GSM/74

The Shoelace Book A Mathematical Guide to the Best (and Worst) Ways to Lace Your Shoes Burkard Polster, Monash University, Clayton, Victoria, Australia Mathematical World, Volume 24; 2006; 125 pages; Softcover; ISBN 0-8218-3933-0; List US$29; All AMS members US$23; Order code MAWRLD/ 24

Enumerative Enumerative Geometry COURSE Geometry and and String Theory ADOPTION Theory String Sheldon Katz, University of Illinois at Urbana-Champaign, IL Shtldon K.J..u Student Mathematical Library, Volume 32; 2006; approximately 214 pages; Softcover; ISBN 0-8218-3687-0; List US$35; All AMS members US$28; Order code STML/ 32

The Cauchy Transform The Cauchy Joseph A. Cima, University of North Carolina, Chapel H ill, NC, Transform Alec L. Matheson, Lamar University, Beaumont, TX, and Wtlliam T. Ross,

Joseph A. Cima University of Richmond, VA Alec Matheson WiiUam T. Ross Mathematical Surveys and Monographs, Volume 125; 2006; 272 pages; Hardcover; ISBN 0-8218-3871-7; List US$75; All AMS members US$60; Order code SURV/ 125

The Theory of Group Characters and Matrix Representations of Groups Second Edition Dudley E. Littlewood AMS Chelsea Publishing; 2006; 310 pages; Hardcover; ISBN 0-8218-4067-3; List US$45; All AMS members US$41; Order code CHEL/ 357.H

Lectures on Quasiconformal Mappings Second Edition Lectures on Quasiconformal Lars V. Ahlfors Mappings University Lecture Series, Volume 38; 2006; approximately 156 pages; Softcover; Second EcllUon ISBN 0-8218-3644-7; List US$35; All AMS members US$28; Order code ULECT/ 38 Lars V . Ahlfors

For the most up-to-date and complete information on all of our newest publications and sales!

• For new titles go to www.ams.org/bookstore/whatsnew • To sign up for email notification for new titles and our Publications Update ®AMS'\. R l 1 H catalog go to www.ams.org/cgi-bin/bookstore/bookstore-maildomo • For our latest sale go to www.ams.org/bookstore/saledirectory.html Editorial Board Aims and Scope Alexander Barg This new journal provides a forum for the publica­ Albrecht Beutelspacher tion of original research articles in the broad area lan Blake of mathematics of communications. It will cover Eimear Byrne work in mathematics and computer science rele­ Yvo Desmedt vant to applications in communications technolo­ Jintai Ding gy. Submissions from many areas of mathematics Steven Dougherty are invited including, coding theory, cryptology, Ivan Duursma combinatorics, finite geometry, algebra and num­ Veerle Fack ber theory. This journal also aims to cover the algo­ Patrick Fitzpatrick rithmic and computational aspects of these disci­ Heide Gluesing-Luerssen plines. Marcus Greferath (EIC) Submissions should be sent directly to the EIC Tom Hoeholdt Marcus Greferath ([email protected]) Thomas Honold Neal Koblitz A number of gift subscriptions, print and electronic, Ivan Landjev are offered to individuals and institues/libraries where Simon Utsyn AMC is found of great interest. Requests should be Jianya Uu (EIC) sent to the EIC. All AIMS journals can be viewed at: Sergio Lopez-Permouth http:/ /AIMSciences.org Michael O'Sullivan Joachim Rosenthal Renate Scheidler Other Journals at AIMSciences.org Stefan Schmidt Hovav Shacham Neil Sloane Discrete and Conflnuous Dynamical Systems Patrick Sole Andreas Stein DCDS-A and DCDS-B are leading journals in analysis, Douglas Stinson dynamical systems and modelling, ranking top among Mario Szegedy math and applied math journals. Edlyn Teske Networks and Heterogeneous Media Henk van Tilborg Xiaoyun Wang NHM, an applied math journal, is focused on traffic Judy Walker flows, internet network, bio-medical problems,filtra­ Harold Ward tion, granular flow, etc. It is distinct in its interdiscipli­ GillesZemor nary character, specific focus and deep mathemati­ cal content

American Institute of Mathematical Sciences • Shandong University NEW (6 FORTHCOMING from Birkhiiuser

Algebraic Geometry and A Beginner's Guide The Unity of Mathematics Number Theory to Graph Theory In Honor of the Ninetieth Birthday of I. M. Gelfand VICTOR GINZBURG, University of Chicago, IL, (Ed.) Second Edition PAVEL ETINGOF, I. M. SINGER, both, Massachusetts This vo lume of invited articles by several outstanding W.D. WALLIS, Southern Illinois University, Carbondale, !L Institute of Technology, Cambridge, MA; mathematicians in algebra, algebraic geometry, and VLADIMIR RETAKH, Rutgers University, Piscataway, NJ number theory is dedi cated to Vladimir Drinfeld on the "Altogether the book gives a comprehensive introduction occasion of his 50th birthday. These surveys and original to graphs, their theory and their application ... The use Atribut e to the vision and legacy of I. M. Gelfand, the research articles broadly refl ect the range of Drinfeld's of the text is optimized when the exercises are solved .. " invited papers in this vo lume re fl ect the uni ty of wo rk in th ese areas, especially his profound contribu­ -SIMULATION NEWS EUROPE (REVIEW OF THE mathematics as a whol e, with par ticular emphasis on tions to the Langlands program and mathematical FIRST EDITION) the many connections among the fields of geometry, phys ics. This concisely written textbook is intended for an physics, and representation theory. Written by lead in g introductory course in graph theory for undergraduate mathematicians, the text is broadly divided into two Contributors include: V.V. Fock, E. Frenkel, mathematics majors or advanced undergraduate and section s: th e fir st is devoted to developments at the D. Gaitsgory, A.B. Goncharov, E. Hrushovski, graduate st ud ents from the many field s that benefit intersection of geometry and phys ics, and the seco nd to Y. lhara, D. Kazhdan, M. Kisin, I. Krichever, from graph-theoreti c applications. This revised and rep resentation theory and alge braic geometry. G. Laumon, Yu. Manin, and V. Schechtmann. expanded seco nd edition contains new chapters on Contents: H. Cartan: Preface * I. M. Gelfand: labe ling and communications netwo rks and small 2006/APPROX 608 PP., 10 ILLUS./HAR DCOVER Mathematics as an Adequate Language * M. Atiyah: ISBN 0-8176-4471 -7/$99.00 (TENT. ) wo rlds, as we ll as additional beginner's material in the The Interaction between Geometry and Physics * PROGRESS IN MATHEMATIC S ea rly chapters, including more examples, exercises, A. Braverman, M. Finkelberg, and D. Gaitsgory: hints and solution s to key problems. Uhlenbeck Spaces via Affine Li e Algebras * H. Brezis: 2006/APPROX. 300 PP., 120 ILLUS./SOFTCOVER New Questions Related to the Topological Degree * D-Modules, Perverse Sheaves, ISBN 0-8176-4484-9/$39.95 (TENT.) T. Coates and A. Givental: Quantum Cobordisms and and Representation Theory Formal Group Laws *A. Connes: On the Foundations of Noncommutative Geometry * S. DeBacker and RYOSHI HOTTA, Okayama University of Science, Okayama, D. Kazhdan: Stable Distributions Supported on the Japan; KIYOSHI TAKEUCHI, University of Tsukuba, Japan; Harmonic Analysis and TOSHIYUKI TANISAKI, Osaka City University, Osaka, Japan Nilpotent Cone for the Group G2 * V. Drinfeld: Applications Infinite-Dimensional Vecto r Bundles in Algebraic This first English-language translation of the Japanese In Honor of John J. Benedetto Geometry* L. D. Faddeev: Alge braic Lessons from text, D Kagun to Daisugun, translated and updated by CHRISTOPHER HElL, Georgia Institute of Technology, the Theory of Quantum Integrable Models * the original authors, examines in detail the foundation s Atlanta, GA (Ed.} M. Kontsevich andY. Soibelman: Affine Structures of D-module theory and its intersection with cohomo­ and Non-Archimedean Analytic Spaces * B. Kostant logy groups and representation theory. Systematic and This volume, in honor of John J. Benedetto, features and N. Wallach: Gelfand-Zeitlin Th eory from the carefullyw ritten, the wo rk begins with preliminary invited articles cove ring a wide range of topics in areas Perspective of Classical Mechanics II * C-H. Liu, concepts before fo cusing on some basic but important where he has made fundamental and las ting K. Liu, and S-T. Yau: Mirror Symmetry and theories that have emerged in the last few decades. contributions, such as harmonic analysis, number Localizations * G. Lusztig: Character Sheaves and theory, weigh ted-norm inequalities, wavelet theory, Generalizations * D. McDuff: Symplectomorphism 2006/A PPROX. 350 PP., 20 ILLUS./HARDCOVER time-frequency analysis, and sampling theory. Although Groups and Quantum Cohomology* M. Movshev and ISBN 0-8176-4363-X/$99.00 (TENT.) the scope of th e book is broad, chapters are clustered A. Schwarz: Algebraic Structure of Yang-M ills Theory* PROGRESS IN MATHEMATICS, VOL. 236 by topic to provide authoritative expositions that will be N.A. Nekrasov and A. Okounkov: Seiberg-Witten of lasting in terest. Theory and Random Partitions *A. Okounkov, History of Banach Spaces and Contributors: A, Aldroubi, L. Baggett, G. Benke, N. Reshetikhin, and C. Vafa: Quantum Calabi-Yau C. A. Cabrelli, P.G. Casazza, 0 . Christensen, and Classical Crys tals *A.M. Vershik: Gelfand-Tsetlin Linear Operators W. Czaja, M. Fickus, J.-P. Gabardo, K. Griichenig, Alge bras, Expectations, Inverse Limits, Fourier Analysis ALBRECHT PIETSCH, University of Jena, Germany K. Guo, E. Hayashi, C. Heil, H. P. Heinig, J.A. Hogan, J. Kovacevic, D. Labate, J.D. Lakey, 2006/654 PP., 41 ILLU S./HAR DCOVER This work is devoted to a comprehensive treatment of ISBN 0-8176-4076-2/$115.00 D. R. Larson, M. T. Leon, S. Li, W.-Q. Lim, the history of Banach spaces and (abstract bounded) PROGRE SS IN MATHEMATICS, VO L. 244 A. Lindner, U. M. Molter, A.M. Powell, B. Rom, linear operators, fo cusing on developments in the E. Schulz, T. Sorrells, D. Speegle, K. F. Taylor, second half of the 20th century. Banach space theo ry is J. C. Tremain, D. F. Walnut, G. Weiss, E. Wilson, presented as a part of a broad mathematics context, G. Zimmermann 8irkhauser invites yo u to take part in our using tools from se t theory, topology, algebra, combinatorics, probability theory, and logic. festschrift sale. For a limited time, a 30% 2006/APPROX. 400 PP., 13 ILLUS./HARDCOVER discount will be available on selected ISBN 0-8176-3778-8/$89.95 (TENT.) 2006/A PPROX. 880 PP., 82 ILLUS./HARDCOVER APP LIED AND NUMERICAL HAR MO NIC ANALYSIS volumes. Use 88066 to get the discount at ISBN 0-8 176-4367 -2/$ 109.00 (TENT.) www.birkhauser.com

CALL: 1-800-777-4643 • FAX: (201) 348-4505 E-MAIL: [email protected] • www.birkhauser.com Please mention promotion #012377 when ordering. Prices are valid in the Americas only Birkhiiuser and are subject to change without notice. For price and ordering information outside the Americas, please contact Birkhauser Verlag AG by E-mail: [email protected] Boston · Basel · Berlin 012377x Notices of the American Mathematical Society May 2006

Communications

572 WHAT IS ... Percolation? Harry Kesten

580 NSF Fiscal Year 200 7 Budget Request Allyn jackson 528 583 2006JPBM Communications Award

585 MAA Prizes Awarded in San Antonio Features 587 AWM Prizes Awarded in San Antonio 528 Cells in Coxeter Groups Paul E. Gunnells Commentary Cells are subsets of Coxeter groups introduced as a 52 5 Opinion: Technology, tool in representation theory. Here, the author consid­ Education, and the Single ers them as geometric objects in their own right, Salary Schedule including their visualization. C. E. Larson 526 Letters to the Editor 536 Serge Lang 1927-2005 574 DarkHeroofthe jay jorgenson and Steven G. Krantz Information Age: In In this memorial article, the authors and others recall Search ofNorbert Wiener, the life and interests, mathematical and nonmathemati­ the Father of cal, of a mathematician famous for both. Cybernetics-A Book Review Reviewed by Michael B. 5 54 The Life and Works of Raoul Bott Marcus Loring W Tu For this memorial article, the author updates his authorized mathematical biography of his distinguished subject. Notices Departments of the American Mathematical Society Mathematics People ...... 589 Avila, Morel, and Payne Named Clay Research Fellows; Meckes EDITOR: Andy Magid Receives AIM Five-Year Fellowship; Barenblatt Receives ASSOCIATE EDITORS: Timoshenko Medal; National Academy of Engineering Elections. Susanne C. Brenner, Bill Casselman (Graphics Editor), Robert ]. Daverman, Nathaniel Dean, Rick Durrett, Mathematics Opportunities ...... 591 Susan Friedlander, Robion Kirby, Steven G. Krantz, Fellowships, Enhancing the Elliott H. lieb, Mark Saul, Karen E. Smith, Audrey NSF Postdoctoral Research Terras, lisa Traynor Mathematical Sciences Workforce in the Twenty-First Century, SENIOR WRITER and DEPUTY EDITOR: Call for Nominations for the ICTP/ IMU Ramanujan Prize. Allyn Jackson MANAGING EDITOR: Sandra Frost Inside the AMS ...... 592 rs CONTRIBUTING WRITER: Elaine Kehoe AMS Archives journals with Portico, Department Chai Workshop, Deaths of AMS Members. PRODUCTION ASSISTANT: Muriel Toupin PRODUCTION: Kyle Antonevich, Stephen Moye, Reference and Book List ...... 593 Erin Murphy, Lori Nero, Arlene O'Sean, Karen Ouellette, Donna Salter, Deborah Smith, Peter Sykes American Mathematical Society- Contribut ions ...... 598 ADVERTISING SALES: Anne Newcomb Mathematics Calendar ...... 606 SUBSCRIPTION INFORMATION: Subscription prices for Volume 53 (2006) are US$430 list; US$344 institu­ tional member; US$258 individual member. (The sub­ New Publications Offered by the AMS ...... 613 scription price for members is included in the annual dues.) A late charge of 10% of the s ubscription price Meetings & Conferences of the AMS ...... 619 will be imposed upon orders received from nonmem­ bers after January 1 of the subscription year. Add for Meetings and Conferences Table of Contents ...... 631 postage: Surface delivery outside the U nited States and lndia- US$20; in !ndia- US$40; expedited d elivery to destinations in North America- US$35; elsewhere­ US$87. Subscriptions and orders for AMS publications should be addressed to the American Mathematical Society, P.O. Box 845904, Boston, MA 02284-5904 USA. All orders m ust be prepaid. ADVERTISING: Notices publishes situations wanted and classified advertising, and display advertising for publishers and academic or scientific organizations. Advertising material or questions may be faxed to 401-331-3842 (indicate "Notices advertising" on fax cover sheet). SUBMISSIONS: Articles and letters may be sent to the editor by email at n oti ces@math . ou . edu, by fax at 405-325-5765, or by postal m ail at Department of Mathematics, 601 Elm, PHSC 423, University of Okla­ homa, Norman, OK 73019-0001. Email is preferred. From the Correspondence with the managing editor may be sent to noti ces@ams . org. For more information, see the AMS Secretary section "Reference and Book list". NOTICES ON THE AMS WEBSITE: Most of this publi­ cation is available electronically through the AMS web­ Officers of the Society 2005 and 2006 Updates ...... 604 site, the Society's resource for delivering electronic products and services. Use the URL http: I / www. ams. o r g/noti ces/ to access the Notices on the website. Call for Nominations for the David P. Robbins Prize ...... 605

(Notices of the American Mathematical Society (ISSN 0002- 9920) is published monthly except bimonthly in june/jul y by the American Mathematical Society at 201 Charles Street, Prov­ idence, R1 02904-2294 USA, GST No. 12189 2046 RT****. Pe­ riodicals postage paid at Providence, Rl, and additional mail­ ing offices. POSTMASTER: Send address change notices to Noticeso{theAmerican Mathematical Society, P.O. Box 6248, Providence, Rl 02940-6248 USA.] Publication here of the So­ ciety's street address and the other information in brackets above is a technical requirement of the U.S. Postal Service. Tel: 401-455-4000, email: noti ces@ams . org. ©Copyright 2006 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 aut!\ors and db" not necessarily reflect opinions of the editors or policies of the American Mathematical Society. Opinion

do could be more important than preparing math and science Technology, Educatio~ and the teachers for California students."5 California proposes some economic incentives in the form Single Salary Schedule of student loan forgiveness (up to US$19,000) in order to It is due to scientific progress and technological innovation, achieve their goal.6 The National Academies, whose recent re­ more than any other reason, that crop yields have gone up, port on educational reform emphasizes the importance of that starvation has decreased, that human longevity has in­ teacher content expertise, advocates programs like California's creased, and that the material conditions of our lives have con­ as its primary recommendation for increasing the number of tinually improved. Our best hope for addressing resource mathematics and science teachers with degrees in these sub­ scarcity, plagues, and other less forseeable disasters is con­ jects. 7 What these incentives do not address is that, according tinued scientific and technological progress. We must produce to a study by Ingersoll, 39 percent ofK-12 teachers leave teach­ scientists and engineers. Systemic changes are required to pro­ ing altogether within five years (he estimates a slightly higher duce more. And the more the better. Changing the culture of percentage for math/ science teachers). 66 percent of math and mathematics and science education by increasing the per­ science teachers cite "poor salary" as a reason for leaving.8 centage of mathematics and science teachers with more than California's plan may yield more mathematics and science a shallow knowledge of the subjects they are teaching may teachers with degrees-but does not increase their incentive be the solution. to stay in teaching after entering the profession. Higher I regularly hear students tell me that they were "never salaries for these teachers-possibly much higher-are al­ good at math". Almost any student can be good at math-cer­ most certainly required to achieve this goal. This solution, sur­ tainly at primary and secondary school levels. This failure is prisingly, is not discussed in the National Academies' report. not inherent in the subject matter. I believe that my students' Salary differentiation is not new in education-it is standard attitudes about math are transmitted to them by under­ at universities where harder-to-attract positions (such as med­ prepared teachers who were themselves not good at math. ical professors) are paid more than others (for instance, jour­ Richard Ingersoll at the University of Pennsylvania has found nalism professors). that 3 5 percent of high-school mathematics classes are taught There are two significant obstacles to this proposal: its cost by someone without even a minor in mathematics or a math­ and opposition from teachers' unions. Taxpayers will have to ematics-related subject.1 These teachers often have to look pay these salary premiums. Taxpayers must be convinced at solution manuals to solve classroom problems. that the costs of better mathematics and science education Genuine knowlege of the subject matter will not guaran­ will be more than outweighed by the benefits. It is possible tee that a teacher will be successful, much less compelling, that this will not occur until some catastrophic event (such but successful and compelling teaching certainly requires as an energy crisis or plague) inspires the recognition that con­ genuine subject knowledge. Mathematics and science teach­ tinued technological innovation requires better mathematics ers should have degrees in the subjects they teach. The "No and science education. Child Left Behind" act does nothing towards this goal. A The second obstacle is union opposition. Teachers' unions "highly qualified" high-school mathematics teacher, for in­ are not opposed to paying teachers more. What they argue stance, must only pass a certification exam. (In some states, though is that all teachers are equally valuable and all should such as Georgia, you can score less than 50 percent and be paid more. This position is enshrined in the "single salary 9 "pass".2 ) The majority of mathematics-instruction certifica­ schedule" used in 96 percent of public schools : under this tion exams, according to a study by the Education Trust, were system, teacher pay is determined by longevity and by the at­ dominated by high-school level material (mostly tenth to tainment of any advanced degrees. What is proposed here is eleventh grade material).3 A teacher who passes a certifica­ a bifurcated salary schedule- secondary school science and tion exam but does not have a mathematics degree is unlikely mathematics teachers should be paid on a different schedule. to have a confident, much less deep, knowledge of the sub­ The relative "value" of teachers of different subjects is not in ject matter. question. The only issue addressed here is how to address so­ According to the Center for the Study of Teaching, the best ciety's (our) technological needs. predictor of student acheivement in science and mathemat­ A direct benefit of this proposal would be an increase in ics is the presence of a teacher with a bachelor's degree in the the production of science and mathematics degrees. Some of subject taught and who is fully certified.4 California is at­ these degree earners, originally motivated to teach, will likely tempting to put more teachers with mathematics and science be drawn to business, government, and the pursuit of advanced degrees in the classroom; the state's university system just degrees. It is reasonable to believe that an indirect benefit will inaugurated an accelerated program to prepare mathematics in time be a measurable change in our cultural attitudes to­ and science majors for the classroom. In June 2005 Califor­ wards mathematics and the sciences. nia State University Chancellor Charles Reed said, "Math and - C. E. Larson science is tied to California's economic future. Nothing we can University of Houston, cl arson@math. uh. edu 5 Tanya Schevitz, San Francisco Chronicle, 1 june 2005. 1 Craig D. jerald, All Talk, No Action, The Education Trust, 2002. 6 ibid. 2 "Not good enough: A content analysis of teacher licensing examinations", 7 Rising Above the Gathering Storm, p. 5-1 . Thinking K- 16, Volume 3, Issue 1, The Education Trust, 1999. 8 R. Ingersoll, "Turnover among mathematics and science teachers in the 3 ibid. U.S.", prepared for the National Commision on Mathematics and Science 4 The National Academies' Committee on Science, Engineering, and Public Teaching for the 21st Century, 2000. Policy, Rising Above the Gathering Storm, The National Academies Press, 9 K. Gruber, S. Wiley, S. Broughman, G. Strizek, M. Burian-Fitzgerald, U. S. 2005, p. 5-5. Department of Education, 1999-2000 Schools and Staffing Surveys, 2002.

MAY 2006 NOTICES OF THE AMS 525 letters to the Ed itor means differences in wording, then Jensen's, we have very good elemen­ this is nonsense. If she means the dif­ tary education students who work Definitions of Fractions as a ference between how these four texts diligently to succeed and, for the most Discriminator present fractions and how it is pre­ part, learn what we try to teach them. The timely review "Mathematicians sented in one of the traditional texts My argument goes to how they will be that she quotes at the and mathematics textbooks for top of the right able to use that knowledge when con­ column of page 25, she prospective elementary teachers" by is absolutely fronted with a different version of right. Raven McCrory (Notices 53, No. 1) is fractions. These prospective teachers a start at making a critical appraisal will see numerous treatments of frac­ -Gary R. Jensen of the recent texts written by mathe­ tions: their own elementary, middle Washington University, St. maticians for a math Louis and high school textbooks; the math­ content course [email protected] given to prospective elementary ematics books and classes they take school teachers. How do these texts in college; and then the wide-ranging, (Received January 13, 2006) compare to the traditional texts and sometimes inconsistent materials to each other? As a means to address with which they teach; as well as dis­ trict, state, and national these questions, McCrory proposes ResponsetoJensen standards to focus on the definition of fractions for K-8 mathematics. We must pay at­ I appreciate Gary Jensen's thought­ in each of the four tention to giving them the "profound mathematician­ ful response to my recent article (No­ authored texts (one by S. Beckmann, understanding of fundamental math­ tices 53, No. 1) and want to apologize one by T. H. Parker S.]. ematics" (Ma, L., Knowing and teach­ and Baldridge, for my error in citing his book. I have one by H. H. ing Elementary Mathematics: Teach­ Wu, and one by me [AMS personally owned the book (Jensen, 2003]). She concludes that ers' Understanding of Fundamental the defin­ G. R., Arithmetic for Teachers: With Ap­ itions are not Mathematics in China and the United word-for-word identical, plications and Topics from Geometry, even States, Lawrence Erlbaum, Mahway, though it is evident that they American Mathematical Society, Prov­ are NJ, 1998) that will enable them to see logically equivalent. Rather than idence, RI, 2003) since the first week and understand the logical and prac­ compare the explanations for clarity, it was published and have shared it tical equivalence of the many versions completeness, and depth, she dwells with many people. This was an over­ of fractions (and on the fact that the definitions are sight on my part. other mathematical ideas) they will encounter. Present­ not literally identical. This is the The problem that Jensen points to ing correct wrong emphasis. with my article suggests that I have mathematics in their un­ dergraduate In her conclusions, McCrory writes, not made clear an essential point. It textbooks and courses "The problems with definition of frac­ is not that the definitions of fractions is the beginning, but not the end, of tions illustrate the complexity of this in these books fail to be identical. No this effort. endeavor, and suggest that we have a one would expect several different long way to go before we reach con­ books to contain identical language in - Raven McCrory clusive answers to the questions of their definitions. Rather, the question Michigan State University what mathematics we should teach is whether the definition in a given [email protected] prospective elementary teachers and book will help future teachers make how it should be presented." Yet all mathematical sense of other ap­ (Received January 27, 2006) four mathematician-authored t exts proaches or definitions he or she en­ include fractions. She continues in counters as a student and teacher. Moliere and the next paragraph, " ... there is no sin­ Jensen says that the definitions are Mathematics gle 'correct' version of this mathe­ logically equivalent, and he is no From time to time we hear of non­ matics." There certainly is. Its essen­ doubt right. My point is that learning mathematicians being averse to math­ tial points and difficulties are written a single, correct definition (especially ematics. The writer Moliere can help out in detail in Book VII of Euclid's El­ one that is full of subtlety) may not nonmathematicians appreciate math. ements. That leaves us with the ques­ equip a teacher to understand the log­ He tells of a person who wants to tion of how fractions should be pre­ ical equivalence of other definitions. learn prose. But as soon the tutor sented to elementary teachers. There These books, especially those by starts teaching prose to the person, is much more to this than the word­ mathematicians, include nuances the learner realizes h e had been ing of definitions. Prospective ele­ across definitions. While perfectly speaking and writing prose all his life. mentary school teachers can learn clear to the mathematically sophisti­ Similarly, nonmathematicians do not this material, in the depth it is pre­ cated, such subtleties are beyond the (wish to) realize that they have been sented in my text, as I've observed ken of most students preparing to be doing certain mathematics since they year after year in my course. McCrory elementary teachers. I am not sug­ started learning their nonmathemat­ continues, "and we do not know what gesting that these students could not, ical disciplines. I consider how a tax­ confusion is generated over time by or do not, understand the presenta­ onomy of function and sets is iso­ the small but significant differences tion of the mathematics in a given morphic to (expressing) four non­ in what teachers are taught." If she book. At my own institution, as at mathematical fields.

526 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Letters to the Editor

First, social science says the state Selected Reviews in the Bulletin mathematics is currently handled by may be democratic or dependent on The new features in the January Bul­ the media. citizen participation, isolated from letin are excellent, and I especially en­ citizens, or can be anarchic. Mathe­ joyed the "Selected Mathematical Re­ -Annette Emerson and Mike Breen matics would say democracy means views". But I suggest that these AMS Public A ware ness Officers the state is a function of citizens, iso­ reviews be looked at critically and [email protected] lation means state and citizens are clarifying comments be appended disjoint sets without either being a where appropriate. (Received February 14, 2006) function of the other, and anarchy A case in point is the reprinted re­ denotes there is only one set con­ view of "The ergodic theoretical proof taining individuality alone. Second, of Szemeredi's theorem" (Fursten­ religion speaks of asceticism where in­ berg, Katznelson, and Ornstein, ]. stitutions may be dependent on a Analyse Math. 31, 1977). I found the transforming individual spirituality, reviewer's paraphrase of the main re­ dualism means institutions isolated sult, Theorem 1.4, is extremely con­ from spiritual individuals, and indi­ fusing. The following clarification may viduals escaping the world mean one help readers who were as puzzled as set exists with the sole member as I was. The result in question is this: the person. Mathematics would say in­ If Tis a measure-preserving trans­ stitutions can be a function of spiri­ formation in a probability measure tuality transforming the world, insti­ space and A is a set of positive mea­ tutions and spirituality as mutually sure, then for any integer k > 1 there exclusive are disjoint sets, and indi­ is an integer n > 0 such that the in­ viduals as fleeing the world mean tersection of the sets TJn(A), there is one set containing spiritual­ (j = 0, ... , k- 1), has positive mea­ ity alone as the member. Third, in sure. philosophy, phenomenology says The reviewer added the true but words depend on culture or values, pointless conclusion that A contains dualism denotes that words are ex­ a set B of positive measure (which is clusive of values, or we have only val­ never mentioned again). And he con­ ues and existentialism. Mathematics siderably weakened the theorem by in­ would say that phenomenology means serting the unnecessary hypothesis words are a function of values, dual­ that Tis invertible. ism means words and values are dis­ -Morris W. Hirsch joint sets, and existentialism means Cross Plains, T1t7 we have only a set containing values [email protected] and no words or reasoning. Fourth, in theology, theism says God is depen­ (Received February 7, 2006) dent on our historical acts, deism means God and the world are mutu­ ally exclusive, while atheism says only Mathematics in the Media the world exists and there is no God. Philip Davis's perspectives on math­ In mathematics, theism would mean ematics and the media are insightful God is a function of history, deism de­ ("The media and mathematics look at notes God and the world as disjoint each other", Opinion, by Philip J. Davis, Submitting Letters to the sets, and atheism means only one set March 2006 Notices of the AMS, Vol. Editor with one member as persons. 53, No. 3, p. 317). The AMS Public The Notices invites readers to The above implies that religion, so­ Awareness Office appreciates his in­ submit letters and opinion pieces cial science, theology, and philoso­ terest and experience, and his articu­ on topics related to mathematics. phy do mathematics as soon as they lation on the challenges, and wishes Electronic submissions are pre­ articulate their own fields. to point readers to "Math in the ferred (notices-l ette rs@ams . Media", a collection of monthly post­ org); see the masthead for postal -Michael M. Kazanjian ings including "Tony Phillips' Take mail addresses. Opinion pieces are Triton College on Math in the Media", "Math Digest", usually one printed page in length mkazanji@depaul .edu and "Reviews" of books, plays, and (about 800 words). Letters are nor­ films with mathematical themes, at mally less than one page long, and (Received January 30, 2006) www. ams. o rg/mathmedi a. The re­ shorter letters are preferred. source presents an overview of how

MAY 2006 NOTICES OF THE AMS 527 Cells in Coxeter Groups Paul E. Gunnells

Introduction Let p, q, r E i"::l u {oo} satisfy p-1 + q- 1 + r - 1 Cells-left, right, and two-sided- were introduced :5 1, where we put 1 I oo = 0. Let 6. = 6.pqr be a tri­ by D. Kazhdan and G. Lusztig in their study of the angle with angles (rr /p,rr/q,rr/ r). If p- 1 +q-1 1 representation theory of Coxeter groups and Heeke +r- = 1, then 6. is Euclidean and can be drawn in 2 algebras [22]. Cells are related to many disparate IR\ ; otherwise 6.lives in the hyperbolic plane. In ei­ ther case, the edges of 6. can be extended to lines, and deep topics in mathematics, including singu­ and reflections in these lines are isometries of the larities of Schubert varieties [2 3], representations underlying plane. The subgroup W = Wpqr of the of p-adic groups [24], characters of finite groups group of isometries generated by these reflections of Lie type [25], the geometry of unipotent conju­ gacy classes in simple complex algebraic groups [5,6], composition factors of Verma modules for semisimple Lie algebras [21], representations of Lie algebras in characteristic p [19], and primitive ideals in universal enveloping algebras [32].1 In this article we hope to present a different and often overlooked aspect of the cells: as geometric objects in their own right, they possess an evoca­ tive and complex beauty. We also want to draw at­ tention to connections between cells and some ideas from theoretical computer science. Cells are subsets of Coxeter groups, and as such can be visualized using standard tools from the the­ ory of the latter. How this is done, along with some background, is described in the next section. In the meantime we want to present a few examples, so that the reader can quickly see how intriguing cells are.

Paul Gunnells is assistant professor of mathematics at the Figure l. Generating a tessellation' of the University of Massachusetts at Amherst. His email ad­ hyperbolic plane by reflections. The central dress is gunne ll s@math. umass. edu. white tile is repeatedly reflected in the red, We thank M. Belolipetsky, W Casselman,]. Humphreys, and green, and blue lines. E. Sommers for helpful conversations. Some computations to generate the figures were done using software by w Cas· is an example of a Coxeter group. Under the action selman, F. du Cloux, and D. Holt. In particular, the basic W, Postscript code to draw polygons in the Poincare disk is due of the images of 6. become a tessellation of the to W. Casselman, as is the photo of G. Lusztig. plane, with tiles in bijection with W (Figure 1). Hence we can picture cells by coloring the tiles of The author is partially supported by the U.S. National Sci­ ence Foundation. this tessellation. · For example, the triangle 6. 236 is Euclidean, and 1 D. Vogan [32] also introduced cells for Weyl groups like the associated group W236 is also known as the those of Kazhdan-Lusztig.

528 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Coxeter groups are certainly the Weyl and affine Weyl groups, which play a vital role in geometry and algebra. In fact, the symmetric group §n is also known to cognoscenti as the Weyl group An- l, while the three Eu­ clidean triangle groups w333' w234' and W23 6 are exam­ ples of affine Weyl groups. The first step to- Figu re 3. G2 = w236 · wards a geometric picture of a Coxeter group is its standard geomet­ ric realization. This is a way to exhibit W as a sub­ Figure 2. George Lusztig delivering an group of GL(V), where V is a real vector space of Aisenstadt lecture at the Centre National de Ia dimension lSI. Suppose we have a basis Recherche Mathematique during the workshop 6. = {()(s Is E S} of the dual space V * . For each Computational Lie Theory in spring 2002. This t E S, there is a unique point ()( : E V such that year marks Lusztig's 60th birthday; a (()(s, ()( () = -2 cos(rr / ms,rl for all s E S, where the conference in his honor will be held at MIT from brackets denote the canonical pairing between V* May 30 to June 3, 2006. and V. Each ()(s determines a hyperplane Hs, namely the subspace of V on which ()(s vanishes. For each affine Weyl group G2. Figure 2 shows George Lusztig s, let O"s E GL(V) be the linear map O"s(V) = sporting a limited edition T-shirt emblazoned with v - ( ()(s , v)()(_i'. Note that O"s fixes Hs and takes()(: the two-sided cells of G2 [26], also reproduced in to - ()(: (Figure S(a)). One can show that the maps Figure 3. Figure 4 shows two hyperbolic examples, {a-s Is E S} satisfy (O"sifr)ms.r = Id, which implies the groups W237 and W23"". The latter group is also that the maps .... O"s extends to a representation of known as the modular group PSL2(1'). We invite W. It is known that this representation is faithful, the reader to ponder how the three pictures are geo­ and thus we can identify W with its image in GL(V) .3 metrically part of the same family. Next we need the Tits cone C c V. Each hyper­ plane Hs divides V into two halfspaces. We let Hi Visualizing Coxeter Groups be the closed halfspace on which ()(s is nonnegative. By definition, a Coxeter group2 W is a group gen­ The intersection ~ 0 = nH; , where s ranges overS, erated by a finite subsetS c W where the defining is a closed simplicial cone in V. The closure of the relations have the form (st)ms,r = 1 for pairs of gen­ union of all W-translates of ~ 0 is a cone C in V; this is the Tits cone. It is erators s, t E S. The exponents m s 1 are taken from known that C = V exactly f':::J u {oo}, and we require m s,s = 1. 'Hence each gen­ when W is finite. Usually in fact Cis much less than erator s is an involution. Two generators s, t com­ all of V. Hence the Tits cone gives a better picture mute if and only if ms,r = 2. for the action of W on V. The most familiar example of a Coxeter group Under certain circumstances we can obtain a more succinct picture is the symmetric group §n; this is the group of all of the action of Won C. For certain permutations of ann-element set {1 , .. . , n}. We can groups Wit is possible to take a nice "cross­ section" of the simplicial cones C take S to be the set of simple transpositions S;, tiling to obtain a manifold M tessellated where S; is the permutation that interchanges i by simplices. An exam­ ple can)e seen and i + 1 and fixes the rest. It's not hard to see that in Figure S(b) for the affine Weyl group A2. This group has three generators s, S generates §n, and that the generators satisfy r, t, (s;s;+Il3 = 1 and commute otherwise. 3 This construction allows us to define Wey The triangle groups Wpqr from the introduction l and affine Weyl groups. A Weyl group are also Coxeter groups, for which the generators W is a finite Coxeter group gen­ erated by a setS of real reflections and also preserving are reflections through the a lines spanned by the certain Euclidean lattice L in its geometric realization. edges of the fixed triangle 6.pqr· The most important The associated affine Weyl fi!OUp W is the extension ofW by L. As a Coxeter group W is generated by S and one 2 For mdre about Coxeter groups, we recommend [8, 20]. additional affine reflection.

MAY 2006 NOTICES OF THE AMS 529 polynomials. To introduce them we require a bit more notation. The Coxeter group (W, S) comes equipped with a length function .f!: W- f:::J u {0}, and a partial order ~ , the Chevalley-Bruhat order. Any w E W can be written as a finite prod­ uct s1 · · · sN of the generators s E S. Such an expression is called reduced if we cannot use the relations to pro­ duce a shorter expression for w. Then the length .f!( w) is the length N of are­ duced expression s 1 · · · sN = w. The partial order ~ can also be charac­ Fi gure 4. (a) W 23 7 (b) W 23 oo terized via reduced expressions. Given an expression s1 · · · sN, a subexpres­ with the product of any two distinct generators hav­ sion is a (possibly empty) expression of the form ing order three. Thus V = 111!. 3 , and the Tits cone C S;1 · · · s;M , where 1 ~ i 1 < · · · < iM ~ N. Then is the upper halfspace {(x,y,z) E 111!. 3 I z ~ 0}. It y ~ w if an expression for y appears as a subex­ turns out that the action of A2 preserves the affine pression of a reduced expression for w. Although hyperplane M := {z = 1}, and moreover the inter­ it is not obvious from this definition, this partial sections of M with translates of Lo are equilateral order is well-defined. The left descent set L(w) c S of w E W is sim­ triangles. This reveals that A2 is none other than ply the set of all generators s such that our triangle group W 333 . A similar picture works for any affine Weyl group, except that the triangles .f!(sw) < .f!(w). There is an analogous definition for right descent set. The definition of the Kazhdan­ must be replaced by higher-dimensional simplices Lusztig polynomials, on the other hand, is too whose dihedral angles are determined by the ex­ lengthy to reproduce here, although it can be ponents ms,t• phrased in completely elementary terms. For each For more examples we can consider the hyper­ pair y, w E W satisfying y ~ w, there is a Kazhdan­ Wpqr. p - 1 q- 1 bolic triangle groups where + Lusztig polynomial P y ,w E ~[t]. By definition 1 +r- < 1. In this case the Tits cone is a certain P y ,y = 1; otherwise Py,w has degree at most round cone in 111!. 3, and the manifold M is one sheet d(y, w) := (.f!(w)- .f!(y)- 1)/2. These subtle poly­ of a hyperboloid (Figure 6). Then M can be identi­ nomials are seemingly ubiquitous in representation fied with the hyperbolic plane; under this identifi­ theory; they encode deep information about vari­ cation the intersections M n wL0 become the tri­ ous algebraic structures attached to (W, S). More­ angles of our tessellation. over, computing these polynomials in practice is daunting: memory is rapidly consumed in even the W-g raphs and Cell s simplest examples. In any case, for our purposes There are two main ingredients needed to define we only need to know whether or not Py,w actually cells: descent sets and Kazhdan-Lusztig attains the maximum possible degree d(y, w) for

Figure 5. (a) u5 negates 01: and fi xes (b) Slicing the Tits cone for A2• Hs.

530 NOTICES OF THE AMS VOLUME 53, NUMBER 5 a given pair y < w. We write y-w if this is so; when w < y we write y - w if w-y holds. We are finally ready to define cells. The left W-graph r£ of W is the directed graph with vertex set W, and with an arrow from y to w if and only if y-w and L(y) ct. L(w). The left cells are extracted from the left W -graph as follows. Given any directed graph, we say two vertices are in the same strong connected com­ ponent if there exist directed paths from each vertex to the other. Then the left cells of Ware exactly the strong connected components of the graph r£. The right cells are defined using the analogously constructed right W -graph fn, while y, ware in the same two-sided cell if they are in the same left or right cell. Figure 7 illustrates all the computations Figure 6. Slicing the Tits cone for a hyperbolic triangle group. necessary to produce the cells for the sym­ metric group § 3 = (s, t I s 2 = t 2 = (st)3 = 1). Fig­ in Chapter 6 of the re­ ure 7(a) shows § 3 with its partial order and with the cently published (8]. left descent sets in boxes. For this group one can The latter is the work of compute that P y,w = 1 for all relevant pairs (y, w). ].-Y. Shi [29]. To describe [I) ts Thus all the information needed to produce f£ is some of his results, recall st~ contained in the left descent sets. Figure ?(b) shows that we can associate to the resulting graph r£, and Figure ?(c) shows the the group An a tiling of IRI. n four left cells. Computing by simplices. The sim­ right descent sets shows ~s that there are three two-sided cells, with the blue plices can be further t [J and green cells forming a single two-sided cell. grouped into certain con­ Now we can explain the coloring scheme used in vex sets called sign-type 1[@ Figures 3 and 4. All regions of a given color com­ regions. Figure 8(a) shows prise a two-sided cell. Moreover, the left cells are the sixteen sign-type re­ exactly the connected components of the two-sided ~ons for A2; in general for cells, in the following sense. Let us say two trian­ An there are nn+Z sign­ gles are adjacent if they meet in an edge. Then by type regions. One of Shi's definition, a set T of triangles is connected if for main results is that each any two triangles ~. ~' E Tit is possible within T left cell is a union of sign­ to build a sequence ~ * = ~1. ~ 2 •... of triangles type regions. Moreover, with each~; adjacent to ~i + l• and such that these­ Shi also gave an explicit quence~ * contains~ and~'. Note a significant dif­ algorithm that allows one ference between the Euclidean group W236 and the to determine to which left two hyperbolic groups. For the former, each two­ cell a given region belongs. sided cell contains only finitely many left cells, The algorithm requires too whereas this is not necessarily the case in general. much notation to state The latter phenomenon was first observed by here, but it is completely R. Bedard [2], who also showed [3] that there are in­ elementary and involves finitely many left cells for all rank 3 crystallographic no computation of Kazh­ hyperbolic Coxeter groups (see the last section for dan-Lusztig polynomials. the definition of crystallographic). M. Belolipetsky Figure 8(b) shows the two­ proved that each Coxeter group in a certain infinite sided cells for Az [26]; one family has infinitely many left cells [4]. can clearly see how the re­ gions are joined into cells. More Examples Figures 9(a) and 9(b) de­ There are two families of Coxeter groups for which pict the cells of A3•4 These imag we have a good combinatorial understanding of es were computed di­ their cells: the symmetric groups .5in and the affine rectly from the data in Weyl groups An. For the former, left cells appear naturally in the combinatorics literature in (c) Left cells the 4 To keep the pictures unc/ut· study of the Robinson-Schenst ed correspondence. tered, we have omitted the Figure 7. (a) top, (b) center, (c) A lucid exposition of this connection can be found edges of the simplices. bottom.

MAY 2006 NOTICES OF THE AMS 531 the most colorful way to describe them is through the permutahe­ dron, which is a polytope Ilw at­ tached to a Weyl group Was fol­ lows. Let x E V be a point in the standard geometric realization of W such that the W-orbit of x has size IWI . Then Ilw is defined to be the closed convex hull of the points {w · x I w E W}. It turns out that the combinatorial type of Ilw is independent of the choice of x, and moreover the structure of Ilw is easy to un­ derstand: its faces are isomor- (b) cells. phic to lower-rank permutahe- Figure 8. (a) A2 sign·type regions. Az dra Ilw·, where W' c W is the subgroup generated by any sub­ set S' c S (such subgroups are called standard parabolic sub­ groups). For example, the poly­ tope underlying Figure 9(a) is the permutahedron for the symmetric group § 4 . The eight hexagonal (respectively, six square) faces correspond to parabolic subgroups isomorphic. to § 3 (respectively, §z x §z). Now the relationship between cells of affine groups of different ranks is conjectured to be as fol­ lows. For any finite Weyl group W, let Wbe the associated affine Weyl group. Then the intersec­ tion of the cells of W with the Figure 9. Two views of the cells (a) (b) face of Ilw corresponding to the of A3. standard parabolic subgroup P should produce the picture for [29, §7.3].5 In the ex­ the cells of the affine group P. This is clearly vis­ ploded view we have ible in Figure 9(a): the cells for A2 (respectively, omitted the red cells, A1 x Ad appear when one slices the cells for A3 which are all simplicial with hexagonal (respect!Yely square) faces of l1A3 . cones. Figure 10 shows Comparing the cells for C3 (Figure ll(b)), originaqy the left cells up to con­ computed by R. Bedard [2], with the cells of Cz gruence. All left cells in (Figure ll(a), [26]) shows another example of this. a given two-sided cell For more along these lines see [17]. · are congruent, except Cells and Automata for the yellow two-sided Simple examples show that W-graphs can be quite cell, which contains two complicated. However, despite this complexity lurk­ distinct types of left ing in their construction, the cells themselves ap­ cells up to congruence pear to be very regular. In fact, for many groups (an S and a U). one can prove that the cells can be built using a rel­ These figures also in­ atively small set of rules, rules that involve no Figure 10. A3 left cells up to dicate relationships be­ Kazhdan-Lusztig polynomial computations at all congruence. tween cells in different [13], [14]. . rank groups. Perhaps Computer scientists have a formal way to work 5 Unfort'unately 'this data is incomplete due to a publisher error: four left with this phenomenon, the theory of regular lan­ cells are missing. guages and finite state automata [1]. One starts with

532 NOTICES OF THE AMS VOLUME 53, N UMBER 5 a finite set A, called an alphabet. Words over the alphabet are se­ quences of elements of A, and any set L of words over A is called a language. Informally, a language is regular if its words can be recog­ nized using a finite list of finite patterns in the alphabet, patterns that are familiar to anyone who has ever used a Unix shell (e.g., l s *. tex). A finite state automaton F over A is a finite directed graph with edges labelled by elements of A. The vertices of F are called states. All vertices are designated as either accepting or nonaccept- Figure 11. (a) C2 ing, and one vertex is set to be the initial state. initial state is the encircled light purple vertex and Such an automaton determines a language over is nonaccepting; all other vertices are accepting. To A as follows. One starts at the initial state and fol­ make the connection between the automaton and lows a directed path terminating at an accepting the cell, start at the bottom grey triangle. Then if state. Such a path determines a word (one simply while following a directed path we encounter an el­ concatenates the labels of the edges along the path ement of S, we flip the indicated vertex to move to to produce a word). We say that this word is rec­ a new triangle in the cell. For another example for ognizedby the automaton. The set of all words rec­ a cell in the hyperbolic group W343 , as well as more ognized by an automaton is hence a language over information about the role of automata in the con­ A . A basic theorem is that a language is regular ex­ text of cells, we refer to [11], [12]. actly when it can be recognized by a finite state au­ For W =An. the existence of automata for tomaton. Reducedw(C) follows easily from the work of For a Coxeter group W, the alphabet is the set P. Headley [18] and Shi. Headley proved that one of generators S, and the language is the set can construct an automaton F recognizing Reducedw of all reduced expressions. By a result Reducedw in which the vertices are the sign-type of B. Brink and R. Howlett [10]. the language regions, and in which all vertices are accepting. Reducedw is regular. Any left cell C determines a Hence to recognize Reducedw(C) one merely takes sublanguage Reducedw(C) := {w E Reducedw I w F and makes a new automaton Fe by designating is a word in C}. W. Casselman has conjectured only the vertices corresponding to regions in C as that the language Reducedw(C) is always regular. accepting. In fact Headley's automaton makes sense Figure 12 illustrates these ideas for one of the for all Coxeter groups,6 although the examples of yellow left cells in A2 (Figure 8(b)). This cell has the Cz and Gz already show that the above argument property that every element in it has a unique re­ for Reducedw(C) breaks down. However, for affine duced expression; such cells were first considered by G. Lusztig [24, Proposition 3.8]. The automaton 6 An exposition can be found in Chapter 4 of [8], where F has edges labelled by elements of {r, s, t} . The is called the canonical automaton.

Figure 1,2 (a) (b)

MAY 2006 NOTICES OF THE AMS 533 Weyl groups, we have conjectured that a closely re­ for an algebra-geometric picture. However, com­ lated automaton works for Reducedw(C) [17]. putations with many examples (cf. Figures 3 and 4) indicate that certain structures vary "continu­ Further Questions ously" in families containing both crystallographic The pictures in this paper certainly raise more and non-crystallographic groups and that these questions than they answer. For example, in the structures are apparently insensitive to whether or case of affine Weyl groups, for all known examples not the underlying group is crystallographic. the left cells are of "finite-type," in the sense that The situation is analogous to that of convex they can be encoded by finitely much data. Here polytopes. In the 1980s many difficult theorems we have in mind descriptions of the cells using such about polytopes were first proven using the geom­ tools as patterns among reduced expressions [2, 13, etry of certain projective complex varieties- toric 14], sign-types [29], or similar geometric struc­ varieties- built from the combinatorics of rational tures [2, 17]. polytopes. Deep properties of the intersection co­ The cells for general Coxeter groups, on the homology of these varieties led to highly nontriv­ other hand, appear to be fractal in nature, and ial theorems for rational polytopes; for some of thus cannot be described in the same way. Au­ these theorems no proofs avoiding geometry were tomata provide one convenient way to treat such known. structures, but they are not the only way. What are By definition rational polytopes are those whose other techniques, and which are natural? vertices have rational coordinates. However, not The situation becomes even more intriguing every polytope is rational, and for irrational poly­ when one considers relationships between cells topes no toric variety exists. Yet irrational polytopes and representation theory. For instance, Lusztig seem to share all the nice properties of their rational conjectured [24, 3.6] and proved [27] that an affine cousins. Weyl group W contains only finitely many two­ Today we have a much better understanding of sided cells. In fact, he proved much more: he this story. Recently several researchers have de­ showed [28] that there is a remarkable bijection be­ veloped purely combinatorial replacements for the tween two-sided cells and the unipotent conjugacy toric variety associated to a rational polytope and classes in the algebraic group dual to that of W. using these replacements have extended various Moreover, each two-sided cell contains only finitely difficult results from the rational case to all poly­ many left cells. Lusztig also conjectured [24, 3.6] topes; see [9] for a recent survey of these results. that the number of left cells in a two-sided cell can For Coxeter groups, the analogy suggests de­ be explicitly given in terms of the cohomology of veloping combinatorial tools to take the role of the Springer varieties [31]. algebra-geometric constructions that seem essen­ For general Coxeter groups our knowledge is tial in the study of crystallographic groups.7 Re­ much more impoverished. First of all, it is not cently there has been significant progress in this known if there are always only finitely many two­ effort [15, 16, 30]. Nevertheless, understanding sided cells, although in all known examples it is ev­ the geometry behind cells for general groups, if it idently true. Perhaps the only general result is due exists, remains an intriguing and difficult problem. to M. Belolipetsky, who showed that right-angled hyperbolic Coxeter groups have only 3 two-sided References cells [4]. Furthermore, in joint work with [l] A. V. AHo, ]. E. HOPCROFT, and]. D. ULLMAN, The Design M. Belolipetsky we have conjectured that the Cox­ and Analysis of Computer Algorithms, Addison· Wesley Publishing Co., Reading, MA-London-Amster­ eter group associated to a hyperbolic n-gon with dam, 1975, Second printing. n distinct angles has (n + 2) two-sided cells. [2] R. BEDARD, Cells for two Coxeter groups, Comm. Alge· The connection with geometry is even more ten­ bra 14 (1986), no. 7, 1253-1286. uous. If a Coxeter group W is crystallographic, [3] __ , Left V-cells for hyperbolic Coxeter groups, which by definition means msr E {2, 3, 4, 6, oo} for Comm. Algebra 17 (1989), no. 12, 2971-2997. all distinct generators s, t, then there is associated [4] M. BELOLIPETSKY, Cells and representations of right­ to W an infinite-dimensional Lie group G called a angled Coxeter groups, Selecta Math. (N.S.) 10 (2004), Kac-Moody group. In principle, G provides a set­ no. 3, 325-339. ting to study geometric questions about cells, since [5] R. BEZRUKAVNIKOV, On tensor categories attached to cells in affine Weyl groups, Representation Theory of many of the standard constructions (e.g., flag va­ Algebraic Groups and Quantum Groups, Adv. Stud. rieties, Schubert varieties) make sense there. Of Pure Math., vol. 40, Math. Soc. Japan, Tokyo, 2004, course, at the moment the connections with geom­ pp. 69-90. etry are poorly understood. For instance, the fact that a two-sided cell can contain infinitely many left 7 In fact, the analogies between convex polytopes and Cox· cells [2-4] is somewhat sobering. eter groups go much further than what is suggested in If W is not crystallographic, then there is no these paragraphs [7} and deserves a lengthy exposition of such group G. For such W we have no candidate its own.

534 NOTICES OF THE AMS VOLUME 53, NUMBER 5 [6] R. BEZRUKAVNIKOV and V. 0STRIK, On tensor categories [28] __ , Cells in affine Weyl groups IV,]. Fac. Sci. Univ. attached to cells in affine Weyl groups II, Represen­ Tokyo Sect. IA Math. 36 (1989), no. 2, 297-328. tation Theory of Algebraic Groups and Quantum [29] ]. Y. SHI, The Kazhdan-Lusztig Cells in Certain Affine Groups, Adv. Stud. Pure Math., vol. 40, Math. Soc. Weyl Groups, Lecture Notes in Math., vol. 1179, Japan, Tokyo, 2004,pp. 101-119. Springer-Verlag, Berlin, 1986. [7] A. BJORNER, Topological combinatorics, lecture delivered [30] W. SoERGEL, Kazhdan-Lusztig-Polynome und unzer­ at lAS conference in honor of Robert MacPherson, Oc­ legbare Bimoduln tiber Polynomringen, a rXi v: tober 2004. Notes available at www. math. kth. se/ math. RT /0403496. -bjorner/files/MacPh60.pdf. [31] T. A. SPRINGER, Trigonometric sums, Green functions [8] A. BJORNER and F. BRENTI, Combinatorics of Coxeter of finite groups and representations of Weyl groups, Groups, Springer-Verlag, 2005. Invent. Math. 36 (1976), 173-207. [9] T. BRADEN, Remarks on the combinatorial intersection [32] D. A. VOGAN JR., A generalized T -invariant for the prim­ cohomology of fans, arXi v: math . C0/0511488. itive spectrum of a semisimple Lie algebra, Math. Ann. [10] B. BRINK and R. B. HOWLETT, A finiteness property and 242 (1979), no. 3, 209-224. an automatic structure for Coxeter groups, Math. Ann. 296 (1993), no. 1, 179-190. [11] W. A. CASSELMAN, Automata to perform basic calcu­ lations in Coxeter groups, Representations of Groups (Banff, AB, 1994), CMS Conf. Proc., vol. 16, Amer. Math. Soc., Providence, Rl, 1995, pp. 35-58. [12] __ , Regular patterns in Coxeter groups, talk at CRM, notes available from www. math. ubc. ca/-cass/ crm. ta 1 k/toc. html, January 2002. [13] C. D. CHEN, The decomposition into left cells of the affine Weyl group of type D4 ,]. Algebra 163 (1994), no. 3, 692-728. [14]]. Du, The decomposition into cells of the affine Weyl group of type B3, Comm. Algebra 16 (1988), no. 7, 1383-1409. [15] P. FIEBIG, Kazhdan-Lusztig combinatorics via sheaves on Bruhat graphs, arXi v: math. RT /0512 311. [16] __ , The combinatorics of Coxeter categories, arXiv:math.RT/0512176. [17] P. E. GuNNELLS, On automata and cells in affine Weyl groups, in preparation. [18] P. HEADLEY, Reduced expressions in infinite Coxeter groups, Ph.D. thesis, University of Michigan, 1994. [19] ]. E. HUMPHREYS, Representations of reduced enveloping algebras and cells in the affine Weyl group, arXiv:math.RT/0502100. [20] __ , Reflection Groups and Coxeter Groups, Cam­ bridge Stud. Adv. Math., vol. 29, Cambridge Univ. Press, Cambridge, 1990. [21]]. C. JANTZEN, Moduln mit einem hdchsten Gewicht, Lec­ ture Notes in Math., vol. 750, Springer-Verlag, Berlin, 1979. [22] D. KAZHDAN and G. LUSZTIG, Representations of Cox­ eter groups and Heeke algebras, Invent. Math. 53 (1979), no. 2, 165- 184. [23] __ , Schubert varieties and Poincare duality, Geom­ etry of the Laplace Operator (Univ. Hawaii, Honolulu, Hawaii, 1979), Proc. Sympos. Pure Math., XXXVI, Amer. Math. Soc., Providence, Rl, 1980, pp. 185-203. [24] G. LuszTIG, Some examples of square-integrable rep­ resentations of p-adic semisimple groups, Trans. Amer. Math. Soc. 277 (1983), 623-653. [25] __ , Characters of finite groups of Lie type, Ann. of Math. Stud., vol. 107, Princeton Univ. Press, 1984. [26] __ , Cells in affine Weyl groups, Algebraic Groups and Related Topics (Kyoto/Nagoya, 1983), Adv. Stud. Pure Math., vol. 6, North-Holland, Amsterdam, 1985, pp. 255-287. [27] __ ,Cells in affine Weyl groups II,]. Algebra 109 (1987), no. 2, 536-548.

MAY 2006 NoncEs OF THE AMS 535 Serge Lang, 1927-2005 jay jorgenson and Steven G. Krantz

Editor's Note: This is the first part of a two-part article. In part two, which will ap­ pear in a later issue, the authors discuss the mathematical accomplishments of Serge Lang and the impact of those achievements.

n September 12, 2005, the mathemat­ informed through the presentation of original doc­ ics community lost Serge Lang, who umentation. We have sought to bring out a full 0 passed away in his apartment in Berke­ picture of Serge's life by inviting contributions ley, California. Lang was well known as from a large number of individuals who knew him a mathematician, and also as an edu- well. For the editors, it was fascinating to witness cator and political activist. The main force in Serge's the diversity of these reminiscences; they represent life was his enthusiasm for mathematics. In a world a broad range of interests and achievements. It is of vagaries and irrational passions, he saw math­ clear that, with Lang's passing, we have lost some­ ematics as a noble pursuit that represented hon­ one unique and irreplaceable. esty and goodness. Within mathematics alone, After Lang's passing, Yale University president Serge had many facets- a researcher, an expositor, Richard C. Levin wrote about Serge, "While having a popularizer, and a teacher. Generations of math­ someone like this in the community is not always ematicians around the world know the name Serge easy, it is salubrious." It is entirely possible Serge Lang through his numerous books and articles. would have agreed with this assessment, perhaps For those individuals who knew Serge, one strik­ even assigning a letter grade for President Levin's ing feature most everyone noted was the com­ summary. partmentalized m anner in which he showed him­ To repeat, our article is an attempt to follow self to anyone: His mathematical colleagues were Lang's insistence for an h onest and complete rep­ told virtually nothing about his personal life, his resentation, allowing readers to draw their own family knew very little abut his mathematical re­ conclusions. With this said, we have no doubt that search, his political allies were only slightly in­ a common judgment will be drawn by everyone: formed of his mathematical interests, and even With Lang's death, the mathematical world, and be­ his closest friends were unaware of each other's yond, has lost someone without equal, and in time presence in his life. we will better understand the significance of Lang's As we prepared this article discussing the many life. aspects of Serge's life, we chose to follow Serge's On Serge Lang's retirement from Yale University method of "file-making", where the reader is in the spring of 2005, Yale president Richard C. Levin honored him with these words: j ay j orgenson is professor of mathematics at City College of New York and Graduate Center. His email address is Serge Lang, A.B., California Institute of jjorge ns on@mi ndspri ng. com. Technology, Ph.D. Princeton University, Steven G. Krantz is professor of mathematics at Washington faculty member at Yale since 1972: Your University, St. Louis. His email address is sk@math. primary love has always been number wu s tl. edu. theory and you h ave written, by one With the assistance of numerous contributors. colleague's estimate, over 50 books and

536 NOTICES OF THE AMS VOLUME 53, NUMBER 5 monographs, many of them concerned with this topic. Several of your mono­ graphs are the only, or nearly the only, book treatments of their important sub­ jects. Your famous theorem in Dio­ phantine equations earned you the dis­ tinguished Cole Prize of the American Mathematical Society. Your textbooks also have garnered accolades. Your cal­ culus for undergraduates went through many editions in the seventies and eighties, and your algebra textbook is a standard reference in the field. So prodi­ gious are you as a scholar that there are actual jokes in your profession about you. One joke goes: "Someone calls the Yale Mathematics Department, and asks for Serge Lang. The assistant who an­ swers says, 'He can't talk now, he is writing a book. I will put you on hold."' In your character, you are uncompro­ mising in your insistence on what you perceive as logical consistency and rhetorical honesty, and you have ques­ tioned much received wisdom and many attention to mathematics. That attention never de­ authorities in the external world as well viated (except occasionally for his politics) for the as here at Yale. You are an excellent rest of Serge Lang's life. and deeply caring teacher, and in honor At Princeton Serge Lang fell under the spell of of this several years ago you received the great algebraic number theorist Emil Artin. the Dylon Hixon Prize for teaching in Along with John Tate, a fellow student of Artin, Lang Yale College. Your students keep in developed a passion for algebra and algebraic num­ touch with you years after they gradu­ ber theory. In later years, Lang and Tate co-edited ate and one has created an endowed the collected works of Artin. Lang earned his Ph.D. fund in your honor. Among your many in 1951. monographs there is one called The Lang's first academic position was as an in­ Beauty of Doing Mathematics, a collec­ tructor at Princeton. Lang also had an instructor­ tion of three dialogues you gave in Paris ship at the University of Chicago from 1953 to in the '80s. Yale is grateful to you for 195 5. Lang's first permanent position was at Co­ the passion with which you understand, lumbia University beginning in 1955. In addition practice and profess the mathematical to producing some terrific mathematics and di­ arts, and wishes you well as you con­ recting five Ph.D. theses, Lang became passion­ tinue your lifelong engagement with ately involved with the politics of the time (in their illimitable splendors. protest against the Vietnam war). Serge ultimately resigned his position at Columbia in 1971 (without Serge Lang was born near Paris on May 19, 1927. yet having arranged for another job) in protest His family lived in St. Germain en Laye. Serge's against Columbia's treatment of anti-war protest­ mother was a concert pianist and his father was a ers. It is also remarkable that, during his tenure at businessman. His sister, with whom Serge main­ Columbia, Lang directed two Princeton Ph.D. stu­ tained an affectionate relationship all his life, cur­ dents: Marvin Greenberg (1959) and Newcomb rently lives in Los Angeles and is a stage and film Greenleaf (1961). actor. Serge's twin brother was a college basketball After leaving Columbia University, Serge Lang coach. landed a job at Yale University (beginning in 1972), The family decided when Serge was a teenager where he spent the remainder of his career. Lang to move to Los Angeles, California. Serge attended directed nine additional Ph.D. degrees while at Caltech as an undergraduate and finished with a Yale. He was awarded the AMS Frank Nelson Cole B.A. degree in physics in 1946. After spending 1.5 Prize (1959) for his mathematical research and the years in the U.S. Army, Serge entered graduate AMS Leroy P. Steele Prize (1999) for his writing. He school at Princeton University in philosophy. He was elected to the National Academy of Sciences abandoned that study after one year and turned his in 1985.

MAY 2006 NOTICES OF THE AMS 537 Although Lang's first mathematical loves were put it, he "put scholarship in the service of action algebra and number theory, his interests rapidly ex­ to stop the nonsense." panded to cover an astonishing panorama of mod­ Serge also was a prolific writer. He wrote more ern mathematics. Areas that he influenced include than 120 research articles and sixty-one books number theory, algebraic geometry, diophantine (and this does not count multiple editions and for­ geometry (in which he was a pioneer), diophantine eign translations). In fact he has 198 citations on approximation, differential geometry, analysis, hy­ Math Sci Net. It is amazing to examine the range of perbolic geometry, Arakelov theory (in which he was mathematical topics covered by Lang's opus: cal­ a pioneer), modular forms, and many other areas culus, real analysis, complex analysis, differential as well. The scope of Lang's books and papers is geometry, algebra, algebraic geometry, diophan­ astonishing not only for its magnitude but for its tine geometry, hyperbolic geometry, math talks breadth. for undergraduates, the heat kernel, and much, Serge Lang resigned from the AMS in 1996 in a much more. Perhaps Lang's most famous and most dispute concerning an article in the AMS Notices by influential book is Algebra, now in its third edition. Denise Kirschner. He retired from Yale in the spring In it, Lang single-handedly reorganizes and revi­ of 2005. talizes this fundamental and central subject. The It gives a sense of Serge Lang to quote from his book has had an enormous impact. formal note of acceptance for the Steele Prize Serge Lang was a man with incredible focus and (which in fact had to be heavily edited because it self-discipline. Mathematics and politics (which he was formulated in such strong language): called "troublemaking") were his primary inter­ ests, and everything else was secondary. As he I thank the Council of the AMS and the grew older, he felt that he had to conserve Selection Committee for the Steele Prize, his en­ ergy and he set other interests aside. He made which I accept. It is of course reward­ hard decisions and stuck by them. As an example, ing to find one's works appreciated by when he decided to stop listening to music, people such as those on the Selection he put all his recordings on the shelf, never to be Committee. At the same time, I am very picked up again. uncomfortable with the situation, be­ It is astonishing how Lang's books affected cause I resigned from the AMS in early peo­ ple at all levels. One high school teacher who 1996, after nearly half a century's mem­ reg­ ularly used Lang's calculus book in his teaching bership. On the one hand, I am now un­ said this: comfortable with spoiling what could have been an unmitigated happy mo­ As a high school teacher, I used this ment, and on the other hand, I do not text with great success several times want this moment to obscure impor­ for both AP Calculus BC and AP Calcu­ tant events which have occurred in the lus AB courses. It is my favorite calcu­ last two to three years, affecting my re­ lus text to teach from, because it is very lationship with the AMS. user-friendly and the material is pre­ sented in such an eloquent way. There are no gratuitous color pictures of peo­ Torn in various directions, sadly but ple parachuting out of airplanes here. firmly, I do not want my accepting the Opening this book is like entering a Steele Prize to further obscure the his­ temple: all is quiet and serene. Epsilon­ tory of my recent dealings with the AMS. delta is banished to an appendix, where (in my opinion) it belongs, but all of the Serge Lang was a remarkably energetic individ­ proofs are there, and they're presented ual with eclectic and broadly ranging tastes. In ad­ in a simple (but not unsophisticated) dition to his passion for mathematics he loved way, with a minimum of unnecessary music and the arts. He himself was an accom­ jargon or obtuse notation. plished pianist and lutenist, and he enjoyed play­ ing in public. He took a keen interest in politics, A somewhat recondite joke is the query "Why especially as it manifested man's inability to face did Bourbaki stop writing?" The answer is that the truth. Lang loved to bring down individuals who they discovered that Serge Lang is one person. obfuscated, who hid behind their rank, or who Lang's output of text connected to his many po­ abused power. He engaged in a great many rather litical disputes was voluminous. He also has some public battles with a wide-ranging collection of unpublished books of a political nature (others of people, from social scientists at Harvard to re­ his political tracts were actually published). Lang searchers at the National Institutes of Health to ed­ liked to say that the best way to learn a new topic ucation researchers at Stanford. As Lang himself is to write a book about it.

538 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Perhaps Serge Lang's greatest passion in life was learning. For Serge, learning manifested itself Ph.D. Students of Serge Lang in many guises; but one of the most important of Marvin Greenberg Princeton 1959 these was his teaching. He saw himself as a role Newcomb Greenleaf Princeton 1961 model for his students, and he spent a great deal Warren May Columbia 1963 of time with them. He often said that the best way Stephen Schanuel Columbia 1963 to learn about a university was to eat in the stu­ William Adams Columbia 1964 dent cafeteria. He did so frequently. He often took Bernard Berlowitz Columbia 1966 his students out to eat, or invited them to his res­ Allen Altman Columbia 1968 idence to listen to music. Although he did so qui­ Joseph S. Repka Yale 1975 etly and discreetly, Serge was known to provide fi­ David E. Rohrlich Yale 1976 nancial assistance to students and mathematicians Donald T. Kersey Yale 1980 who were in need. Serge is remembered fondly for Jing Yu Yale 1980 entreating his students, cajoling his students, Minhyong Kim Yale 1990 screaming at his students, and especially for throw­ William A. Cherry Yale 1993 ing chalk at his students. Michael]. Nakamaye Yale 1994 Serge's graduate courses frequently followed Lisa A. Fastenberg Yale 1996 the track of the book he was currently writing. His Eliot P. Brenner Yale 2005 undergraduate courses could be more freewheel­ ing. An important point to note is the joy that Serge Lang derived from all things mathematical. latter would not accede to the self-evident asser­ It can certainly be said that most of us mathe­ tion that the Beatles were greater musicians than maticians experience some sort of "high" when we Beethoven. learn to tackle and tame new ideas. As we get older, At Yale in 2001, Serge was invited to be the we become more jaded; as a result, this "high" is keynote speaker at a Pierson College Master's Tea. harder and harder to achieve. Not for Serge. He was He dressed in a courtier's outfit and regaled the truly engaged and fulfilled when he discovered packed room with his theory of the similarities be­ new ideas on any level, be that an illuminating tween Elizabethan music and classic rock of the problem for one of his undergraduate texts or an 1960s. Lang illustrated his points by playing (clas­ insight into a new mathematical landscape. As are­ sical LP records of) Ding Dong the Witch is Dead, a sult, Serge Lang always remained mathematically 1969 hit by the Fifth Estate, a 1612 piece by Michael young. Praetorius, and We Can Work it Out by the Beatles Serge Lang spent the fall of 2004 at U. C. Berke­ (1965). ley as a Miller Visiting Professor. He gave a num­ Serge loved to challenge people- friend and foe ber of lectures and made his presence known in alike- just for the sake of challenging them. As an many other ways. As an example, budget cutbacks instance, James Borger recalls had caused severe curtailment of departmental I remember one time when I was a grad teas. People now had to pay daily for their bever­ student, I was standing next to him at ages and cookies. Serge quietly contributed a sub­ tea while he was explaining to a first­ stantial amount of money so that the Monday teas year student that analysis is just num­ would be lavish: many fine cakes and pastries and ber theory at infinity. I said, "Come on, lots of nice things to drink. Certainly this had a very that's not true." He immediately turned positive effect on departmental life, and Serge up the volume, challenging me to stop this gesture. asked for no particular credit for bullshitting and give an example. I said, had wide-ranging interests. He visited Serge "OK, p-adic analysis," and then walked ey every summer for the past several decades Berkel away. But I've always wished I had (in fact he kept an apartment there) and he would stayed to see what his reaction would ranging from attend colloquia in departments have been. We need more trouble mak­ ics to history to political science to medicine phys ers like him. to mathematics. Of course he did not simply attend. His habit was to confront the speaker with detailed In 1998 Serge Lang published a book called and probing questions. Frequently the sessions Challenges. This editor (Krantz) found the volume would become so heated and protracted that in­ to be particularly inspiring, for it recounted, from tervention was necessary. Lang's personal perspective, some of his most in­ One memorable incident- just to illustrate the volving and exciting political battles. The book is eclecticism and vehemence of Serge's interests­ truly outstanding for its honesty and incisiveness. has Serge threatening to clobber with a bronze Two particular battles that stand out are bust a very distinguished Princeton mathemati­ The Case of Ladd and Lipsett. In the late 1970s cian in the Fine Hall Professors' Lounge because the the distinguished social scientists Everett Carll

MAY 2006 N OTICES OF THE AMS 539 Ladd Jr. and Seymour Martin Lipsett set out to creates tension, and it may be inter­ evaluate the American professoriate. They con­ preted as carrying out a "personal cocted a questionnaire to be distributed across the vendetta" ... I regard such an interpre­ country, asking professors detailed questions about tation as very unfortunate, and I reject how they plied their trade, what values they held it totally. as members of the academic profession, and so forth. Their results were published as The 1977 Sur­ Serge spent hours every day on the telephone, vey of the American Professoriate in 1979. Lang wheedling, cajoling, instructing, and most often found the questionnaire, and the premises for the yelling. His collaborators relate that Serge would study, to be repugnant. He conducted a massive ef­ often phone several times a day- every day. He fort to discredit their work. In fact Lang published would learn what was the best time to phone and a rather massive tome, The File (Springer-Verlag, then phone regularly at that time. Often one would 1981), containing all his correspondence and in­ pick up the phone and hear "Serge! Let me continue formation about the battle. In the end, Lang caused to instruct you about .. . " But it should be stressed Ladd/Lipsett to lose much of their funding and a that Serge was disciplined to the extreme. He did great deal of their credibility. not waste time. It was amazing to watch him eat The Case of Samuel P. Huntington. In 1968 lunch in five minutes and dash back to his office Samuel P. Huntington wrote a book entitled Polit­ to resume his writing. ical Order in Changing Societies. In it Huntington In the last twelve years of his life Serge Lang de­ uses what might charitably be characterized as veloped a deep and energetic program to fight the pseudomathematical hucksterism to "prove" that current directions of research on the disease AIDS South African society in the 1960s was a "satisfied (Acquired Immune Deficiency Syndrome). A naive society". Serge Lang decided that nothing could be further from the truth, and in any event Hunting­ assessment of Serge's position is that HIV does not ton's methodology was suspect if not corrupt. He cause AIDS. But this would be an injustice to Serge. conducted a vigorous campaign to derail Hunt­ First of all, he was very careful. He very rarely ington's credibility, and he twice successfully made an error of fact. Secondly, he was quite a sub­ blocked Huntington's election to the National Acad­ tle thinker. His cause and his complaint, in fact, was emy of Sciences. that the search for a cure to AIDS had become politicized. At a certain point, the federal govern­ Serge Lang was quite proud of his efforts to in­ ment simply commanded the National Institutes of still a sense of truth and honor into our public dis­ Health to declare that HIV caused AIDS . The causal course. For years after his battle with Huntington, mechanism had not been identified, and the con­ he would give his students "Huntington tests" to nection not logically established. To be sure, there ascertain their ability to think critically. Serge's is considerable ad hoc evidence of a link between battle cry was to demand whether his listeners knew "a fact from a hole in the ground". Evidently HIV and AIDS. Certainly many of the modern treat­ Serge did. In a particularly earthy moment, Serge ments for AIDS are premised on that link. But liked to say that "he was inside the tent pissing in" Serge's assessment was that the existing data analy­ (with allusion to Lyndon Johnson commenting sis does not support the conclusion that HIV causes about J. Edgar Hoover). For each of his battles, AIDS . Lang would create what he called a "File". This was The present article is a celebration of the life of a detailed and copious collection of all his corre­ Serge Lang. We present a number of vignettes, con­ spondence and all his data connected with any tributed by mathematicians, former students, col­ given case. Often a file would consist of several hun­ leagues, and friends. These are divided into pieces dred pages of closely knit text. Lang would, at his about Serge the man, pieces about Serge the writer, own expense, send copies of his files to mathe­ pieces about Serge the tilter at windmills, and maticians and other interested parties all over the pieces about Serge the mathematician. Our aim is world. The Serge Lang files have been a staple of to give a well-rounded picture of what a diverse and mathematical life for over forty years. multi-faceted person we have lost. He was in many Serge Lang said of himself ways a thorn in our collective sides, but he was a I personally prefer to live in a society friend to us all. where people do think independently and clearly. One of my principal goals is therefore to make people think. When faced with persons who fudge the is­ sues, or cover up, or attempt to rewrite history, the process of clarifying the is­ sues does lead to confrontation, it

540 NOTICES OF THE AMS VOLUME 53, NUMBER 5 for a few days. After his European visit he went to Berkeley where he enjoyed the cooler climate and Memories of Serge Lang where we met him in a number of years. During his visits to Bonn he gave many lectures, in seminars on his own research and for students of beginning and advanced level. In an official re­ port to me (13 February 1997) he wrote as follows: Friedrich Hirzebruch, Max-Planck-Institut "While at the Max-Planck, I also visit other mathe­ fii.r Mathematik maticians, both in Germany and elsewhere such as Serge Lang was a close friend of my wife and me, Holland. I have substantial con­ of our three children, and even of some of our tacts with students. I used to grandchildren. We miss his frequent telephone lecture every year in your analy­ calls- "It's me"- the last one was on September 10, sis course. Last year I lectured to 2005. We shall miss his visit next summer and all the high school class of the following summers. Karcher's son. Thus my days at My wife and I met Serge 53 years ago in Prince­ the Max-Planck, regularly for one ton when he and I were 25 years old. We became month in June every year, and good friends. It was old Europe that all three of us once for four months in fall of liked. Serge rarely spoke about his personal past, 1993, have been important pe­ but by asking questions we slowly learnt the basic riods in providing proper envi­ facts. He came with his father and his sister from ronment for establishing math­ Paris to the United States after France had been oc­ ematical contacts at all levels, as cupied by Germany. He was a soldier in the U.S. well as learning and doing math­ Army from 1946 to 194 7 and was stationed in Italy ematics." and Germany for part of the time. The fact that his Serge also lectured once to life was disturbed by the Nazi war was not a bar­ my algebra course (120 stu­ rier between us. How little Serge spoke of himself dents) where I used his algebra can also be seen from the Curriculum Vitae in his book. When I came to the lecture Collected Papers. The CV has thirteen brief lines, hall, I saw that the official stu­ from the first one, "1927 born" to the last one dent representatives were sell- · Ma·E~c=~!Ji~ "1972-present Yale" . ing cheap photocopied versions We kept close contact with Serge, also after our of Serge's book. I told them that return to Germany. In the summer of 1955 he vis­ this was illegal. The students ited us in the house of my parents in Hamm (West­ said "The author is far away." I falen) (see top photo, right). replied "You are wrong. He will I was appointed to the University of Bonn in be here in two minutes, because 1956 and began the series of Arbeitstagungen he is taking over my lecture where the speakers are chosen by "public vote" at today." Serge came and did not the beginning of the meeting. With very few ex­ make a great fuss about it. He ceptions Serge attended all Arbeitstagungen until even signed some of the copies 2003. During the thirty Arbeitstagungen I orga­ before he began his lecture. nized from 19 57 to 1991, Serge gave thirteen lec­ Serge also lectured to the pub­ tures. The second photo from the top, right, shows lic. He wanted not only to teach Serge lecturing at one of the Arbeitstagungen. The mathematics, but also how to be next photo down shows him at some other Ar­ critical and responsible. "I want beitstagung activity. to make people think." Among During each of the twenty-five years from 1979 the public lectures I mention to 2003, Serge spent one month in Bonn, usually the "Beauty of Mathematics". He June; in addition, he came for three sabbatical fall explained the hyperbolic 3- ~ terms in 1993, 1997, and 2000. He financed his June dimensional manifolds with 0 visits from 1984 to 1989 by the funds of his Hum­ cuspidal ends which are like the arms of an octo- ~ boldt Prize. pus. The bottom photo above shows Serge sitting ~ He had a stable routine: In the summer he went at my desk in our home drawing octopuses. Behind ~ from Yale to Europe. For many years he visited his back there is one shelf with 40-50 of his books. ~ .c When Paris for a month until he stopped. "To everything he presented me a new edition of one of his :SE.b :5 there is a season," he used to say. In other years books, he threw the old edition into the was tepa- ~ - ~ he went Zurich or Berlin. He never omitted Bonn per basket from where I retrieved it later. 8 ~ until the season also ended for the Max-Planck-In­ Mathematics was the most important part of ~ ~ stitute.ln 2004 and 2005 he only visited us privately Serge's life. He worked with great self-discipline for £

MAY 2006 NOTICES OF THE AMS 541 many hours seven days a week. I admired the way century who can lay claim to this epithet. If col­ he could turn courses into books and how he con­ leagues sometimes felt he was overdoing things, tinuously did fundamental research. In the early this may actually confirm what he represented. years there was time for the piano (including com­ But Serge Lang lived and acted in the U.S. where posing), for playing guitar and lute, for going to con­ no heritage of intellectuals exists, in spite of liter­ certs, to the theatre and opera, and to enjoy liter­ ary figures like Arthur Miller. So Lang had to cut ature. We could often do all this together with him one out for himself, as the Yale professor who when visiting him in his apartment in New York, made The New York Times by blocking Samuel where he played his piano compositions for us, and Huntington's admission to the National Academy when we went out in New York to the theatre. Sim­ of Sciences. In doing so he was surely helped by ilar activities took place in Bonn during his visits. the ambient climate of the late 1960s and 1970s, He enjoyed the musical life around Bonn. But then the Free Speech Movement, etc. But his personal de­ came the time of the file making. He was able to vice, "the file", was his own creation. give up things he loved and to concentrate on the Let me add something more personal: The most two parts of his later life (mathematics first and wonderful thing about Serge was that he was always then political work). All other things had to go. "To around, and meeting him would always matter. I everything there is a season." The file concerning first saw him as a young student in Bonn during The 1977 Survey of the American Professoriate by the Arbeitstagungen of 1970 and 1971; I went to Ladd and Lipsett developed from 1977 to 1979. his talks because I knew the name from his Serge put a lot of time and energy into it. We al­ Algebra book. At the time I did not understand the ways received Serge's mailings in installments of · least bit of the mathematics he was talking about; 20-30 pages. It was exciting reading, full of sus­ but I distinctly remember the presentation: his talk pense. Other files followed. The mailings came reg­ seemed to be about presenting things from the ularly, the last one on the day of his death. It was right point of view, which others working in the field certainly not easy to discuss the files with him in had failed to see or to adopt. such a way that he did not begin to yell. "Ich bin I kept meeting him over the years in many places, ein unbequemer Mensch," he said. We admired his and each time I was greeted by his charm and sincere way to rely only on facts, "to distinguish a thrilled by his intensity. The last two summers I in­ fact from an opinion." He fought for honesty and vited him for talks to Darmstadt and Strasbourg. precision in research and in journalism. He hated Even though he would never complain about "big shots who throw their weight around". He ob­ mediocre accommodation or food offered to him, Serge enjoyed being taken out to good restaurants. jected to covering up because of collegiality. At least by the time we got to dessert he invariably His heritage is his Collected Papers, his scientific had to raise his voice (for instance, because I still and his political books. But we miss Serge as a would not understand his way of putting to rest friend. the philosophy of the Vienna circle, that he had fig­ ured out after a few months as a student of phi­ losophy), and posh people around us would start Norbert Schappacher, University of raising their eyebrows. I loved this kind of scene Strasbourg (with him, not in general), and I will now miss it a lot. Two years ago, I gave a seminar in Zurich on the topic of intellectuals among twentieth-century mathematicians. My list included the Englishman G. H. Hardy, the Germans E.]. Gumbel and E. Kamke, Barry Mazur, Harvard University the Frenchman L. Schwartz- and the French-born In 1958, at Princeton, I had accidentally slipped into American Serge Lang. the room in which Serge was giving his seminar in The term intellectual (intellectuel) used here is Abelian Varieties; I was transfixed by the metallic a French invention of the Dreyfus affair, from the urgency, the vitality, of the voice of this chalk­ final years of the nineteenth century. Emile Zola, wielding person; I understood absolutely nothing Anatole France, Marcel Proust, and others were the of the subject, but was instantaneously convinced, first self-declared intellectuals. The expression has with that utterness of conviction that is the gift of a built-in partiality: it is only used for people whose ignorance, that abelian varieties- whatever they opinions you sympathize with, and whose opinions were- were of breathtaking importance, and fur­ and ways of expressing them are loathed by those thermore, of breathtaking importance to me. That who are on the other side. Serge (a "mathematical grown-up") would, shortly Serge Lang was an intellectual in this European afterwards, collar me and request a series of pri­ sense of the word, and he was one of the rare math­ vate lectures in differential topology was as­ ematicians of the second half of the twentieth tounding to me. I treasure the halting lectures I gave

542 NOTICES OF THE AMS VOLUME 53, NUMBER 5 him, as a rite of passage, of immense importance. memory is watching him in classes by Andre Weil And Serge did this sort of thing, through the on abelian varieties. Whereas the rest of us were decades, with many of the young: he would prof­ totally cowed by Weil's personality, Lang seemed fer to them gracious, yet demanding, invitations to to be able to follow anything and even to make cor­ engage as a genuine colleague- not teacher to stu­ rections and amendments to Weil's presentation. dent, but mathematician to mathematician; he did This seemed to me to be nothing short of miracu­ all this naturally, and with extraordinary generos­ lous. After Serge left Chicago to pursue his very il­ ity and success. Serge was a gadfly with formida­ lustrious career, I had only intermittent contact with ble tenacity. That we are personally responsible for him. When we did meet, the memory of those early the web of compromises that we have all come to days in Chicago would come flooding back, and in accept, and to think are inevitable, is something he a way, he was a powerful force in my life. From a would never let us forget. That we, as editors or ref­ distance, of course, I watched his erratic battles and erees of journals, make our judgments based on even was very rarely a victim of some of his out­ some presumed social, or sociable, contract (e.g., bursts. But his textbooks, his successes, gave me no political articles in a math journal) does not let great pleasure. I regard him as a great mathemati­ us off the hook when asked to examine without cian, much more than an expositor, as I believe prejudice the underpinnings of our (usually only some regard him. I don't know if we will ever see implicit) social contracts. Serge seemed to be, over his like again, with anything approaching the enor­ the decades, of one age, and that age was young mous energy and insight which he brought to every­ (with its virtues and drawbacks). He had, when he thing he touched. played the piano, something of a brilliant French When I learned of his death, a profound sadness articulation to his style, and there was a hint of this came upon me. It was a feeling of incompleteness, in everything he did, from his walking gait (stac­ that somehow I could not express my closeness to cato) to the way in which he pronounced certain him personally, nor help him avoid some of his key words in mathematics, like idea which, from more acrimonious disputes. But Serge would have Serge, would sound like EYE-dee, which has a kind scoffed at such thoughts, and said that he managed of platonic zing to it. perfectly well. Over decades of mathematics Lang was led, more specifically, by an over-arching vision, which he pursued through the agency of various fields of mathematics. The vision, baldly put, is that geom­ Stephen Smale, University of California, etry is an extraordinarily striking dictator of qual­ Berkeley itative diophantine behavior. The still open Con­ jecture of Lang in higher dimensions continues to I had met Serge Lang by 1960. In fact in that year serve as a guiding principle to the way in which the he initiated a very nice offer to me to leave Berke­ grand subjects of geometry and number theory ley to join the faculty at Columbia University, which meet, just as Serge himself served as an inspiror I accepted. During the three years I spent at Co­ of generations of mathematicians, and a spokesman lumbia I became close friends with Serge and it was for intellectual honesty. his support and friendship that helped make the Columbia years such a memorable period. During my first years back at Berkeley, I became involved in the Free Speech Movement and espe­ Paul Cohen, Stanford University cially the Vietnam anti-war protests. In this period I was a graduate student when Serge Lang arrived I invited Serge for a visit and we shared an office in Chicago as an assistant professor. My interests for a year (1966-67, I believe). I introduced him to were tending towards number theory, but were some local activists and Serge himself became a po­ not very focused. The arrival of Serge made a huge litical activist, writing a book on the Bob Scheer cam­ difference to me and to many other graduate stu­ paign for political office. dents. He immediately gave courses in algebraic During the following decades we kept in touch geometry and in algebraic number theory, also I be­ especially during his summer visits to Berkeley. I lieve, accompanied by a constant output of notes. tried without success to get a permanent appoint­ Suddenly I had a different idea of what mathe­ ment in the mathematics department for Serge. matical research was. One could just attack prob­ We got along well and I seemed to have some im­ lems without a huge background of knowledge. munity from his occasional outbursts of anger. I His lectures were entertaining, of course, but also saw much less of Serge in the last decade partly be­ a little intimidating to the poor souls who might cause of my life in Hong Kong and Chicago. ask silly, or too elementary, questions. It was ad­ While leaving to others assessments of Serge's visable to be nimble enough to dodge flying chalk mathematical research, I want to make some brief coming from his direction. Another powerful remarks on other contributions.

MAY 2006 NOTICES OF THE AMS 543 Serge's Books: My opinion is that these books We only wrote one small joint paper together (on were a great contribution to mathematics. He gave diophantine approximation on abelian varieties), copies of almost all to me, and I frequently used but his encouragement and friendship came at a some for textbooks in courses I was teaching. They crucial time in my own mathematical evolution, and were characterized by economy and elegance and I have always been immensely grateful for it. The were written from a broad point of view, mathe­ second striking quality of Serge's was that he re­ matics as a whole. I especially enjoyed his gradu­ ally did live for mathematics, and somehow his be­ ate analysis book. This was a book written for stu­ lief in the goodness of the endeavour to do math­ dents in mathematics, (and mathematicians, not ematics was profoundly moving. just analysts) and reached the heart of the subject quickly. In contrast, other texts I have encountered spend a semester or even a year on foundational material as a step in the training of an analyst. On Dorian Goldfeld, Columbia University the other hand, his book on differential geometry had the virtue of giving infinite dimensional foun­ Of the many people who had serious interactions dations to the subject, which I found important in with Serge, I am one of those who came away with my own research. fierce admiration and loyalty. In the mid-1960s, I Serge's Teaching and Inspiration to Students: was an undergraduate in the Columbia engineering Serge made a special and constructive effort to school on academic probation with a C-average . In reach out to students of all ages. For example, he my senior year I had an idea for a theorem which wrote a mathematics book for high school stu­ combined ergodic theory and number theory in a dents and gave annual lectures to math club stu­ new way, and I approached Serge and showed him dents at Berkeley. His contributions included mak­ what I was doing. Although I was only a C-level stu­ ing elementary expositions of topics in current dent in his undergraduate analysis class he took research. On two different occasions Serge gave lec­ an immediate interest in my work and asked Lorch tures- in German!-to Don Zagier's wife's son Bern­ if he thought there was anything in it. When Lorch hard's school class. The talks were enjoyed by all, came back with a positive response, Lang imme­ and greatly increased Bernhard's prestige with both diately invited me to join the graduate program at his teachers and his classmates. Columbia the next year, September 196 7. In the fall Serge's Files: These files and their accounts con­ of 1967 I found an unfixable error in my ergodic­ tained extensive documentation of hypocrisy of number theoretic theorem. Lang was not at all per­ "The Establishment", in science, in and outside of turbed. He said these things happen all the time mathematics. Although acknowledging their pos­ and encouraged me to move on to something else. itive role in science I sometimes disagreed with him The following year Serge refused to discuss math­ in these matters. In particular, though his critique ematics with me or with anyone else. He said he of the AIDS bureaucracy sometimes made sense, was taking a year off from mathematics and doing it was hard to go along with his attack on the HN politics instead, but he kept encouraging me to theory of AIDS. prove theorems and told me to talk to Gallagher, In ending, let me emphasize how big an influ­ who became my official advisor. ence Serge has been in my life and how much I will Despite his kindness to students, everyone close miss him. to Serge has seen him explode, and this happened to me on several occasions. For example, in 1992 I organized a special year on number theory at Co­ lumbia University. I invited some of the well-known john Coates, Cambridge University established leaders of the field such as Bombieri, I think Serge's most remarkable quality as a col­ Lang, Mazur, Mallin, Schmidt, Szpiro, as well as league was his unstinting support for young math­ many younger people. Lucien Szpiro was chief ed­ ematicians. I personally benefited from this myself itor at Asterisque and invited me to submit a pro­ when I was a young postdoc at Harvard in about ceedings of the conference to Asterisque. Lang be­ 1970, and Serge came as a visitor for a semester, come utterly infuriated and blew up at me when shortly after he had resigned from Columbia. We Asterisque refused to accept his paper with Jor­ tend to forget when we are more established in the genson, which I previously had invited him to sub­ mathematical world how precious it is when one mit. I ultimately told Asterisque that I would resign is trying to make one's way in research to have the as editor of the proceedings and withdraw my own support and encouragement of an older mathe­ submitted paper unless they accepted the matician. Serge was not at all distant to young peo­ Jorgenson-Lang paper. Asterisque refused to budge, ple, but went out of his way to find out what one and I immediately followed through on my threat. was thinking about, and took time to discuss his The proceedings were, nonetheless, eventually pub­ own ideas and feelings about the subject with one. lished. My paper and the Jorgenson-Lang paper

544 NOTICES OF THE AMS VOLUME 53, NUMBER 5 were published in 1994 by Springer-Verlag as a During the summer of 1990, Lang called me book. every day, several times, as he traveled through Eu­ I was very shaken for several days when I heard rope and to Berkeley. We always spoke about math­ that Lang is with us no more. He has had a profound ematics, and he challenged me on the same point influence on my life and I will miss him enor­ I attempted to make to him earlier: Why did I be­ mously. lieve that one can use heat kernels and heat ker­ nel techniques, perhaps formally in a way to be de­ veloped, in a wide range of mathematical questions? We discussed, argued, and debated, as only Serge jay jorgenson, City College ofNew York could, until he returned to Yale that fall. He invited During the past fifteen or so years, Serge Lang and me to lunch one of the first days he was back. Dur­ I were colleagues, co-authors, and close friends. ing lunch he asked if I would be willing to work with Since Serge was known to never discuss his personal him on the mathematics we discussed. One cannot life, it is my inclination to not comment on our imagine my thoughts at that moment. To have a friendship beyond the mathematical collaboration. senior mathematician express interest in one's For instance, the reporter for the New York Times ideas is remarkable enough, but to have Lang say who interviewed me for Lang's obituary did not un­ he wanted to work with me simply cannot be de­ derstand why I would not answer questions re­ scribed. I found out later a more touching aspect garding Lang's family. Whereas I do feel compelled to that conversation with Serge. Apparently, in to respect his privacy after his passing, I have de­ 1988 Lang had told some faculty at Yale that his cided to accept the invitation of Steven Krantz and mathematical abilities were gone, and he couldn't include comments regarding my own interactions continue; after we began working together, he then with Serge. would comment to others that our work was keep­ As with so many others, I first learned the name ing him alive. Serge Lang as an undergraduate mathematics major During the first few years of our joint investi­ when I purchased textbooks for my mathematics gations, we spent countless hours developing a course. I met Serge for the first time in 1987, dur­ long-term program of research. Serge was a very ing my second year in graduate school at Stan­ private person, more so than I have seen with any­ ford. I remember the level of excitement among the one else. Although I knew his private telephone graduate students in anticipation of Serge's talk at numbers, I never called him at home, though he Stanford. It was thrilling to see his energy during would not hesitate to call me at any time. It was his lecture. Lang seems to be someone, I remem­ very rare that he mentioned his family or his non­ ber thinking, who has discovered what will give him mathematical, or non-trouble-making, interests. the most out of life, namely his mathematics and We became friends, and we made a point of talk­ his politics (what he himself called "trouble­ ing, perhaps quite briefly, each morning when he making"), and he is doing it. arrived in his office, each evening when he left to In 1990 I joined the faculty at Yale as a Gibbs go home, and many times during the day. instructor, and during that spring semester I gave I left Yale in December 1996. Serge was hospi­ a graduate course which Lang attended. As I had talized in December and later in February, and he expected, from the audience Serge directed my lec­ insisted that I visit him each day to continue our tures for himself, insisting on immediate changes mathematical conversations. His February stay in in notation and topics. In one particular lecture, I the hospital was reported in the Yale student news­ presented an evaluation of spectral determinants paper, and the dean of Yale College, a person Serge on elliptic curves which avoided the usual ap­ very much disliked, was quoted as saying that Lang proach, namely Kronecker's limit formula, and in­ was an excellent mathematician. I showed Lang stead relied on a trick I developed in my thesis. Lang the article while he was still in the intensive care was silent during the entire presentation. When I unit, and, when he read the dean's comments, he finished, he insisted that I wait in the classroom screamed, "That * * * * * * * * isn't qualified to so that he could go to his office. When he returned, judge." Serge was stunned when his outburst re­ he had two papers with him, one by Artin from 1923 sulted in my expulsion from his hospital room and another by himself from 19 56. He pointed out until the next day. As in so many other instances, that the technique I presented in the setting of Serge was right, but perhaps his message could have heat kernels on elliptic curves was conceptually been delivered differently. identical to Artin's ideas in the setting of Later in the spring of 1997, as I was seeking em­ L-functions of number fields and Lang's ideas in ployment for the upcoming year, Serge asked me topology, in the context of the characteristic poly­ to visit him at Yale so we could discuss our pro­ nomial in linear algebra. That conversation grew jects. At the time, I considered leaving academic into our first joint paper which was published in mathematics and seeking a career change away Crelle's journal in 1994. from a university environment. We discussed the

MAY 2006 NOTICES OF THE AMS 545 Stanford during my graduate school years. How­ ever, when we returned to the office, he needed to rest, and that evening during dinner he directed the conversation away from the specific ideas for our next project. Instead, he spoke of his wish that I always pursue our program of research, hopefully arriving at the goals we set for ourselves. During the next weeks, we spent time revising our wish list and reviewing our original mathematical plan. We continued to speak several times each day, up to and including September 12, 2005, the day Serge died. Having spent so much time with Serge, there are Lang at Berkeley. many stories I could tell. Serge had many sides matter in detail, and Serge pointed out to me the which affected everyone, including me, many dif­ effect it would have on him if I were to quit math­ ferent ways. He had friends and enemies, and per­ ematics. At the end of the day, I honored his request haps I inherited some of both. As he pointed out to continue our program of study and promised to to me quite some time ago, it is possible that my not seek a nonacademic position. association with him was both positive and nega­ We spent less time together during the next two tive for me. Serge once told me that he made cer­ years when I was at Oklahoma State during 199 7-98 tain personal decisions early on in his life, and he and in Greece during 1998-99. In the fall of 1999, stuck by those decisions. For me, I stand by my de­ I accepted a position at the City College of New cision to work with Serge. He was a close and loyal York, in part because it allowed me to visit Serge friend, and I believe that I was for him. Serge was frequently, which I did on most weekends. At that a part of my daily life for nearly fifteen years, and time, we began focusing our ideas, even going so for me his absence is great. far as writing a document for ourselves, estab­ lishing a "wish list" for our mathematical program. We created an outline of the articles, monographs, and books which must be completed in order to ful­ Paul Vojta, University of California, fill the steps of the program which we envisioned. Berkeley One of the earliest words my son knew was "Serge", Although I had seen Serge at Harvard once or twice which he said each time the phone rang; frequently, in the Common Room (invariably arguing with my son was right, and indeed the call was from someone), I first met him when I went down to Yale Serge. to ask him to write a letter of recommendation for It was evident to me, even iri 1999, that at age me. At one point in the discussion, he asked me 72 Serge was growing tired. Even though the learn­ ing process did keep him alive, time was catching about my definition of integral point. I started to up with him. He was always very sharp, with fas­ describe Serre's idea of an infinite set of points with cinating ideas and insight. However, he left the of­ bounded denominators, but he angrily interrupted fice earlier in the evening, and he required rest me: "I don't know what an integral point is, and nei­ during the day. Without acknowledging the act, ther do you!!!" However, later on that day he added we altered the pace of our work, finishing what we a flattering paragraph to his new edition of Dio­ could given the energy he had. When he was at Yale, phantine Geometry describing work in my thesis I visited him most weekends, and when he was in on integral points relative to a divisor with suffi­ Berkeley, I traveled there to continue our work. ciently many irreducible components. Most everyone saw Serge as the forever young, Later, upon hearing about my conjectures, he highly energized individual. For the most part, he called me up and invited me to come to Yale. I was did not allow many people to see that time was af­ a little hesitant about this, given his propensity for fecting him. The fact that hardly anyone noticed anger, but my advisor Barry Mazur convinced me that Serge was aging was, I believe, another mani­ that I should accept. festation of his level of privacy. During my time at Yale, I gave two or three grad­ In August 2005, we completed another book. On uate courses. Serge always sat in the front row, pay­ the day we submitted the manuscript to Springer­ ing close attention to the point of interrupting me Verlag, Serge was excited. To me, he momentarily midsentence: "The notation should be functorial regained his youth as I knew it when we delivered with respect to the ideas!" or ''This notation sucks!" the manuscript to the post office. That day I saw But, after class he complimented me highly on the again the dynamic person I remember lecturing at lecture.

546 NOTICES OF THE AMS VOLUME 53, NUMBER 5 While on sabbatical at Harvard, he sat in on a of his life during his youth with his parents were course Mazur was giving and often criticized the enduring, in particular vacations on the Mediter­ notation. Eventually they decided to give him a ranean seashore. Later in his life, he became a T-shirt which said, "This notation sucks" on it. So member of the Bourbaki group, stayed in France one day Barry intentionally tried to get him to say and lectured at length, in Paris and elsewhere. it. He introduced a complex variable B, took its com­ In some respects, Serge was rather austere. But plex conjugate, and divided by the original B. This he had a cheerful and authentic enthusiasm for was written as a vertical fraction, so it looked like mathematics: this is reflected in his books, and this eight horizontal lines on the blackboard. He then enthusiasm was contagious. did a few other similar things, but Serge kept At the beginning of 2005, we exchanged our quiet- apparently he didn't criticize notation un­ wishes by phone. As usual, he was infuriated, this less he knew what the underlying mathematics time by his own illness: he was conscious of his bad was about. Eventually Barry had to give up and health, and desperate not to be able to fulfill be­ just present him with the T-shirt. fore his death the program on zeta functions he Once, close to the end of my stay at Yale, I was had in mind, after twenty years of work on analy­ in his office discussing some mathematics with sis on groups. him. He was yelling at me and I was yelling back. We miss a great intellectual figure, and I miss a At the end of the discussion, he said that he'd miss friend. me (when I left Yale). Now that he has left, I will miss him, too. Roger Howe, Yale University

Gilles Lachaud, Institut de Mathematiques Books: Serge wrote an extraordinary array of books, from widely used texts, including calculus deLuminy texts, and even a high school geometry text, to I met Serge Lang in 1972, during the AMS sympo­ standard references, to monographs which are the sium on Harmonic Analysis on Homogeneous only treatment of their subject in book form. Spaces, held in Williamstown. He had just left Co­ The topics tend to cluster around algebra, and lumbia for Yale, and I was with Paris 7. especially Diophantine equations, which were his At that time, French and American universities great love in mathematics, but they span a re­ were a hotbed for the antiestablishment ideas, and markable range. Serge was involved in the Free Speech Movement. For a long time, his practice was to give each year Also, he was writing his book on SL(2, JR ): thus, in a graduate course on a new topic, and at the end, Williamstown, we discussed both alternative poli­ to turn the notes into a book. He had an amazing tics and spherical functions. capacity to boil a subject down to its essence, In my mind, Serge was a hunter, in mathemat­ which he often formulated with a few axioms or ics as in his political and social struggles. He was properties. He was a consummate axiomatizer. As chasing a precise game and nothing was able to he reached his late 70s, he gave little sign of slow­ make him deviate from this goal. ing. His last several books presented joint work with As a polemicist, he was very proud of the File Jay Jorgenson on applications of the heat equation process, consisting of bringing lies on some topic to analysis on symmetric spaces, with a view to­ to full light by sending letters to opponents, to wait wards automorphic forms. for contradictions in the answers, and to send Files: Besides books, a lot of Serge's literary ef­ Xerox copies to all the people involved in the con­ fort went into making files which chronicled his test. He was sure that victory would emerge from fights. Serge loved a good fight and he didn't have this confrontation. trouble finding them. He was especially concerned The special feature was that he wanted to work with honesty, especially honesty in public rhetoric. along scientific lines and to prove the statements Serge would coordinate multi-party correspon­ he was defending, an uncommon and irritating po­ dence, organizing sets of letters into packets, and sition outside the mathematical community. His circulating them with supplementary documents to model was in mathematics, as opposed to social sci­ a "cc list" of parties to the correspondence and ences, about which he used to say: "In mathemat­ other interested readers. At the end of the fight, ics you cannot say, 'I disagree with this statement.' he collected the whole into a file, and gave it a title. You can say, 'this is false' or 'this is of poor inter­ Serge thought of his files as documentation of est,' but there is no disagreement to express what­ the way life works today, and especially of his First ever.'' Law of Sociodynamics: the power structure does Among American mathematicians, Serge was one what it wants when it wants, and looks around of those who were closest to France: the souvenirs later for justification. Serge was willing to follow

MAY 2006 NOTICES OF THE AMS 547 his principles and beliefs (almost) wherever they to enter into another area which had been opened led. by Serge: he had started in 1962 with a series of Students and Teaching: A third focus of Serge's papers in transcendence theory out of which an­ energy was teaching and students. Generations of other book resulted in 1966 which turned out- at Yale undergraduates benefited from his teaching least in my eyes-to be the most influential book in the broad sense. He spent hours outside of class (Introduction to Diophantine Approximations) that talking with undergraduates, about mathematics, he ever wrote. politics, music, anything. He frequently ate meals Even the abc conjecture, one of the favorite with undergraduates in the Yale dining halls. He was subjects of Serge, has now been incorporated into especially concerned with promoting clear think­ transcendental context, and this indicates how im­ ing. He often found that students who came to portant Serge's impact into diophantine geometry him had been confused by poor education, but and transcendence has become. He had created that by appropriate challenges he could help them the frame of a very active and broad area to which to become independent thinkers. He referred to this he substantially contributed, he had the right math­ process as "Recycling their brains". ematical visions and supported enthusiastically Serge's attention to undergraduates was a part any progress. of his concern for the advancement of younger For many years Serge visited me at Zllrich. He people generally. It was a habit with Serge to en­ gave numerous talks in my seminar and to under­ courage younger mathematicians and be interested graduates, and he enjoyed visiting us at our house. in their work. That was certainly so for me. I recall Only one thing I never forgive him: when he to­ with gratitude his wholehearted enthusiasm for gether with Schinzel once had been guests at our my work, and his help in promoting it, and I know house in Bonn he came after the main course into of many others who similarly benefited. the kitchen where I had started to prepare a souf­ fle Grand Marnier. Without stopping he talked to me and distracted me so much that at the end the souffle did not rise in the oven. I did not want to Cis bert Wiistholz, Eidgenossisches offer it to the guests but Schinzel forced us to eat Technische Hochschule, Zurich it since he would not agree to throw away food. After this I essentially stopped cooking souffles. Presumably my most expensive investment in math­ ematics as a student was a book with the title Algebra by an author whose name was Serge Lang. A former schoolmate had recommended it to me ]iirg Kramer, Humboldt University as a very modern new tract in algebra. He had en­ I had my first encounter with Serge Lang as a stu­ tered university one year before me and just started dent in an indirect way: after having just finished with a course in algebra where the book had been my first two years as an undergraduate with the recommended. Certainly the top of my list of fa­ "Vordiplom" in mathematics at the University of vorite mathematics books would be Algebra by Basel (Switzerland), Martin Eichler proposed a sem­ Serge Lang. The reason I like it so much is that it inar on Lang's new book on modular forms. Since had a clear vision for modern and conceptual math­ I was just a beginner in the subject, the book made ematics and this was put together with much math­ me work quite hard, but anyhow, as a result I be­ ematical taste. came strongly interested in the subject. Six years It was exactly ten years later at the Arbeitstagung later, just before completing my Ph.D., Eichler in 1978 in Bonn when I first met Serge personally. asked me to accompany him to the "Arbeitsta­ At the time everybody talked about "Bombieri­ gung" taking place in June 1984 in Bonn. After ar­ Lang", a paper which influenced enormously the re­ riving at the entrance hall in front of the big lec­ search in transcendence. For us young students in ture hall at Wegelerstrasse 10, one of the first number theory it was a big challenge to try to un­ persons to meet was Serge. At the time, it was derstand the difficult methods from geometric quite impressive for me to have been personally in­ measure theory, the theory of plurisubharmonic troduced to this world-known mathematician. What functions and L 2-estimates. It took us away from I didn't know at the time was that this was the start the classical methods in transcendence theory and of a relationship lasting for more than twenty years. taught us that you need mathematics as a whole In fact, when I returned to Bonn in 198 5 to visit to formulate and to prove interesting new results the Max-Planck-Institut fur Mathematik (MPIM) , I in transcendence theory. was caught by surprise when I met Serge again dur­ At that Arbeitstagung I talked quite a bit to ing his regular trip to Europe in June and that he Serge and he eventually helped me to find a post­ immediately remembered me; as a consequence, doc position in Wuppertal. There I got into contact we started to talk about mathematics, first on a quite with people in algebraic groups and this helped me "innocent" level. Our mathematical communication

548 NOTICES OF THE AMS VOLUME 53, NUMBER 5 intensified in the years 198 7/88, when I was giv­ stance here was based on principle- in the present ing a course on "arithmetic surfaces" at MPIM, case, a defense of an author's right to self-expres­ while at the same time Serge was preparing his book sion free from gratuitous editorial intervention­ Introduction to Arakelov Theory. and what converted integrity into eccentricity was After having completed my "Habilitation" at ETH simply his stubborn insistence on continuing to do Zurich (where Serge was also visiting on a regular battle far beyond the point where the battle seemed basis since the late 1980s), I moved to Humboldt worth fighting. But you had to hand it to Serge: he University (HU) in Berlin in 1994. From 1995 until had the courage of his convictions. We can proba­ 2003, Serge regularly visited HU in late May/early bly all learn something from his example. June for one week. During this period of almost ten years, we got to know each other more closely and our relationship deepened. In Berlin, aside from his traditional talk in our number theory seminar Marvin jay Greenberg, University of (demonstrating the ubiquity of the heat kernel in California, Santa Cruz the last years), he was intensively arguing, dis­ cussing, and interacting with our graduate stu­ I was Lang's first Ph.D . student. Officially, Emil dents. In addition, he always generously offered Artin is listed as my thesis advisor, but he left talks to gifted high school students. In particular, Princeton for Hamburg three years before I wrote the high school students consider his unexpected my thesis. What Artin did was ask Lang to commute death as an infinite loss, and it is very sad that this from Columbia to Princeton in academic year tradition has come to such a sudden end. 1956-57 to continue teaching algebraic geometry, Although Serge tried to stay away from close per­ which Artin began toward the end of his graduate sonal relationships, it seemed to me that in his last algebra course- an extraordinary request and an years when coming to Berlin the ties between him, extraordinary acceptance on Lang's part. So Lang my wife Ruth, and myself got somewhat closer. We taught a course at Princeton on Abelian varieties, will surely miss him. in the style of A. Weil, and after a few weeks, I was his only student. Lang went away to Paris the following year, and when he returned after that, we continued meet­ David E. Rohrlich, Boston University ing informally at Princeton. He told me a conjec­ ture of his about Abelian varieties he wanted me Shortly after Serge's death a few people sug­ to prove for my thesis. I had no idea how to do so, gested to me that I write something about his math­ ematical contributions. The suggestion apparently and I was extremely busy teaching four elementary stemmed from a concern that obituaries would courses at Rutgers that year. Then he confronted focus on Serge's eccentricities and temper tantrums me, as he is famous for doing, and shouted at me rather than on the highlights of his career. Later, that if I did not show some progress with his con­ when I was invited to contribute something to the jecture in the next two weeks, I would no longer present article, I decided to intertwine my personal be his student. reminiscences with some glances at Serge's math­ The following weekend, in my attic room in New ematics, partly because I was mindful of the con­ Brunswick, after rereading Lang's thesis, I sud­ cern that had been expressed to me, but partly denly had a flash of insight on how to solve that also because Serge's passion for mathematics was problem. I needed a few more weeks to write out in my view an essential part of his persona. How­ all the technical details, but when I told him I had ever, the length of my submission ended up ex­ the solution, he was delighted. He took me out to ceeding the stipulated limit, and the editors excised a fine Spanish restaurant in NYC and treated me large portions of it, including all of the mathemat­ to paella, which I'd never eaten before. He invited ics. For that I had only myself to blame, but the prob­ me to his apartment overlooking the Hudson River lem was that the mathematics had been the glue and played Bach's dramatic Partita #6 (which I'd which joined one paragraph to the next, and with never heard before) and a Brahms Rhapsody for me the glue removed, the entire piece fell apart like a on his grand piano. I felt as if he had lifted me into Brunnian link. So I withdrew my submission. But in an exalted new world of excellence. doing so I realized that this little episode was some­ I remained on good terms with Lang for quite a how a fitting memorial to Serge, for one of his note­ few years after I left Princeton for Berkeley. Many worthy eccentricities was his proclivity for quarrels people were turned off by his aggressive person­ with editors. He could be very hard on editors, and ality, but I always enjoyed that immensely, partic­ he certainly would not have withdrawn a submis­ ularly his brutal honesty and taunting sense of sion without a fight; in fact, his fights with editors humor. He once told me bluntly that I would never were a significant component of some of his files. become a great mathematician because I was afraid As with so many of Serge's "eccentricities", his of making mistakes. He certainly made plenty of

MAY 2006 NOTICES OF THE AMS 549 them in the first editions of the many books he While I had known about Serge since my un­ dashed off so quickly. John Rhodes once told me: dergraduate days at Columbia when he was still "Serge Lang writes great books badly." with that institution, my first meeting with him was in Berkeley around 1987, when he called to talk about Nevanlinna theory. At the time, he was still campaigning against Huntington's election to the Shoshichi Kobayashi, University of National Academy of Sciences. In any case, Serge California, Berkeley and I began political discussions with increasing fre­ With no more than amateurish interest in number quency after that. He was at Yale, of course, and I theory, my mathematical contact with Serge Lang was at Berkeley, but Serge was never shy about is mostly through hyperbolic complex analysis. using the phone. While I sometimes had reserva­ With his several conjectures and his introductory tions about the tone of his writing, there was never book on hyperbolic complex manifolds, he was the any doubt in my mind about its substance. It may best promoter of the subject. surprise some that a firebrand like Serge could In 1985 I participated in the AMS Summer use any encouragement, as it did me, but I dis­ Research Institute on number theory at Humboldt covered that fighting the kind of lonely battle that State University, Arcata, in northern California. he did, he probably found it easier to listen to Thanks to Serge, I was invited to give an introduc­ someone sympathetic to his views than to engage tory talk on hyperbolic complex analysis, including in a shouting match twenty-four hours a day. From Noguchi's partial answers to the function-theoretic time to time, he would sound me out on his strate­ analogue of the higher dimensional Mordell con­ gies. One consequence of all these phone conver­ jecture as formulated by Lang. sations was that I was privy to all his fights since Since Serge was the only one driving back to about 1990, including the denouement of the Hunt­ Berkeley on the last day of the workshop, I ac­ ington case, the Gallo case, the Baltimore case, and cepted his offer of ride with trepidation. As I had of course the still-ongoing HIV controversy. expected, his driving was like his typing- fast. He Because I had no strong mathematical connec­ drove German-style, flashing the headlights when­ tion with Serge, our relationship could afford to be ever slow-moving cars blocked us, which happened more relaxed. Each time he called my home when to be all cars ahead of us. We were back in Berke­ my wife and I were out, he would leave a message ley by 10:00 p.m. As a result, the only thing Ire­ reprimanding us for "goofing". Serge was famous member from the workshop is this experience. for getting along with young people. He and my son In 1987, Serge published Introduction to Com­ were great pals, and each time my son would as plex Hyperbolic Spaces with Springer-Verlag. While much as ask a question about mathematics, a few writing this book, he called me, not day and night, days later a book or two on that subject would ar­ but 9 in the morning. Quickly he found out my rive from Springer-Verlag. The author of those morning routine and the most convenient time. So books was of course Serge Lang. usually I answered when the telephone rang around In the last year of his life, his big fights were still 9. When my wife answered, he invariably said "Is over HIV. One involved his submission of a survey Kobayashi out of the shower yet?" When he stopped to the Proceedings of the National Academy of Sci­ calling after several months, my wife said "I guess ences on the state of HIV research and government Serge finished his book." actions on the so-called anti-HIV drugs. He and I Without Serge, the summer in Berkeley will no knew from the beginning that it would get nowhere, longer be the same. I miss him. but the cavalier way in which his survey was re­ jected was stunning. I went to his office when word had just come and, perhaps owing to similar frus­ Hung-Hsi Wu, University of California, trations in my own work in mathematics education, Berkeley I lost it and said in less than polite language that I had had it. In an instant, his role and mine were Serge Lang's life is easy to characterize: it was 90 reversed and he soothed me with the philosophi­ percent mathematics and 10 percent scientific pol­ cal observation that life was hard and that we just itics. I happen to be one of the few mathematicians move on. who forged a friendship with him through politics, The last I heard from Serge was a phone call on thereby getting a glimpse of a side of him that was September 3, 2005, after he had left Berkeley for perhaps denied other mathematicians. New Haven. Four days after he passed away, his last For a mathematician to have any kind of friend­ file on HIV and the Proceedings of the National ship with Serge, it is a given that it could not have Academy of Sciences arrived. It was addressed to been completely divorced from mathematics. I was my son. He lived his life on his own terms to the no exception. very end.

550 NOTICES OF THE AMS VOLUME 53, NUMBER 5 the Daily Cal staff members. As far as he was con­ Lisa R. Goldberg, Morgan Stanley cerned, they may as well have been running The New York Times, or the world for that matter. High Expectations Serge was a champion of youth, but he had Serge Lang believed that young people have a spe­ plenty of energy left over for grownups. The Daily cial ability to see the truth. He was a champion of Cal file closes with a letter to the dean of the U.C. youth, and his students loved him. In January 2005, Berkeley School of Journalism. This time, the topic Serge submitted a brief op-ed piece and a dense four was journalistic responsibility. Serge stated some page advertisement to the The Daily Californian, of his concerns, drawing examples from the media the U.C. Berkeley campus newspaper. The topic coverage of HIV and AIDS. As always, his high ex­ was a Serge standard: growing pectations were in evidence. He dissent against the orthodox po­ asked, non-rhetorically, "How ..., sition that AIDS is a disease and does one make up for defective ~ it is caused by the HIV virus. The reporting over two decades?" -s And he signed the missive "In- § ad contained supporting docu­ ~ ~ mentation for the op-ed piece, formatively yours, Serge Lang." 0 ~ which had previously been re­ ~ !3 jected. The submission was ac­ Allyn jackson, Notices 8 companied by a personal check Deputy Editor ""'g. to pay for the ad. i;J, Serge had been disseminat­ "It's me": This is what some of 8 ing information on HIV and AIDS us in the AMS headquarters of- ~ for roughly twelve years. The fice would hear every now and Daily Cal was a natural outlet again when our phones rang. No for Serge's challenges. After greeting, no name: Of course it some back and forth, Serge re­ was Serge Lang. Probably the ceived a rejection letter explain­ very first time he called me he ing why his material could not be announced his name, but never published in The Daily Cal. Here again. When I got that first call, are some excerpts that are I mentioned it to a mathemati­ pieced together from The Daily cian acquaintance, who said, Cal file, which contains a copy of "Watch out, you are in the room the rejection letter and Serge's with the snake." Indeed, it was response. with some trepidation that we Daily Cal: We are confident that you under­ AMS staffers would take these calls. Usually Lang stand all newspapers must have the flexibility as would rail on about whatever issue he was currently a business to reserve the right to refuse any ad­ campaigning about, and we would give noncom­ vertisement, letter to the editor, op-ed or press re­ mital replies to avoid getting drawn into a debate lease at the discretion of their publishers. with this tireless debater. But after a while he would Serge: In fact, I know that newspapers have the soften up and crack a joke or tell a story; once he power to refuse advertisements, letters to the ed­ related to me the plot of a play he had seen. Some­ itor, op-ed or press releases, and I have known it times he would end the call with a whimsical clos­ for a long time. It's not exactly secret information. ing along the lines of "Toodle-oo". Carol McConway, Daily Cal: These are the things we do to protect a former AMS employee with whom I worked on our readership. the Notices many years back, received calls from Serge: I call this position Nannyism. Lang on a regular basis. She told me that in one con­ Daily Cal: Clearly it would have been a serious versation he remarked that she was very smart ethical error if we had elected to publish all or a and asked where she had gone to college. Carol, like portion of any op-ed letter to the editor that ref­ many of the highly intelligent and capable women erenced an advertisement specifically designed to on the AMS staff, had never gone to college. "Aha," clarify or provide back-up data for the op-ed. Lang replied. "That's why you are so smart. You Serge: What you find "clear" I do not. In fact, I were not ruined by the educational system." Lang take an opposite view. You might have stated more befriended another former AMS employee, Terry accurately "clearly to us" to make your assertion Drennan, who worked in the editorial department. more precise, instead of pretending to a universal When Terry got into a serious scrape with her boss, ethical standard, applicable to others, and imply­ Lang stood up for her. ing that I asked you to do something unethical. I was one of the many recipients of Lang's in­ The reply is pointed and funny throughout. Ev­ numerable and lengthy "files". He sent me masses ident in every line are Serge's high expectations of of documentation about his campaign about

MAY 2006 NOTICES OF THE AMS 551 HN1 AIDS. As I recall, he never stated that he be­ lieved that HN does not cause AIDS. Rather, he ad­ joseph Gerver, Rutgers University vocated the need to view this hypothesis with skep­ I met Serge Lang in 1967, my sophomore year at ticism and rigor. He poked holes in research papers Columbia, when I took his multivariable calculus and other writings that uncritically assumed the hy­ class. This was before the days of unified calculus. pothesis to be true. At some point I looked care­ All of us were math majors and many of us were fully at one of the AIDS research papers that Lang spoiled by our high school experience of learning had denounced; this particular one had been writ­ math with very little effort. So Lang would fre­ ten by the prominent AIDS researcher David Ho, to­ quently throw chalk at us, or yell. gether with some colleagues, and had been much I often ate dinner at the Gold Rail with Richard cited in the subsequent literature. I became con­ (now Susan) Bassein and Eli Cohen, and if Lang vinced that Lang was right in saying that the paper was also eating there he would always join us and lacked rigor and used mathematics inappropri­ usually pick up the tab. Sometimes we would talk ately. This is how it was with Lang's campaigns: He about math. Lang did not think logicians were true always had a valid point. Yes, he was obsessive, he mathematicians, because no real mathematician could be antagonistic, he had a bulldog-like at­ would worry about whether a proof made use of tachment to his causes and sometimes lacked a the axiom of choice. Why shouldn't you use the sense of proportion. And the solutions he pro­ axiom of choice? It's obviously true! Think about posed to the problems he identified were often to­ it! How could you not be able to construct a set by tally impractical and naive. Still, whenever I took choosing one element from each set in a collection the time to examine his analysis of those problems, of sets? Just do it! I found myself concluding that he was basically We also talked about politics, music, life. He right. shared with us his growing unease about the Viet­ I do not believe Lang pursued his causes out of nam war and what he viewed as Columbia's com­ a desire for fame or notoriety. Rather, he was hor­ plicity in it, although he made a point of never dis­ rified by the falsity he found all around him, and cussing politics in class. He offered advice about he would not let others turn their eyes from it. We unrequited love: If at first you don't succeed, try, have lost a person with a highly attuned sense of try again, but after the third time, if you still don't what is truthful and what is sham, and it is a pro­ succeed, give it up! found loss indeed. Rarely, he took someone seriously who was merely pulling his leg. Dorian Goldfeld, at that time a postdoc at Berkeley, once reported glee­ fully that Lang had posed him a problem which he john Ewing, Executive Director, American had in fact been working on for months and had Mathematical Society already solved. Goldfeld told him with a straight face that he would think about it, and the next day I had contacts with Serge over many years, begin­ presented Lang with the solution. Astounded, Lang ning when I was editor of the Mathematical Intel­ asked him how he had managed to find the answer ligencer (and he insisted on publishing a long, long so quickly, and Goldfeld explained that he had article). My kids got to know him by phone- he used ginseng, which greatly enhanced his mental called me at home on and off for more than a year. powers. Lang excitedly promised to try it himself, Later as executive director I came in contact but reported disappointedly a few days later that with him, larg ely because of his dispute with the ginseng had had no discernable effect on his brain. Notices of the AMS (about HN). Serge always called In the summer of 1968, I was on vacation with the office and simply said, "It's me." Everyone here my parents in Berkeley. Lang was there as usual, knew who that was. When I explained that the ED so my folks invited him for dinner. Naturally the doesn't make editorial decisions, he insisted that conversation turned to the demonstrations.at Co­ the "higher ups" at the AMS could do whatever lumbia and the other student protests around the they wanted. He often sent material related to var­ country, many of them against the war. Although ious things, and always mentioned that he was my mother was opposed to the war in Vietnam, she sending it to the "higher ups". Phone conversa­ deplored the excesses of some of the demonstra­ tions were always long, protracted affairs. tors, which she attributed to their permissive up­ But even with long phone calls and huge quan­ bringings. "But what about France?" Lang objected. tities of written material (we have a giant file here), "And Poland? Even in Poland students are protest­ Serge was really a charming guy whom I instinc­ ing against the government, and you can be sure tively liked and admired. The world is better off hav­ they weren't raised permissively! Young people ing had him fight for his causes, with passion and don't want to be used by their governments. All over indignation. the world, students are fighting for their freedom!"

552 NOTICES OF THE AMS VOLUME 53, NUMBER 5 like nitrite inhalants and amphetamines and are Peter Duesberg, University of California, prescribed DNA chain-terminators. Berkeley 4) The epidemic would have formed a classical About twelve years ago I received a package of bell-shaped time curve, increasing exponentially "files" from Serge Lang. The files objected to the and then declining exponentially owing to natural Washington-style cover up of the scandal sur­ immunity within weeks to months, like a seasonal rounding the American discovery of the hypo­ flu. Instead AIDS increased slowly over a decade and has since leveled off, without ever inducing im­ thetical AIDS virus. Impressed by the thoroughness munity against itself. and mathematical logic of his case, I sent my own Thus AIDS fits a lifestyle-but not a viral epi­ file in response: "Dr. Lang," I wrote, "the political demic. I hope, therefore, that those who saw Lang's scandal about who discovered the hypothetical AIDS engagement as an Achilles heel might re­ AIDS virus is from a scientific point of view no more consider. than a distraction-a catchy story about who stole I already miss Lang as an ally in the politically whose fake diamonds. The scientific challenge, incorrect debate on the cause of AIDS. And I hope however, is whether AIDS is a viral or a chemical that you all let me join you in missing the Mensch alias lifestyle epidemic, caused by the long-term that was hidden behind the machine. Au revoir consumption of recreational drugs and anti-viral Serge Lang! drugs such as the inevitably toxic DNA chain­ terminator AZT." Not much later Lang and I became allies in the AIDS debate. Lang gave seminars on AIDS, wrote for the Yale Scientific, included two AIDS chapters in his book Challenges (1998) and generated a steady flow of AIDS files, the last of which arrived here only after his death. But only now, on the sad occasion of his death, is Lang's AIDS engagement presented as an Achilles heel of a mind that seemed otherwise irrefutable in its high standards of ac­ curacy and precision not only by the politically correct New York Times and Yale Daily News, but even by several of his mathematical peers. In view of this I take a last stand on behalf of our colleague, who cannot do this anymore, trying to inform his survivors with "primary evidence", rather than "condition them" with government handouts, as Lang would have said. Even if one virus could cause the twenty-six infectious and non­ infectious (!) diseases that are now defined as AIDS, the following would be true: 1) AIDS would be contagious. But, there is no case report in the peer-reviewed literature of even one doctor who ever contracted AIDS from one of the 929,985 (2004) American AIDS patients in twenty­ three years. Moreover, not even one of the thou­ sands of AIDS virus researchers ever contracted AIDS from their "deadly virus", as the New York Times calls it. 2) AIDS should appear within days to weeks after infection, because the AIDS virus, like other viruses, replicates with multiplication rates of 100 to 1,000 .within twenty-four hours. But AIDS is said to appear only five to ten years after infection by its hypothetical viral cause. 3) The epidemic would spread randomly like all viral epidemics. But AIDS cases in the U.S. and Eu­ rope are highly nonrandom, 80% are males, of which 1/3 are intravenous drug users and 2/3 are male homosexual users of toxic, recreational drugs

MAY 2006 NOTICES OF THE AMS 553 The Life and Works of RaoulBott Loring W. Tu

Editor's Note: This is the first part of a two-part article. In part two, which will ap­ pear in a later issue, the author presents some personal reminiscences of Raul Bott from colleagues, students, and others.

aoul Bott passed away on December 20, Austrian and Catholic. His parents divorced soon 2005. In a career spanning five decades, after his birth, so he grew up with his mother and he has wrought profound changes on stepfather. Raised as a Catholic, Raoul spent his Rthe landscape of geometry and topol­ childhood and adolescence in Slovakia, which sev­ ogy. It is a daunting task to improve enty years later, after alternating between Hun­ upon his own reminiscences [B3], [B4], [B1] and gary and Czechoslovakia, is today an independent commentaries on papers [B5], punctuated as they country. are by insight, colorful turns of phrases, and amus­ In the first five years of school Raoul was not a ing anecdotes. This article is an updated reprint of good student. This should give comfort to all par­ one that first appeared in the book The Founders ents of late bloomers. In fact, he did not earn a sin­ of Index Theory: Reminiscences of Atiyah, Batt, gle A except in singing and in German. Nonethe­ Hirzebruch, and Singer (edited by S.-T. Yau, Inter­ less, he showed an early talent for breaking rules national Press, 2003). Taking a personal interest in and for generating sparks-electrical sparks, that the project, Raoul Bott introduced me to some of is, rigged up with wires, fuse boxes, vacuum tubes, his friends and gave me access to his files. After and transformers. The schools were formal and the original article was completed in June 2000, he strict, and one could get slapped or have one's read it and verified the essential correctness of ears pulled for misbehaving. For a budding origi­ the accounts of both his life and his works. In the nal thinker, Raoul survived the schools relatively interest of preserving its official imprimatur as an unscathed. He recalls a friar hitting him on the authorized biography, the only changes I made to hand once and a teacher cuffing his ear another the 2003 version consist of this introductory para­ time, for horsing around too much. graph, an update of the awards he received, and the It was by all accounts an idyllic existence, com­ suppression of a bibliography of his works not ref­ plete with a family villa, English governesses, and erenced in the article. music lessons. This world came to an abrupt halt in 1935, when his mother died of cancer. In time Early Years his stepfather remarried. Raoul Bott was born in Budapest in 1923. His lin­ Raoul's experimental talent found its full flow­ eage fully reflects the geopolitical complexity of ering in adolescence. He and a kindred 'spirit Tomy the region at the time. His mother's family was Hornak built a small box with a slit for coins. When Hungarian and Jewish, while his father's side was someone dropped a coin through the slit, a display lit up saying "Thank you." In this way they funded Loring W. Tu is associate professor of mathematics at Tufts University. His email address is loring. tu@ their early experiments. tufts. edu. Raoul struggled with some subjects in school and This article is reprinted with permission of Interna­ a tutor was hired to help him a few hours a week tional Press. in his house. At the time Raoul and To my had built

554 NOTICES OF THE AMS VOLUME 53, NUMBER 5 a gadget to communicate by Morse code. As he was that, Batt could go to all the shows at this theater being tutored, he would hold the gadget under the for free. table and Tomy would be sitting in the basement. Batt's roommate Rodolfo, equally penniless, also Raoul received the code by getting short and long loved the opera. But Rodolf a did not have the nerve electric shocks in his hand. He then responded by to sneak into the theater. When the opera Carmen pressing a button to light up a bulb in the basement. was playing, Rodolfo was very eager to attend. Batt While the tutor believed that his student was lis­ magnanimously invited him. By then, the ticket tening intently to the lesson, Raoul was chatting taker knew Batt very well, but he stopped Rodolf a away in Morse code under the table. In retrospect, at the entrance. Batt turned around and intoned in Batt calls this his first attempt at email. his authoritative voice, "It's all right. He can come in." Without any hesitation the ticket taker obeyed Canada the order of this "nephew" of Ezio Pinza. In 1938, with Hitler's ascendancy and Germany's One New Year's Day, Raoul, Rodolfo, and some march into Czechoslovakia, Batt's stepparents flew friends went to Mont Tremblant, a winter resort him to the safety of England and enrolled him in north of Montreal. In the most prominent and ex­ an English boarding school. Since they had only pensive hotel, a big celebration was going on. Some­ transit visas for England, the following year they how, to the envy of his friends, Raoul sneaked in. headed for Canada, a country that to this day has A little later, Raoul was standing on the balcony, been extraordinarily welcoming to refugees and looking down contemptuously at his friends and immigrants from around the world. showing them a chicken leg he was eating. After In the fall of 1941, after a rigorous year of he finished it, he threw the bone, with disdain, to preparatory studies in Ontario, Raoul Batt found his hungry friends. himself at McGill University in Montreal. Given his (Old habits die hard. In 1960 Batt, by then a full electrical know-how, he chose, not surprisingly, professor at Harvard, was in India with Michael electrical engineering as his major. His grades were Atiyah, both giving lectures as guests of the Tata respectable, but as he recalls in [B4], he was more Institute of Mathematics. One day, as they walked interested in upholding the "engineering tradition in the streets of New Dehli, they passed by a big of hard drinking, loud, boisterous, mischievous, and celebration. Batt decided to slip in uninvited, drag­ macho behavior." Mathematics was his best sub­ ging Atiyah along with him. Atiyah, a professor at ject; still, it was mathematics in the engineering Oxford who was later anointed Sir Michael by the sense, not the kind of pure reasoning for which he Queen and elected President of the Royal Society, became so well known years later. was at first discomfited, but soon joined whole­ With his European flair, his six-foot two-inch heartedly in the festivities. They had a rousing frame, and the conspicuous fur cap he often wore, time, sharing in the general merriment of com­ Batt stood out from the crowd at McGill. When plete strangers.) friends asked him where he was from, he said from Upon graduation, Batt joined the army, but the Dioszeg, Czechoslovakia, and he added facetiously, atomic bomb at Hiroshima put an end to his mili­ where he "was a Count". After that, everyone called tary career after only four months. He entered a him the Count. one-year master's program in the engineering de­ The Count sometimes spoke a very foreign partment at McGill. Gradually it dawned on him that tongue. In the streetcars of Montreal, Raoul and his his interest lay more in mathematics than in engi­ roommate Rodolfa Gurdian would occasionally en­ neering, and he produced a very mathematical gage in a deliberately loud and animated conver­ master's thesis on "impedance matching", which sation. Nothing they said made sense, for they he said, "the department accepted with some mis­ were making up the language as they went along. givings and about whose mathematical rigor I have From the corners of their eyes, they enjoyed watch­ doubts to this very day." ing the quizzical expressions on the faces of the At McGill Raoul met his future wife, Phyllis, an surrounding passengers, who were trying hard to English literature major from the West Indies. figure out what language the two of them were Today, Phyllis remembers Raoul's first marriage speaking. proposal. At the time he was doing his short stint Batt loved the opera, but as a penniless student in the army. In full uniform, he said, "Would you how was he to afford it? One time the famous marry me? Because if you do, the army will pay me tenor Ezio Pinza came to sing in His Majesty's The­ more money." And then pointing through the win­ ater, the opera house of Montreal in the 1940s. For dow to his little room, he added, "And we could be this occasion, Batt dressed up in his Sunday best living there." The proposal was not accepted. But and went to the theater. When the man at the en­ two years later, they married. The Botts have been trance stopped him, Batt told him he couldn't do together ever since, and now have four children and this because he was Ezio Pinza's nephew. Batt said eight grandchildren. They celebrated their golden it with such assurance that the man let him in. After anniversary in 1997.

MAY 2006 NOTICES OF THE AMS 555 Sermon me a little about yourself. Did you ever While in the master's program in engineering at have any interest in botany, say, or bi­ McGill, Bott floundered in trying to decide on the ology? Well, not really, I had to admit. general direction of his career. Thirty years later, How about chemistry- Oh, I hated that Bott was asked to deliver a sermon at Harvard's course. And so it went. After a while he Memorial Church. As he discussed the biblical pas­ said, "Well, is it maybe that you want to sage of Eli, the wise man who counseled the young do good for humanity?" And then, while Samuel (1 Samuel3: 3-6, 8-10), he reflected on the I was coughing in embarrassment, he pivotal moment in his life that launched his math­ went on, "Because they make the worst ematical career. His description of his own Eli de­ doctors." serves to be read in the original: I thanked him, and as I walked out of And so when I saw the two readings we his door I knew that I would start afresh just heard juxtaposed in a Scripture and with God's grace try and become a Service, I could not resist them. For they mathematician. are appropriate to all of us, whether called to high causes or to lowly ones. Mathematical Career And they are maybe especially appro­ Initially Bott wanted to stay at McGill to do a math­ priate to the young people of today in ematics Ph.D. Because of his sketchy background, their search of their destiny. however, the McGill mathematics department rec­ ommended that he pursue a bachelor's degree in For surely there never has been a time mathematics first. It would have taken another when our young people have been given three years. Sensing his disappointment, Professor such freedom and therefore such re­ Williams of McGill then suggested Carnegie Tech sponsibility to find this destiny. (now Carnegie-Mellon University) to Bott, where But how are we to know where we are John Synge was just forming a new graduate pro­ called? And how are we to know who is gram and would need some students. calling us? These are questions beyond Synge received Bott warmly at Carnegie Tech, but a mathematician's ken. There are some as they read the rules of the program together, they who seem to have perfect pitch in these found that Bott would have to spend three years matters. There are many more who taking courses in the newly minted master's pro­ might think that they have. But with gram. In a flash of inspiration, Synge said, "Let's most of us, it is as it was with Samuel, look at the Ph.D. program." It turned out to have and we are then truly blessed to have hardly any requirements at all! Normally the mas­ an advisor such as Eli. He stands for all ter's program is a prerequisite to the Ph.D. program, of us Teachers as an example. For apart but perhaps recognizing a special gift in Bott, Synge from communicating our call to our stu­ put him in the Ph.D. program. In just two years Bott dents, we should try and help them would walk out with his degree. above all to discern theirs. Bott found the Carnegie Tech atmosphere ex­ ceedingly supportive. The small coterie ·of mathe­ I well remember my Eli. He was the Dean matics students included Hans Weinberger, now at of the Medical School at McGill and I ap­ the University of Minnesota, and John Nash, an ad­ proached him for help in entering the vanced undergraduate who after a thirty-year bat­ medical school there, when in 194 5 the tle with schizophrenia received the Nobel prize in atomic bomb unexpectedly put an end 1994. In later years Bott said of Carnegie Tech, to the war and to my four-month old ca­ "Being a brand new graduate program, they hadn't reer in the Canadian Infantry. learned yet how to put hurdles in front of gradu­ The Army very wisely decided to get ate students." Bott considers himself very fortunate rid of such green recruits as soon as pos­ to have an advisor in R.]. Duffin, for Duffin treated sible, and so we all again found our­ him as an equal from the very outset and together selves quite unexpectedly in charge of they published two papers on the mathematics of our own lives. I had graduated in engi­ electrical networks. neering earlier that year but had already The first of these two papers, on impedance decided against that career. functions [1], so impressed Hermann Weyl that he invited Bott to the Institute for Advanced Study in The Dean greeted me very cordially and 1949. Thus began Bott's initiation into the mysteries assured me that there was a great ne.ed of algebraic topology. Apart from Weyl, among his for technically trained doctors. But, he main teachers were N. Steenrod, E. Specker, K. Rei­ said, seating me next to him, first tell demeister, and M. Morse. Of Ernst Specker, Bott said

556 NOTICES OF THE AMS VOLUME 53, NUMBER 5 in [B2], "I bombarded Ernst with so many stupid that hat!" Now, this is the sort of order under­ questions that in desperation he finally imposed graduates love to obey. In no time the Dunster stu­ a fine of 2 5 cents on any conjecture he could dis­ dents had paddled to the Lowell raft. A struggle en­ prove in less than five minutes. This should give sued, and like any good pirates, the Dunster you some idea of the inflation of the past thirty contingent captured the admiral's hat. It was later years and also help to explain Ernst's vast fortune hung, as a trophy, high in the ceiling of the Dun­ at this time." ster House Dining Hall. At the time Norman Steenrod was writing his Showing true House spirit, the Dunster House classic book on the topology of fiber bundles and Crew Team had its official team T-shirt embla­ teaching a course based on it. This course greatly zoned with "Dunster House," a pair of oars, and the influenced Bott's mathematical development. exhortation: "Raoul, Raoul, Raoul your Bott." Bott describes Steenrod with admiration as The Harvard Houses have counterparts at Yale, someone who treated high and low alike, with equal where they are called Colleges. A friendly rivalry respect. At Princeton, the graduate students could has always existed between these two august in­ be intimidating, because they knew so much, and stitutions, and it extends to the Houses and Col­ they let you know it. Steenrod, on the other hand, leges. Some of the Houses at Harvard even have "sis­ was different. In spite of his stature in the mathe­ ter Colleges" with which they are loosely affiliated. matical community, he put everyone at ease. In They would, for example, visit each other during seminars Steenrod did not hesitate to ask the most the Harvard-Yale football games. basic questions. This was quite often a boon to the In the aftermath of the 1960s, many of the tra­ others in the audience, too intimidated and too be­ ditions at the Ivy League universities, such as the fuddled to ask the questions themselves. dress code and the parietal rules, have gone by the After two years at the Institute, Bott went to the wayside, and for a number of years Dunster House University of Michigan. In 1959 he became a pro­ had not had contact with Berkeley College, its sis­ fessor at Harvard, where he has remained since. In ter College at Yale. One year the Berkeley College 1999 Bott formally retired from teaching. He was Master, a distinguished historian, decided to revive William Casper Graustein Research Professor at the tradition. He wrote to Bott suggesting a visit to Harvard. Dunster House during the weekend of the Har­ vard-Yale football game. Bott readily agreed, but de­ Dunster House cided to make the occasion a memorable one. Why An unusual item in the curriculum vitae of Raoul not fool the Yalies into thinking that Harvard has Bott, for a mathematician at least, was his tenure kept up, at least to a certain point, the Oxbridge as the Master of Dunster House in 1978- 84. At tradition of High Table and academic gowns at Harvard the undergraduates live in social units dinner? Why not show that, perhaps, Dunster House called "Houses", modelled somewhat after the Col­ was more "civilized" than its Yale counterpart? leges at Oxford and Cambridge. A House is more With enthusiasm, the Dunster House undergradu­ than a place to sleep; it is a way to create a sense ates all supported the idea. of a small community within a large university. On the appointed day, the Dunster House Din­ Each House has its own dining hall, dormitories, ing Hall was transformed from a cafeteria into a social activities, and a staff headed by a Master. The hallowed hall, complete with linen, waiters and academic staff consists of a bevy of resident and waitresses, and even a wine steward wearing a non-resident tutors. large medal. Unlike on a normal day, there were no Whether out of a lack of interest or a perceived T-shirts or cut-offs in sight. Every tutor was attired mismatch of temperament, pure mathematicians in a black academic gown. An orchestra sat in wait­ are rarely called to be Masters of the undergradu­ ing. When the Yale Master and his tutors arrived, ate Houses. In 1978, in a break with tradition, the Bott asked, with a straight face, "Where are your president of Harvard University appointed Bott the gowns?" Of course, they didn't have any. "Well, no Master of Dunster House. This entailed living in the problem, you could borrow some of ours." So the Master's Residence in the midst of three hundred Dunster tutors led them to some gowns that had undergraduates. Bott's gregariousness was a good just been lent from Harvard's Chapel. As Bott en­ match for the post. tered the Dining Hall with his guests, trumpets Every year the Houses compete in a water-raft blared forth and the orchestra started playing. The race on the Charles River. This is no gentleman's undergraduates were already seated, looking prim, canoe race as practiced in England. In the Harvard proper, and serious. Bott and his tutors dined with version, attacks on other Houses' rafts are con­ the Yale visitors at a High Table, on a stage espe­ doned, even encouraged. One year the Lowell House cially set up for this occasion. The orchestra sere­ team had its Master at the helm, resplendent in an naded the diners with music. Everything went ac­ admiral's hat. Bott, commanding the Dunster House cording to plan. But the Yale Master, ever sharp, had armada, saw the beautiful hat. He hollered, "Get me the last laugh. He opened his speech by saying, ''I'm

MAY 2006 NOTICES OF THE AMS 557 Year Ph.D. Student Dissertation Title

1957 Smale, Stephen Regular Curves on Riemannian Manifolds 1961 Edwards, Harold Application of Intersection Theory to Boundary Value Problems Curtis, Edward The Lower Central Series for Free Group Complexes 1963 Conlon, Lawrence Spaces of Paths on a Symmetric Space Zilber, Joseph Abraham Categories in Homotopy Theory 1964 Holzsager, Richard Allan Classification of Certain Types of Spaces Quillen, Daniel Formal Properties of Over-Determined Systems Of Linear Partial Differential Equations 1965 Landweber, Peter S. Kiinneth Formulas for Bordism Theories Lazarov, Connor Secondary Characteristic Classes inK-theory 1969 Brooks, Morris William The Cohomology of the Complement of a Submanifold Brown, Richard Lawrence Cobordism Embeddings and Fibrations of Manifolds 1970 Blass, Andreas R. Orderings of Ultrafilters MacPherson, Robert D. Singularities of Maps and Characteristic Classes 1973 Miller, Edward Y. Local Isomorphisms of Riemannian Hermitian and Combinatorial Manifolds 1974 Garberson, John Dayton The Cohomology of Certain Algebraic Varieties 1975 Mostow, Mark Continuous Cohomology of Spaces with Two Topologies Perchik, James Cohomology of Hamiltonian and Related Formal Vector Field Lie Algebras 1976 Weiss, Richard Simon Refined Chern Classes for Foliations 1977 Brooks, Robert On the Smooth Cohomology of Groups of Diffeomorphisms 1981 Hingston, Nancy Equivariant Morse Theory and Closed Geodesics 1982 Gunther, Nicholas Hamiltonian Mechanics and Optimal Control Laquer, H. Turner Homogeneous Connections and Yang-Mills Theory on Homogeneous Spaces 1984 Ticciati, Robin Singular Points in Moduli Spaces of Yang-Mills Fields 1985 Forman, Robin Functional Determinants and Applications to Geometry 1986 Corlette, Kevin Stability and Canonical Metrics in Infinite Dimensions 1987 Block, Jonathan Excision in Cyclic Homology of Topological Algebras 1989 Kocherlakota, Rama Integral Homology of Real Flag Manifolds and Loop Spaces of Symmetric Spaces Morelli, Robert Hilbert's Third Problem and the K-Theory of Toric Varieties Bressler, Paul Schubert Calculus in Generalized Cohomology 1991 Grossberg, Michael Complete Integrability and Geometrically Induced Representations 1992 Weinstein, Eric Extension of Self-Dual Yang-Mills Equations across the Eighth Dimension Szenes, Andras The Verlinde Formulas and Moduli Spaces of Vector Bundles 1993 Tolman, Susan Group Actions and Cohomology 1994 Teleman, Constantine Lie Algebra Cohomology and the Fusion Rules Castes, Constantine Some Explicit Cocycles for Cohomology Classes of Group Diffeomorphisms Preserving a G-Structure 2000 Bernhard, James Equivariant de Rham Theory and Stationary Phase Expansions

glad to see that culture has finally migrated from encapsulates the essence of the problem. Often, as New Haven to Harvard." if by magic, a concrete formula with transparent significance appears. Bott as a Teacher At a reception for new graduate students at Har­ Batt's lectures are legendary for their seeming ease vard, he once shared his view of the process of writ­ of comprehension. His style is typically the an­ ing a Ph.D. thesis. He said it is like doing a home­ tithesis of the Definition-Theorem-Proof approach work problem; it's just a harder problem. You try so favored among mathematical speakers. Usually to understand the problem thoroughly, from every he likes to discuss a simple key example that conceivable angle. Much of the thesis work is

558 N OTICES OF THE AMS VOLUME 53, N UMBER 5 perseverance, as opposed to inspiration. Above all, Mathematical Works "make the problem your own." The bibliography in Raoul Bott's Collected Papers Many of his students testify to his warmth and [B5]lists his publications, with some omissions, up humanity, but he also expects the students to meet to 1990. an exacting standard. He once banned the word "ba­ When asked to single out the. top three in the sically" from an advisee's vocabulary, because that manner of an Olympic contest, he replied, "Can I word to Bott signifies that some details are about squeeze in another one?" But after listing four as to be swept under a rug. the tops, he sighed and said, "This is like being This insistence on thoroughness and clarity ap­ asked to single out the favorites among one's chil­ plies to his own work as well. I. M. Singer remarked dren." In the end he came up with a top-five list, that in their younger days, whenever they had a in chronological order: mathematical discussion, the most common phrase [15] Homogeneous vector bundles, Bott uttered was "I don't understand," and that a [24] The periodicity theorem, few months later Bott would emerge with a beau­ [51 ] Topological obstruction to integrability, tiful paper on precisely the subject he had repeat­ [81] Yang-Mills equations over Riemann sur­ edly not understood. faces, Seminar speakers at Harvard tend to address [82 ] The localization theorem in equivariant co­ themselves to the experts in the audience. But like homology. Steenrod, Bott often interrupts the speakers with To discuss only these five would not do justice the most basic questions, with the salutary effect to the range of his output. On the other hand, it is of slowing down the speakers and making them evidently not possible to discuss every item in his more intelligible to lesser mortals. ever-expanding opus. As a compromise, I asked him At Michigan and Harvard, Bott directed over 36 to make a longer list of all his favorite papers, Ph.D. theses. Some of his students have become lu­ without trying to rank them. What follows is a minaries in their own right: Stephen Smale and leisurely romp through the nineteen papers he Daniel Quillen received the Fields Medal in 1966 and chose. My goal is to explain, as simply as possible, 1978 respectively, and Robert MacPherson the Na­ the main achievement of his own favorite papers. tional Academy of Science Award in Mathematics For this reason, the theorems, if stated at all, are in 1992. The accompanying sidebar presents what often not in their greatest generality. is, I hope, a complete list of his Ph.D. students. Impedance The subject of Raoul Bott's first paper [1] dates back Honors and Awards to his engineering days. An electrical network de­ Throughout his career, Bott has been showered termines an impedance function Z(s), which de­ with honors, awards, and prizes. The more note­ scribes the frequency response of the network. worthy awards include: Sloan Fellowship (1956-60), This impedance function Z(s) is a rational function the AMS Veblen Prize (1964), Guggenheim Fellow­ of a complex variables and is positive-real (p.r.) in ship (1976), National Medal of Science (1987), the the sense that it maps the right half-plane into it­ AMS Steele Prize for Lifetime Achievement (1990), self. An old question in electrical engineering asks and the Wolf Prize in Mathematics (2000). whether conversely, given a positive-real rational He was twice invited to address the Interna­ function Z(s), it is possible to build a network with tional Congress of Mathematicians, in Edinburgh Z(s) as its impedance function. In some sense 0. in 1958 and in Nice in 1970. Brune had solved this problem in 1931, but Brune's He was elected vice president of the AMS in solution assumes the existence of an "ideal trans­ 1974- 75, Honorary Member of the London Mathe­ former", which in practice would have to be the size matical Society (1976), Honorary Fellow of St. of, say, the Harvard Science Center. The assump­ Catherine's College, Oxford (1985), Honorary Mem­ tion of an ideal transformer renders Brune's algo­ ber of the Moscow Mathematical Society (199 7), and rithm not so practical, and it was Raoul's dream at Foreign Member of the Royal Society (2005). He has McGill to remove the ideal transformer from the so­ been a member of the National Academy of Sciences lution. since 1964 and the French Academy of Sciences At his first meeting with his advisor Richard since 1995. Duffin at Carnegie Tech, he blurted out the prob­ In 1987 he gave the Convocation Address at lem right away. Many days later, after a particularly McGill University. He has also received Honorary fruitless and strenuous discussion, Raoul went Degrees of Doctor of Science from the University home and realized how to do it. He called Duffin. of Notre Dame (1980), McGill University (1987), The phone was busy. As it turned out, Duffin was Carnegie Mellon University (1989), and the Uni­ calling him with exactly the same idea! They wrote versity of Leicester, England (1995). up the solution to the long-standing problem in a

MAY 2006 NOTICES OF THE AMS 559 joint paper, which amazingly took up only two polynomial is pages. Morse Theory As mentioned earlier, the paper on impedance so For a Morse function f on a compact manifold impressed Hermann Weyl that he invited Bott to the M, the fundamental results of Morse theory hinge Institute for Advanced Study at Princeton in 1949. There Bott came into contact with Marston Morse. on the fact that M has the homotopy type of a CW Morse's theory of critical points would play a de­ complex with one cell of dimension i\ for each crit­ cisive role throughout Bott's career, notably in his ical point of f of index i\. This realization came work on homogeneous spaces, the Lefschetz hy­ about in the early 1950s, due to the work of Pitcher, perplane theorem, the periodicity theorem, and Thorn, and Bott. the Yang-Mills functional on a moduli space. Two consequences follow immediately: In the 1920s Morse had initiated the study of the i) The weak Morse inequalities: critical points of a function on a space and its re­ # critical points of index i 2. i 1hBetti number. lation to the topology of the space. A smooth func­ tion f on a smooth manifold M has a critical point If at pin M if there is a coordinate system (x 1, ... , Xn) Pc(M) = I dimH;(M)t; at p such that all the partial derivatives off van­ ish at p: is the Poincare polynomial of M, the Morse in­ equalities can be restated in the form of - (p) = O for all i = 1, ... , n. :M.r(fl 2. Pc(M), OX; meaning that their difference :M. 1(f) - P1(M) is a Such a critical point is nondegenerate if the matrix polynomial with nonnegative coefficients. This of second partials, called the Hessian off at p, inequality provides a topological constraint on analysis, for it says that the i 1h Betti number of 2 the manifold sets a lower bound on the number Hpf = [ OX;OXjof (p) J ' of critical points of index i that the function f must have. is nonsingular. The index A(p) of a nondegenerate ii) The lacunary principle: If no two critical points critical point p is the number of negative eigen­ of the Morse function f have consecutive in­ values of the Hessian Hpf; it is the number of in­ dependent directions along which f will decrease dices, then from p. (1) If a smooth function has only nondegenerate crit­ ical points, we call it a Morse function. The behav­ The explanation is simple: since in the CW com­ ior of the critical points of a Morse function can plex of M there are no two cells of consecutive be summarized in its Morse polynomial: dimensions, the boundary operator is automat­ ically zero. Therefore, the cellular chain complex :M.r(fl :=I tt\(p)' is its own homology. A Morse function f on M satisfying (1) is said to where the sum runs over all critical points p. A typical example of a Morse function is the be perfect. The height function on the torus above height function f of a torus standing vertically on is a perfect Morse function. a table top (Figure 1). Classical Morse theory deals only with functions all of whose critical points are nondegenerate; in particular, the critical points must all be isolated index 2 points. In many situations, however, the critical points form submanifolds of M. For example, if the index 1 torus now sits flat on the table, as a donut usually would, then the height function has the top and bot­ tom circles as critical manifolds {Figure 2). index 1 maximum C) index 1 index 0 Figure 1. Critical points of the height function. minimum C) index 0 The height function on this torus has four critical points of index 0, 1, 1, 2 respectively. Its Morse Figure 2. Critical manifolds of the height function.

560 NOTICES OF THE AMS VOLUME 53, NUMBER 5 One of Batt's first insights was to see how to ex­ tend Morse theory to this situation. In [9] he in­ troduced the notion of a nondegenerate critical manifold: a critical manifold N is nondegenerate if at any point p inN the Hessian off restricted to the normal space toN is nonsingular. The index A.(N) of the nondegenerate critical manifold N is then defined to be the number of negative eigen­ values of this normal Hessian; it represents the number of independent normal directions along which f is decreasing. For simplicity, assume that the normal bundles of the nondegenerate critical manifolds are all orientable. To form the Morse Figure 3. The diagram of SU(3). polynomial off, each critical manifold N is counted with its Poincare polynomial; thus, For G = SU(2) and Jvlc(fl : = L Pc(N)tA(N), _ { [ e2rrix summed over all critical manifolds. T- 0 With this definition of the Morse polynomial, Bott proved in [9] that if a smooth function f on a the Lie algebra t is ~ , the roots are ±2x, and the smooth manifold M has only nondegenerate crit­ adjoint representation of Gong= ~ 3 corresponds ical manifolds, then the Morse inequality again to rotations. The root plane is the origin. holds:

Jvlc(fl ~ Pc(M). -2 - ~ - 1 -.!. 1 3 2 2 0 2 1 2 2 · Lie Groups and Homogeneous Spaces In the 1950s Bott applied Morse theory with great Figure 4. The diagram of SU(2). success to the topology of Lie groups and homo­ geneous spaces. In [8] he showed how the diagram A point B in t is regular if its normalizer has min­ of a compact semisimple connected and simply con­ imal possible dimension, or equivalently, if its nor­ nected group G determines the integral homology malizer is T. It is well known that a point B in t is of both the loop space QG and the flag manifold regular if and only if it does not lie on any of the G IT, where T is a maximal torus. hyperplanes of the diagram. If B is regular, then the Indeed, Morse theory gives a beautiful CW cell stabilizer of B under the adjoint action of G is T, structure on G I T, up to homotopy equivalence. To and so the orbit through B is G I T. explain this, recall that the adjoint action of the Choose another regular point A in t , and define group G on its Lie algebra g restricts to an action the function f on Orbit(B) = GI Tto be the distance of the maximal torus Ton g. As a representation from A ; here the distance is measured with re­ of the torus T, the Lie algebra g decomposes into spect to the Killing form on g. Let {B;} be a ll the a direct sum of irreducible representations points in t obtained from B by reflecting about the root planes. Then Batt's theorem asserts that f is g = tffi L.Ea. a Morse function on G IT whose critical points are where tis the Lie algebra ofT and each Ea is a 2- precisely all the B;'s. Moreover, the index of a crit­ dimensional space on which T acts as a rotation ical point B; is twice the number of times that the e 2rria(x), corresponding to the root cx(x) on t. The di­ line segment from A to B; intersects the root planes. agram of G is the family of parallel hyperplanes This cell decomposition of Morse theory fits in in t where some root is integral. A hyperplane that with the more group-theoretic Bruhat decomposi­ is the zero set of a root is called a root plane. tion. For example, for the group G = SU(3) and max­ For G = SU(3) and T the set of diagonal matri­ imal torus ces in SU(3), the orbit G 1Tis the complex flag man­ ifold Fl?(1, 2, 3), consisting of all flags

2 V1 V2 3 , e TT ixz . ] I X ] + X z + X 3 = 0, X; E oc }. c c ( dime V; = i . e27TIX3 Batt's recipe gives 6 critical points of index the roots are ± (x1 - x2), ± (xl - x3), ± (x2 - x 3), and 0, 2, 2, 4, 4, 6 respectively on G I T (see Figure 5). By the diagram is the collection of lines in the plane the lacunary principle, the Morse function f is per­ x1 + x 2 + X3 = 0 in ~ 3 as in Figure 3. In this figure, fect. Hence, the flag manifold F-8( 1, 2, 3) has the ho­ the root planes are the thickened lines. motopy type of a CW complex with one 0-cell, two

MAY 2006 N OTICES OF THE AMS 561 2-cells, two 4-cells, and one 6-cell. Its Poincare p polynomial is therefore

Pc(Fe(1, 2, 3)) = 1 + 2t2 + 2t4 + t 6. q

index 2 index 4 B3 B2

B=B1 index 6 p'

Figure 6. Geodesics on a sphere. Now consider the space D.M of all smooth loops Bs • • B6 in M, that is, smooth functions J1 : S 1 - M. The index 2 index 4 critical points of the energy function on D.M are again the geodesics, but these are now closed ge­ Figure 5. The flag manifold F.f'(1, 2, 3). odesics. A closed geodesic is never isolated as a crit­ ical point, since for any rotation r : S 1 - S 1 of the circle, J1 o r: S 1 - M is still a geodesic. In this way, Index of a Closed Geodesic any closed geodesic gives rise to a circle of closed For two points p and q on a Riemannian manifold geodesics. When the Riemannian metric on M is M, the space D.p,q (M) of all paths from p to q on generic, the critical manifolds of the energy func­ M is not a finite-dimensional manifold. Nonethe­ tion on the loop space D. will all be circles. less, Morse theory applies to this situation also, with Morse had shown that the index of a geodesic a Morse function on the path space D. p,q given by is the number of negative eigenvalues of a Sturm the energy of a path: differential equation, a boundary-value problem of the form Ly = Ay, where Lis a self-adjoint sec­ b dJ1 dJ1 E(J1) = fa(dt'dt)dt. ond-order differential operator. For certain bound­ ary conditions, Morse had expressed the index in The first result of this infinite-dimensional Morse terms of conjugate points, but this procedure does theory asserts that the critical points of the energy not apply to closed geodesics, which correspond function are precisely the geodesics from p to q. to a Sturm problem with periodic boundary con­ Two points p and q on a geodesic are conjugate ditions. if keeping p and q fixed, one can vary the geodesic In [1 4] Batt found an algorithm to compute the from p to q through a family of geodesics. For ex­ index of a closed geodesic. He was then able to de­ ample, two antipodal points on ann-sphere are con­ termine the behavior of the index when the closed jugate points. The multiplicity of q as a conjugate geodesic is iterated. Batt's method is in fact ap­ point of p is the dimension of the family of geo­ plicable to all Sturm differential equations. And so desics from p to q . On then-sphere sn, the multi­ in his paper he also gave a geometric formulation plicity of the south pole as a conjugate point of the and new proofs of the Sturm-Morse separation, north pole is therefore n - 1. comparison, and oscillation theorems, all based If p and q are not conjugate along the geodesic, on the principle that the intersection number of two then the geodesic is nondegenerate as a critical cycles of complementary dimensions is zero if one point of the energy function on D.p,q· Its index, ac­ of the cycles is homologous to zero. cording to the Morse index theorem, is the num­ Homogeneous Vector Bundles ber of conjugate points from p to q counted with Let G be a connected complex semisimple Lie multiplicities. group, and P a parabolic subgroup. Then G is a prin­ On the n-sphere let p and p' be antipodal points cipal P-bundle over the homogeneous manifold and q f. p' . The geodesics from p to q are X = G/P . Any holomorphic representation pq,pp'q,pqp'pq,pp' qpp' q, ... , of index O,n - 1, 1:> : P - Aut(£) on a complex vector space E in­ 2(n - 1), 3( n - 1), . .. , respectively. By the Morse duces a holomorphic vector bundle [ over X: index theorem the energy function on the path space D. p,q (Sn) has one critical point each of index [ := G Xq, E := (G x E) / ~ , 0, n- 1, 2(n - 1), 3(n- 1), .... It then follows from where (gp, e) ~ (g, ¢>(p)e). Then [ is a holomorphic Morse theory that D.p,q (Sn) has the homotopy type vector bundle over X = G I P . A vector bundle over of a CW complex with one cell in each of the di­ X arising in this way is called a homogeneous vec­ mensions 0, n - 1, 2(n- 1), 3(n- 1), .... tor bundle. Let (')([) be the corresponding sheaf of

562 NOTICES OF THE AMS VOLUME 53, NUMBER 5 holomorphic sections. The homogeneous vector 2 _ U(2n) bundle [ inherits a left G-action from the left mul­ G(n, n) - U(n) x U(n) tiplication in G: By Morse theory, the loop space Q SU(2n) has the h.(g, e) = (hg, e) for h,g E G,e E E. homotopy type of a CW complex obtained from the Thus, all the cohomology groups Hq(X, <9([)) be­ Grassmannian G(n, 2n) by attaching cells of di­ come G-modules. mension ~ 2n + 2: In [15] Bott proved that if the representation <:P Q SU(2n) ~ G(n, 2n) u e" u ... , dime" ~ 2n + 2. is irreducible, the cohomology groups Hq(X, <9([)) all vanish except possibly in one single dimension. It follows that Moreover, in the nonvanishing dimension q, Hq(X, <9([)) is an irreducible representation of G whose highest weight is related to ¢. This theorem generalizes an earlier theorem of for n > k. Borel and Weil, who proved it for a positive line bun­ It is easily shown that dle. rrk(Q SU(2n)) = TTk +l (SU(2n)) = TTk+ r(U(2 n)). In Bott's paper one finds a precise way of de­ termining the nonvanishing dimension in terms Using the homotopy exact sequence of the fiber­ of the roots and weights of G and P. Thus, on the ing one hand, Bott's theorem gives a geometric real­ U(n) ~ U(2n)/U(n) ~ G(n, 2n), ization of induced representations, and on the other hand, it provides an extremely useful van­ one gets ishing criterion for the cohomology of homoge­ neous vector bundles. The Periodicity Theorem Homotopy groups are notoriously difficult to com­ Putting all this together, for n large relative to k, pute. For a simple space like the n-sphere, already, we get the higher homotopy groups exhibit no discernible TTk - l(U(n)) = rrdG(n, 2n)) = patterns. It was therefore a complete surprise in rrk(Q SU(2n)) = rrk+l (U(2n)). 1957, when Raoul Bott computed the stable ho­ motopy groups of the classical groups and found Thus, the stable homotopy group of the unitary a simple periodic pattern for each of the classical group is periodic of period 2: groups [24]. We first explain what is meant by the stable ho­ TTk - l (U) = TTk +l (U). motopy group. Consider the unitary group U(n + 1). It acts transitively on the unit sphere S2n+l in l(n+l, Applying the same method to the orthogonal with stabilizer U(n) at the point (1, 0, ... , 0). In this group and the symplectic group, Bott showed that way, the sphere S2n+l can be identified with the ho­ their stable homotopy groups are periodic of pe­ mogeneous space U(n + 1) I U(n), and there is a riod 8. fibering U(n + 1) ~ S 2n+l with fiber U(n). By the ho­ motopy exact sequence of a fibering, the following Clifford Algebras sequence is exact: The Clifford algebra Ck is the algebra over IP!. with · · · ~ TTk+l(S2n+l) ~ rrdU(n)) k generators e1 , ... , ek and relations

~ rrk( U(n + 1)) ~ rrk(S 2n+I) ~ .... ef = - 1 fori=l, ... ,k, Since TTk(S m) = 0 for m > k, it follows immediately e;eJ = -eJe; for all i !- j. that as n goes to infinity (in fact for all n > k/2), the kth homotopy group of the unitary group sta­ The first few Clifford algebras are easy to de­ bilizes: scribe TTk(U(n)) = TTk(U(n + 1)) = rrdU(n + 2)) = .... Co= IP!., C1 = 1(, C2 = IHl = {quaternions}. This common value is called the kth stable homo­ topy group of the unitary group, denoted rrk(U). If IF is a field, denote by IF(n) the algebra of all n x n In the original proof of the periodicity theorem matrices with entries in IF . We call IF( n) a full ma­ [24], Bott showed that in the loop space of the spe­ trix algebra. It turns out that the Clifford algebras cial unitary group SU(2n), the manifold of minimal are all full matrix algebras or the direct sums of geodesics is the complex Grassmannian two full matrix algebras:

MAY 2006 NOTICES OF THE AMS 563 k ck k ck k ck In this way, the 8-fold periodicity of the Clifford algebras reappears as the 8-fold periodicity of the 0 IRI. 8 IR/. (16) 16 IRI. (2 8) stable homotopy groups of the orthogonal group. 1 «:: 9 «:: (16) 17 «::(2 8) The Index Theorem for Homogeneous 2 IHl 10 IHI (16) 18 Differential Operators 3 IHIEBIHI 11 IHI(l6) EB IHI(16) The 1960s was a time of great ferment in topology 4 IHI(2) 12 IHI(32) and one of its crowning glories was the Atiyah­ 5 «::(4) 13 «::(64) Singer index theorem. Independently of Atiyah and 6 IRI. (8) 14 IR/. (128) Singer's work, Bott's paper [37] on homogeneous 7 IRI. (8) EB IR/. (8) 15 IR/. (128) EB IR/.(128) differential operators analyzes an interesting ex­ ample where the analytical difficulties can be avoided by representation theory. This table exhibits clearly a periodic pattern of Suppose G is a compact connected Lie group and period 8, except for the dimension increase after H a closed connected subgroup. As in our earlier each period. The 8-fold periodicity of the Clifford discussion of homogeneous vector bundles, a rep­ algebras, long known to algebraists, is reminiscent resentation p of H gives rise to a vector bundle of the 8-fold periodicity of the stable homotopy G X p H over the homogeneous space X= GIH. groups of the orthogonal group. Now suppose E and Fare two vector bundles over In the early 1960s Michael Atiyah, Raoul Bott, and G I H arising from representations of H. Since G acts Arnold Shapiro found an explanation for this tan­ on the left on both E and F, it also acts on their talizing connection. The link is provided by a class spaces of sections, [(E) and [(F). We say that a dif­ of linear differential operators called the Dirac op­ ferential operator D : f(E) ~ [(F) is homogeneous erators. The link between differential equations if it commutes with the actions of G on [(E) and and homotopy groups first came about as a result [(F). If D is elliptic, then its index of the realization that ellipticity of a differential op­ index(D) = dimker D - dimcoker D erator can be defined in terms of the symbol of the differential operator. is defined. Suppose we can find k real matrices e1, ... , ek of Atiyah and Singer had given a formula for the size n x n satisfying index of an elliptic operator on a manifold in terms ef=-1, e;ej =-eje; fori =/= ). of the topological data of the situation: the char­ acteristic classes of E, F, the tangent bundle of the This corresponds to a real representation of the Clif­ base manifold, and the symbol of the operator. In ford algebra Ck. The associated Dirac operator [3 7] Raoul Bott verified the Atiyah-Singer index D = Dk,n is the linear first-order differential oper­ theorem for a homogeneous operator by intro­ ator ducing a refined index, which is not a number, but a character of the group G. The usual index may be obtained from the refined index by evaluating at the identity. A similar theorem in the infinite­ where I is the n x n identity matrix. Such a differ­ dimensional case has recently been proven in the ential operator on [RI.k +l has a symbol o- (~) ob­ 0 context of physics-inspired mathematics. tained by replacing aI ox; by a variable ~;: Nevanlinna Theory and the Bott-Chern Classes o-v(~) =no+ el~l + ... + ek~k· Nevanlinna theory deals with the following type of questions: Let f : «:: ~ «::P 1 be a holomorphic map. The Dirac operator D is readily shown to be ellip­ Given a in «::P 1 , what is the inverse image f - 1 (a)? tic; this means its symbol o-0 (~) is nonsingular for Since «:: is noncompact, there may be infinitely all~ =I= 0 in [RI. k+l. Therefore, when restricted to the many points in the pre-image f - 1(a). Sometimes unit sphere in [RI.k +l, the symbol of the Dirac oper­ 1(a) will be empty, meaning that f misses the ator gives a map r- point a in «:: P 1 . o-v(~) : Sk ~ GL(n, IRI. ). The exponential map exp : «:: ~ «:: P 1 misses ex­ actly two points, 0 and oo , in «::P 1. According to a Since GL(n, IRI.) has the homotopy type of O(n), this classical theorem of Picard, a nonconstant holo­ map given by the symbol of the Dirac operator de­ morphic map f : «:: ~ «::P 1 cannot miss more than fines an element of the homotopy group two points. rrk(GL(n, IRI.)) = rrk(O(n)). Nevanlinna theory refines Picard's theorem in a The paper [33] shows that the minimal-dimen­ beautiful way. To each a E «::P 1 , it attaches a real sional representations of the Clifford algebras give number b(a) between 0 and 1 inclusive, the defi­ rise to Dirac operators whose symbols generate the ciency index of a. The deficiency index is a nor­ stable homotopy groups of the orthogonal group. malized way of counting the number of points in

564 NOTICES OF THE AMS VOLUME 53, NUMBER 5 the inverse image. If f -1 (a) is empty, then the de­ equal to the number of zeros of a vector field on ficiency index is 1. the manifold, each counted with its index. In [41] In this context the first main theorem of Nevan­ and [4 3], Bott generalized the Hopf index theorem linna theory says that a nonconstant holomorphic to other characteristic numbers such as the Pon­ map f : ( - (P1 has deficiency index 0 almost tryagin numbers of a real manifold and the Chern everywhere. The second main theorem yields the numbers of a complex manifold. stronger inequality: We will describe Bott's formula only for Chern numbers. Let M be a compact complex manifold L 6(a) :o: 2. of dimension n, and c1 (M), ... , Cn(M) the Chern a E(P l classes of the tangent bundle of M. The Chern numbers of M are the integrals JM 1>(c1 (M), Ahlfors generalized these two theorems to holo­ ... , Cn(M)), as 1> ranges over all weighted homo­ morphic maps with values in a complex projective geneous polynomials of degree n. Like the Hopf space ( pn. index theorem, Bon's formula computes a Chern In [38] Bott and Chern souped up Nevanlinna's number in terms of the zeros of a vector field X hard analysis to give a more conceptual proof of on M, but the vector field must be holomorphic and the first main theorem. the counting of the zeros is a little more subtle. A by-product of Bott and Chern's excursion in For any vector field Yand any c oo function f on Nevanlinna theory is the notion of a refined Chern M, the Lie derivative Lx satisfies: class, now called the Bott-Chern class, that has since been transformed into a powerful tool in Lx(fY) = (Xf)Y + f Lx Y. Arakelov geometry and other aspects of modern It follows that at a zero p of X, number theory. Briefly, the Bott-Chern classes arise as follows. (LxfY)p = f(p)(Lx Y)p. On a complex manifold M the exterior derivative Thus, at p, d decomposes into a sum d = i3 + and the the Lie derivative Lx induces an endo­ a, morphism smooth k-forms decompose into a direct sum of (p, q)-forms. Let AP.P be the space of smooth (p, p)­ Lp: TpM- TpM forms on M. Then the operator oa makes ffiAP.P into a differential complex. Thus, the cohomology of the tangent space of M at p. The zero p is said H * {AP·P, oJ} is defined. to be nondegenerate if Lr is nonsingular. A Hermitian structure on a holomorphic rank n For any endomorphism A of a vector space V, vector bundle E on M determines a unique con­ we define the numbers c;(A) to be the coefficients nection and hence a unique curvature tensor. If K of its characteristic polynomial: K' and are the curvature forms determined by two det(I + tA) = L c;(AJti. Hermitian structures onE and 1> is a GL(n, ()-in­ variant polynomial on gf(n, (),then it is well known Bott's Chern number formula is as follows. Let that 1>(K) and 1>(K') are global closed forms on M M be a compact complex manifold of complex di­ whose difference is exact: mension n and X a holomorphic vector field hav­ ing only isolated nondegenerate zeros on M. For 1>(K) - 1>(K') = dOi any weighted homogeneous polynomial 1>(xJ, ... ,Xn), degx; = 2i, for a differential form Oi on M. This allows one to define the characteristic classes of E as cohomol­ ogy classes in H * (M). (2) L4>(c1 (M), ... , Cn(M)) = In the holomorphic case, 1>(K) and 1>(K') are (p, p)-forms closed under oCi. Bott and Chern found L 4>(c1 (Lp), . .. , Cn(Lp)) that in fact, P Cn(Lp) ' 1>(K) - 1>(K') = oCi(3 summed over all the zeros of the vector field. Note for some (p - 1, p - 1)-form (3. For a holomorphic that by the definition of a nondegenerate zero, vector bundle E, the Bott-Chern class of E associ­ Cn(Lp), which is detLp. is nonzero. ated to an invariant polynomial1> is the cohomol­ In Bott's formula, if the polynomial1> does not ogy class of 1>(£), not in the usual cohomology, but have degree 2n, then the left-hand side of (2) is zero, in the cohomology of the complex {AP.P, oCi}. and the formula gives an identity among the num­ bers C;(Lp). For the polynomial 1>(x 1 , ... , Xn) = Xn, Characteristic Numbers and the Bott Bott's formula recovers the Hopf index theorem: Residue 'Cn(Lp) According to the celebrated Hopf index theorem, I Cn(M) = L -(L) = # zeros of X. the Euler characteristic of a smooth manifold is M P Cn p

MAY 2006 NOTICES OF THE AMS 565 Bott's formula (2) is reminiscent of Cauchy's residue formula and so the right-hand side of (2) M may be viewed as a residue of¢ at p. Graph({) In [43] Bott generalized his Chern number for­ mula (2), which assumes isolated zeros, to holo­ morphic vector fields with higher-dimensional zero sets and to bundles other than the tangent bundle (a vector field is a section of the tangent bundle). The Atiyah-Bott Fixed Point Theorem A continuous map of a finite polyhedron, f : P ~ P, M has a Lefschetz number: Figure 7. A transversal map f.

where f * is the induced homomorphism in coho­ D = L A"'(x)~, x E U, mology and tr denotes the trace. According to the IO< Ism uX"'

Lefschetz fixed point theorem, if the Lefschetz 1 0 ( 0 ) "' ( 0 ) Ochart (U, x1 , ... , Xn) and triv­ for all i. Such a collection T = {Td induces maps ializations for E and F relative to which D can be in cohomology T/ : Hi ~ Hi. The Lefschetz number written in the form of Tis then defined to be

566 NOTICES OF THE AMS VOLUME 53, NUMBER 5 L(T) = .2.:< - lhrT;*. By the Frobenius theorem, often proven in a first-year graduate course, a subbundle E of the tan­ A map f : M ~ M induces a natural map gent bundle TM is integrable if and only if its space of sections [(E) is closed under the Lie rr : f(El ~ 1 El nr- bracket. The Pontryagin ring by composition: fr(s) = s o f. There is no natural way Pont(V) of a vector bundle to induce a map of sections: [(E) ~ [(E). However, V over M is defined to be the subring of the co­ if there is a bundle map cf> : f - 1 E ~ E, then the homology ring H * (M) generated by the Pontryagin composite classes of the bundle V. In [51] Bott found an ob­ r . struction to the integrability of E in terms of the [(E) .!. f(f- 1 E) i'. [(E) Pontryagin ring of the quotient bundle Q : = T M 1E. is an endomorphism of [(E). Any bundle map More precisely, if a subbundle E of the tangent cf> : f - 1 E ~ E is called a lifting of f to E. At each bundle TM is integrable, then the Pontryagin ring point x E M, a lifting cf> is nothing other than a lin­ Pont(Q) vanishes in dimensions greater than twice ear map cf> x : Ef(x ) - Ex. the rank of Q. In the case of the de Rham complex, a map What is so striking about this theorem is· not only f : M - M induces a linear map fx* : T/(x)M ~ Tx* M the simplicity of the statement, but also the sim­ and hence a linear map plicity of its proof. It spawned tremendous devel­ opments in foliation theory in the 1970s, as re­ Aqfx*: i\.qT/(x)M ~ i\.Tx* M, counted in [C] and [Hl]. which is the lifting that finally defines the pullback The Cohomology of Vector Fields on a Manifold of differential forms f*: f(AqT* M)- [(A.qT* M). For a finite-dimensional Lie algebra L, let Aq(L) be the space of alternating q-forms on L. Taking cues Theorem 1. (Atiyah-Bott fixed point theorem). from the Lie algebra of left-invariant vector fields Given an elliptic complex (3) on a compact mani­ on a Lie group, one defines the differential fold M, suppose f: M ~ M has a lifting cf>i : f -1 Ei - Ei for each i such that the induced d: Aq(L) - Aq+1(L) maps Ti : f(Ei) - [(Ei) give an endomorphism of by the elliptic complex. Then the Lefschetz number of Tis given by (4) (dw)(X0 , ... ,Xq) = L(T) = L 2:( -l)i tr cf>i. x . I(-l)i+Jw([Xi,X1l.Xo, ... • xi .... ,x1, ... ,Xq). f( x ) ~ x I det(l - f * .x) I i

As evidence of its centrality, the Atiyah-Bott As usual, the hat ' over Xi means that Xi is to be fixed point theorem has an astonishing range of ap­ omitted. This makes A * (L) into a differential com­ plicability. plex, whose cohomology is by definition the co­ Here is an easily stated corollary in algebraic homology of the Lie algebra L. geometry: any holomorphic map of a rational al­ When L is the infinite-dimensional Lie algebra gebraic manifold to itself has a fixed point. L(M) of vector fields on a manifold M, the formula Specializing the Atiyah-Bott fixed point theo­ (4) still makes sense, but the space of all alternat­ rem to the de Rham complex, one recovers the ing forms A * (L(M)) is too large for its cohomology classical Lefschetz fixed point theorem. When they to be computable. Gelfand and Fuks proposed applied the theorem to other geometrically inter­ putting a topology, the C"" topology, on L(M), and esting elliptic complexes, Atiyah and Bott obtained computing instead the cohomology of the contin­ new fixed point theorems, such as a holomorphic uous alternating forms on L(M). The Gelfand-Fuks Lefschetz fixed point theorem in the complex an­ cohomology of M is the cohomology of the com­ alytic case and a signature formula in the Rie­ plex {A ~ (L(M)), d} of continuous forms. They mannian case. In the homogeneous case, the fixed hoped to find in this way new invariants of a man­ point theorem implies the Weyl character formula. ifold. As an example, they computed the Gelfand­ Obstruction to Integrability Fuks cohomology of a circle. A subbundle E of the tangent bundle T M of a man­ It is not clear from the definition that the ifold M assigns to each point x of the manifold a Gelfand-Fuks cohomology is a homotopy invariant. subspace Ex of the tangent space TxM. An inte­ In [71] Bott and Segal proved that the Gelfand­ grable manifold of the subbundle E is a submani­ Fuks cohomology of a manifold M is the singular fold N of M whose tangent space TxN at each point cohomology of a space functorially constructed x inN is Ex. The subbundle E is said to be integrable from M . Haefliger [H] and Trauber gave a very dif­ if for each point x in M, there is an integrable man­ ferent proof of this same result. The homotopy in­ ifold of E passing through x. variance of the Gelfand-Fuks cohomology follows.

MAY 2006 NOTICES OF THE AMS 567 At the same time it also showed that the Gelfand­ on a symplectic manifold (M, w) of dimension of Fuks cohomology produces no new invariants. 2n: Localization in Equivariant Cohomology 1) If a torus action on M preserves the sym­ Just as singular cohomology is a functor from the plectic form and has a moment map f, then the category of topological spaces to the category of push-forward f*(wn) of the symplectic volume rings, so when a group G acts on a space M, one under the moment map is piecewise polyno­ seeks a functor that would incorporate both the mial. topology of the space and the action of the group. 2) Under the same hypotheses, the stationary The naive construction of taking the cohomol­ phase approximation for the integral ogy of the quotient space MIG is unsatisfactory be­ e- itf wn cause for a nonfree action, the topology of the I M n! quotient can be quite bad. A solution is to find a contractible space EG on which G acts freely, for is exact. In case the vector field on the manifold is gen­ then EG x M will have the same homotopy type as erated by a circle action, the localization theorem M and the group G will act freely on E G x M via specializes to Bott's Chern number formulas [41] the diagonal action. It is well known that such a of the 1960s, thus providing an alternative expla­ space is the total space of the universal G-bundle nation for the Chern number formulas. EG ~ BG, whose base space is the classifying space of G. The homotopy theorists have defined the Yang-Mills Equations over Riemann Surfaces homotopy quotient Me of M by G to be the quo­ In algebraic geometry it is well known that for any tient space (EG x M)/G, and the equivariant co­ degree d the set of isomorphism classes of holo­ homology H(; (M) to be the ordinary cohomology of morphic line bundles of degree d over a Riemann its homotopy quotient Me. surface M of genus g forms a smooth projective The equivariant cohomology of the simplest G­ variety which is topologically a torus of dimen­ space, a point, is already quite interesting, for it is sion g. This space is called the moduli space of holo­ the ordinary cohomology of the classifying space morphic line bundles of degree dover M. of G: For holomorphic vector bundles of rank k ~ 2, the situation is far more complicated. First, in order H(;(pt) = H*((EG x pt)/G) = to have an algebraic structure on the moduli space, H*(EG/G) = H * (BG). it is necessary to discard the so-called "unstable" bundles in the sense of Mumford. It is then known Since equivariant cohomology is a functor of that for k and d relatively prime, the isomorphism G-spaces, the constant map M ~ pt induces a ho­ classes of the remaining bundles, called "semi­ momorphism H(;(pt) ~ H(;(M). Thus, the equi­ stable bundles", form a smooth projective variety variant cohomology H(;(M) has the structure of a N(k, d). module over H * (BG). In [N] Newstead computed the Poincare poly­ Characteristic classes of vector bundles over M nomial of N(2, 1). Apart from this, the topology of extend to equivariant characteristic classes of equi­ N(k, d) remained mysterious. variant vector bundles. In [81] Atiyah and Bott introduced the new and When M is a manifold, there is a push-forward powerful method of equivariant Morse theory to map rr!: : H(;(M) ~ H(;(pt), akin to integration study the topology of these moduli spaces. along the fiber. Let P = M x U(n) be the trivial principal U(n)­ Suppose a torus T acts on a compact manifold bundle over the Riemann surface M, .Jt = .Jt(P) the M with fixed point set F, and ¢ E H;(M) is an affine space of connections on P, and (j = (j(P) equivariantly closed class. Let P be the connected the gauge group, i.e., the group of automorphisms components ofF and let lp : P ~ M be the inclu­ of P that cover the identity. Then the gauge group sion map, Vp the normal bundle of P inM, and e(vp) (j(P) acts on the space .Jt(P) of connections and the equivariant Euler class of Vp. In [82] Atiyah there is a Yang-Mills functional L on .Jt(P) invari­ and Bott proved a localization theorem for the ant under the action of the gauge group. equivariant cohomology H;(M) with real coeffi- Equivariant Morse theory harks back to Bott's ex­ cients: tension of classical Morse theory to nondegener­ M ri, " P ( [~ 1J ) ate critical manifolds three decades earlier. The key TT* '+' = 7 TT* e(vp) . result of Atiyah and Bott is that the Yang-Mills functional L is a perfect equivariant Morse func­ It should be noted that Berline and Vergne [BV] in­ tion on .Jt(P). This means the equivariant Poincare dependently proved the same theorem at about the series of .Jt(P) is equal to the equivariant Morse se­ same time. ries of I: This localization theorem has as consequences the following results of Duistermaat and Heckman (5)

568 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Once one umavels the definition, the left-hand the rigidity theorem as a consequence of the Atiyah­ side of (5) is simply the Poincare series of the clas­ Bott fixed point theorem. sifying space of (j(P), which is computable from ho­ The idea of [91) is as follows. To decompose a motopy considerations. The right-hand side of (5) representation, one needs to know only its trace, is the sum of contributions from all the critical sets since the trace determines the representation. By of L. By the work of Narasimhan and Seshadri, the assumption, the action of G on the elliptic complex minimum of L is precisely the moduli space N(k, d). D : r(E) ~ [(F) commutes with D. This means each It contributes its Poincare polynomial to the equi­ element g in G is an endomorphism of the elliptic variant Morse series of L. By an inductive proce­ complex. It therefore irlduces an endomorphism g * dure, Atiyah and Bott were able to compute the con­ in the cohomology of the complex. But H 0 = ker D tributions of all the higher critical sets. They then and H 1 = coker D. The alternating sum of the trace solved (5) for the Poincare polynomial of N(k, d). of g * in cohomology is precisely the left-hand side Witten's Rigidity Theorem of the Atiyah-Bott fixed point theorem. It then Let E and F be vector bundles over a compact man­ stands to reason that the fixed poirlt theorem could be used to decompose the index of D into irre­ ifold M. If a differential operator D : [(E) ~ [(F) is elliptic, then ker D and coker D are finite­ ducible representations. dimensional vector spaces and we can define the Papers of Raoul Bott Discussed in this Article index of D to be the virtual vector space [1] (with R. ]. Duffin) Impedance synthesis without use D = ker D - coker D. of transformers,]. Appl. Phys. 20 (1949), 816. Now suppose a Lie group G acts on M, and E and [8] On torsion in Lie groups, Proc. NAS 40 (1954), 586-588. [9] Nondegenerate critical manifolds, Ann. of Math. 60 Fare G-equivariant vector bundles over M. Then (1954), 248-261. G acts on [(E) by [1 2] (with H. Samelson) The cohomology ring of G I T, Proc. (g.s)(x) = g.(s(g- 1 .x)), NAS 41 (1955), 490-493. [14] On the iteration of closed geodesics and the Sturm for g E G, s E [(E), x E M. The G -action is said to intersection theory, Comm. Pure Appl. Math. IX (1956), preserve the differential operator D if the actions 171-206. of G on [(E) and [(F) commute with D. In this case [1 5] Homogeneous vector bundles, Ann. of Math. 66 ker D and coker D are representations of G, and (1957), 933-935. soD is a virtual representation of G. We say that [24] The stable homotopy of the classical groups, Ann. the operator D is rigid if its index is a multiple of of Math. 70 (1959), 313-337. the trivial representation of dimension 1. The rigid­ [33] (with M. F. Atiyah and A. Shapiro) Clifford modules, ity of D means that any nontrivial irreducible rep­ Topology 3 (1965), 3- 38. resentation of G in ker D occurs in coker D with [37] The index theorem for homogeneous differential the same multiplicity and vice versa. operators, in: Differential and Combinatorial Topology: If the multiple m is positive, then m.1 = A Symposium in Honor of Marston Morse, Princeton, 1 EB • • • EB 1 is the trivial representation of dimen­ (1964), 167-186. sion of m. If m is negative, the m.1 is a virtual rep­ [38] (with S. Chern) Hermitian vector bundles and the resentation and the rigidity of D implies that the equidistribution of the zeroes of their holomorphic sec­ trivial representation 1 occurs more often in tions, Acta Mathematica 114 (1964), 71-112. coker D than in ker D. [41] Vector fields and characteristic numbers, Mich. Math. For a circle action on a compact oriented Rie­ ]. 14 (1967), 231-244. mannian manifold, it is well known that the Hodge [42] (with M. F. Atiyah) A Lefschetz fixed point formula operator d + d * : n even ~ ooctd and the signature for elliptic complexes: I, Ann. of Math. 86 (1967), operator ds = d + d* : Q + ~ n- are both rigid. 374-407. An oriented Riemannian manifold of dimension [43] A residue formula for holomorphic vector fields,]. n has an atlas whose transition functions take val­ Differential Geom. 1 (1967), 311-330. ues in SO(n). The manifold is called a spin mani­ [44] (with M. F. Atiyah) A Lefschetz fixed point formula fold if it is possible to lift the transition functions for elliptic complexes: II, Ann. of Math. 88 (1968), to the double cover Spin(n) of SO(n). 451-491. Inspired by physics, Witten discovered infinitely [51] On a topological obstruction to integrability, in: many rigid elliptic operators on a compact spin Global Analysis, Proceedings of Symposia in Pure Math. manifold with a circle action. They are typically of XVI (1970), 127-131. the form ds 0 R, where ds is the signature opera­ [71] (with G. Segal) The cohomology of the vector fields tor and R is some combination of the exterior and on a manifold, Topology 16 (1977), 285-298. the symmetric powers of the tangent bundle. In [91) [81] (with M. F. Atiyah) The Yang-Mills equations over Rie­ Bott and Taubes found a proof, more accessible to mann surfaces, Ph il. Trans. R. Soc. Land. A 308 (1982), mathematicians, of Witten's results, by recasting 524-615.

MAY 2006 NOTICES OF THE AMS 569 I I

CAMBRIDGE [82) (with M. F. Atiyah) The moment map and equivari­ I I ant cohomology, Topology 23 (1984), 1-28. I [91) (with C. Taubes) On the rigidity theorem of Witten, The Latest Research in ]. of the Amer. Math. Soc. 2 (1989), 137-186. References [BV) N. BERLINE and M. VERGNE, Classes caracteristiques equi­ Applied Mathematics variantes, Formule de localisation en cohomologie equivariante, C. R. Acad. Sci. Paris Ser. I Math. 295 ... from Cambridge (1982), No.9, 539-541. [B1) R. Borr, Some recollections from 30 years ago, Con­ structive FREE online samples from our prestigious list Approaches to Mathematical Models, Acade­ mic Press, 1979, 33- 39. Sample the latest research from Cambridge journals. [B2) __ , An equivariant setting of the Morse theory, High lights for 2006 include the following titles L'Enseignement mathematique XXVI (1980), 68-75. [B 3) __ , The Dioszeger Years (1923-1929), Raoul Batt: Journal of Fluid Mechanics­ Collected Papers, Vol. 1, Birkhauser, Boston, 1994, digital archive coming soon! 11-26. The leading international journal in the [B4) __ , Autobiographical sketch, Raoul Batt: Collected field JFM publishes authoritative articles Papers, Vol. 1, Birkhauser, Boston, 1994, 3-9. in all aspects of the mechanics of fluids. [B5) __ , Collected Papers, Vol. 1-4, Birkhauser, Boston, We are delighted to announce the 1994, 1995. imminent launch of an online digital [C) L. CoNLON, Raoul Bott, foliations, and characteristic archive. Email [email protected] to classes: An appreciation, Raoul Batt: Collected Papers, register an interest! Vol. 3, Birkhauser, Boston, 1995, xxiv-xxvi. [H) A. HAEFLIGER, Sur la cohomologie de Gel'fand-Fuks, Visit: Ann. Scient. Ec. Norm. Sup. 9 (1976), 503-532 . www.journals.cambridge.org/jid_FLM [Hl) __ , Raoul Bott and foliation theory in the 1970s, for a FREE sample Raoul Batt: Collected Papers, Vol. 3, Birkhauser, Boston, 1995, xxvii-xxxi. [N) P. E. NEWSTEAD, Characteristic classes of stable Probability in the Engineering . PROBABILITY bundles in the over and Informational Sciences hginu ri~g an algebraic curve, Trans. Am. Math. Soc. 169, 'lnhmn"' atioul 337- 345. is the primary focus of the journal is on Stiencu stochastic modelling in the physical and engineering sciences. Visit: www.journals.cambridge.org/jid_PES for a FREE sample

European Journal of Applied Mathematics -now incorporating 'Surveys on Mathematics for Industry' ! EJAM is a journal for original work in areas of mathematics in which an understanding of the application requires the use of new and interesting mathematical ideas. Visit: www.journals.cambridge.org/jid_EJM for a FREE sample

Visit: www.journals.cambridge.org/subjects for our full list To subscribe please contact: Cambridge University Press, The Edinburgh Building, Cambridge CB2 2RU, UK, Phone +44 (0)1223 326070, Fax +44 (0)1223 325150, [email protected]

www.cambridge.org a CAMBRIDGE ~ UNIVERSITY PRESS

570 NOTICES OF THE AMS VOLUME 53, NUMBER 5 -.• To mak e an animated tube plot Animated Tube Plot 1 Type an expression in one or :,;;,two ~~J ~~~~~~~~~~i~~~~~~~~~~~~~=~~ 2 With the insertion point in the 3 Open the Plot Properties dialog the same variables The next animation shows a knot i Plot 3D Animated .o. Tube

Scientific Word

Scientific Word® offers the same features as Scientific WorkPlace, without the computer algebra ______... system. W H A T I S Percolation? Harry Kesten

Percolation is a simple probabilistic model which v E z Z as the collection of points connected to v exhibits a phase transition (as we explain below). by an open path. The clusters C(v) are the maxi­ The simplest version takes place on Z2 , which we mal connected components of the collection of view as a graph with edges between neighboring open edges of zZ, and e(p) is the probability that vertices. All edges of Z2 are, independently of each C(O) is infinite. If p < Pc, then e(p) = 0 by defini­ other, chosen to be open with probability p and tion, so that C(O) is finite with probability 1. It is closed with probability 1 - p. A basic question in not hard to see that in this case all open clusters this model is "What is the probability that there ex­ are finite. If p > Pc. then e(p) > 0 and there is a. ists an open path, i.e., a path all of whose edges are strictly positive probability that C(O) is infinite. open, frcim the origin to the exterior of the square An application of Kolmogorov's zero-one law shows Sn := [ -n, n]2 ?" This question was raised by Broad­ that there is then with probability 1 some infinite bent in 1954 at a symposium on Monte Carlo meth­ cluster. In fact, it turns out that there is a unique ods. It was then taken up by Broadbent and Ham­ infinite cluster. Thus, the global behavior of the sys­ mersley, who regarded percolation as a model for tem is quite different for p < Pc and for p > Pc· a random medium. They interpreted the edges of Such a sharp transition in global behavior of a sys­ z Z as channels through which fluid or gas could tem at some parameter value is called a phase tran­ flow if the channel was wide enough (an open edge) sition or a critical phenomenon by statistical physi­ and not if the channel was too narrow (a closed cists, and the parameter value at which the edge). It was assumed that the fluid would move transition takes place is called a critical value. wherever it could go, so that there is no random­ There is an extensive physics literature on such phe­ ness in the behavior of the fluid, but all random­ nomena. Broadbent and Hammersley proved that ness in this model is associated with the medium. 0 < Pc < 1 for percolation on Z2 , so that there is We shall use 0 to denote the origin. A limit as indeed a nontrivial phase transition. Much of ~he n - oo of the question raised above is "What is the interest in percolation comes from the hope that probability that there exists an open path from 0 one will be better able to analyze the behavior of to infinity?" This probability is called the percola­ various functions near the critical point for the tion probability and denoted by e(p). Clearly simple model of percolation, with all its b'uilt-in in­ e(O) = 0 and e(l) = 1, since there are no open edges dependence properties, than for other, ~ore com­ at all when p = 0 and all edges are open when plicated models for disordered media. Indeed, per­ p = 1 . It is also intuitively clear that the function colation is the simplest one in the family of the p ~ e(p) is nondecreasing. Thus the graph of e as so-called random cluster or Fortuin-Kasteleyn mod­ a function of p should have the form indicated in els, which also includes the celebrated Ising model Figure 1, and one can define the critical probabil­ for magnetism. The studies of percolation and ran­ = e(p) = . ityby Pc sup{p: 0} dom cluster models have influenced each other. Why is this model interesting? In order to answer Percolation can obviously be generalized to per­ we define the (open) cluster C(v) of the vertex this colation on any graph (j, even to (partially) directed Harry Kesten is emeritus professor of mathematics at Cor­ graphs. One can also consider the model in which nell University. His e-mail address is kesten@math. the vertices are independently open or closed, but · ·cornell.edu. all edges are as s~med open. This version is called

572 NOTICES OF THE AMS VOLUME 53, NUMBER 5 site percolation, in contrast to the version we con­ Loewner Evolutions (SLE) and e (p) (1,1) sidered so far, and which is called bond percolation. on Smirnov's beautiful proof Initially research concentrated on finding the pre­ of the existence and confor- cise value of Pc for various graphs. This has not mal invariance properties of been very successful; one knows Pc only for a few certain crossing probabilities. planar lattices (e.g., Pc = 1 I 2 for bond percolation Roughly speaking this says the on Z2 and for site percolation on the triangular lat­ following: Let 'D be a "nice" tice). The value of Pc depends strongly on geo­ domain in IR2 and let A and B metric properties of (j. Attention has therefore be two arcs in the boundary. shifted to questions about the distribution of the For.\> 0, let P;>..('D,A,B) be 0 p number of vertices in C(O) and geometric proper­ the probability for p = Pc that ties of the open clusters when p is close to Pc· It there exists an open path of is believed that a number of these properties are site percolation on the trian- Figure 1. Graph of e. Many aspects universal, that is, they depend only on the dimen­ gular lattice in .\'D from .\A to of this graph are still conjectural. sion of (j, and not on details of its structure. .\B. In fact it is neater to take In particular, one wants to study the behavior P;>.. ('D, A, B) as the probability at Pc of an open con­ of various functions as p approaches Pc. or as nection in 'D from A to B on (1 1,\) times the tri­ some other parameter tends to infinity, while p is angular lattice. Conformal invariance says that kept at Pc· It is believed that many functions obey Q('D, A, B) := lim;>.. - oo P;>..('D, A, B) exists and that so-called power laws. For instance, it is believed that Q('D, A, B) = Q ( ('D), (A), (B)) for every confor­ the expected number of vertices in C(O), denoted mal map from 'D onto ('D). Further crucial in­ by x(p), behaves like (Pc- p)-Y asp I Pc. in the gredients are characterizations by Lawler and sense that -log x(p)/log(pc - p) - ;y for a suit­ Werner of some SLE process on a domain by means able constant ;y. Similarly one believes that e(p) be­ of properties of its evolution before it hits the haves like (p - Pc)/3 for some f3 asp I Pc. or that boundary. Conformal invariance had earlier been the probability that there is an open path from 0 conjectured by physicists, and Cardy had given a to the exterior of Sn for p = Pc behaves like n- 1/P formula for Q('D, A, B). Smirnov's work gives a rig­ for some p. Even though such power laws have been orous proof of Cardy's formula for percolation on proven only for site percolation on the triangular the triangular lattice. Further work (see Camia and lattice or on high-dimensional lattices, it is be­ Newman (2005)) also has led to a description of the lieved that the exponents {3, ;y, p, etc. (usually called limit (in a suitable sense) as .\ - oo of the full pat­ critical exponents), exist, and in accordance with the tern of the random configuration of open paths at universality hypothesis mentioned above depend criticality, i.e., for p = Pc· Since their discovery, SLE only on the dimension of (j. For instance, bond and processes have led to exciting new probability the­ site percolation on Z2 or on the triangular lattice ory in their own right, for instance, to power laws should all have the same exponents. Physicists in­ for the intersection probabilities of several Brown­ vented the renormalization group to explain and/ or ian motions (see Lawler (2005)). prove such power laws and universality, but this So far conformal invariance results have been has not been made mathematically rigorous for per­ achieved only for site percolation on the triangu­ colation. lar lattice. It is perhaps the principal open problem zd for large d behaves in many respects like a of the subject to prove conformal invariance for per­ regular tree, and for percolation on a regular tree colation on other two-dimensional lattices. An­ one can easily prove power laws and compute the other related major problem is to establish power relevant critical exponents. For bond percolation laws and universality for percolation on d-dimensional on zd with d ~ 19 Hara and Slade succeeded in lattices with 2 .:-:; d .:-:; 6. Finally, an proving power laws and in showing that the expo­ unsolved problem of fifteen years' standing is whether nents agree with those for a regular tree. They there is an infinite open cluster for criti­ cal percolation have even shown that their theory applies down to on zd' d ~ 3. I thank G d > 6 when one adds edges to zd between any two eoffrey Grimmett for several helpful suggestions. sites within distance Lo of each other for some Lo = Lo(d) . Further Reading Due to this theory we have a reasonable under­ [1] F EDERICO (AMIA and C HARLES M. N EWMAN , The full scal­ standing of high-dimensional percolation. 1n the last ing limit of two-dimensional critical percolation, few years Lawler, Schramm, Smirnov, and Werner arXiv:math.PR/ 0504036. have proven power laws for site percolation on [2] G EOFFREY GRIMMm , Percolation, second edition, Springer, the triangular lattice and confirmed most of the val­ 1999. ues for the"critical" exponents conjectured by physi­ [3] G REGORY F . LAWLER, Conformally Invariant Processes in cists. Their proof rests on Schramm's invention of the Plane, Arner. Math. Soc., 2005. the Stochastic Loewner Evolutions or Schramm

MAY 2006 NOTICES OF THE AMS 573 Book Review

Dark Hero of the Information Age: In Search of Nor bert Wiener, The Father of Cybernetics Reviewed by Michael B. Marcus

Dark Hero of the Information Age: In Search of I'm sure that every Norbert Wiener, The Father of Cybernetics reader of the Notices DARK HERO Flo Conway and]im Siegelman knows that Norbert 2005, Basic Books, New York OF THE Wiener was a child $27.50,423 pages, ISBN 0-738-20368-8 INFORMATI ON AGE prodigy. He was "home schooled" for a The thesis of this book is that Norbert Wiener, few years, by his over­ 1894-1964, was unknown outside the mathemat­ bearing father, a pro­ fessor of Slavic ical community until shortly after World lan- IN SEARCH OF guages at Harvard, War II. Then he invented cybernetics, which has the NORBERT who continued to capacity WIENER to enormously transform the world for the teach and torment better. The authors believe that since the promises THE F ATHER OF' CYO£fl N £TIC$ him until he gradu­ of cybernetics have not been realized, Wiener is not ated from high school the recognized genius of the Age of the Informa­ P \.0 CONWAY I. J I M '!SII!O t:H .. MA ~ at eleven. Young Nor­ tion, but its dark hero. And what, according to the - ...... _...... _..,.,...... ,...._ bert graduated from authors, was the greatest of the forces that pre­ Tufts in 1909, at fif­ vented the realization of the cybernetics utopia? It teen, with a B.A. in mathematics. He received his was a single person, Wiener's wife Margaret, nee En­ Ph.D. in philosophy from Harvard in 1913 when he gemann. was eighteen years old. There are many points to be examined here. Wiener spent his first postdoctoral year in Cam­ First of all, was the significance of Wiener's math­ bridge, England, studying mathematical logic with ematical contributions really secondary compared Bertrand Russell. As P. R. Masani writes in his bi­ to his latter work in naming and championing cy­ ography of Wiener ([4], p. 55), "Russell urged Wiener bernetics? More significantly, what is cybernetics, to approach mathematical philosophy from the and what does its implementation promise? Also, broadest standpoint, to concentrate not just on what did Margaret do and what good does it do us foundations but also to look at the frontiers of and Wiener's memory to dwell on it? And, finally, mathematics and theoretical physics. This advice not only brought Wiener why is there such a continuing fascination with Nor­ into contact with G. H. Hardy ... but it also exposed Wiener to Bohr's atomic bert Wiener? theory, the work of ]. W. Gibbs on statistical me­ Michael B. Marcus is professor of mathematics at the City chanics, and the Einstein-Smoluchowski papers on College of New York and the CUNY Graduate Center. His Brownian motion." (Most of the biographical details email address is mbmarcus@opton line. net. of Wiener's life in this review are taken from

574 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Masani's biography.) Because Russell was to be Wiener's mathematical contributions prior to World away from Cambridge in the spring of 1914, he sent War II. In 1933 he was elected to the National Acad­ Wiener off to Gottingen, where he took courses with emy of Sciences. David Hilbert, Edmund Husserl, and Edmund Lan­ During World War II Wiener worked with dau. One must marvel at the education Wiener re­ ]. Bigelow on predicting the future position of air­ ceived and that he was able to absorb it. This must craft, so that anti-aircraft guns would know where account for his enormous breadth later on and his to aim. This led to his work in prediction theory willingness to consider questions in so many dif­ and the closely related questions of filtering and ferent areas of science. extrapolation of stochastic processes. Moreover, In 191 5 Wiener returned to Harvard as an as­ beginning with his work with Lee, Wiener was also sistant and docent lecturer in the philosophy de­ interested in constructing electrical devices to partment. But he didn't continue with cushy ap­ perform the operations he was analyzing. He had pointments at prestigious universities. After this developed an interest in computers, stimulated by academic-year appointment he had a teaching job Vannevar Bush's work on constructing a machine at a minor branch of the University of Maine and to solve differential equations. He and Bigelow subsequently worked as a writer for the Encyclo­ actually built a device to carry out the prediction pedia Americana and as a journalist. During World to be used in anti- aircraft aiming. (In fact Wiener's War I he worked as a "computer" at the U. S. Army theory was not practical. The amount of time an proving grounds in Aberdeen, Maryland, and was airplane could be observed was not long enough even a private in the Army for a short time in 1918. to make his brilliant theory superior to the simple In 1919, largely on the recommendation of W. F. deterministic model then being employed. It is Osgood of Harvard, Wiener was offered a one-year significant that he, himself, pointed this out in instructorship in mathematics at the Massachu­ his final report to the National Defense Research setts Institute of Technology. MIT was not a pres­ Committee.) tigious research institute in 1919. The mathemat­ After the war Wiener devoted himself to apply­ ics department was a service department for the ing his formidable mathematical talents to prob­ engineering school. lems in biology, although not exclusively. He still At this point the miracles began to happen. produced some very good mathematics, perhaps Wiener had become increasingly interested in analy­ the best being his papers with Masani on the pre­ sis during the years between his docent lectureship diction theory of multivariate stochastic processes. at Harvard and his appointment to MIT. After some But he achieved fame and wide recognition outside work in functional analysis, in which he defined and the mathematical community by naming and pop­ studied what are now referred to as Banach spaces, ularizing cybernetics, "the science of control and he made Brownian motion mathematically rigorous communication in the animal and the machine". by obtaining a measure, now called Wiener measure, Wiener didn't leap from pure analysis to physiol­ on the space of continuous functions with the ogy. In 1933 he "became a regular participant in sup-norm that is supported on functions of Lip an interdisciplinary seminar on scientific 1 I 2 - E (for any 0 < E < 1 I 2) and that satisfies the method ... the Philosophy of Science Club" ((4), conditions of independent increments and nor­ p. 197), conducted by Arturo Rosenbleuth, a neu­ mality of Einstein's model for Brownian motion. rophysiologist, who was working at the Harvard Wiener's rigorous development of Brownian motion Medical School. He wrote several papers with was done prior to Kolmogorov's systematic de­ Rosenbleuth immediately following the war and scription of stochastic processes. dedicated his book Cybernetics [5] to him. As Masani Wiener's interest in stochastic process and er­ also reports ([4], p. 218), Wiener first encountered godic theory led him to consider stationary Warren McCulloch at the neurophysiological meet­ processes. Since these processes are not in L2 (R 1) ing in New York in 1942 where Rosenbleuth was and hence not amenable to Fourier analysis he in­ presenting their joint work with Bigelow on tele­ vented generalized harmonic analysis to study ology. (Divulging the secret behind the relationship them. Problems in generalized harmonic analysis between Wiener and McCulloch is the height of required new, deep, Tauberian theorems, which Conway and Siegelman's investigative reporting. themselves required new results in Fourier series, We'll get back to this later). all of which he discovered. Correlation functions I called cybernetics the "science of control and are fundamental in generalized harmonic analysis. communication in the animal and the machine" be­ These were to be his foremost probe in the analy­ cause this is the subtitle of Wiener's famous book sis of random phenomena in biology and commu­ [5], published in 1948. Actually, it is not clear to me nication theory in the years to come. In the mid- what the definition of cybernetics really is, or whether 1930s he teamed up with the electrical engineer it is a science. In his 1956 book [1], W. R. Ashby states Y. W. Lee to essentially create statistical commu­ that "Cybernetics is the general study of mecha­ nication theory. This is only a survey of some of nism from the standpoint of functionality and

MAY 2006 NOTICES OF THE AMS 575 behavior rather than internal structure and mate­ pronouncements were noticed only because of his rial." After presenting Ashby's definition, Masani previous achievements. ([4], p. 256) adds his own, rather unhelpful defin­ People's attitudes in the United States right after ition, "Cybernetics is the extension of the scientific World War II were very different from what they methodology necessitated by the existence of are now. Scientists and scientific achievement were processes for which time is Bergsonian." held in very high regard. It was not only the atomic Perhaps more helpful is what V. M. Glushkov bomb that won the war but sonar, radar, and the wrote in 1969, (as reported in [4], p. 260): brilliance to crack the enemy's codes. For a while it was widely believed that taking an impersonal sci­ It is usual nowadays to define Cyber­ entific approach was a better way to deal netics as the science of the general laws with so­ ciety's problems than by leaving them in the hands of data transformations in complex con­ of self-interested, trol systems and systems of information indebted politicians (as though processing. scientists couldn't also manage to be both self­ interested and indebted). Wiener's book came upon this scene with When defining the subject of Cyber­ a synthesis of all activity based on the ideas of message, netics it is important to avoid two ex­ noise, and control. Without tremes. These are, first, including in Cy­ diminishing the significance of his vision, it is fair bernetics everything which concerns to say that his was the next "new thing". His ideas control, and secondly attempting to re­ were immediately extolled by the influential news duce Cybernetics to a comparative study weeklies. of the relation between control systems Another aspect of the Wiener phenomenon that in engineering and those in living beings. added to his popularity and underscored his sin­ cerity was his morality. At the same time he extolled Unfortunately, the second, too narrow, descrip­ the enormous potential of science to do good, he tion of cybernetics is the subtitle of Wiener's book. also lamented its more likely uses for destruction. Conway and Siegelman finesse the issue of say­ In the preface to Cybernetics he wrote ing what cybernetics is by simply not defining it. They give an enthusiastic and exciting account of Those of us who have contributed to the how Wiener's book came about at the behest of a new science of Cybernetics thus stand French publisher and present a detailed descrip­ in a moral position which is, to say the tion of its contents. They point out that "His new least, not very comfortable. We have communication theory came together from oppo­ contributed to the initiation of a new sci­ site ends of the scientific universe: engineering ence which ... embraces technical devel­ and biology, thermodynamics and homeostasis, opments with great possibilities for information and entropy, computing machines and good and evil. We can only hand it over nervous systems" (p. 173). It is certainly true that to the world that exists about us, and Wiener was very broad. But did he really invent a this is the world of Belsen and Hi­ new science, or did he rather describe in a unified roshima. We do not even have the choice way the direction that research was heading dur­ of suppressing these new technical de­ ing the postwar period? I have always thought of velopments. They belong to the age, cybernetics as a point of view-a recognition that and the most we can do by suppression problems of communication and control through­ is to put the development of the subject out all disciplines of science, even economics and into the hands of the most irresponsible political science, have many common aspects and and most venal of our engineers. The that it is useful for scientists to be aware of them. best we can do is to see that a large Wiener was fifty-four years old when Cybernet­ public understands the trend and bear­ ics was published. He became an international fig­ ings of the present work and confine our ure who explained how the new technologies de­ personal efforts to those fields ... most veloped during the war would change people's remote from war and exploitation. As we lives. He used mathematical concepts as analogies have seen, there are those who hope to ponder religious, social, political, and economic that the good of a better understanding concepts. His years of startling mathematical of man and society which is offered by achievement were pretty much over. Conway and this new field of work may anticipate and Siegelman's thesis rests on their remark, "But his outweigh the incidental contributions greatest work lay ahead" (p. 128). I don't think so. we are making to the concentration of Wiener was a great mathematical analyst. He was power (which is always concentrated, also a very moral and courageous man. He had a by its very conditions of existence, in the great deal to say. But like Einstein's his moral hands of the most unscrupulous). I write

576 NOTICES OF THE AMS VOLUME 53, NUMBER 5 in 1947, and I am compelled to say that MIT. Lettvin recalled that, during a visit with Rosen­ it is a very slight hope. bleuth in Mexico City in 1960, Rosenbleuth told him "that Margaret told Wiener [in a letter written to him The passages I put in italics are deleted from this in 19 51 while he was visiting Rosenbleuth in Mex­ quotation on page 181 of Conway and Siegelman's ico] ...that the boys in McCulloch's group- Wiener's book. The first one is replaced by ellipses. The sec­ boys-had seduced his elder daughter [Barbara] ond, more significant statement is not. I don't un­ during her stay at the McCulloch home in Chicago derstand why, because, from other parts of the four years earlier. ..Margaret alleged that not one book, one gets the impression that Conway and but 'more than one' of the boys had seduced the Siegelman were attracted to Wiener in part by his chaste nineteen-year-old during her first foray away political positions. from home and the protected environment of her Warren McCulloch's educational background boarding school." Conway and Siegelman present was much like Wiener's. He studied philosophy corroborating evidence that leaves little doubt that and mathematics and had a degree in medicine. (Un­ this story is true, although they do not claim to have like Wiener he was also somewhat of a bohemian.) seen the actual letter. "He [McCulloch] became a serious student of math­ Margaret gets bashed in this book. Apparently ematical logic, and investigated the mathe­ she was enamored of Adolf Hitler long after a rea­ matico-logical aspects of schizophrenia and psy­ sonable person of German descent should have chopathia while serving at the Rockland Hospital been. Also she was very troubled by her daughters' for the insane" ([4], p. 218). In 1942, the year Mc­ sexuality and made many apparently false accu­ Culloch met Wiener, he was working with Walter sations about the girls' behavior. I first met Mar­ Pitts, trying to understand the organization of the garet and Wiener together in 1959 in Los Angeles. cortex of the brain. Pitts was a self-taught "ge­ Of course I was very young then and never had more nius", who had had a poor, troubled childhood in than a tangential relationship with them. Never­ Detroit but who nevertheless attracted Bertrand theless they seemed like a loving couple to me. Russell's attention and was encouraged by Russell Wiener wrote, in 19 53, in the dedication to [7], "To to study mathematical logic. In 1942 Pitts was my wife under whose gentle tutelage I first knew twenty years old. Pitts went to MIT in 1943 to study freedom" ([4], p. 94). with Wiener. As Mas ani points out ([ 4], p. 219), My new wife and I had dinner with Margaret in "Both McCulloch and Pitts played an absolutely Cambridge a couple of times in the year after positive role in the evolution of Wiener's ideas in Wiener's death. She gave me his academic gown to neurophysiology, especially on the problems of wear when I received my Ph.D. in 1965. I was fond logical manipulation, Gestalt or pattern-recognition, of her, and I think Masani was also. He filled me in gating, brain rhythms and sensory prosthesis." on what she was doing whenever we saw each other Wiener wrote two papers with Pitts (along with at meetings. Masani says nothing disparaging about Rosenbleuth and]. Garcia Ramos) and none with Margaret in his biography of Wiener. McCulloch. Nonetheless, Conway and Siegelman I knew Wiener well enough to know that he was write, "McCulloch had promoted Wiener's theories fiercely loyal and really very manly, despite his and ideas [on cybernetics] with almost as much en­ awkward appearance. He would have been devas­ thusiasm as Wiener himself" (p. 214). On the same tated by the way Margaret is treated in this book page they also report that [in 1950 (or maybe and fighting mad. I could have lived without know­ 1951)], "Jerome Wiesner, who was now head of the ing all the dirt on my hero's wife. I'm glad Wiener Rad Lab [Radiation Laboratory at MIT], with Wiener's never had to read this book. blessings, invited McCulloch to come to Cambridge To be fair to the authors, their gossip is not to head up a major new research effort on the spurious. They view the breakup of the research brain and its cybernetic connections." McCulloch team of Wiener, Pitts, and McCulloch as a primary did come to MIT, but before he did, Wiener abruptly reason that cybernetics did not achieve the great broke off his relationship with him and Pitts and success that they think was its destiny. But here I didn't even mention them in his otherwise detailed think that they are guilty of a misunderstanding of memoir [6]. the nature of mathematical research that is preva­ During the time I was Wiener's graduate student lent among the nonmathematical public. That is, assistant (1961-1963) I asked a faculty member, I that mathematics, and perhaps scientific research don't remember who, why there was a conflict be­ in general, advances by the achievements of a very tween Wiener and McCulloch. He said it had some­ few extremely gifted individuals-people who are thing to do with McCulloch having had an affair with so deep that even their colleagues don't under­ Wiener's daughter. Beginning on page 225, Conway stand them. This is the viewpoint of the movie and Siegelman paraphrase a recollection of Jerome Good Will Hunting and the play Proof. In this view Lettvin, who as a medical doctor at Boston City Hos­ Wiener's separation from Pitts and McCulloch pital had persuaded Pitts to study mathematics at doomed their effort in using the principles of

MAY 2006 NOTICES OF THE AMS 577 cybernetics to explain the workings of the brain. reflected in the nature of the mathematics that Of course, collaboration with Wiener would have they created. (In Cold War terminology, Wiener was been helpful. But McCulloch and Pitts were not a dove and von Neumann a hawk.) Heims' book con­ dummies. They were tackling a problem that is centrates on Wiener as a man opposed to mili­ still very far from a solution. There was enormous tarism and powerful institutions, Masani's book on enthusiasm in the 1950s and 1960s for the revo­ Wiener the mathematician and philosopher, and the lutionary changes that would be brought about, not book under review on Wiener's work in cybernet­ only by cybernetics, but also by artificial intelli­ ics. I think that a mathematician who is unfamil­ gence. Progress was made, and work is continuing. iar with Wiener's life and work would most enjoy But the mysteries McCulloch and Pitts were trying Masani's book and Wiener's autobiographies [7, 6]. to answer are amongst the deepest that exist. But both Heims' book and this book are well worth John von Neumann was also involved in this re­ reading. Conway and Siegelman have dug up a lot search. Conway and Siegelman point out that he, of interesting material on the early days of cyber­ Wiener, McCulloch, and about twenty others, in­ netics, and they certainly capture the enthusiasm cluding Margaret Mead, met in several closed con­ of the early years of our information age. They ferences sponsored by the Josiah Macy Foundation also uncover many facts about Wiener's life that to explore questions "at the junction between psy­ were not commonly known. chology and brain science" (p. 131). The conference I think that the next biography of Wiener should series ran from 1942 to 1953. Its name evolved to be written by an historian of the mid-twentieth "The Feedback Mechanisms and Circular Systems century who would study Norbert Wiener along in Biology and Social Sciences Meeting". To Wiener's with other scientific public intellectuals, like Linus delight, after his book appeared, the group was Pauling, Leo Szilard, Benjamin Spack, and Phillip happy to simply use the name "Cybernetics" to Morrison. I admire these figures because they spoke describe itself and its proceedings. But it seems out against militarism. However, the other side clear that the goal of describing how the brain had equally eloquent spokesmen, such as Edward functions was too ambitious. Starting on page 243 Teller and John von Neumann. Why are scientists Masani reprints a six-page letter that von Neumann absent from public discourse today? It seems that wrote to Wiener in 1946 that points this out and the only people we read about, other than politi­ suggests that perhaps they should first try to un­ cians and entertainers, are those who either make, derstand how viruses function. lose, or steal a great deal of money. To compare Conway and Siegelman present a lot of inter­ public discourse today with that during Wiener's esting history about research funding in this pe­ prime is to see how drowned our society is by ma­ riod. A great deal of money was being pumped terialism and superstition. into artificial intelligence and very little into cy­ Rather than end on a discouraging note, let us bernetics. This, too, they blame for the absence of return to Wiener himself. He was really a wonder­ a cybernetics revolution. In fact, they are so con­ ful man. This is what he wrote about mathematics vinced that great things would have occurred if only in 1933, when he was thirty-nine years old. cybernetics were vigorously pursued that they deal with the absence of substantial results from the So­ Mathematics is a subject worthy of the viet Union, where after initial hostility the govern­ entire devotion of our lives. We are serv­ ment strongly supported cybernetics research, by ing a useful place in the community by saying that, "In the end, cybernetics did not give our training of engineers, and by our de­ the Soviet Union the winning hand in the Cold War. velopment of the tools of future sci­ ... the socialist system's creed of centralized plan­ ence and engineering. Perhaps no par­ ning and rigid, top-down, authoritarian rule ran ticular discovery that we make may be counter to the most basic principles of self-gov­ used in practice; nevertheless, much of erning cybernetic systems" (p. 331). Rather than th,e great bulk of mathematical knowl­ finding excuses for the limited advances resulting edge will be, and we are contributing to from cybernetics, perhaps the authors might have that bulk, as far as lies in us. recognized that the translation of mathematical re­ sults into concrete social advances takes a very long Moreover, a clearly framed question time and follows devious paths and that it is im­ which we can not answer is an affront possible to predict which discoveries will eventu­ to the dignity of the human race, as a ally have a significant effect on society. race of thinking beings. Curiosity is a Ten years before Masani's biography of Wiener good in itself. We are here but for a day; appeared, Steve Heims [2] wrote a joint biography tomorrow the earth will not know us, of Wiener and von Neumann, with an interesting, and we shall be as though we never but I think fallacious, hypothesis (see [3]). That is, were. Let us then master infinity and that the political positions these men took were eternity in the one way open to us;

578 NOTICES OF THE AMS VOLUME 53, NUMBER 5 through the power of the understand­ ing. Knowledge is good with a good ~ which is above usefulness, and igno­ NOVA rance is an evil, and we have enlisted as NSU SOUTHEASTERN UNIVERSITY good soldiers in the army whose enemy ts~DY\Cl ~ e.J..a_'>'>VOtf'>"'-. is ignorance and whose watchword is Truth. Of the many varieties of truth, *Faculty- Mathematics/Statistics mathematical truth does not stand low­ est. ([4], p. 341) (rank TBD) #994887 There was nothing "dark" about Norbert Wiener's The Division of Math, mathematics or his morals. Science and Technology of Nova Southeastern University invites applications Acknowledgment: I am pleased to acknowledge for two full time, 9.5 month contract, faculty mem­ many discussions with David Isles which helped me bers in the area of mathematics and/or statistics at shape my ideas for this review. the assistant professor level, beginning August 2006, subject to budgetary approval. A Ph.D. in sta­ References tistics or mathematics is required. Teaching respon­ sibilities include 8 courses a year in statistics [1) W. R. ASHBY, An Introduction to Cybernetics, Wiley, and/or math. Candidates should exhibit a strong New York, 1963. commitment to undergraduate teaching and [2) S. ]. HEIMS, john von Neumann and Norbert Wiener: research. Experience in applied statistics and with From Mathematics to the Technologies of Life and online or computer-assisted instruction is a plus. Death, MIT Press, Cambridge, MA, 1980. [3) M. B. MARcus and]. MARcus, Book review: "The prince To apply, please use position number at www.nsu­ and the prodigy; Review of john von Neumann and Nor­ jobs.com and submit a letter of interest and cur­ bert Wiener: From Mathematics to the Technologies of riculum vitae. Life and Death, by Steve]. Heims", Minnesota Review, *Pending final budget approval. 20:143-148, 1983. Nova Southeastern University is an Equal [4) P.R. MASANI, Norbert Wiener 1894- 1964, Vita Matem­ Opportunity/Affirmative Action Employer. atica, Birkhauser, Boston, 1990. [5) N. WIENER, Cybernetics, or Control and Communication in the Animal and the Machine, MIT Press, Cambridge, MA, second edition, 1948. [6) __ , I Am a Mathematician. The Later Life of a Prodigy, MIT Press, Cambridge, MA, paperback edition, 1964. [7) __ , ExProdigy: My Ch ildhood and Youth, MIT Press, Cambridge, MA, second edition, 1965. It[:~ University of Zurich ~ ~

The Institute of Mathematics at the University of Zurich wishes to appoint an Oberassistent (Lecturer) in Pure Mathematics

We are looking for an outstanding young mathematician who intends to pursue an academic career. The position is for 3 years in the first instance, and may be extended for one further period of 3 years: the starting date is negotiable.

Applications are requested in the form of a single pdf-file, to arrive by e-mail at [email protected] at the latest by May 20, 2006.

Information about the Institute can be found on www.math.unizh.ch.

MAY 2006 NOTICES OF THE AMS 579 NSF Fiscal Year 200 7 Budget Request

This article is the 34th in a series of annual reports outlining the president's request to Congress for the budget of the National Sci­ ence Foundation. Last year's report appeared in the June/ July 2005 issue of the Notices, pages 637-41.

The fiscal year 2007 budget request for the National initiative calls for a 7.9% increase for the NSF, to Science Foundation (NSF) is a big disappointment just over US$6 billion. This increase represents a for mathematics. Despite a substantial increase marked shift from FY2006, when the Bush Ad­ for the NSF, as well as for the Mathematical and ministration sought only a 2.4% rise for the NSF. Physical Sciences directorate, the Division of Math­ The final Congressional appropriation for fiscal ematical Sciences (DMS) is slated for an increase of 2006 brought the NSF budget up just 2.0% above only about 3%, a little bit above the expected in­ the fiscal 2005 level. flation rate. What is more, the NSF's "Mathemati­ A preliminary analysis by the American Associ­ cal Sciences Priority Area", which has boosted the ation for the Advancement of Science finds that, DMS budget over the last several years, is sched­ under the terms of the FY2007 request, overall uled to end in 2007. federal funding for research and development The Bush Administration sent its fiscal year would climb by about US$2.6 billion. Even more 2007 budget request to Congress in February 2006. than that amount would be absorbed as increases One component of the request is the "American for weapons development and space exploration Competitiveness Initiative", consisting of sub­ technologies, "leaving declining funding for the stantial increases for the Department of Energy's remainder of the [research and development] port­ Office of Science, for the National Institute of Stan­ folio," the analysis notes. In his State of the Union dards and Technology, and for the NSF. The address, President Bush set out the goal of doubling

Table 1: National Science Foundation (Millions of Dollars) 2003 2004 2005 2006 2007 Actual Change Actual Change Actual Change Plan Change Request (1) Mathematical Sciences Research Support $ 178.8 12.0% $ 200.3 0.0% $ 200.2 -0.4% $ 199.3 3.2% $ 205.7 (2) Other Research Support (Note a) 4054.7 5.5% 4277.0 -1.8% 4 199.7 2.9% 4323.1 8.7% 4700.7 (3) Education and Human Resources (Note b) 934.9 1.0% 944.1 -10.7% 843.5 -5.5% 796.7 2.4% 816.2 (4) Salaries and Expenses (Note c) 201 .0 14.7% 230.6 2 .9% 237.3 10.4% 262.1 13.5% 297.6

(5) Totals $5369.3 5.3% $5652.0 -3 .0% $5480.8 1.8% $5581.2 7.9% $6020.2 (6) (1) as a% of the sum of (1) and (2) 4.22% 4.47% 4.55% 4.41 % 4.19%

(7) (1) as a% of (5) 3.3 3% 3.54% 3.65% 3.57% 3.42% Tables prepared by Notices staff. Totals may not add up due to rounding. Note a: Support for research and related activities in areas other than the mathematical sciences . Includes scientific research facilities and instrumentation. Note b: Support for education in al l fields, including the mathematical sciences. Note c: Administrative expenses of operating the NSF , including the National Science Board and t he Office of the In spector General.

580 NOTICES OF THE AMS VOLUME 53, NUMBER 5 funding for "critical basic research programs in the The DMS budget was flat in fiscal 2005 and de­ physical sciences over the next 10 years". The em­ clined slightly in fiscal2006, so the modest proposed phasis on the physical sciences is reflected in the increase for fiscal 2007 would leave the division proposed budget for the National Institutes of with less, in constant dollar terms, than it had Health, which under the terms of the request would three years earlier. The requested increase for the be flat for the second year in a row. DMS is the lowest among the six divisions within It appears that the Bush Administration's in­ the Mathematical and Physical Sciences (MPS) di­ terpretation of the term "physical sciences" does rectorate, which overall has a requested increase not include mathematics, as the DMS is slated for of 6.0%. "It looks to me as if the NSF or MPS is in­ only a 3.2% increase. And outside of the NSF, there tent on bringing DMS back to its historical position, as far as funding goes-that is, the lowest-funded is less and less grant money available for mathe­ division within MPS," Rankin remarked. "To me matics research, due in large part to declining bud­ this is not recognizing current reality-the many gets in those agencies of the Department of Defense contributions the mathematical sciences makes to­ that have mathematics programs. Indeed, nowadays ward technological innovation and competitive­ the DMS provides 77% of all federal funding for aca­ ness. This is surprising, given that innovation and demic research in the mathematical sciences, up competitiveness are on the minds of Congress and from about 50% a dozen years ago. By contrast, ac­ the Administration." cording to Samuel M. Rankin ill, director of the AMS The NSF's "Mathematical Sciences Priority Area" Washington Office, the NSF is currently the source (MSPA), which officially began in fiscal year 2003, of just 40% of the funding for academic research has had an effect: Between 2002 and the current in the physical sciences. fiscal year of 2006, the DMS budget rose by

Table 2: Directorate for Mathematical and Physical Sciences (Millions of Dollars) 2 0 0 3 2 0 0 4 2 0 0 5 2 0 0 6 2 0 0 7 Actual %of Total Actual %of Total Actual %of Total Plan %of Total Request %of Total

(1) Mathematical Sciences $ 178.8 17.2% $ 200.3 18.3% $ 200.2 18.7% $ 199.3 18.4% $ 205.7 17.9% (2) Astronomical Sciences 187.1 18.0% 196.6 18.0% 195.1 18.2% 199.6 18.4% 2 15.1 18.7% (3) Physics 224.5 21.6% 227.8 20.9% 224.9 21.0% 233.1 21.5% 248.5 21.6% (4) Chemistry 181.6 17.4% 185.1 17.0% 179.3 16.8% 180.8 16.7% 191. 1 16.6% (5) Materials Research 241.4 23.2% 250.6 23.0% 240.1 22.4% 242.9 22.4% 257.4 22.4% (6) Office of Multidisciplinary Activities 27.3 2.6% 31.1 2 .8% 29.8 2.8% 29.7 2.7% 32.4 2.8% (7) Totals $1040.7 100.0% $1091. 6 100.0% $1069.4 100.0% $1085.4 100.0% $1 150.3 100.0%

Table 3: Compilation of NSF Budget, 2001-2007 (Millions of Dollars) 2001 2002 2003 2004 2005 2006 2007 2001-2005 200 1-2007 Actual Actual Actual Actual Actual Plan Request Change Change (1) Mathematical Sciences Research Support $ 121.4 $ 1 51.5 $ 178.8 $ 200.3 $ 200.2 $ 199.3 $ 205.7 64.9% 69.4% Constant Dollars 68.5 84.2 97.2 706.0 102.5 49.6% (2) Other Research Support (Note a) 3370.2 3579.8 4054.7 4277.0 4199.7 4323.1 4700.7 24.6% 39.5% Constant Dollars 1903.0 1989.9 2203.6 2264.2 2150.4 13.0% (3) Education and Human Resources (Note b) 795.4 866.1 934.9 944.1 843.5 796.7 816.2 6.0% 2.6% Constant Dollars 449.1 48 1.4 508.1 499.8 43 1. 9 -3.8% (4) Salaries and Expenses (Note c) 172.9 176.6 201.0 230.6 237.3 262 .1 297.6 37.2% 72.1% Constant Dollars 97.6 98.2 109.2 122.7 727 .5 24.5% (5) Totals $4459.9 $4774.1 $5369.3 $5652.0 $5480.8 $5581.2 $6020.2 22 .9% 35.0% Constant Dollars 25 18.3 2653.8 2918.1 2992.0 2806.3 7 7.4%

Current dollars are converted to constant dollars using th e Consumer Price Index (based on prices during 1982-84). For Notes a, b, and c, see Table 1.

MAY 2006 NOTICES OF THE AMS 581 almost one-third. In fact, one of the original aims Call for Original Research Articles of the MSPA was to double the DMS budget. Al­ Emphasis should be in a) combinatorial results and analytic methods, though the doubling will not have been achieved, b) analytic resuhs and combinatorial methods, or c) a mixture the budget request calls for the MSPA to end in of combinatorics and analysis in the methods or in their applications. FY2007. The NSF budget request document speaks of "mainstreaming" interdisciplinary research pro­ We also encoumge subm~sion of high quality exposilory papers on In pies jects that were funded through the MSPA. "Invest­ of significant interest and short leHers (1-3 pages) describing open research problems ments in formal interdisciplinary partnerships d cunent interest !hat !nil within !he scope d !he joumal. through the MSPA will be redirected to unsolicited For more infonnation on submissions visit http://www.ojac.org/info.html proposals and the fundamental mathematical sci­ ences component of the MSPA," the document states. It also says that in fiscal 2007 the DMS in­ I Ool-ioe douroAl of tends to increase by about US$ 5 million its support . (~' ol\oA lut1e Comb1oAtor1es for core research in order to maintain its proposal success rate of 32% (in 2005, the DMS received Department of Mathematics 2,172 proposals and funded 687). In addition, the ~ University of Missouri-Columbia / Columbia, Missouri 65211 DMS will increase funding for Research Experi­ FAX (573) 882-1869 ences for Undergraduates and for Enhancing the .,y [email protected] Mathematical Sciences Workforce in the 21st Cen­ http://www.ojac.org tury (each has a requested increase of US$ 500,000). Each year in the spring, the AMS Committee on Managing Editors Science Policy (CSP) meets in Washington, DC. At Alex losevich (University of Missouri) its last meeting in 2005, the CSP decided to dispense lzabella Laba (University of British Columbia) with the customary format for its meeting, which Sinai Robins (Temple University) featured a succession of presentations about fed­ eral science policy and funding. Instead, the format for the spring 2006 meeting has CSP members fan­ ning out across Capitol Hill to meet with congres­ sional representatives, senators, and staffers to discuss funding for research. The meetings will be James Joseph Sylvester arranged by the AMS Washington Office, under Rankin's direction. In its dealings with Congress, Jewish Mathematician in a Victorian World the Washington Office has generally pursued a Karen Hunger Parshall strategy of banding together with other science organizations to speak in a unified voice for strong "As the story both of the man in his federal support for NSF overall, rather than seek­ times and the mathematician with his work, it is a fascinating read." ing favored treatment for the DMS specifically. -Joan Richards, Brown University "However, we will be advocating for a better in­ $69.95 hardcover crease to the DMS budget this year," says Rankin. -Allyn jackson Arthur Cayley Mathematician Laureate ofthe

Victorian Age Arrhur Cayle;' Tony Crilly Arthur Cayley's influence still pervades modern mathematics, in group theory (Cayley's theorem) , matrix algebra (the Cayley-Hamil­ ton theorem), and invariant theory. Tony Crilly, the world's leading authority on Cayley, provides the first definitive account of his life. $69.95 hardcover THE JOHNS HOPKINS UNIVERSITY PRESS 1-800-537-5487 • www.press.jhu.edu

582 NOTICES OF THE AMS VOLUME 53, NUMBER 5 2006]PBM Communications Award

The 2006 JPBM Communications Award was pre­ Citation sented at the Joint Mathematics Meetings in San An­ The Joint Policy Board for Math­ tonio in January 2006. ematics presents its 2006 Com­ The Joint Policy Board for Mathematics (JPBM) munications Award to Sir Roger established its Communications Award in 1988 to Pemose for the discovery of Pen­ reward and encourage journalists and mathemati­ rose tilings, which have captured cians who, on a sustained basis, bring mathemat­ the public's imagination, and for ical ideas and information to nonmathematical au­ an extraordinary series of books diences. Presented annually, the award recognizes that brought the subject of con­ a significant contribution or accumulated contri­ sciousness to the public in math­ butions to the public understanding of mathe­ ematical terms. matics, and it is meant to reward lifetime achieve­ Dr. Penrose has acquired a ment. The award carries a cash prize of US$1 ,000. large public following for eight books he has written. A number The JPBM represents the American Mathematical Sir Roger Penrose Society, the American Statistical Association, the of these explore ideas that re­ Mathematical Association of America, and the So­ late fundamental physics, math- ciety for Industrial and Applied Mathematics. ematical logic, and human consciousness. In The Emperor's New Mind (1989) and also in later vol­ Previous recipients of the JPBM Communica­ umes, he has argued that known laws of physics tions Award are: James Gleick (1988), Hugh White­ do not constitute a complete system and that more (1990), Ivars Peterson (1991), Joel Schneider human consciousness cannot be explained until a (1993), Martin Gardner (1994), Gina Kolata (1996), new physical theory of quantum gravity has been Philip]. Davis (1997), Constance Reid (1998), Ian devised. These ideas have stimulated broad pub­ Stewart (1999), John Lynch and Simon Singh (spe­ lic debate. They have brought widespread attention cial award, 1999), Sylvia Nasar (2000), Keith]. to the scientific and philosophical implications of Devlin (2001), Claire and Helaman Ferguson (2002), consciousness. The most recent book of Dr. Pen­ Robert Osserman (2003), and Barry Cipra (2005). rose, The Road to Reality (2005), is a bold and The selection committee for the 2006 award broadly conceived attempt to present the tech­ consisted of: James Arthur, Carl Cowen, Martin niques of modern mathematics and physics be­ Golubitsky, and Elizabeth Halloran. fore a general public audience. This year's JPBM The 2006 JPBM Communications Award was Communication Award is a tribute to the way that presented to SIR ROGER PENROSE. What follows is the Dr. Pemose has made the ideas behind high level award citation, a brief biographical sketch, and the mathematics accessible to large segments of the recipient's response to the award. general public.

MAY 2006 NOTICES OF THE AMS 583 Biographical Sketch in particular, have grown to enormous proportions As a graduate student, Roger Penrose studied math­ over the years, and over the centuries. Semi-pop­ ematics and physics at Cambridge University from ular expositions which give clear and intuitive ac­ the likes of Bondi, Dirac, Hodge, Steen, and Todd. counts of one area of work can be an invaluable aid He was awarded his Ph.D. there in algebraic geom­ to others whose expertise may lie in some area of etry in 1958. After positions at various universities science or mathematics which is far from that in both England and the United States, he was ap­ being explained. In my own experience, such ac­ pointed the Rouse Ball Professor of Mathematics counts can have enormous value. at the University of Oxford in 1973, a position he If, as this award seems to imply, I have con­ held until1998, when he became Emeritus Rouse tributed, in some significant way, to the spreading Ball Professor of Mathematics. of scientific or mathematical knowledge and un­ In his research career Penrose has made funda­ derstanding, then I am indeed well pleased. Thank mental and remarkably diverse contributions to you very much. both mathematics and physics. Many of these con­ cern the interplay between relativity, geometry, and topology, and are related to the attempt to unify relativity with quantum theory. In 1967, Penrose discovered twistor theory, a beautiful mathemati­ cal formalism that combines powerful techniques of algebra and geometry. In 1971, he introduced the theory of spin networks, which later became a part of the geometry of spacetime in loop quantum gravity. In 1974, he discovered what are now known as Penrose tilings, which are formed from two tiles that can only tile the plane aperiodic ally. Such pat­ terns were later found, quite remarkably, to occur in the arrangement of atoms into quasicrystals. Penrose has received many awards and honors. He was elected to fellowship in the Royal Society of London (1972) and as a foreign associate of the National Academy of Sciences (1998). He received the Wolf Foundation Prize in Physics (with Stephen Hawking, 1988) and the DeMorgan Medal of the Lon­ don Mathematical Society (2004). In 1994, he was knighted for his service to science. Response It is a deep and unexpected honour, and a great pleasure for me, to receive the JPBM Communica­ tions Award for 2006. I certainly believe in the importance of convey­ ing to the general public, as far as this is possible, something of the real nature of mathematics, not only for its increasing utility across so many areas of importance to modern society, but also for its beauty and for the inner satisfaction that it brings. Perhaps these latter qualities are even more im­ portant than the more utilitarian ones; for one can­ not really properly understand mathematics with­ out having some kind of appreciation of its aesthetic qualities. Moreover it is a belief (or a faith?) of mine that there are many more out there, among those who claim no appreciation or under­ standing of mathematics, who actually have within themselves some genuine but unrecognized abili­ ties in this direction. And it is certainly not just the general public who can stand to gain from clear expositions of mathe­ matical topics. Science in general, and mathematics

584 NOTICES OF THE AMS VOLUME 53, NUMBER 5 MAAPrizesAwardedinSan Antonio

At the Joint Mathematics Meetings in San Antonio mathematics educator to mathematicians," the ci­ in January 2006, the Mathematical Association of tation states. "He is conducting carefully reasoned, America (MAA) presented several prizes. incremental, foundational research in order that dis­ cussion of educational issues may one day be based Gung and Hu Award for Distinguished on more rigorous scientific findings. His service to Service mathematics and its teaching and learning at all lev­ The Yueh-Gin Gung and Dr. Charles Y. Hu Award els is truly remarkable." for Distinguished Service to Mathematics is the most prestigious award made by the MAA. First Haimo Awards for Teaching given in 1990 the Gung and Hu Award is the suc­ The Deborah and Franklin Tepper Haimo Awards cessor to the Award for Distinguished Service to for Distinguished College or University Teaching of Mathematics, awarded since 1962, and has been Mathematics were established in 1991. These made possible by the late Charles Y. Hu and his wife, awards honor college or university teachers who Yueh-Gin Gung. It is worth noting that Hu was not have been widely recognized as extraordinarily a mathematician but a retired professor of geology. successful and whose teaching effectiveness has He had such strong feelings about the basic nature been shown to have had influence beyond their own of mathematics and its importance in all human en­ institutions. Deborah Tepper Haimo was president deavors that he felt impelled to contribute gener­ of the MAA during 1991-1992. ously to our discipline. The 2005 Haimo Awards were presented to HYMAN BASS of the University of Michigan re­ jACQUELINE DEWAR of Loyola Marymount University, ceived the 2006 Gung and Hu Award for "invest[ing] KEITH STROYAN of the University of Iowa, and JuDY vast energies over several decades to strengthen the LEAVITT WALKER of the University of Nebraska at mathematical community." An outstanding re­ Lincoln. searcher and former member of Bourbaki, Bass The award citation for Jacqueline Dewar states, has made many contributions to the mathematical "In her 32 years at Loyola Marymount University, community. He is a member of the National Acad­ Jackie Dewar's enthusiasm, extraordinary energy, emy of Sciences and has served on many NAS com­ and clarity of thought have left a deep imprint on mittees, including the Mathematical Sciences Edu­ students, colleagues, her campus, and a much cation Board, which he chaired. A past president larger mathematical community." She profoundly of the AMS, he has served on many Society com­ influenced the mathematics curriculum at Loyola mittees. He has been on the boards of trustees of Marymount, helping to shape the biomathematics the Institute for Advanced Study and the Mathe­ program, the mathematics program for prospective matical Sciences Research Institute. In recent years, secondary school teachers, and the Master of Arts Bass has turned his attention to improving school in Teaching program. Her freshman-level work­ mathematics. "Hyman Bass is playing a vital role shop course is credited with improving the reten­ in bringing the insights of a mathematician to tion of mathematics majors. Outside the Loyola mathematics educators and the insights of a Marymount campus, Dewar has been active with

MAY 2006 NOTICES OF THE AMS 585 in-service programs for teachers, and with the Ex­ immune to the limpid charms of a clever counting panding Your Horizons conferences for middle- and argument," the prize citation states. The book by high-school girls. The citation recognized Dewar for Benjamin and Quinn "will charm you over and over "her passionate devotion to the art of teaching". again. The authors claim that counting arguments "Keith Stroyan's name is synonymous with in­ make the most compelling, natural, and memo­ novation in the teaching of calculus," the award ci­ rable proofs. It is hard to disagree with them after tation states. "In more than 30 years of teaching dipping into this lovely volume .. .Proofs That Really at the University of Iowa, he has constantly sought Count illustrates in a magical way the pervasiveness ways in which to combine past knowledge with re­ cent discoveries and technology, and to find the and power of counting techniques throughout I]lental 'hooks' with students' previous experiences, mathematics. It is one of those rare books that current interests, and future aspirations." One of will appeal to the mathematical professional and the keys to his success is his careful training of seduce the neophyte." graduate and undergraduate assistants for his courses, in which he inculcates the assistants with Chauvenet Prize good teaching practices. Long before calculus re­ The Chauvenet Prize recognizes a member of the form projects started receiving grants from the MAA who has written an outstanding expository ar­ National Science Foundation, Stroyan pioneered ticle. First awarded in 192 5, the prize is named for the use of computer programs to help students William Chauvenet, who was a professor of math­ grasp calculus concepts. Then, with several NSF ematics at the United States Naval Academy. grants, he developed materials to integrate com­ The 2006 Chauvenet Prize was awarded to puters into calculus teaching. FLORIAN PFENDER and GONTER M. ZIEGLER, both of the "Judy Walker cares deeply about her students," Technische Universitat the prize citation states. "Her students testify that Berlin, for their article "Kiss­ her courses are among the most demanding they ing Numbers, Sphere Packings, and Some Unex­ ever had, yet consistently praise her ability to guide pected Proofs" (Notices, September 2004, pages the direction of a class through questions. Superb 873-883). According to the citation, this "lucid and at explaining mathematics and communicating the beautifully illustrated paper" discusses the history joy of discovery, she is readily available outside of and progress of three classical packing problems class for special problem sessions, and is in demand in various dimensions: the kissing number prob­ as a doctoral thesis advisor." One of her major in­ lem, the sphere packing problem, and the lattice novations at the University of Nebraska was cre­ packing problem. The immediate backdrop for this ating a freshman honors seminar for nonmajors paper is Thomas Hales's controversial solution in called "The Joy of Numbers: Search for the Big 1998 to Kepler's Conjecture, which is the general Primes", which she also adapted to serve elemen­ sphere packing problem for dimension three. tary and middle school teachers. In 1997, Walker Pfender and Ziegler's paper clarifies the differ­ and a colleague launched ALL GIRLS/ ALL MATH, a program to encourage high-school girls to pursue ences among the problems while also shedding mathematics. She also started the Nebraska Con­ light on recent developments in the kissing num­ ference for Undergraduate Women in Mathematics, ber problem. The authors "strip away all but the which over its first seven years attracted 800 par­ essentials so that novices may appreciate the power ticipants. The citation praised her "dynamic lead­ and beauty of these new approaches to finding an­ ership and passionate commitment to teaching swers to the kissing number problem." mathematics". Certificates of Meritorious Service Becken bach Book Prize Each year the MAA presents Certificates of Meri­ The Beckenbach Book Prize, presented since 1982, torious Service to honor outstanding service to is named for the late Edwin Beckenbach, a longtime sections of the MAA. Those honored in 2006 are: leader in the MAA publications program and a pro­ KAY SoMERS of Moravian College, Eastern Pennsyl­ fessor of mathematics at the University of Cali­ vania-Delaware Section; CALVIN (CAL) VAN NIEWAAL of fornia, Los Angeles. The prize is awarded to an au­ Coe College, Iowa Section; ALAN TuCKER of Stony thor of a distinguished, innovative book published by the MAA. Brook University, Metropolitan New York Section; ARTHUR BENJAMIN, professor of mathematics at IVY KNOSHAUG of Bemidji State University, North Harvey Mudd College and JENNIFER QUINN , executive Central Section; MARJORIE ENNEKING of Portland State director of the Association for Women in Mathe­ University, Pacific Northwest Section; and WILLIAM matics, received the 2006 Beckenbach Book Prize YsLAs VELEZ of the University of Arizona, South­ for their book Proofs that Really Count: The Art of western Section. Combinatorial Proof. "Few mathematicians are -From MAA announcements

586 NOTICES OF THE AMS VOLUME 53, NUMBER 5 AWMAwardsPresentedin San Antonio

The Association for Women in Mathematics (AWM) In particular, her special attention to children and presented two awards at the Joint Mathematics their parents, women, minorities, and the envi­ Meetings in San Antonio in January 2006. ronment, all with respect to mathematics, have been and continue to be of benefit for the mathe­ Louise Hay Award matical community and our society as a whole." The Louise Hay Award for Contributions to Math­ Schafer Prize ematics Education was established in 1990 to honor the memory of Louise Hay, who was widely recog­ The Alice T. Schafer Prize for Excellence in Math­ nized for her contributions to mathematical logic ematics by an Undergraduate Woman was estab­ and for her devotion to students. lished in 1990. The prize is named in honor of The 2006 Hay Award was presented to PATRICIA Alice T. Schafer, one of the founders of AWM and CLARK KENSCHAFT of Montclair State University. The one of its past presidents. award was made "in recognition of her long career ALEXANDRA OVETSKY of Princeton University was of dedicated service to mathematics and mathe­ named the 2006 winner of the Schafer Prize. A se­ matics education". The prize citation states that nior and a Goldwater scholar, Ovetsky is the re­ Kenschaft has "found her true calling in not only cipient of the Princeton mathematics department's Andrew H. Brown Prize for outstanding research teaching university-level mathematics, but also in in mathematics by a junior. The Brown Prize was writing about, speaking about, and working for given for Ovetsky's proof of a generalization of a mathematics and mathematics education in the well known theorem of Claude Shannon from 1948. areas of K-12 education, the environment, affir­ She co-authored a paper, "Surreal dimensions", mative action and equity, and public awareness of which has appeared in Advances in Applied Math­ the importance of mathematics in society." The ci­ ematics. In the summer of 2004 she participated tation also mentions the wide range of books and in the Research Experiences for Undergraduates articles Kenschaft has written and edited, includ­ program at the University of Minnesota, Duluth, ing her latest book, Change is Possible: Stories of where she wrote a paper on well-covered graphs. Women and Minorities in Mathematics (AMS, 2005). The following summer she participated in a pro­ She founded and directed PRIMES, the Project for gram at the National Security Agency, and her work Resourceful Instruction of Mathematics in the El­ there is being published internally by the NSA. The ementary School, which was supported by fourteen prize citation quoted one of Ovetsky's mentors as Eisenhower grants and served teachers in nine saying, "She already has the research capabilities urban and suburban schools. As a result of her work of an advanced graduate student or junior faculty on this project, Kenschaft developed the book Math member." Power: How to Help Your Child Love Mathematics Two other senior mathematics majors were also Even If You Don't (Addison-Wesley-Longman, 199 7). recognized by AWM. ALLISON BISHOP of Princeton The award citation ends with this quotation University was named runner-up for the Schafer from one of Kenschaft's colleagues: "She deserves Prize, and ELLEN GASPAROVIC of the College of the Holy to be recognized for her decades of dedication to Cross received an honorable mention. mathematics and math education and for her in­ novative and unique contributions in these areas. - From A JiVM announcements

MAY 2006 NOTICES OF THE AMS 587 NORTHWESTERN UNIVERSITY

THE FR.EDERJC ESSER. THE EIGHTH NEMMER.S PR.IZE IN NEMMER.S PR.IZE IN MATHEMATICS MATHEMATICS WILL BE AW AR.DED IN 2008 WITH NOMINATIONS DUE $150,000 AW AR.D BY DECEMBER. 1, 2007. PR.ESENTED BY FOR. FUR. THER. NOR.THWESTER.N INFOR.MA TION, UNIVER.SITY CONTACT:

nemmers @northwestern.edu Previous recipients: OR. 2004 MIKHAEL GR.OMOV

2002 SECR.ET AR. Y YAKOV G. SINAI SELECTION COMMITTEE 2000 FOR. THE NEMMER.S PR.IZES' EDW AR.D WITTEN OFFICE OF THE PR.OVOST NOR.THWES TER.N 1998 UNIVERSITY JOHN H. CONWAY 633 CLAR.K STR.EET 1996 EvANSTON, ILLINOIS JOSEPH B. KELLER. 60208-1119 U.S.A. 1994 JVlVJV.IJort/;we.rtem.edu/ provo.rt/ mvardr / nemmers YUR.I I. MANIN Mathematics People

Avila, Morel, and Payne Named Meckes Receives AIM Five-Year Clay Research Fellows Fellowship The Clay Mathematics Institute (CMI) has announced the ELIZABETH MECKES of Stanford University has been named the appointment of three Research Fellows: ARTUR AviLA of recipient of the 2006 American Institute of Mathematics the Centre Nationale de la Recherche Scientifique (CNRS), (AIM) Five-Year Fellowship. SoPHIE MOREL of Universite de Paris Sud, and SAM PAYNE of Meckes is completing her Ph.D. thesis at Stanford under the University of Michigan. They were selected for their re­ the direction of Persi Diaconis. Her research interests in­ search achievements and their potential to make signifi­ clude analysis, convex geometry, and probability theory. cant future contributions to the field. She has developed a new infinitesimal version of Stein's Artur Avila, born in 1979, received his Ph.D. in 2001 at the Instituto Nacional de Matematica Pura e Aplicada method of exchangeable pairs, which she has used in (IMPA) in Rio de Janeiro, Brazil, under the direction of studying random matrices and eigenfunctions of the Lapla­ Welington de Melo. In his thesis Avila generalized the reg­ cian on certain Riemannian manifolds. In addition to her ular or stochastic dichotomy from the quadratic family to thesis work, she has coauthored papers on convex geom­ any nontrivial family of real analytic unimodal maps. Since etry and on Poisson approximation. She earned a B.S. from then he has made numerous outstanding contributions to Case Western Reserve University in 2001 and has been the one-dimensional and holomorphic dynamics, spectral the­ recipient of a Goldwater Fellowship. ory of the Schrodinger operator, and ergodic theory of The runners-up for the AIM Fellowship are Alireza interval exchange transformations and the associated Golsefidy (Yale University), Richard Kent (University of Teichmiiller flow. Texas, Austin), Abhinav Kumar (Harvard University), and Sophie Morel, born in 1979, is completing her Ph.D. at Benjamin Schmidt (University of Michigan, Ann Arbor). the Universite Paris-Sud, Orsay, under the direction of Gerard Lauman. In her thesis she develops a theory of - From an AIM announcement weight truncation on varieties over finite fields, with which she derives a simple description of the intersection com­ plexes on the Baily-Borel compactifications of certain Shimura varieties over finite fields. From this she obtains Barenblatt Receives a formula for the trace of the Frobenius endomorphism on the Euler characteristic of the intersection cohomology. Timoshenko Medal Sam Payne, born in 1978, is completing his Ph.D. at the GRIGORY I. BARENBLA TT of the University of California, Berke­ University of Michigan under the direction of William Ful­ ley, has received the Timoshenko Medal of the American ton. His thesis gives a surprising and simple construction Society of Mechanical Engineers (ASME). He was honored of complete toric varieties on which there are no nontriv­ "for seminal contributions to nearly every area of solid an~ ial equivariant bundles of rank two. In other work, Payne fluid mechanics, including fracture mechanics, turbulence, has given counterexamples to conjectures of Fujino and stratified flows, flames, flow in porous media, and the the­ of Hibi (with Mircea Mustata), as well as a complete, ele­ ory and application of intermediate asymptotics." gant description of the equivariant Chow cohomology of toric varieties: it is the ring of integral piecewise linear poly­ · The Timoshenko Medal was established in 1957 and is nomial functions. conferred in recognition of distinguished contributions to Current Clay Research Fellows include Daniel Biss, Maria the field of applied mechanics. Instituted by the Applied Chudnovsky, Ben Green, Sergei Gukov, Bo'az Klartag, Mechanics Division of ASME, it honors Stephen P. Timo­ Ciprian Manolescu, Maryam Mirzakhani, David Speyer, shenko, world-renowned authority in the field, and it com­ Andras Vasy, and Akshay Venkatesh. memorates his contributions as an author and teacher.

- From a CMI announcement - From ASME announcements

MAY 2006 NOTJCES OF THE AMS 589 Mathematics People

Congratulation~ National Academy of to Patrick Gilmer of Louisiana State University who Engineering Elections submitted the winning entry in the AMS T-Shirt The National Academy of Engineering (NAE) has announced Captioning contest held at the JMM in San Antonio, Texas the election of seventy-six new members and nine foreign associates, including six whose work involves the mathe­ matical sciences. Their names, institutions, and the re­ search for which they were elected follow. EGON BALAs, Carnegie Mellon University, for contributions to integer programming and its applications to the sched­ uling and planning of industrial facilities; MANUEL BLUM, Carnegie Mellon University, for contributions to abstract complexity theory, inductive inference, cryptographic protocols, and the theory and applications of program checkers; LESLIE GREENGARD, Courant Institute of Mathe­ matical Sciences, New York University, for work on the de­ velopment of algorithms and software for fast multipole methods; ALVY RAY SMITH of Seattle, a consultant, for the development of digital imaging, compositing, and paint­ ing that have led to fundamental changes in the graphic arts and motion picture industries; and VLADIMIR N. VAPNIK, ,, 1 HOPE '(oU ARE WCJR.Io;!IJ(j MoOV'-0 /k>O.'" NEC Laboratories, for insights into the fundamental com­ plexities of learning and for inventing practical and widely . T-Shir:s· can be purch:sed through the AMS Online applied machine-learning algorithms. Elected as a foreign Bookstore at www.ams.org/bookstore or by associate was CHARLES ANTHONY RICHARD HOARE, Microsoft calling AMS Customer Service at 1-800-321-4267. Call for availibility. Research, Cambridge, United Kingdom, for fundamental contributions to computer science in the areas of algo­ rithms, operating systems, and programming languages.

-From an NAE announcement

590 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Mathematics Opportunities

and includes a broadened VIGRE activity, as well as addi­ NSF Postdoctoral Research tional components for Research Training Groups in the Fellowships Mathematical Sciences (RTG) and for Mentoring through Critical Transition Points (MCTP) in the Mathematical Sci­ The National Science Foundation (NSF) awards Mathe­ ences. matical Sciences Postdoctoral Research Fellowships (MSPRF) VIGRE grants are designed to allow departments in the for research in areas of the mathematical sciences, in­ mathematical sciences to carry out innovative educational cluding applications to other disciplines. Awardees are per­ programs in which research and education are integrated mitted to choose research environments that will have and in which undergraduates, graduate students, post­ maximal impact on their future scientific development. doctoral fellows, and faculty are mutually supportive. In­ Awards are made in the form of either Research Fellow­ tegrating research and education for graduate students and ships or Research Instructorships. The Research Fellow­ postdoctoral associates, involving undergraduates in sub­ ship option provides full-time support for any eighteen aca­ stantial learning by discovery, and developing a team demic-year months in a three-year period, in intervals not approach are keys to successful VIGRE projects. VIGRE stu­ shorter than three consecutive months. The Research In­ dent and postdoctoral associates and their mentors may structorship option provides a combination of full-time and participate in international research and education col­ half-time support over a period of three academic years, laborative activities, including activities in other countries usually one academic year full-time and two academic that are integrated into and that benefit the overall VIGRE years half-time. Under both options, the award includes program at the institution. six summer months; however, no more than two summer The DMS expects to make nine to fifteen awards under months of support may be received in any calendar year. this program in 2006. The deadline for proposals is Under both options, the stipend support for twenty-four june 6, 2006. For more information about the program and months (eighteen academic-year months plus six summer all of its components, see the website http: I lwww. nsf. months) will be provided within a forty-eight-month govlpublicationslpub_summ.jsp?ods_key=nsf05595. period. The program solicitation and deadline for the 2006 fel­ -From an NSF announcement lowships will be available in July 2006; see the DMS web­ site, http: I lwww. nsf. gov ldi v li ndex. j sp?di v=DMS.

-From an NSF announcement Call for Nominations for the IClP /IMU Ramanujan Prize Enhancing the Mathematical The Abdus Salam International Centre for Theoretical Physics (ICTP), in conjunction with the International Math­ Sdences Workforce in the ematical Union (IMU), is seeking nominations for the 2006 Ramanujan Prize for Young Mathematicians from Devel­ Twenty-First Century oping Countries. Researchers under forty-five years old who In an effort to increase the number of U.S. citizens, na­ work in any branch of the mathematical sciences are eli­ tionals, and permanent residents who are well prepared gible. The prize carries a cash award of US$10,000 and a in the mathematical sciences and who pursue careers in travel and subsistence allowance to deliver a prize lecture the mathematical sciences and other scientific disciplines, at the ICTP. The deadline for receipt of nominations for the Division of Mathematical Sciences (DMS) of the National the 2006 prize is july 31, 2006. For further information, Science Foundation (NSF) has instituted a program titled seehttp:llwww.ictp.trieste.itl-sci_infolawardsl Enhancing the Mathematical Sciences Workforce in the RamanujaniRamanujan.html . Twenty-First Century. This program builds on the Verti­ cal Integration of Research and Education (VIGRE) program -From an ICTP announcement

MAY 2006 NOTICES OF THE AMS 591 Inside the AMS

The Department Chairs Workshop is an annual event AMS Archives Journals with hosted by the AMS prior to the start of the Joint Meetings. Portico Workshop sessions have focused on a range of issues fac­ ing departments today, including personnel issues (staff The AMS has begun electronic archiving of eleven of its and faculty), long-range planning, promotion and tenure, journals with Portico, a long-term archive of digital schol­ budget management, assessments, outreach, and junior fac­ arly e-journals. ulty development. Portico began in 2002 as an outgrowth of JSTOR. The Those interested in attending a future workshop should initial impetus was the need for a sustainable model for look for registration information sent out in advance of electronic archiving; JSTOR, by contrast, is a service that the Joint Meetings or contact the AMS Washington Office offers Internet access to journals. After extensive devel­ at amsdc@ams. org. opment of appropriate technology as well as many dis­ cussions with publishers and librarians, Portico was -AMS Washington Office launched in 2005. It receives support from JSTOR, Ithaka Harbors, the Library of Congress, and the Andrew W. Mel­ lon Foundation. Correction Portico will provide archival storage for the AMS jour­ The article "The Incompleteness Theorem" by Martin nals and will also migrate the journals to upgraded sys­ Davis in the April 2006 Notices included an error on the tems as needed. Other participating publishers are Berke­ last page. The sentence "For A<; V(G) we write & ley Bectronic Press, Elsevier, John Wiley Sons, Symposium GA = {y I :Jx E A say that (x, y) is an edge of G} ."should Journals, and United Kingdom Serials Group. For more in­ have read "For A<; V(G) we write GA = {y I :Jx E A such formation on Portico, visit the website http: 1 jwww. that (x, y) is an edge of G} ." The word "say" was erro­ portico. org. neously substituted for the word "such". -Sandy Frost -Allyn jackson Deaths of AMS Members Department Charrs Workshop ADHIR K. BAsu of Calcutta University, died in April 2005. The AMS hosted a one-day workshop for mathematical sci­ Born on September 10, 1941, he was a member of the So­ ences department chairs at the 2006 Joint Mathematics ciety for 2 5 years. Meetings in San Antonio, Texas. This year's workshop fo­ ]AAK LoHrvrus , research associate, Tartu University, Esto­ cused on a number of areas of importance to department nia, died on February 23, 2006. Born on September 28, 1937, chairs, including: utilizing resources in tight budget envi­ he was a member of the Society for 8 years. ronments, departmental assessment and long-range plan­ PERE RODRIGU EZ MUMBRU , assistant professor, University ning, evaluation and development of faculty and staff, of Barcelona, died in July 2005. Born on November 19, 1955, and department engagement within and outside the in­ he was a member of the Society for 15 years. stitution. Over thirty-five department chairs and leaders CHANDRA S. SHARMA, retired, from London, England, died came together to share ideas and experiences in a form of in December 2005. Born on June 17, 1933, he was a mem­ department chair therapy, thus creating an environment ber of the Society for 30 years. that enabled attendees to address departmental matters ]AMES G. WENDEL, retired, from Portland, OR, died on from new perspectives. January 16, 2006. Born on April18, 1922, he was a mem­ ber of the Society for Workshop leaders included: Krishnaswami Alladi, chair 60 years. BENJAJviiN YANDELL, a writer from Pasadena, CA, at the University of Florida; Deanna Caveny, chair at Col­ died on August 25, 2004. Born on March 16, 1951, he was a mem­ lege of Charleston; David Manderscheid, chair at the Uni­ ber of the Society for 5 years. versity of Iowa; and Peter March, chair at the Ohio State University.

592 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Reference and Book List

The Reference section of the Notices See the CBMS website, http: I lwww. May 1, 2006: Applications for AWM is intended to provide the reader with cbmsweb.orgiNSFI2007_call .htm, Travel Grants. See http: I lwww. awm­ frequently sought information in or contact: Conference Board of the math. orgltrave l grants. html; tele­ an easily accessible manner. New Mathematical Sciences, 1529 Eigh­ phone 703-934-0163; email: awm@ information is printed as it becomes teenth Street, NW, Washington, DC math. umd. edu; or contact Associa­ available and is referenced after the 20036; telephone: 202-293-1170; tion for Women in Mathematics, first printing. As soon as information 11240 Waples Mill Road, Suite 200, fax: 202-293-3412; email: l kol be@ is updated or otherwise changed, it Fairfax, VA 22030. [email protected]. will be noted in this section. May 31, 2006: Registration for the April 20, 2006: Proposals for NSF Thirteenth International Mathemat­ Contacting the Notices Program in Mathematical, Social, and ics Competition for University Stu­ The preferred method for contacting Behavioral Sciences. See http: I lwww. dents (IMC). See the website http: I I the Notices is electronic mail. The nsf.govlpublicationslpub_summ. www. i me-math. org or contact John E. editor iS the person to whom to send jsp?ods_key=nsf06531&org=NSF. Jayne, Department of Mathematics, articles and letters for consideration. Articles include feature articles, memorial articles, communications, Where to Find It opinion pieces, and book reviews. The A brief index to information that appears in this and previous issues of the Notices. editor is also the person to whom to AMS Bylaws- November 2005, p. 1239 send news of unusual interest about AMS Email Addresses- February 2006, p. 251 other people's mathematics research. AMS Ethical Guidelines-june/July 2004, p. 675 The managing editor is the person AMS Officers 2005 and 2006 (Council, Executive Committee, to whom to send items for "Mathe­ Publications Committees, Board of Trustees)- May 2005, p. 604 matics People", "Mathematics Op­ AMS Officers and Committee Members-October 2005, p. 1073 portunities", "For Your Information", Conference Board of the Mathematical Sciences-September 2005, "Reference and Book List", and "Math­ p. 892 ematics Calendar". Requests for permissions, as well as all other Information for Notices Authors-june/July 2005, p. 660 inquiries, go to the managing editor. Mathematics Research Institutes Contact Information-A ugust 2005, The electronic-mail addresses are p. 770 noti ces@math. ou. edu in the case of National Science Board-january 2006, p. 62 the editor and noti ces@ams. org in New journals for 2004-june/july 2005, p. 662 the case of the managing editor. The NRC Board on Mathematical Sciences and Their Applications- March 2006, fax numbers are 405-325-7484 for p. 369 the editor and 401-331-3842 for the NRC Mathematical Sciences Education Board-April 2006, p. 488 managing editor. Postal addresses NSF Mathematical.and Physical Sciences Advisory Committee-February may be found in the masthead. 2006,p. 255 Program Officers for Federal Funding Agencies-October 2005, Upcoming Deadlines p. 1069 (DoD, DoE); November 2005, p. 1223 (NSF) April 7, 2006: Proposals for 2007 Stipends for Study and Travel- September 2005, p. 900 NSF-CBMS Regional Conferences.

MAY 2006 NOTICES OF THE AMS 593 Reference and Book List

University College London, Gower may be sent to noti ces-bookl i st@ Princeton University Press, November Street, London WClE 6BT, United ams.org. 2004. ISBN 0-691-11530-3. Kingdom; telephone +44-20-7679 ''Added to "Book List" since the '' The Coxeter Legacy: Reflections 7322; fax +44-20-7419 2812; email: list's last appearance. and Projections, edited by Chandler j. j ayne@i me-math. or g. Davis and Erich W. Ellers. AMS, March june 1, 2006: Applications for fall A 3 & His Algebra: How a Boy from 2006. ISBN 0-8218-3722-2. program of the Christine Mirzayan Chicago's West Side Became a Force in The Curious Incident of the Dog in Science and Technology Policy Grad­ American Mathematics, by Nancy E. the Night-time, by Mark Haddon. Vin­ uate Fellowship Program of the Na­ Albert. iUniverse, Inc., January 2005. tage, May 2004. ISBN 1-400-03271-7. tional Academies. See the website ISBN 0-595-32817-2. (Reviewed De­ (Reviewed March 2006.) http:llwww7.nationalacademies. cember 2005.) Dark Hero of the Information Age: o rgl pol i cyfe ll ows, or contact The Beyond Coincidence: Amazing Sto­ In Search of Norbert Wiener, by Flo National Academies Christine Mirza­ ries of Coincidence and the Mystery Conway and Jim Siegelman. Basic yan Science and Technology Policy and Mathematics behind Them, by Books, December 2004. ISBN 0-738- Graduate Fellowship Program, 500 Martin Plimmer and Brian King. 20368-8. (Reviewed in this issue.) 5th Street, NW, Room 508, Washing­ Thomas Dunne Books, December Decoding the Universe: How the New ton, DC 20001; telephone: 202-334- 2005. ISBN 0-312-34036-2. Science of Information Is Explaining 2455; fax: 202-334-1667. The Book of Presidents, by Susan Everything in the Cosmos, from Our June 6, 2006: Proposals for NSF En­ Oakes, Alan Pears, and Adrian Rice. Brains to Black Holes, by Charles Seife. hancing the Mathematical Sciences London Mathematical Society, 2005. Viking Adult, February 2006. ISBN 0- Workforce in the Twenty-First Century. ISBN 0-950-27341-4. 670-03441-X. See "Mathematics Opportunities" in A Brief History of Infinity, by Paolo Divine Proportions: Rational this issue. Zellini. Penguin Books (paperback), Trigonometry to Universal Geometry, June 22, 2006: Full proposals for March 2005. ISBN 0-141-00762-l. by N. J Wildberger. Wild Egg Books, September 2005. ISBN 0-9757492-0-X. NSF Program in Informal Science Ed­ The Calculus Gallery: Masterpieces The Equation That Couldn't Be ucation. See http: I lwww. nsf. gov I from Newton to Lebesgue, by William Solved (How Mathematical Genius Dis­ publicationslpub_summ.jsp?ods_ Dunham. Princeton University Press, covered the Language of Symmetry), key=nsf06520. December 2004. ISBN 0-691-09565-5. by Mario Livio. Simon and Schuster, july 31,2006: Nominations for the Chance: A Guide to Gambling, Love, September 2005. ISBN 0-743-25820-7. ICTP I IMU Ramanujan Prize. See the Stock Market and just About Every­ M. C. Escher's Legacy: A Centen­ "Mathematics Opportunities" in this thing Else, by Amir D. Aczel. Thun­ nial Celebration, edited by Doris issue. der's Mouth Press, October 2004. ISBN Schattschneider and Michele Emmer. October 1, 2006: Applications for 1-56858-316-8. (Reviewed August Springer, September 2005 (paperback 2005.) AWM Travel Grants. See http: I lwww. edition). ISBN 3-540-20100-9. awm-math.orgltravelgrants.html; Change Is Possible: Stories of Women ''Euler through Time: A New Look telephone 703-934-0163; email: and Minorities in Mathematics, by Pa­ at Old Themes, by V. S. Varadarajan. awm@math. umd. edu; or contact As­ tricia Clark Kenschaft. AMS, Septem­ AMS, June 2006. ISBN 0-8218-3722-2. sociation for Women in Mathematics, ber 2005. ISBN 0-8218-3748-6. Experimentation in Mathematics: 11240 Waples Mill Road, Suite 200, Coincidences, Chaos, and All That Computational Paths to Discovery, by Fairfax, VA 22030. Math jazz: Making Light of Weighty Jonathan Borwein, David Bailey, and Ideas, by Edward B. Burger and Michael Roland Girgensohn. A K Peters, March Book List Starbird. W. W. Norton, August 2005. 2004. ISBN 1-56881-136-5. (Reviewed The Book List highlights books that ISBN 0-393-05945-6. September 2005.) have mathematical themes and are The Colours of Infinity: The Beauty The Fermat Diary, by C. J Mozzochi. aimed at a broad audience potentially and Power of Fractals, by Michael AMS, August 2000. ISBN 0-8218- including mathematicians, students, Barnsley, Nigel Lesmoir-Gordon, 2670-0. and the general public. When a book Benoit B. Mandelbrot, Ian Stewart, The Fermat Proof, by C.]. Mozzochi. has been reviewed in the Notices, a Gary Flake, Robert Prechter, and Trafford Publishing, Inc., February reference is given to the review. Gen­ Arthur C. Clarke. Clear Press, March 2004. ISBN 1-412-02203-7. erally the list will contain only books 2004. ISBN 1-904-55505-5. Geometry and Meaning, by Dominic published within the last two years, Complexities: Women in Mathe­ Widdows. Center for the Study of Lan­ though exceptions may be made in matics, edited by Bettye Anne Case guage and Information, November cases where current events (e.g., the and Anne M. Leggett. Princeton Uni­ 2004. ISBN 1-5 75-86448-7. death of a prominent mathematician, versity Press, January 2005. ISBN God Created the Integers, by Stephen coverage of a certain piece of mathe­ 0-691-11462-5. Hawking. Running Press, October 2005. matics in the news) warrant drawing Converging Realities: Toward a ISBN 0-762-41922-9. readers' attention to older books. Sug­ Common Philosophy of Physics and Godel's Theorem: An Incomplete gestions for books to include on the list Mathematics, by Roland Omnes. Guide to Its Use and Abuse, by Torkel

594 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Reference and Book List

Franzen. A K Peters, May 2005. ISBN The Man Who Knew Too Much: Alan Numbers, the Language of Science, 1-56881-238-8. Turing and the Invention of the Com­ by Tobias Dantzig. Pi Press, fifth edi­ Graphic Discovery: A Trout in the puter, by David Leavitt. Great tion, March2005. ISBN 0-131-85627-8. Milk and Other Visual Adventures, by Discoveries series, W. W. Norton, The Oxford Murders, by Guillermo Howard Wainer. Princeton University December 2005. ISBN 0-393-05236-2. Martinez. Abacus, January 2005. ISBN Press, October 2004. ISBN 0-691- The Math Instinct: Why You're a Math­ 0-349-11721-7. (Reviewed November 10301-1. ematical Genius (Along with Lobsters, 2005.) Hiding in the Mirror: The Mysteri­ Birds, Cats, and Dogs), by Keith Devlin. The Pea and the Sun: A Mathe­ ous Allure of Extra Dimensions, from Thunder's Mouth Press, March 2005. matical Paradox, by Leonard M. Plato to String Theory and Beyond, by ISBN 1-5602 5-672-9. Wapner. A K Peters, April2005. ISBN Lawrence M. Krauss. Viking Adult, Mathematical Adventures for Stu­ 1-56881-213-2. October 2005. ISBN 0-670-03395-2. dents and Amateurs, David F. Hayes Fiero della Francesca: A Mathe­ Incompleteness: The Proof and and Tatiana Shubin, editors. Mathe­ matician's Art, by]. V. Field. Yale Paradox of Kurt Gddel, by Rebecca matical Association of America, 2004. University Press, August 2005. ISBN Goldstein. W. W. Norton & Company, ISBN 0-88385-548-8. 0-300-10342-5. February 2005. ISBN 0-393-05169-2. Mathematical fllustrations: A Manual PopCo, by Scarlett Thomas. Har­ (Reviewed April 2006.) ofGeometry and PostScript, by Bill Cas­ . ''Infinite Ascent: A Short History of selman. Cambridge University Press, vest Books, October 2005. ISBN 0-156- Mathematics, by David Berlinski. December 2004. ISBN 0-521-54788-1. 03137-X. (Reviewed February 2006.) Modern Library, September 2005. ISBN Mathematical Musings: A Collection Probability Theory: The Logic of 0-679-64234-X. ofQuotes, edited by Dan Sonnenschein. Science, by E. T. Jaynes. Edited by G. The Infinite Book: A Short Guide to Clarium Press, November 2005. ISBN Larry Bretthorst. Cambridge Univer­ the Boundless, Timeless and Endless, 0-9697688-8-5. sity Press, April 2003. ISBN 0-521- by John D. Barrow. Pantheon, August Mathematics by Experiment: Plau­ 592 71-2. (Reviewed January 2006.) 2005. ISBN 0-375-42227-7. sible Reasoning in the 21st Century, by Reality Conditions: Short Mathe­ Introducing Game Theory and Its Jonathan Borwein and David Bailey. matical Fiction, by Alex Kasman. Math­ Applications, by Elliott Mendelson. A K Peters, December 2003. ISBN ematical Association of America, May CRC Press, July 2004. ISBN 1-584- 1-56881-211-6. (Reviewed September 2005. ISBN 0-88385-552-6. 88300-6. 2005 .) Reflections: V I. Arnold's Reminis­ j(inos Bolyai, Euclid, and the Nature Mathematics in Nature: Modeling ofSpace, by Jeremy]. Gray. MIT Press, cences, byV. I. Arnold. Springer, April May 2003. ISBN 0-262-57174-9. (Re­ Patterns in the Natural World, by 2006. ISBN 3-540-28734-5. viewed October 2005 .) John A. Adam. Princeton University Science in the Looking Glass, by john Pell (1611-1685) and His Press, November 2003. ISBN 0-691- E. Brian Davies. Oxford University Correspondence with Sir Charles 11429-3. (Reviewed june/July 2005.) Press, August 2003. ISBN 0-198- Cavendish: The Mental World of an Meta Math! The Quest for Omega, by 52543-5. (Reviewed December 2005.) Early Modem Mathematician, by Noel Gregory Chaitin. Pantheon, October Sneaking a Look at God's Cards: Malcolm and Jacqueline Stedall. 2005. ISBN 0-3 75-42313-3. Unraveling the Mysteries of Quantum Oxford University Press, second R. L. Moore: Mathematician and Mechanics, by Giancarlo Ghirardi, edition, January 2005. ISBN 0-198- Teacher, by John Parker. Mathemati­ translated by Gerald Malsbary. Prince­ 56484-8. cal Association of America, 2004. ISBN ton University Press, revised edition, The Knot Book: An Elementary 0-88385-550-X. January 2005. ISBN 0-691-12139-7. More Mathematical Astronomy Introduction to the Mathematical Space/and, by Rudy Rucker. Tor Morsels, by Jean Meeus. Willmann­ Theory ofKnots, Colin C. Adams. AMS, Books, June 2002. ISBN 0-765-30366- Bell, 2002. ISBN 0-943396-743. September 2004. ISBN 0-8218-3678-1. 3. (Reviewed August 2005.) (Reviewed September 2005.) Musings of the Masters: An An­ thology of Miscellaneous Reflections, The Three Body Problem, by Cather­ Knots and Links, by Peter R. ine Shaw. Allison and Busby, March Cromwell. Cambridge University Press, edited by Raymond G. Ayoub. 2005. ISBN 0-749-08347-6. October 2004. ISBN 0-521-83947-5. Mathematical Association of Amer­ A Tour through Mathematical Logic, Luck, Logic, and White Lies: The ica, June 2004. ISBN 0-88385-549-6. Mathematics of Games, by Jorg New Mexico Mathematics Contest by Robert S. Wolf. Mathematical Bewersdorff. Translated by David Problem Book, by Liong-shin Hahn. Association of America, January 2005. Kramer. A K Peters, November 2004. University of New Mexico Press, No­ ISBN 0-88385-036-2. ISBN 1-56881-210-8. vember 2005. ISBN 0-8263-3534-9. Using the Mathematics Literature, Saunders Mac Lane: A Mathemati­ The Newtonian Moment: Isaac New­ by Kristine K. Fowler. Marcel Dekker, cal Autobiography, by Saunders ton and the Making ofModern Culture, June 2004. ISBN 0-824-75035-7. Mac Lane. A K Peters, May 2005. ISBN by Mordechai Feingold. New York The Visual Mind II, edited by 1-56881-150-0. (Reviewed December Library and Oxford University Press, Michele Emmer. MIT Press, May 2005. 2005.) December 2004. ISBN 0-195-17735-5. ISBN 0-262-05076-5.

MAY 2006 NOTICES OF THE AMS 595 Read Something Different

Nonlinear Classical Dr. Euler's Optimizatio~ CHARLES R. MACCLUER Mathematical Fabulous Formula ANDRZEJ RUSZCZYNSKI This is the first modern Cures Many Mathematical Ills calculus book to be organized Logic . Nonlinear Optimization is the The Semantic Foundations PAUL J. NAHIN axiomatically and to survey the first comprehensive treatment of Logic subject's applicability to science In the mid-eighteenth century, of optimization that will and engineering. A cha ll enging RICHARD L. EPSTEIN Swiss-born mathematician allow graduate students and exposition of calculus in the With contributions by Leonhard Euler developed a researchers to understand its European style, it is an excellent Lesfaw W Szczerba formula so innovative and modern ideas, principles, and complex that it continues text for a first-year university Richard L. Epstein relates the methods within a reasonable to inspire research and honors course or for a third­ systems of mathematical logic time, but without sacrificing discussion. Dr. Euler's F,abulous year analysis course. The to their original motivations mathematical precision. The Formula shares the fas

American Mathematical Society Contributions

Dear Friends and Colleagues,

During 2005 your generous support helped the Society and our profession in many ways. I thank each of you for that support.

The Young Scholars program is in its sixth year, supporting summer workshops for talented high school students-the future of our profession. We are building an endowment, the Epsilon Fund, to support this program far into the future, and we hope to reach our goal of two million dollars over the next few years. Supporting such programs is important for mathematics.

The Centennial Fellowships play a key role for outstanding young mathematicians at the formative stages of their careers, from three to twelve years beyond the degree. These fellowships are funded by contributions from mathematicians throughout the world.

We use contributions to the General Fund to support all of our activities, including survey work, public awareness, and outreach to mathematicians in the developing world.

Your generosity allows the Society to carry out all these programs and shows that mathematicians care deeply about our profession. Thank you for that expression of caring.

John H. Ewing

Thomas S. Fiske Society

The Executive Committee and Board of Trustees have established the Thomas S. Fiske Society to honor those who have made provisions for the AMS in their estate plans. For further information contact the Development Office at 800-321-4AMS, or deve l opment@ams. o rg.

Roy L. Adler KyFan Ralph Mansfield Hemy M. Schaerf Pedro B. Barquero Isidore Fleischer Trevor McMinn Margaret W. Taft Kathleen Baxter Ramesh Gangolli Cathleen Synge Morawetz B. A. Taylor Shirley and Gerald Bergum Rosalind Guaraldo Franklin P. Peterson Steven H. Weintraub Shirley Cashwell Yanguang Li Moshe Rosenfeld Sally Whiteman Carl Faith Joseph S. Mamelak Theda Salkilld

Bequests Received

Barbara ]. Beechler

598 NOTICES OF THE AMS VOLUME 53, NUMBER 5 AMS Contributions

Gifts in Memory and Gifts in Honor The American Mathematical Society welcomes gifts made in memory or honor of members of the mathematical com­ munity or others. Unless directed toward a special fund or program, such gifts are used to support the general mission of the Society.

Gifts were made in memory of the following individuals: Eugene A. Pflumm by Mollie Pf/umm Joseph L. Doob by Paul R. and Virginia Halmos Irving Reiner by Irma M. Reiner Paul Erdos by john Morse David P. Robbins by Ann and Charles Aknin Murray Klamkin by Susan Schwartz Wildstrom David P. Robbins by Gary and Laurie Aknin Saunders Mac Lane by the Communications Staff at the Rockefeller David P. Robbins by Marjorie Friedlander Foundation with a matching gift from the Rockefeller Foundation David P. Robbins by Deborah Robbins Vincent 0 . McBrien by Wa lter F. Brady David P. Robbins by Matthew Robbins Vincent 0 . McBrien by Marie L. Carbone David P. Robbins by Peter Robbins Vincent 0 . McBrien by Daniel G. Dewey David P. Robbins by Sheila johnson Robbins Vincent 0 . McBrien by Friends at The Louis Berger Group, Inc. Arnold Ross by jim Stasheff Vincent 0 . McBrien by Friends of Dr. Stephen McBrien at MITRE Hans Samelson by Steven E. Blasberg Corporation Hans Samelson by Nancy Samelson Vincent 0 . McBrien by joseph W Paciorek Albert Leon Whiteman by Sally Whiteman Vincent 0. McBrien by Peter Perkins Vincent 0. McBrien by William F. Reynolds Gifts were made in honor of the following individuals: Vincent 0 . McBrien by Patrick Shanahan Donna R. Claire by Katherine Pardee Vincent 0. McBrien by Me lvin C. Tews William]. LeVeque by ]ames W. Maxwell Vincent 0 . McBrien by Susan Zaverucha and Family Philip Rosenau by julian R. Karelitz

Contributors to the AMS During 2005 '' Donors who have given for three years consecutively. £ Donors who have given to the AMS Epsilon Fund, the endowment for the support of young scholars programs. The names of donors who have given $1,000 or more in a single year are affixed to a plaque that is prominently displayed in the Society's headquarters.

PRESIDENT'S e'•Samuel Murray Rankin III PATRONS '' Donald L. Burkholder e Lincoln K. Durst Societe Mathematique e"Thomas R. Butts e John W. Duskin Jr. (G ifts of $100 and above) ASSOCIATES de France Karl E. Byleen e''Ernest ]. Eckert (G ifts of $5,000 and above) '~ No rton Starr • Roy L. Adler e'' j ames L. Calvert David Albert Edwards Ann L. & Charles Aknin e'' jean E. de Val pine e" Alfred Aeppli e" Bengt G. Carlson Mitchell Wayne Ehrlich Alan & Katherine Stroock e"Steven H. Weintraub e"Michael l. Aissen e" Robert G. Cawley Elmer Eisner Fund €"Susan Schwartz Wildstrom e j. Ralph Alexander Jr. Paul G. Chauveheid <'' Philip G. Engstrom Marjorie Friedlander Anonymous (2) * Neal R. Amundson e*Stuart Citrin e John M. Erdman Paul R. & Virginia Halmos e"Kenneth I. Appel e'' William A. Clee e" Leonard Evens e* Harry Lucas Jr. James G. Arthur e'' Daniel I. A. Cohen john C. Fenley Deborah Robbins SPONSORS e*Donald G. Babbitt E'' Charles A. Cole E Aurino Ribeiro Filho Matthew Robbins (Gifts of $500 and above) e'' George Bachman E Paul Dana Cole <''Benjamin Fine joseph A. Ball E Vincent E. Co li Jr. E'' Gerald B. Folland Peter Robbins '' Arielle & jerald Brodkey e'' Donald H. Ballou £·'Pierre E. Co nner Jr. ··• Paul Fang Sheila Johnson Robbins Philanthropic Fund E*Thomas R. Savage Arienne S. Balser E''Peter S. Constantin Friends of Dr. Stephen Gary & Laurie Aknin e'• Robert and Maria W. E'' john A. Beekman * Arthur H. Copeland Jr. McB rien Walter F. Brady Steinberg E Gene A. Berg " H. Cornet E Robert A. Fuller * William L. Duren Jr. '' Sally Whiteman E Alan E. Berger * Louis ]. Co te £'' Joseph M. Gani E''Tirnothy & Laurie Francis- * Gerald E. Bergum E'' Albert W. Currier " Irving E. Gaskill Wright ASSOCIATES E··• Leonard D. Berkovitz * Philip C. Curtis Jr. E Pedro]. Geraldo Cabrera .., Andrew M. Gleason E Manuel P. Berriozabal E'.'Everett C. Dade E"S tephen P. Gill (G ifts of $1,000 and above) Kazuyuki Hatada e Gerald G. Bilodeau john P. Dalbec e Richard M. Gillette E"Richard D. Anderson E" Joseph Kist c.''' Joan S. Birman e" Paul L. Davis e'' Anthony A. Gioia E Abraham Arad Ho chman E''Robert V. Kohn e* Richard L. Bishop Franklin D. Demana <'' Richard P. Goblirsch <''Nathaniel Chafee " Gary ]. Kurowski E* Peter B. Bjorkiund '' Ralph E. DeMarr Elliot C. Gootman E*William Craig ·• Ralph Mansfield E* Jerome Blackman €"Guy M. De Primo e.''' Curtis Greene E'' John H. Ewing E'' james W. Maxwell Raul Bravo e Charles A. Desoer E Wilfred Martin Greenlee * Carl E. Harrell E''Ernest A. Michael E"David M. Bressoud E'' Robert L. Devaney e Robert Louis Griess Jr. john M. Hosack e"M. Susan Montgomery * Jerald S. Brodkey E'' Charles E. Dickerson e Phillip A. Griffith '' Phyllis & Donald Kahn E" Donald S. & Shari Ornstein E" john Bromback e David Diminnie E Elizabeth Grossman & Philanthropic Fund E Aruna & David Spencer John E. Brothers E''Gerald P. Dinneen Joshua Boorstein e'•George F. Leger e'' joel H. Spencer E"Robert L. Bryant E"Robert L. Druet e''Victor W. Guillemin E''William James Lewis E" Buck Ware E.., joseph T. Buckiey E Krishan La! Duggal E Susan C. Gurney * jacques Neveu E Hermann Zapf e··•S tephen S. Bullock E Loyal Durand * Geo rge F. Haddix E* Joseph W. Paciorek Anonymous (3) E'' james E. Burke E* Peter L. Duren E''William D. Hahn

MAY 2006 NOTICES OF THE AMS 599 AMS Contributions

£"Ruth M. Hailperin e* David B. Massey e Tina H. Straley E Dinu D. Amzar E* Aldridge K. Bousfield e''Deborah Tepper Haimo e Ronald M. Mathsen * G. Ralph Strohl Jr. E Fredric Davis Ancel E*Ward D. Bouwsma e''Mary-Elizabeth Hamstrom e* Jacob R. Matijevic e"' Richard W. Sullivan ··· Allan G. Anderson Paul j. Bowron e Carsten Hansen R. Daniel Mauldin E Communications Office Staff E* Douglas R. Anderson E* Mike Boyle £"Bill Hassinger Jr. e··• Stephen B. Maurer at Rockefeller Foundation E* Joel H. Anderson e Brian D. Bradie e'' Adam O'Neill Hausknecht ,., Dusa McDuff E Barbara L. Tabak E john D. Anderson ·• John S. Bradley e Mikihiro Hayashi E''T. G. McLaughlin B. A. Taylor E'' Michael T. Anderson e·• Richard C. Bradley e David R. Hayes E Brockway & Audrey McMillan jean E. Taylor E Susan Andima E Robert E. Bradley E'' Sigurdur Helgason * Robert F. McNaughton Jr. Melvin C. Tews E'' Peter P. Andre Wray G. Brady ··• Henry Helson Rosemary J. Menger Gilmore E Peter j. Thomas £*George E. Andrews * Louis R. Bragg

E Sandra Larelle Hendon E Richard C. Metzler 1' james G. Timourian £* Michael M. Anshel £*Kenneth A. Brakke e*Gerald A. Heuer * Kenneth S. Miller John Todd E''Peter H. Anspach E Robert L. Brandon e* Michael ]. Heumos £*Guido Mislin * Hing & Mary Powderly Tong * Stuart S. Antman e* George U. Brauer * Gloria C. Hewitt E*Peter L. Montgomery Selden Y. Trimble V E* Myla M. Archer E'' Richard P. Brent e*Peter]. Hilton E''Richard A. Moore £* Timothy Guy Trucano £*Richard F. Arenstorf * James G. Bridgeman * Roger E. Howe * Cathleen S. Morawetz E Masayoshi Tsutsumi '' Susumu Arlki E* judith E. Broadwin e'~w. L. Hoyt E Alberto Cezar Moreira University of Helsinki £*Thomas E. Armstrong E Shirley T. Bromberg , •• ] ames G. Huard * Yasuhiro Morita E Donovan H. Van Osdol Robert L. Amberg E'' Edgar H. Brown Jr. '' Joseph A. Hughes E Ronald G. Mosier E*Wolmer V. Vasconcelos E Masahiko Atsuji '' Kenneth S. Brown e'' George W. Hukle E Paul S. Muhly Sophia Vassiliadou & e Luchezar L. Avramov £''Lawrence G. Brown E'' Craig L. Huneke E'' Albert A. Mullin Thomas ]. Haines E joel D. Avrln e'' Robert F. Brown e Walker E. Hunt e Tadashi Nagano * David A. Vogan Jr. E* Sheldon Axier E* Russell M. Brown Franklin T. Iha Masaomi Nakata E*Karen Vogtmann E Edward A. Azoff e Andrew M. Bruckner Yulij Sergeevich Ilyashenko Mie Nakata £* Evelyn K. Wantland E'' Kiyoshi Baba E Robert R. Bruner E'' I. Martin Isaacs E'' Peter E. Ney * Frank W. Warner III E Victor Adekola Babalola Barry W. Brunson e* Eugene Isaacson e Eugene D. Nichols £''William E. Warren ··· William G. Bade ··· Billy F. Bryant e''Lloyd K. Jackson e* Charles A. Nicol * Ellen Westheimer e* Richard ]. Bagby E Rebecca A. Buchanan e Arthur M. Jaffe Paul ]. Nikolai ·• Roger A. & Sylvia Margaret E Patrick Bahls £*Nicholas P. Buchdahl ··· Melvin F. Janowitz E''Nobuo Nobusawa Wiegand E* Joni E. Baker E Edwin 0. Buchinan e''George A. Jennings E 0. Timothy O'Meara e'' John F. Wilkinson E"john T. Baldwin E~'Royce E. Buehler ··· David Lee johnson E Peter P. Orlik £''Clinton Curtis Williams '' S. F. Barber E Sebastian loan Buhai E''Donald G. Johnson E"Scott C. Otermat Ruth J. Williams & ]. William '' Carlo Bardaro E Joe P. Buhler Bjarni JOnsson e'''lstvan Ozsvath Helton E* Julio Edgardo Barely * Robert Bumcrot E* Henry Price Kagey ··• Jingyal Pak E Christopher Wolfe ··· Bruce H. Barnes * Daniel Willis Bump E Thomas Kailath Hiram Paley E''Scott A. Wolpert E Domingo Barrera-Rosillo E Almut Burchard e~'Joji Kajiwara E* Alberto Parmeggiani '' jay A. Wood E* jose Barros-Neto Krzysztof Burdzy E*Herbert M. Kamowitz ··· Henry ]. Passerini E* Alan C. Woods E* Philip R. Bartick E John Brian Burghduff Eberhard Kaniuth e*Charles M. Patton E'' Arthur Wouk E* Alexander Barvinok E Daniel M. Burns Jr. E* Juli an R. Karelitz E Mollie Pflumm E Tsu C. Wu E* Hyman Bass £*Warren T. Burns Jr. E* Jerome Karle E* Mark A. Pinsky E Tatsuhlko Yagasaki Ji.irgen 0. Batt Colin ]. Bushnell E Louis H. Kauffman E* George Piranian E*Masayuki Yamasaki E Margaret M. Bayer E* Ralph Stevens Butcher e Jerry L. Kazdan * Aleksey Popelyukhin E Paul C. Yang Jack Bazer Charles L. Byrne e* John F. Kellaher E''M. M. Rao E* ]. Michael Yo he E*]. Thomas Beale E Robert Lawrence Byrom E*Wayne G. Kellner £*Robert]. Reynolds E Roy Young E*R. Michael Beals Luciano Caccianotti E*Harry Kesten Jae Chul Rbo * Michel M. Zarka £*William A. Beck e ]ames C. Cantrell e* Barbara Lee Keyfitz & * john F. Richards * Allen D. Ziebur £* Edward Beckenstein Marie L. Carbone Martin Golubitsky E*Marc A. Rieffel E Steven M. Zucker £*William H. Beckmann e Corrado Cardarelli ,.,,Allan M. Kirch E''Hans P. Rogosinski Anonymous (48) E" Robert Beig e'' Jon F. Carlson e·• Jane & James Kister E*Vijay K. Rohatgi e Sarah-Marie Belcastro Francis W. Carroll e'tMaria Margaret Klawe £*Emilio 0. Roxin Steven R. Bell e'' David W. Carter E''Roland R. Kneece Jr. £''Bernard D. Rudin FRIENDS £''Wolfgang Bell E Andre jean Casadevall e George H. Knightly David Ryeburn (Gifts of less than $100) E David P. Bellamy ·• Alfred S. Cavaretta e Nicholas Jacob Korevaar * Alexander A. Rylov jean M. Abadie e'' Katalin A. Bencsath E Afton H. Cayford E'' Antoni A. Kosinski E'' Jeffrey R. Sachs E William Ablkoff E'' Carlos Benitez E*Gulbank D. Chakerian Adnah G. Kostenbauder Nancy Samelson e* Clarence M. Ab low E'' Georgia Benkart Stefan M. Chaladus E''Thomas R. Kowalski E j. M. Sanz-Serna e Thundiyil Samuel Abraham e George Benke E Bruce Chandler E Ralph M. Krause ··· Paul T. Schaefer e''William P. Abrams E Irving 0. Bentsen E* Jag dish Chandra e Adererni 0. Kuku E Michael Schlessinger E Robert D. M. Accola E Gary D. Berg E''C hao-Ping Chang E Masatake Kuranishi David I. Schneider E' Colin C. Adams E'' George M. Bergman E Weita Chang E S. T. Kuroda £''Richard M. Schoen £''William W. Adams E Elwyn R. Berlekamp E'' Paul jackson Channell Yukio Kusunoki Richard M. Schori E'' lain T. A. C. Adamson * Salvatore D. Bernardi E'' Ruth M. Charney E Carole B. Lacampagne Mark Schroder E Arnold M. Adelberg E*Chris Bernhardt E* Pak Soong Chee E" jeanne LaDuke E Jerry D. Schuur ··· Irving Adler E John A. Berton E'' Concordia C. Chen E'' Joseph A. Langsam Robert T. Seeley e Jeffrey D. Adler '' ]ames S. Bethel Hwa-Nien Chen e* H. Blaine Lawson Jr. Stuart A. Seligson E Alan C. Adolphson £"Gerhard Betsch '' Kwan-Wei Chen e*Walter R. Lawson £''Norman E. Sexauer E''Max K. Agoston e* Meempat Bhaskaran * Robert Chew ··· Alan C. Lazer E''Freydoon Shahidi N. U. Alnned '' Kiaus D. Bierstedt * Peter H. S. Chiang Earl E. Lazerson E Patrick Shanahan E T. M. G. Ahsanullah '' Martin Billik E Theodore S. Chihara e*William G. Leavitt E*Yuji Shirnizuike E"T. Alkou e Terrence Paul Bisson E Sunday C. Chikwendu e'' joan R. Leitzel E William Ivan Shorter e Michael Aizenman E*Denls L. Blackinore * Choong Yun Cho E'' H. W. Lenstra Jr. E'' Allan ]. Silberger E'' Ethan]. Akin David E. Blair E* Jal R. Choksi William]. LeVeque " David B. Singmaster £*Michael 0. Albertson £*William D. Blair £*C harles C. Chouteau e'' H. L. Lewis E* John R. Smart Akram Aldroubi E*Paul E. Bland E Yuan S. Chow * Elliott H. Lieb * Joel A. Smaller E Kenneth S. Alexander E* John D. Blanton £''William E. Christilles e Gary M. Lieberman E Frank Sottile E Roger K. Alexander E* Steven E. Blasberg e'~ Demetrios Christodoulou E Robert J. Lipshutz ·• Steven Robert Spencer E''Gerald L. Alexanderson E Patricia M. Blitch * Phllip T. Church e·• Ming Chit Liu £*Stephen E. Spielberg e''M. Kursheed Ali E''David S. Bloom £*Richard C. Churchlll * Albert T. Lundell * Ivar Stakgold JohnS. Alin E* Gebhard Boeckle E Erhan Cinlar £'' Russell D. Lyons e Friedemann W. Stallmann e* Anne H. Allen E* S. Elwood Bohn E Dana Dwight Clahane e Yoshiaki Maeda Peter F. G. Stanek E Ibrahim H. Al-Rasasi Miklos Bona E Kevin F. Clancey e'' Joseph S. Mamelak * Robert L. Stanley E John P. van Alstyne £* Francis Bonahon E'' Alfred Clark Jr. E Edward Manougian E* Jim Stasheff E Alexander Anthony Ambrioso E William M. Boothby £*Chester Dodge Clark E M. H. Martin e'''Lav.,rence D. Stone * Alain Amiouni £* Rupert D. Boswell Jr. E'' Robert A. Clark

600 NOTICES OF THE AMS VOLUME 53 NUMBER 5 AMS Contributions

E Sharon M. Clarke E Stephen C. Dumars • Jane P. Gilman Christopher M. Herald E Charles H. Jepsen •.., Jack D. Clayton •''William Dart Dunbar Jr. e Gloria Gilmer E* James B. Herreshoff e David j erison

E"Ed Cline 1' Kanat Durgun E'' Maurice Eugene Gilmore E Michael jay Herringshaw Arnold A. johanson E PhJJip A. Cobb £'' Timothy R. Eaton •''Samuel Gitler E Diane L. Herrmann £'' Paul]. Johanson •* Alan Cobham ... Patrick Barry Eb erlein e·' Earl C. Gladue Patricia Hersh David]. John •.. ,John C. Cock • Allan L. Edelson E Milton Alfred Glass Takeyuki Hida £'' Eu gene C. Johnsen E* james Wesley Cogdell Monica jeanne Eder James G. Glimm E Yasunari Higuchi ··• Trygve johnsen E Kurt D. Cogswell E Allan L. Edmonds £* Raoul F. Gloden E Stephen R. Hilbert E Bradford W. Johnson •*Amy Cohen •*Charles H. Edwards Jr. £* ]. D. Go ddard James Porter Hill •*Charles N. Johnson Ralph L. Cohen Robert D. Edwards " Kazimierz A. Goebel £''Shirley A. Hill E D. Randolph Johnson • Henry Lee Cohn •*William I. Eggers Valentina Gogovska E Theodore P. Hill E David Copeland Johnson , .• George Cole E David M. Einstein £'' Robert Gold AJexandrou A. Himonas •*Guy Johnson Jr. E Donald B. Coleman * Staniey Mamour Einstein- •* Samuel Goldberg £''Nancy Hingston E Harold H. Johnson e''' Daniel Comenetz Matthews E'' Seth I. Goldberg F. E. P. Hirzebruch * jon L. Johnson

E Jack Frederick Conn * Sylvan H. Eisman •'' Dorian Goldfeld £ 1' Peter David Hislop E Kenneth David Johnson £''Frank F. Connor ···· David Eliezer £*William Mark Goldman '' Chungwu Ho e*Norman W. johnson E Bruce N. Cooperstein £* joanne Elliott •" Daniel A. Goldston £*Arthur M. Hobbs E Theodore D. Johnson ... Thomas A. Cootz E Deborah Lee Ellis Luis Arbey Gomez '' Harry Hochstadt •* Jeffrey P. Jones '' Heinz 0. Cordes •*Steven P. Ellis E Jose Luis Gomez Pardo E'' Scott H. Hochwald •*William B. Jones ... Thomas Carney Corrigan * Richard S. Elman • Jose M. Gonzalez-Fernandez E'' Jonathan P. E. Hodgson E*Troels jorgensen E Steven R. Costenoble •*Paul W. Eloe £''Kenneth R. Goodearl E'' Helmut H. W. Hofer •*Virginia V. ]ory Ovidiu Costin E Rod Elsdon •''Roe W. Goodman •" Michael E. Hoffman E ]oaquim]. A. Judice £''Malcolm A. Coulter •*Thomas ]. Emerson •''Richard P. Gosselin Detlev W. Hoffmann •* ]ames H. justice Carl C. Cowen E Jessie Ann Engle E John A. Goth E Tom H0holdt £'' Daniel S. Kahn E Robert H. Cowen •* Hans P. Engler E''Yasuhiro Goto •''Charles S. Holmes •* Jeffry N. Kahn E Petru Craciunas E Linda L. Eroh •* Claude Goutier •*Philip John Holmes • Yuichir6 Kakihara •*Stephen H. Crandall E Luis Estrada E Todor V. Gramchev E Fred B. Holt Diana Kalish E Annalisa Crannell e* Kumar Eswaran e''' Kevin A. Grasse E Roger H. Homer £''Richard A. Kanner

E Bruce C. Crauder •*Carl Faith £ 1' Larry K. Graves • Raymond T. Hoobler £*Stanley Kaplan E T. Arleigh Crawford •* Edson de Faria E Richard M. Green E Jennifer L. Hopkins •*Wilfred Kaplan £" Thomas M. Creese e Bruno Farina ··· William L. Green E'' Jean MacGregor Horn e* Joannis Karatzas E Colin W. Cryer '' George F. Feeman E Vladimir A. Greenberg E'' james E. Householder e* Martin Lewis Karel ··• Helen F. Cullen E Howard D. Fegan e''' Peter H. Greene e''' Fredric T. Howard Anatole Katok E AJ Cuoco •'' Mark E. Feighn £'' Frederick P. Greenleaf ··· Henry C. Howard * Svetlana R. Katok Donald L. Curlovic E Arnold D. Feldman Staniey ]. Greif E Gary W. Howell £*Arthur Kaufman •''Charles W. Curtis E Mikhail Feldman Alain A. Grigis ··· j. S. Hsia E William E. Kaufman •* Herbert]. Curtis E1'Norman Feldman E Charles Grimm •" Pao-sheng Hsu •'' Anthony]. Keeping E Michael Cwikel E Jose Humberto Ferreira Rosa Helmut Groemer E1'Tiao-Tiao Hsu E'' James E. Keisler

, ..• David Scott Cyphers E1' Ian M. Ferris E Kenneth I. Gross E'' Archibald Perrin Hudgins • Bernhard M. KeLler e* james N. Damon t'''Victor G. Feser e'~ Leonard Gross c.''' Denise Huet •* Edward L. Keller e*Martin P. Dana " Maurice C. Figueres Edward H. Grossman E Louie C. Huffman E John F. Kennison * John P. D'Angelo £'' David V. Finch E'' Robert Andrew Grossman £" Ruth L. Hughes • Patricia Clark Kenschaft •'' Boris A. Datskovsky E'' Benji N. Fisher E'' Edward A. Grove ,., Mark E. Huibregtse E Candace Marie Kent E Leonardo D'Attorre •" Uri Fixman Gerd Grubb E'' Birge K. Huisgen- E''Efim Khalimsky Donald M. Davis * Mary Elizabeth Flahive ··· Rosalind ]. Guaraldo Zimmermann £ Dima Khavinson e'''Martin D. Davis •*Leopold Flatto £" Esther E. Guerin E Thomas C. Hull * Rudger W. Kieboom £''Anthony T. Dean Bernard A. Fleishman Craig R. Guilbault '' James E. Humphreys • Peter C. K.iessler E''Daniel G. Degiorgi •"Richard]. Fl eming ··· Robert D. Gulliver II E''Thomas W. Hungerford Michael K- H. Kiessling Ronald W. DeGray • Russell Alan Foltz-Smith E Gary G. Gundersen £'' Karen C. Hunt £'' Jun Kigami •'' Percy AJec Deift '' Julie A. Fondurulia • Thomas E. W. Gunnarsson • Paul M. Hunt E Ann Chi Kim E'' Dominick Del Casale E Robert A. Fontenot ··· Wynne Alexander Guy '' Beryl E. Hunte Tatsuo Kimura , ..• Aristide Deleanu • S. Ashby Foote •"Peter Hagis Jr. •"Marcel Hupperich • Donald R. King E Sergei S. Demidov E Hans-Bjorn Foxby James Haglund E Thomas Robert Hurd • Kelley H. Kirklin E joseph B. Dennin E Simon John Fraser E Gerhard E. Hahne •"Michael G. Hurley •* Ellen E. Kirkman £'' Jochen Denzler •"Michael W. Frazier E Theodore Hailperin '' Taqdir Husain E''Kouichi Kiyokawa E John E. Derwent •* Daniel S. Freed •"Richard M. Hain •"Edward L. Hutton E David S. Klatman Dennis DeTurck Marianne Smith Freundlich E Andras Hajnal E'' Jang C. Hwang E Martin Klazar Daniel G. Dewey •*Peter]. Freyd £" David R. Hale £''John jew-Chen Hwang •*Peter H. Kleban £''Fred I. Diamond •*Eugene M. Friedman •'• Robert Joseph Halk Francesco Iachello * Bruce A. Kleiner • Charles R. Diminnie •*Merwyn M. Friedman £'' Brian C. Hall £'' Tom llmanen E Frank B. Knight E Dragomir i . Djokovic E Bent Fuglede E james E. Hall E Stanko lndihar •* Julia Knight £"Glen Allen Dobbs Ryoh Fuji-Hara £'' David Handel e''' Hiroshi Inoue E Jon Kniss •'' Heinz Deitrich Doebner £* Nobuhiko Fujii Robert M. Hardt E Arnold ]. Insel ,., Alan P. Knoerr

E Ogun Dogru lchiro Fujimoto £ 1' Beverly Bailey Hargraves 1' Ron Irving E K. R. K. Knorr •"Pierre E. Dolbeault e* Hisanori Fujita Dorothee D. Haroske • Richard E. Isaac * Richard M. Koch Igor Dolgachev £''William Fulton £'' joe T. Harris Jr. E Tadashige Ishihara * John G. Koelzer • James A. Donaldson •* Jeffry B. Fuqua e''' Dennis E. Harrison £* Noburo Ishii e*Yoshiharu Kohayakawa E Simon K. Donaldson e* Steven Allen Gabriel £''Fred F. Harrop Sahidul Islam E Se men Koksal E" James P. Donaly E'~ Andrei Gabrielov E Deirdre Haskell * Mourad E. H. Ismail E'' Kurt Siegfried Kolbig E Donald R. Doolittle E Peter Gacs E Ralph H. Hautau e*Masanori ltai E Hikosaburo Komatsu • Jay Robert Dorfman Marvin C. Gaer E Jane M. Hawkins e·• Noboru Ito e'''Yasuo Komori •* Jim Douglas Jr. •'' Michael E. Gage E Mary Stephanie Hawkins E Yoshiaki Itoh Heinz J. Konig E'' Ronald G. Douglas E Joseph A. Gallian e··~F umio Hazama E N. M. lvochkina Ralph D. Kopperman E Karl Heinz Dovermann •''Richard L. Gantos E Melvin L. Heard E'' Eric Robert Jablow E'' Gabor Korchinaros E Keith Dow * Howard Garland E Ann M. Heath ... William Burkley Jacob '' Eric]. Kostelich •'' Robert E. Dowd e'''Constantinos Georghiou E William S. Heck E Herve M. Jacquet E Jerald]. Kovacic •" Alex ]. Dragt * John C. Georgiou Dennis A. Hejhal E'' Robert E. Jaffa ····zdislav v. Kovarik •'' Arthur A. Drisko E Claus Gerhardt Leo Hellerman E Martin Jakob £ Erwin Kozakiewicz e''' Bruce K. Driver * Stephen R. Gerig • LaRae C. Helliwell • John Antone jaksich * Irwin Kra ... Thomas L. Drucker Noel Germay Robert L. Hemniinger E David M. James •* Jurg Kramer E Thomas E. Duchamp E Murray Gerstenhaber •''Judson Hendelman • jens Carsten Jantzen •* Raymond F. Kramer Jr. •''Steve N. Dulaney £''Tepper L. Gill £'' Francis McVey Henderson e:''' Trevor M. Jarvis • Bernd Krauskopf

MAY 2006 NOTICES OF THE AMS 601 AMS Contributions

E*Gary R. Krumpholz E''Manohar L. Madan E Robert Laurens Moore •* Robert G. Payton E* Joseph B. Roberts E Peter Kuchment James Joseph Madden E"W. Keith Moore Erik Kjaer Pedersen •*Lois]. Roberts Sharon Kunoff E* Adolf Mader E Frank Morgan E* Lambertus A. Peletier E. Arthur Robinson Jr. E*Yoshiki Kurata E Duraid Madina Motohiro Morii E Richard P. Pembroke '' John Roe E''Robert P. Kurshan E Franco Magri Larry ]. Morley E John W. Pennisten E Hartley Rogers Jr. •*Herbert Kurss E*Richard B. Magruder E L. E. Morris Paul M. Pepper E* David E. Rohrlich E* John F. Kurtzke Jr. * Mehran Mahdavi E Peter C. Morris <''Juan C. Peral E* Judith Roitman E''Ruishi Kuwahara E Robert S. Maier * John A. Morrison Peter Perkins E Dale P. 0. Rolfsen <''Nosup Kwak George Nick Makrakis E John Morse E Robert V. Perlis E Josephine Jardin Romero Jeffrey C. Lagarias * Athena Makroglou •* Joseph G. Moser E'' Peter A. Perry E'' Guillermo Romero Melendez E Kee Y. Lam <''Peter Malcolmson E Lawrence Stuart Moss '' William L. Perry E''William L. Root E John Patrick Lambert E* Joseph Maikevitch Pierre Marie Moussa E Frank E. Peseckis E Colin Raymond Rose William A. Lampe E''David M. Malon E Tomasz S. Mrowka E'' Justin R. Peters E Michael I. Rosen E''Peter S. Landweber E Kenneth L. Manders * Marvin G. Mundt <''Troels Petersen * Robert A. Rosenbaum •* Oscar E. Lanford III E Haria Mantellini E*Nobuyuki Murai E Gary L. Peterson E'' David Rosenberg <''Leo]. Lange E Hrant Babken Marandjian E Grattan P. Murphy <''Holger P. Petersson E* Jonathan M. Rosenberg <''David C. Lantz E Peter D. March E''Takasi Nagahara E W. V. Petryshyn •* Gerald Rosenfeld * Arnold Lapidus E*Margaret 0 . Marchand <*Alexander Nagel E Gotz Eduard Pfander E Roger D. Rosenkrantz E Michel L. Lapidus E Eugene A. Margerum E*Kuniaki Nakamitsu E Christine Melle Phili E Joachim Rosenthal E Michael ]. Larsen * Scott A. Markel e*Kazumi Nakano E I. Piatetski-Shapiro Michael A. Roth E Bruce R. Larson * David E. Marker E* Joseph Neggers E Derk Roelof Pik E Linda Preiss Rothschild E Richard G. Larson E*Charles Michel Marie E Arnold L. Neidhardt Anand Pillay •*Mitchell]. Rothstein

'' Margaret M. LaSalle E David Imler Marshall John D. Nelligan e1' Steven Pincus E*Cecil C. Rousseau * George Laush E*Murray Angus Marshall * Edward 0 . Nelson E'' Gilles Pisier E*Virginia G. Rovnyak •*Lorraine D. Lavallee Mario Umberto Martelli E Patrick W. Nelson •* Sergio Plaza E*Melvin Glenn Royer E Albert F. Lawrence E Jeremy Leander Martin E Robert R. Nelson ··· VeraS. Pless E Wimberly C. Royster E* John W. Lawrence E Senisho Philip Mashike E* Csaba Nemethi E* John William Poduska Sr. e Herman Rubin * Robert F. Lax •*Frank H. Mathis Christoph]. Neugebauer * John C. Polking •* Robert ]. Rubin Arthur Layzer * Yasuo Matsushita •*Walter D. Neumann E Daniel Pollack E Irving Rubinstein E Ian]. Leary E*Farley Mawyer E Lee P. Neuwirth E Robert J. Pollack Joachim H. Rubinstein E George Ledin Jr. E J. Peter May E*Charles W. Neville E Harriet S. Pollatsek E Alexei N. Rudakov * Robert N. Leggett Jr. E* John C. Mayer ·• James A. Nickel E John A. Poluikis ··• Robert S. Rumely E* Shawn Pierre Legrand E*Rafe Mazzeo E Liviu I. Nicolaescu •*Robert T. Powers E Bernard Russo ]. Larry Lehman E* Stephen ]. McAdam E'' Lance W. Nielsen E Narahari Umanath Prabhu <*Leon W. Rutland Jr. Gerald M. Leibowitz * Michael ]. McAsey E'' Gerhard Nikiasch E* Dipendra Prasad E Donald K. Ryan E Roy B. Leipnik E Dennis ]. McCaughan E Louis Nirenberg E John K. Prentice E Cihan K. Saclioglu <*Cecil E. Leith Gregory L. McColm E Seiki Nishikawa E*Dean L. Preston E* Anthony Sacremento E*Manoel Jose M. S. Lemos Robert M. McConnel Zbigniew H. Nitecki E* David S. Protas E Albert W. Saenz E Andrew Lenard E Mark W. McConnell Ricardo H. Nochetto <* J6zef H. Przytycki E'' Salem A. Sahab e*Hernan Leon E Marjorie Frost McCracken <''Scott R. Noller E Paul R. Pudaite E Kazuyuki Saito E HenryS. Leonard Jr. Michael M. McCrea E Virginia A. Noonburg E Francis Quayson E''Takashi Sakai E Nikolay Nikolayevich E'' John G. McDonald F. Alexander Norman E Michael F. Quinn E Toshio Sakata Leonenko E Michael ]. McGraw * John W. Norris E Adbeel N. Quinones E Susana Alicia Salamanca-Riba <*James I. Lepowsky E*William D. Mcintosh E Olav Kristian Nygaard E*Paul H. Rabinowitz E Hector N. Salas E*Steven C. Leth E''Thomas G. McKay E* Duane Q. Nykamp E Serban Raianu E'' Habib Salehi * Robert]. Levit E* Lionel W. McKenzie E* Serge Ochanine <''Louis B. Rail * Salim W. Salem E* Andrew M. Lewis E George Joseph McNinch Mitsuyuki Ochiai E*Melapalayam S. Ramanujan E Scott Anthony Sallberg E'' George M. Lewis E* Robert C. McOwen * Andrew M. Odlyzko ··· R. Michael Range •*Laurent Saloff-Coste E*L Gaunce Lewis Jr. <* John C. Meakin E* Hajimu Ogawa E Salvatore Rao E Hatem Abdul-Mohsin E Roger T. Lewis E David Meier •* Andrew P. Ogg '' John Elton Rawson Samman E''Frederick W. Leysieffer E*Morris]. Meisner * Yong-Geun Oh E Dwijendra K. Ray-Chaudhuri E1'Pierre Samuel ·• Yanyan Li E Ljudnula Meister E Michael L. O'Leary E Frank Raymond •*Robert W. Sanders •• Stephen Lichtenbaum Aramanovich E* Robert F. Olin E*Maxwell 0 . Reade * Angel San Miguel E Paulo Figueiredo Lima * Anders Melin E Paul D. Olson * Douglas C. Reber •*Donald E. Sarason <*Shen Lin •* Jose M. R. Mendez-Perez <*Pe ter]. Olver •*Don Redmond E*Chelluri C. A. Sastri E* John E. Lindgren E*Bruce E. Meserve E John Arthur Oman E"Timothy Redmond E* Hiraki Sa to E''Peter A. Linnell E Siavash Meshkat E''Philip]. O'Neil * Christopher L. Reedy e Ken~iti Sa to •*Miriam A. Lipschutz-Yevick * Herman Meyer * Takashi Ono E* David E. Reese E C. Sauerbier * Sally Irene Lipsey <*Jean-Pierre G. Meyer ··· Yoshitsugu Oono * Ernestine Reeves-Hicks William G. Saunders E*William G. Lister E Lung-Chi Miao · Kent Orr E'' Eugenio Regazzini * Stanley A. Sawyer Leigh]. Little * Michael H. Millar E" Mason S. Osborne E A. ]. Reichwein <*Richard C. Scalzo E*Chiu-Chu Melissa Liu Ellen Rammelkamp Miller E'' Steve G. Oslon <*William H. Reid E* Michelle Schatzman E*George W. Lofquist E Haynes R. Miller E''James M. Osterburg e* lrma M. Reiner * Gideon Schechtman Terry M. Lohrenz E Hugh Miller E Javier Oral E*Robert B. Reise! E Ernest C. Schlesinger E*Walter L. Lok E*Michael]. Miller E'' James C. Owings Jr. <*Peter A. Rejto E Maria Elena Schonbek E Mahir Lokvancic E Russell G. Miller E''Michio Ozeki * Richard S. Rempel E''Roberto H. Schonmann John M. Long <''Thomas Len Miller E''Cardenas Agustin Pacheco William F. Reynolds ·• John Schue <*William C. Lordan E Victor Saul Miller E* Judith A. Packer E*Fazlollah Reza * George W. Schueller E George G. Lorentz E William David Miller E* Lowell ]. Paige E''S tephen ]. Ricci e'' Paul E. Schupp E*Michael P. Loss Kenneth C. Millett e Emiliano Palacios-Silva E Barbara Slyder Rice E Charles Freund Schwartz E WarrenS. Loud * Milton A. Mintz E*Diethard Ernst Pallaschke E" Richard S. Rich * Gerald W. Schwarz * Laszlo Lovasz E*Norman D. Mirsky E John H. Palmieri E Franklin B. Richards E Willi Schwarz E Benedikt Lowe e*Michal Misiurewicz E Katherine Pardee E Erik Conrad Richardson E*Eric Schweitzer E Tsu-Ming Lu •*Theodore Mitchell E Kyoo-Hong Park E''Norman]. Richert E*Michael]. Schwietzer e Miriam Laura Lucian E William ]. Mitchell E''George D. Parker E Michael M. Richter E Stanley L. Sclove E''Eduardo A. Luna Boris Mityagin e·'Walter R. Parry E''Benjamin Rickinan •*Leonard L. Scott Jr. Clement Henry Lutterodt Ismail ]. Mohamed E* Antonio Pasini E'' Eleanor G. Rieffel •* Ridgway Scott E Richard N. Lyons E Martin James Mohlenkamp <*Donald S. Passman E* Ronald Edgar Rietz E Warner Henry Harvey E Gennady Lyubeznik E Paul H. Monsky E* John]. Pastor E''Pete E. Riley Scott III E Kirill C. H. Mackenzie E Hugh L. Montgomery E Nicholas]. Patterson <'' Jose Rio E George F. Seelinger E Diane Maclagan •*Barbara B. Moore Walter M. Patterson III E*Thomas W. Rishel E'' George Seifert E''Moray S. Macphail * Hal G. Moore E*Sandra 0 . Paur Joel L. Roberts E Bart Selman

602 NOTICES OF THE AMS VOLUME 53 NUMBER 5 AMS Contributions

e Francesco Serra Cassano E Daniel Louis Tancreto E" John E. Wetzel E John Shafer e Leon H. Tatevossian E'' E. A. Whelan E Richard J. Shaker E Michael D. Taylor E NeilL. White About the Cover E Pe ter B. Shalen E S. James Taylor E'' Kathleen B. Whitehead Automata in Coxeter groups e Priti Shankar E Silviu Teleman E Charles K. Williamson E Gerald Peter Shannon E'' Paul M. Terwilliger Richard E. Williamson Many interesting subsets of infinite Coxeter Mich ael Shapiro II Andrew ]. Terz uoli E" A.Jan Stephen Wilmshurst groups, and algorithms involved in compu­ E'' Henry Sharp Jr. Friends at The Louis Berger E Richard Wilson Zhongwei Shen Group, Inc. •· Robert Lee Wilson tation within those groups, are related to au­ T. Shintani E Diana M. Thomas e John W. Wingate tomata or finite state machines, as the ar­ e* Kenichi Shiraiwa e''' Lawrence E. Thomas E''Eric ]. Wingler ticle b y Paul Gunnells in this issue £*Steven E. Shreve P. Emery & Jean C. Thomas E" F. Wintrobe E Chi-Wang Shu E Abigail A. Thompson E'.' Heinz-Wolfgang Wissner demonstrates. One automaton that arises is E'.'D avid S. Shucker E Edward G. Thurber Sarah j. Witherspoon that which generates elements of the group E'' Frederic W. Shultz E Dan Timotin James ]. Woeppel E''Stuart ]. Sidn ey E'' Kathryn B. Toll E" Peter John Wolfenden one by one, the ShortLex automaton, as An tonio Carlos M. Silva * Daniel B.]. Tomiuk Stephen D. Wolthusen proven by Brigitte Brink and Bob Howlett in e''' Daniel S. Silver E'' Jun Tomiyama E James S. W. Wo ng a classic paper many years ago. Others, more E Evelyn rvrarie Silvia E Michael E. Town send E* John W. Wood ·~< Anne-Marie Si.mon e Lisa Gail Townsley E N. M. ]. Woodhouse conjectural, are those which seem to de­ E Lloyd D. Simons Craig A. Tracy E''George V. Woodrow Ill scribe the Kazhgdan-Lusztig cells of an ar­ E Charles C. Sims E" Charles R. Traina E Zhen Xiang Xiong bitrary Coxeter group. The cover illustrates * Iakov G. Sinai Long-Yi Tsai e''Michael Yanowitch Ivan Singer E Kaz6 Tsuji '' Fawzi M. Yaqub both of these types for the Coxeter group E Aj it Iqbal Singh E" Ra!ph P. Tucci E'''Ivlitsuru Yasuhara with Coxeter numbers 2, 3, 7. In the back­ E* Dev P. Sini1a E Howard G. Tucker E Suresh Yegnashanikaran ground the alcoves are colored in pastels ac­ E Hardiv H. Situmeang E'' Johan Tysk Hisashi Yokota E'.'Walter S. Sizer E'' Karen Uhlenbeck E Yosef N. Yomdin cording to the state of the ShortLex au­ E*Thomas Skill E Douglas L. Ulmer E Ki-Jo Yoo tomaton they are associated to. In darker, E Michael Slattery E'~To mi o Umeda E Donald F. Young E'' Richard A. Smith Yasushl Unai Unai * Noriko Yui primary colors are three left cells of the Tara L. Smith E Michael Ungs E Radu Zaharopol group, with the repetitiveness characteriz­ E''Wilbur L. Smith E'' John A. W. Upton E* Charles T. Zahn ing the structures determined by finite state E''Emilio del Solar-Petit E John H. Ursell e:*Nobuo Zama e''' Baorui Song E juan Luis Valle * j ean-Claude Zambrini machines singled out. E'' Linda R. Sons E* R. Lee Van de Wetering e:* Fran<;ois Zara The relationship between subsets of Cox­ E Alexia Henderson Sontag E Aernout C. D. Van Enter E Thomas Zaslavsky E jonathan P. Soren son E* A. H. Van Tuyl Su san Zaverucha & Family eter groups and finite state machines is just E Michael]. Sormani E Werner Varnhorn E'' Karl Zeller a small part of a theory not yet clearly per­ E'' Birgit Sp eh Anatoly M. Vershlk Meijun Zhu ceived in which very complicated infinite E''Dennis Spellman E Charles l. Vinsonhaler E Lee L. Zia E David H. Spring E''Michael Voichick E Jennifer Joy Ziebarth patterns of allldnds are described by finite E Richard H. Squire E''Dan-Virgil Voiculescu e Michael Francis Zielinski data structures related to the theory of lan­ E Ram P. Srivastav Paul S. Vo igt David E. Zitarelli guages. Here the languages are the regular E'' Ross E. Staffeldt E'' Paul A. Vojta Paul Zo rn * ]. T. Staffo rd E'' Hans W. Volkmer Yun Zou languages, those recognized by finite state E Paul H. Stanford E'' Philip D. Wagreich E'' John A. Zweibel machines. Even in Coxeter groups, struc­ E* Lee James Stanley E'' Jona than M. Wa hl Paul F. Zweifel E*Richard P. Stanley E Masato Wakayama Anonymous (289) tures related to the level of complexity in lan­ E*Walter C. Stanl ey E"David B. Wales guage theory above regular languages, some­ * Charles S. Stanton E Justin Clement Walker thing like of context free languages, appear. E Michael Starbird E''William Wall ace E*Christopher W. Stark E'' John Thomas Walsh The explicit finite state machines required E* John Q. St Clair * Lesley A. Ward to draw the Kazhdan-Lusztig cells on the E Arthur Steger E* Seth L. Warner cover were supplied by Gunnells. E''C harles I. Steinhorn E* Bette L. Warren Ell en M. Stenson Arthur G. Wasserman E'' Ronald ]. Stern E'~ Ro b e rt H. Wasserman - Bill Casselman, Graphics Editor E Katherine F. Stevenson E*lvlichiaki Watanabe ([email protected]) E Paul K. Stockmeyer E*Sh 6ji Watanab e E Ke nneth B. Stolarsky E Shuji Watanabe E Stephan A. Stolz William C. Waterhou se ·•· Emil]. Straube E*D avid S. Watkins e'''Walter A. Strauss E*Mark E. Watkins E Gerhard 0 . Strohmer E"Greg M. Wa tson E Garrett James Stuek Edward C. Waymire E Eric A. Sturley E*Cary H. Webb E Francis ]. Sullivan Glenn F. Webb E Kelly John Suman Joseph A. Wehlen Jr. E Lawrence A. Susanka E Clifford E. We il E Ue li Suter E Peter .f. Weinberger E Daniel B. Swearingen E*Michael l. Weinstein E*William ). Sweeney E" Michael S. We iss E Glen Dewane Swiggart E" Ri chard M. We iss E Roman Sznajder * John Weissman ' Margaret W. Taft E Ll oyd R. Welch E*Kazuaki Taira E'' David M. Wells E* Lajos F. Tak<\cs E'' Raymond 0 . Wells Jr. E'.' Shigeo Takenaka E.,, John C. We nger E Laj os Tam assy E*Henry C. Wen te E'' Hi sao Tanaka E" Aric ]. We rl y '' Yoshihiro Tanaka Robert J. Wernick

MAY 2006 NOTICES OF THE AMS 603 From the AMS Secretary

Officers of the Sodety 2005 and 2006 Updates

Except for the members at large of the Paul]. Sally Jr. journal oftheAMS Editorial Council, the month and year of the Paul Zorn Committee first term and the end of the present 2006 Ingrid Daubechies 2/ 04-1/ 07 term are given. For members at large James W. Cannon Mathematical Reviews Editorial of the Council, the last year of the Sylvain E. Cappell Committee present term is listed. Beverly E. ]. Diamond B. A Taylor 2/02- 1/ 06 Jonathan I. Hall 2/ 06- 1/09 Council Mark Goresky Alejandro Uribe Mathematical Surveys and President Monographs Editorial Committee 2007 James G. Arthur 2/ 05-1/07 ]. Tobias Stafford 2/ 05- 1/ 08 Sara Billey President elect Carolyn S. Gordon Mathematics ofComputation Editorial Committee James G. Glimm 2/06-1/07 Sheldon H. Katz Chi-Wang Shu 2/02- 1/ 08 Immediate Past President Michael F. Singer David Eisenbud 2/05- 1/ 06 Catherine H. Yan Proceedings Editorial Committee Vice Presidents Eric D. Bedford 2/ 01- 1/06 2008 Ronald Fintushel 2/ 06- 1/10 Halm. Brezis 2/ 05-1/ 08 William M. Goldman Ruth M. Charney 2/06-1/09 Transactions and Memoirs Editorial Craig L. Huneke Committee Vaughan F. R. Jones 2/04- 1/ 07 Judy Anita Kennedy - 1/06 Robert Guralnick 2/ 05- 1/09 Karen Vogtmann 2/03 Ken Ono Secretary Judy L. Walker Board of Trustees Robert ]. Daverman 2/99- 1/ 09 James G. Arthur (ex officio) 2/05- 1/07 Associate Secretaries Members of Executive John B. Conway 2/ 01-1/11 Susan]. Friedlander 2/96-1/08 Committee John M. Franks (ex officio) 2/ 99- 1/ 09 MichelL. Lapidus 2/02- 1/08 Members of the Council, as provided Eric M. Friedlander 2/ 00-1/10 Matthew Miller 2/05- 1/09 for in Article 7, Section 4 (last sen­ Linda Keen 2/99- 1/ 09 Lesley M. Sibner 2/93- 1/ 09 tence), of the Bylaws of the Society. Donald E. McClure (ex officio) Treasurer 2/ 03- 1/09 Walter L. Craig 2/03-1/ 07 John M. Franks 2/99- 1/09 Jean E. Taylor 2/03- 1/ 08 Robert Guralnick 2/05- 1/ 09 Associate Treasurer Carol S. Wood 2/ 02- 1/07 Hugo Rossi 2/ 02-1/06 Donald E. McClure 2/ 03-1/ 09 Paul]. Sally Jr. 2/04-1/ 08 Members at Large All terms are for three years and ex­ Publications Committees pire on January 31 following the year given. Bulletin Editorial Committee 2005 Donald G. Saari 2/98- 6/ 05 Susan Hermiller Susan]. Friedlander 7/05-1/ 09 Brian Marcus Colloquium Editorial Committee John E. McCarthy Paul]. Sally Jr. 2/05- 1/ 08

604 N OTICES OF THE AMS VOLUME 53, NUMBER 5 • Com-binatorics

This prize was established in 2005 in memory of David P. Robbins by members of his family. Robbins, who died in 2003, received his Ph.D. in 1970 from MIT. He was a long-time member of the Institute for Defense Analy­ sis Center for Communications Research and a prolific mathematician whose work (much of it classified) was in discrete mathematics. The prize is for a paper with the following characteristics: it shall report on novel research in algebra, combinatorics or discrete mathematics and shall have a significant experimental component; and it shall be on a topic which is broadly accessible and shall provide a simple statement of the problem and clear exposition of the work. The US$5,000 prize will be awarded every three years. It is expected that the first award will be made in January 2007.

Nominations should be submitted to the AMS Secretary, Robert J. Daverman, American Mathematical Society, 312D Ayres Hall, University of Tennessee, Knoxville, TN 37996-1330. Include a complete bibliographic citation for the work that is the basis of the nomination, supplemented with brief remarks explaining what aspects of the work make it particularly suited for this prize. The nominations will be forwarded by the Secretary to the Prize Selection Committee, which will make the final decision on the award. Deadline for nominations: August 15, 2006 ®AMS AMERICAN MATHEMATICAL SOCIETY www.ams.org I Mathematics Calendar

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

May 2006 Conference Description: This conference will explore new devel­ opments in applied analysis as they relate to fluid flow and wave '' 5-6 Operator Algebra Workshop 2006, Queen's University Belfast, motion, both from a theoretical and applied point of view. It will Belfast, Northern Ireland. intensify scientific interactions between the two different groups Information: A two-day workshop dedicated to all aspects of of researchers, expose junior mathematicians to state-of-the-art operator algebras, both selfadjoint and non-selfadjoint, will be developments in analysis and its applications and provide a stage held in the Department of Pure Mathematics of Queen's University for the dissemination of research results. Belfast on Friday, 5 May and Saturday, 6 May 2006. Organizers: Martin Mathieu and Ivan Todorov. Organizers: Fernanda Botelho, Thomas Hagen, Jim Jamison. Details: http: I /www. qub. ac. uk/ opaw2006; email: opaw2006©qub. Information: http: I /www .msci .memphis. edu/fluidsandwaves/. ac.uk. ,., l 3 Graph Theory Day 51, Montclair State University, Montclair, '' 5- 7 2006 Midwest Geometry Conference, The University of New Jersey. Oklahoma, Norman, Oklahoma. Host: The Department of Mathematical Sciences, Montclair State Main topics include {but not limited to): P-harmonic geometry; University. Geometric flows; Complex and Riemannian geometry; Conformal Sponsor: The Mathematics Section of the New York Academy of geometry; Convex Geometry; Minimal varieties, symmetric criticality Sciences. This one day meeting is to stimulate activity among graph and algebraic geometry; PDEs, geometric measure theory and theorists. Papers for contributed presentations (15 minutes) are mathematical physics. invited. Organizers: Chair: Shihshu Walter Wei (University of Oklahoma); Information: http: I /www. csam .montclair. edu;-lia/gtday51/. co-chairs: Thomas Branson (University of Iowa), Marilyn Breen (Uni­ versity of Oklahoma), Andrzej Derdzinski (Ohio State University), ,., 14-1 8 3rd International Conference: Chebyshev's Mathematical Robert M. Hardt (Rice University), Ralph Howard (University of Ideas and Applications to the Modern Science, Obninsk State South Carolina), Weiping Li (Oklahoma State University), Gerard University, Russia. Walschap (University of Oklahoma), and Meijun Zhu (University of Description: On May 16, 2006 we shall celebrate the 185th birthday Oklahoma). of the Great Russian mathematician P. L. Chebyshev. Chebyshev Information: More information will be posted on the conference was a famous scientist, and on a historical level we can say that homepage as it becomes available at http: I /www. math. ou. edu;­ he was really the scientific "father" of the Russian St. Petersburg wwei/mgc06 .html. Mathematical School. Topics: Functions theory and spectral theory of operators. Theory '' ll - 1 3 Fluids and Waves-Recent Trends in Applied Analysis, of approximations. Probability methods and theory of numbers. University of Memphis, Memphis, Tennessee. Mathematical physics. Small parameter methods, inverse and ill-

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 to allow them to appear in more than one issue of the Notices prior to An announcement will be published in the Notices if it contains a call 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 appeared. Asterisks (") mark those announcements containing new or the twelve-month period 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/.

606 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Mathematics Calendar

posed problems. Difference equations and methods. Mathematical Description: This Congress continues the tradition of biennial simulation. Computer simulation in problems of nuclear power meetings focused on nonstandard methods. While the fields of engineering and physics. Problem of modern natural-scientific application of nonstandard analysis are diverse, the common education and scientific inheritance of the academician P. L. methodologies and ideas make it appropriate to consider nonstan­ Chebyshev. Ordinary differential equations. dard methods as a unified mathematical field of research. Applications: Please submit your application form to the Con­ Invited Speakers: S. Albeverio'', R. Anderson,]. Bell, V. Benci, I. van ference via the website http://www. amath. ru. Additionally you den Berg, N. Cutland, A. Enayat, M. Forti, E. Gordon, K. Hrbacek, can write directly to the Organizing Committee: CHEBYSHEV-2006, R. Jin, ]. Keisler, R. Kossak, S. Leth, T. Lindstrom, P. Loeb, W. Professor V. A. Galkin, Head of Department of Applied Mathematics, Luxemburg'', A. Macintyre, T. Nakamura, V. Neves'', D. Ross, T. Obninsk State University, Studgorodok, l, Obninsk, 249040, Russia; Sari, Y. Sun, K. Tanaka, M. Wolff", A. Zemanian ['' denotes that the email: cheb©amath. ru; phone: +(48439)37908; fax: +(48439)70822. acceptance is provisional]. Languages: Russian and English. Information: http://www. dm. unipi. i trnsm2006. Information: http://www. amath.ru; http://www .mathsoc. spb. ru/pantheon/chebyshe/index.html. '' 29-june 2 International School on Partial Differential Equa­ tions, Depto. Matematicas y Meca.nica, liMAS Universidad Nacional '' 17-20 Combinatorial and Additive Number Theory (CANT 2006), Aut6noma de Mexico (UNAM), Mexico City, Mexico. CUNY Graduate Center, New York, New York. Description: This one-week school is organized around 6 short Description: This is the fourth in a series of annual workshops courses on various problems involving nonlinear PDEs and it is sponsored by the New York Number Theory Seminar on problems intended that young researchers and graduate students will be in combinatorial and additive number theory and related parts well-informed on the trend and current states of research in the of mathematics. A list of invited and confirmed lecturers will be fields. Short presentation by other participants are also encouraged. posted on the conference website. Mathematicians who wish to Invited speakers: Luis Caffarelli, Irene Gamba, Frank Morgan, speak at the meeting should submit a title and abstract by email Takayoshi Ogawa, Peter Sternberg, Eiji Yanagida. Organizer: Padilla to:NewYorkNumberTheory©gmail.com. Pablo (UNAM). Support: It is expected that there will be some financial support, Information: http://www. fenomec. unam.mx/. especially for graduate students and young faculty. Organizer: Mel Nathanson. June 2006 Information: http: //theoryofnumbers. com/CANT /2006/ cant_2006 .htm, or email: NewYorkNumberTheory©gmail . com. * 2-14 Approximation Algorithms, Centre de Recherches Mathe­ matiques, Montreal, Canada. '' 1 9-20 Groups in Galway 2006, National University of Ireland, Description: The workshop will include lectures on the latest Galway, Ireland. developments in the field of approximation algorithms, on both Scope: Covers all areas of group theory, applications, and related the approximability and the inapproximability sides. fields. Organizers: Joseph Cheriyan (Waterloo), Michel Goemans (MIT). Provisional List of Speakers: Cedric Bonnafe (Univ. de Franche­ Invited Speakers: Please see website below. Comte, France), Peter Cameron (Queen Mary, Univ. of London, Information: http : //www. crm. umontreal. ca/ UK), Rod Gow (UC Dublin, Ireland), John Murray (NUl, Maynooth, Approxi mation06/. Ireland), Shane O'Rourke (Cork Inst. of Tech., Ireland), Gretchen Ostheimer (Hofstra University, USA), Gotz Pfeiffer (NUl, Galway, ''5-9 Workshop on Fourier Analysis, Geometric Measure Theory Ireland), Martyn Quick (St. Andrews, UK), Sarah Rees (Univ. of and Applications, Centre de Recerca Matematica, Barcelona, Spain. Newcastle, UK), Chiara Tamburini (Univ. Cattolica del Sacro Cuore, Co-ordinators: jose Maria Martell (Universidad Aut6noma de Italy). Madrid), Joan Mateu (Universitat Autonoma de Barcelona), Alberto Information: Details of the talks and their scheduling will be posted Ruiz (Universidad Aut6noma de Madrid), Xavier Tolsa (Universitat at http ://www.maths.nui gal way.ie/gig06.html closer to the Autonoma de Barcelona), Ana Vargas (Universidad Aut6noma de event. For further information, please contact one of the conference Madrid), Joan Verdera (Universitat Autonoma de Barcelona). organizers, Rachel Quinlan (rachel. quinlan©nuigalway. ie) or Information: http://www.crm.es/Conferences/0506/Fourier/ Dane Flannery (dane. flannery©nuigalway. ie). default . htm. * 24-26 DIMACS Workshop on Polyhedral Combinatorics of Ran­ '' 9-11 Logic and Mathematics 2006, University of Illinois at dom Utility, DIMACS Center, CoRE Bldg, Rutgers University, Pis­ Urbana-Champaign, Urbana, Illinois. cataway, N]. Organizers: C. Ward Henson and Slawomir Solecki. Organizers: Jean-Paul Doignon, Univ. Libre de Bruxelles, email: doignon©ulb . ac . be; Aleksandar Pekec, Fuqua School of Business, Meeting Topics: The focus of the meeting will be on descriptive set Duke University, email: pekec©duke. edu. theory and its connections (with algebra, topology, measure theory, topological dynamics, combinatorics, etc). In part, the meeting is Local Arrangements: Workshop Coordinator, DIMACS Center, organized to honor Alexander S. Kechris of Cal Tech on the occasion email: workshop©dimacs. rutgers. edu; 732-445-5928. of his 60th birthday. Description: Utility functions have a long history in economics and Meeting Webpage:http: //www. math. uiuc. edu/Bulletin/lm2006 . psychology but have recently caught the attention of computer html. Check here for later information including titles of talks and scientists in various applications. Random utility approaches have been extensively used in the soccial sciences. The fundamental abstracts. idea is that utilities of agents could be hard or even impossible '' 1 0-16 Discontinuous change in behavior issues in partial to precisely assess or elicit, so one should model these utilities differential equations, Anogia Acdademic Village, Crete, Greece. as random variables. This modeling approach could turn out to be useful in developing and solving optimization problems and Organizers: I. Athanasopoulos, L.A. Caffarelli. algorithms for which there is no time to or where it is is impossible Main Speakers: D. Christodoulou, I. Gamba, R. Nochetto, S. Salsa, to assess/obtain input data precisely. M. Soner, P. Souganidis, T. Zariphopoulou. Support: In part by European Commission under FP6. Available ,.,2 5-31 NSM2006 "Nonstandard Methods and Applications in grants: at least 43. Mathematics", Pisa, Italy. Information: E. Kafatos and T. Pheidas; math. uoc. gr.

MAY 2006 NOTICES OF THE AMS 607 Mathematics Calendar

,., 1 2-1 5 (REVI SED) Conference on 3-manifold topology in honour Hilbert's 16th problem; Sergei Yakovenko (The Weizmann Institute of Peter Shalen's 60th birthday, Centre de Recherches Mathema­ of Science, Israel): Algebraic Solutions of Polynomial Vector Fields; tiques, Montreal, Canada. (Feb. 2006, p. 286) Co-ordinators: Armengol Gasull (Universitat Autonoma de Barce­ Organizers: Steve Boyer, Dick Canary, Marc Culler, Nathan Dunfield, lona), Jaume Ll ibre (Universitat Autonoma de Barcelona). Benson Farb. Reg istration and payment: Fee: 200 euros; Deadline: May 26, 2006. Speakers (*tentative): lanAgol (Univ. of Ill inois at Chicago), Mladen Grants: The CRM offers a limited number of grants for registra­ Bestvina (Univ. of Utah), Marc Culler (Univ. of Illinois at Chicago), tion and/ or accommodation addressed to young researchers. The Nathan Dunfield (Cal tech), Cameron Gordon (Univ. of Texas), ''Alex deadline for application is April 26, 2006. Lubotzky (Hebrew Univ. of Jerusalem), Yair Minsky (Yale Univ.), Further information: http: I lwww . crm. esl ACDifEquations, email: ''Maryam Mirzakhani (Princeton Univ./Clay Institute), John Morgan [email protected]. (Columbia Univ.), ''Lenhard Ng (Stanford Univ. I AIM), Peter Ozsvath (Columbia Univ.), Jake Rasmussen (Princeton Univ.), Michah Sageev '' 28-30 Workshop From Lie Algebras to Quantum Groups, Uni­ (Technion). versidade de Coimbra, Portugal. Information: http: I /www. crm. umontreal. ca/Shalenfest/. Speakers and Members of the Sc ientific Committee are: Helena Albuquerque, Universidade de Coimbra (Portugal), Georgia Benkart '' 12-16 Permutation Patterns 2006, Reykjavik University, Reyk­ (University of Wisconsin-Madison), Alberto Elduque (Universidad javik, Iceland. de Zaragoza, Spain), George Lusztig (Massachusetts Institute of Conference Themes: Include (but are not limited to) enumeration Technology), Shahn Majid (University of London (UK)), Michael questions, excluded pattern questions, study of the involvement Semenov-Tian-Shansky, Universite de Bourgogne (France). order, algorithms for computing with permutation patterns, appli­ Talks: There will be plenary talks (50 min), section talks (20- cations and generalisations of permutation patterns, and others. 25 min) and a poster session. Proceedings will be published by Information: http: I /www. cs. otago. ac. nz/ staffpri v /mike/ the International Center of Mathematics (CIM) and will contain PP2006/Home. html. contributions by all the participants, after revision by the Scientific Committee. '' 1 2-1 7 Boltzmann Equation and Flu idodynamic Limits, SISSA­ Deadline: April30, 2006. For more details, including the Conference ISAS, Trieste, Italy. Program and information on accommodation in Coimbra, please re­ Aims and Scope: The conference wishes to provide an up-to-date fer to http: I lwww. aim. estt. ipt. pt;-jmmpiCI MILielindex. htm. overview of the recent results and open problems related to the We regret to inform that we cannot offer the participants any study of the Boltzmann equation and of the connections between financial support. the macroscopic and mesoscopic description of gas dynamics. Information: Persons interested in participating are kindly asked Organizing Committee: Fabio Ancona (University of Bologna, to register at the conference website http : I lwww. aim. estt. ipt . Italy), Stefano Bianchini (SISSA-ISAS, Trieste, Italy), Camillo DeLellis pt;-jmmpiCIMILielindex .htm as well as to submit an abstract for (University of Zurich, Switzerland), Andrea Marson (University of a section talk or a poster. The registration fee is 100 Euros, and Padova, Italy). there will be a reduced student fee of 50 Euros. Deadlines: Registration: May 25, 2006. Financial support: April 30, 2006. July 2006 Informations: http : I lwww. sissa. i tlbol tzmannl. '' 3-7 lwasawa 2006 Congres s, Universite de Limoges, Limoges, '' 2 1-24 "Views on ODEs" Conference in Honor of Arrigo Cell ina France. and James A. Yorke on the Occasion of their 65th Birthdays, Program: Recent advances in Iwasawa Theory. Aveiro University, Aveiro, Portugal. Organizing Committee: Fran<;:ois Laubie, Abbas Movahhedi, Alain Conference: "Views on ODEs" will celebrate the 65th birthdays Salinier, Stephane Vinatier. of Professors Arrigo Cellina and james A. Yorke and aims to Information: http: I lwww. unilim. frll acoliwasawa2006. bring together those enrolled in research activities related with ordinary differential equations, differential inclusions and their '' 9-1 5 Which Mathematics for Biology?, Anogia Academic Village, applications. Crete, Greece. Main Topics: Dynamical systems; Bifurcations; Invariant measures; Organizers: A. Deutsch, D. Manoussaki, B. Perthame. Chaotic attractors; Prevalence; Population dynamics; Markov oper­ Main Speakers: A. Deutsch, E. Grenier, Y. Kevrekidis, H. Metz, P. ators; Semigroups; Viscosity solutions; Hamilton-Jacobi equations; Maini, C. j. Weijer. Hyperbolic systems; Optimal control and differential inclusions; Support: In part by European Commission under FP6. Available Variational and topological methods. grants: at least 43. · Information: http: I lww. di vp-proj. orgl. Information: E. Kafatos and T. Pheidas: math. uoc. gr.

'' 26-29 Special session on "Coding theory and cryptography", '' 1 0-1 5 6th Czech-Siovak International Symposium on Combina­ Varna, Bulgaria. torics, Graph Theory, Algorithms and Applications: Honoring Organizers: Stefan Dodunekov, (Bulgarian Academy of Sciences), the 60th Birthday of J. Nesetril, Prague, Czech republic .. Tony Shaska, (Oakland University). Description: The program will be devoted (but not restricted) Overview: As technology becomes increasingly involved in commu­ to subjects in contemporary Combinatorics and Graph Theory nication, coding theory and cryptography also become increasingly involving also relationships and applications in Algebra, Algorithms, important. The goal of this session is to bring together researchers Topology, Probabil ity and Statistics, Mathematical Logic, Computer in all aspects of coding theory, cryptography and related areas and Science and other fields. explore the use of computational algebra in such areas. Invited Speakers: Martin Aigner, Noga A! on, Peter Cameron, Shalom Contact: T. Shaska ( shaskal!loakland. edu); http: I lww. oakland. Eli ahou, Isidoro Gitler, Ron Graham, Pavol Hell, Laszlo Lovasz, ji0i edu/-shaskalcod_06.html. Matouek, Patrice Ossona de Mendez, Andre Raspaud, Vojtich Rod!, Gert Sabidussi, Oriol Serra, Jozef Fira6, Paul Seymour, Endre ,., 26-July 8 Advanced Course on Limit Cycles of Differential Szemeredi, Claude Tardif, Robin Thomas, Carsten Thomassen, Vera Equations, Centre de Recerca Matematica, Barcelona, Spain. T. S6s, Herbert Wilf, Peter Winkler, Xuding Zhu. Speakers: Colin Christopher (University of Plymouth, United King­ Organizer: DlMATIA Charles University, Prague. dom): Around the Center-Focus Problem; Chengzhi Li (Peking Deadlines: Early registration: May 15, 2006. Abstract submission: University, China): Abelian integrals and application to weak May 30, 2006.

608 N OTICES OF THE AMS VOLUME 53, N UMBER 5 Mathematics Calendar

Information: http: I /kam. mff. cuni. cz/ cs06. exciting new developments in the Geometric Group Theory. We expect around twenty talks during the workshop week. '' 1 0-1 5 Conference on Recent Developments in the Arithmetic of Information: http: I /www. crm . umontreal. ca/Geometric06/ Shimura Varieties and Arakelov Geometry (An EMS Marie Curie index_e. html. Conference, supported by the European Commission), Centre de Recerca Matematica, Bellaterra, Spain. '' 1 6-21 Recent Advances in Nonlinear Partial Differential Equa­ Speakers: Ahmed Abbes (Univ. de Paris XIII, France), Pascal Boyer tions: A celebration of Norman Dancer's 60th birthday, University (Univ. de Paris VI, France), Jan H. Bruinier (Univ. Koln, Germany), of New England, Armidale, NSW, Australia. Laurent Clozel (Univ. de Paris XI), Henri Darmon (McGill Univ., Workshop Topics: Nonlinear Partial Differential Equations, Cal­ Canada), jiirg Kramer (Humboldt Univ. zu Berlin, Germany), Elena culus of Variations, Topological Methods in Nonlinear Analysis, Mantovan (Harvard Univ., USA), Sophie Morel (Univ. de Paris XI, Harmonic Analysis and Dynamical Systems. France), Bao Chau Ngo (Univ. de Paris XI, France), Michael Rapoport Organizers: Daniel Daners, Yihong Du, Chris Radford, Shusen Yan. (Univ. Bonn), Damian Roessler(Univ. de Paris VII, France), Christophe Deadline: Registration deadline: March 31, 2006. Soule (CNRS-IHES, France). Information: http: I /www .maths. usyd. edu. au/u/daners/ Information: http: I /www. crm. es/SVAG; email: SVAG@crm. es. une2006/.

* 11-12 DIMACS Workshop on Machine Learning Techniques in ,., 16-22 Horizon of Combinatorics, Lake Balaton, Hungary. Bioinformatics, DIMACS Center, CoRE Bldg, Rutgers University, Topics: Horizon of Combinatorics intends to gather researchers Piscataway, New jersey. from all areas related to combinatorics. These include amongst Description: Bioinformatics aims to solve biological problems by others: Combinatorial structures (graphs, hypergraphs, matroids, using techniques from mathematics, statistics, computer science, designs, permutation groups), Combinatorial optimization, Com­ and machine learning. Recent years have observed the essential binatorial aspects of geometry and number theory, Infinite com­ use of these techniques in this rapidly growing field. Examples of binatorics, Algebraic combinatorics, Algorithms in combinatorics such applications include those to gene expression data analysis, and related fields. gene-protein interactions, protein folding and structure prediction, Information: http: I /www. renyi. hu/ conferences/horizon. genetic and molecular networks, sequence and structural motifs, genomics and proteomics, text mining in bioinformatics, and so '' 1 7-21 Workshop on Singularities in POE and the Calculus on. of Variations, Centre de Recherches Mathematiques, Montreal, Organizers: Dechang Chen, Uniformed Services University of the Canada. Health Services, email: dchen@usuhs. mil; Xue-Wen Chen, University Focus: The development and structure of singular structures in of Kansas, email: xwchen@ku . edu; Sarin Draghici, Wayne State solutions to nonlinear partial differential equations. University, email: sod@cs. wayne. edu. Participants: Amandine Aftalion (Paris VI), Giovanni Alberti (Pisa), Local Arrangements: Workshop Coordinator, DIMACS Center, Yaniv Almog (LSU), Leonid Berlyand (Penn State), Fabrice Bethuel email: workshop@dimacs. rutgers. edu; 732-445-5928. (Paris VI), Rustum Choksi (SFU), Manuel DelPino (U. de Chile), Car­ los Garcia-Cervera (CaL- Santa Barbara), Stephen Gustafson (UBC), Information: http: I I imacs. rutgers . edu/Workshops/ Robert Jerrard (Toronto), Shuichi Jimbo (Hokkaido), Bernd Kawohl MLTechniques/ . (Koln), David Kinderlehrer (Carnegie-Mellon), Robert V. Kahn (NYU), ,., 1 2-16 Anomalous Transport: Experimental Results and Theoret­ ChunLiu (Penn State), Andrea Malchiodi (SISSA, Trieste), Vincent Mil­ ical Challenges, Physikzentrum Bad Honnef near Bonn, Germany. lot (CMU), Alberto Montero (Toronto), Yoshihisa Morita (Ryukoku, Scope: Anomalous transport phenomena such as sub- and su­ Japan), Pablo Padilla (UNAM, Mexico), Daniel Phillips (Purdue), Xi­ aofeng Ren (Utah State), Maria Reznikoff(Princeton), Etienne Sandier perdiffusion, non-Gaussian probability distributions, aging and dynamical localization form a rapidly growing research area within (Paris-12), Sylvia Serfaty (NYU), Daniel Spirn (Minnesota), Edward Stredulinsky (Wisconsin-Richland), Gabriella Tarantella (Rome II). nonequilibrium statistical physics. The seminar will provide a unique opportunity to learn about topics ranging from mathe­ Information: http: I /www. crm. umontreal. ca/Singulari ties06/ matical foundations of anomalous dynamics to the most recent index_ e . html. experimental results in this field. Invited Speakers: R. Artuso (Como), E. Barkai (Bar- ll an), C. Beck August2006 (London), A. V. Chechkin (Kharkov), D. Del-Castillo-Negrete (Oak '' 7-11 Effective Randomness, AIM Research Conference Center, Ridge), P. Dieterich (Dresden), T. Geisel (Goettingen), R. Gorenflo Palo Alto, California. (Berlin), R. Hi! fer (Stuttgart),]. Kaerger (Leipzig), R. Kimmich (Ulm), Description: This workshop, sponsored by AIM and the NSF, will ]. Klafter (Tel Aviv), W. Kob (Montpellier), A. Kusumi (Kyoto), E. Lutz bring together researchers who have studied effective randomness (Ulm), R. Metzler (Copenhagen), M. ]. Saxton (Davis), M. Shlesinger at different times, with different motivations, and drawing from (Arlington), S. Tasaki (Tokio), G. Vogl (Vienna), A. Vulpiani (Rome), different academic backgrounds, with an aim toward increasing S. Yus te (Badajoz). communication and collaboration, and developing broad shared Registration: Applications are welcome and should be made by research goals and a coherent research community. using the application form on the conference web page, however, Topics: For the workshop will include effective notio'ns of ran­ the number of attendees is limited. The seminar's registration fee domness such as Martin-Lof randomness; measures .of relative is EUR 200 and will cover accommodation and meals. randomness; effective dimension; Kolmogorov complexity and Deadline: For applications is April 30, 2006. other concepts from algorithmic information theory; and interac­ Information: For further information please visit the conference tions with computability theory and complexity theory. webpagehttp://anotrans.physik.hu-berlin.de,orcontactone Organizers: joseph Miller and Denis Hirschfeldt. of the organizers. Deadline: May 7, 2006. Information: http: I I aimath. org/ ARCC/workshops/randomness. '' 1 3-14 Conference on Geometric Group Theory, Centre de Re­ html. ·cherches Mathematiques, Montreal, Canada. Description: From july 3-7:There will be five mini-courses focusing ,., 14- 1 6 Network Design: Optimization and Algorithmic Game on emerging ideas in Geometric Group Theory that will be especially Theory, Centre de Recherches Mathematiques, Montreal, Canada. aimed at graduate students, but are sure to be of wider interest. Description: As the network infrastructure keeps changing and new From jUly 1 0-1-4: There will be a workshop featuring some of the applications are emerging, .the mathematical models themselves

MAY 2006 NOTICES OF THE AMS 609 Mathematics Calendar

must be adapted constantly. The workshop will explore recent L6ira (Andorra), Madrid, Spain. developments in the field and especially the relationship between Plenary Speakers: Bard Ermentrout (University of Pittsburgh), combinatorial optimization and the models used in distributed Nancy Kopell (Boston University), John Rinzel (New York University). network design. Invited Speakers: Ad Aersten, Janet Best, Alla Borisyuk, Amithaba Organizers: Shie Marmor (McGill) and Adrian Yetta (McGill). Bose, Paul Bressloff, Eric Brown, Nicolas Brunel, Carson Chow, Participants: Kamal Jain (Microsoft Research Center), Ramesh Stephen Coombes, Gustavo Deco, Jean-Pierre Franoise, Boris Gutkin, Johari (Stanford University), George Karakostas (McMaster Univer­ David Hansel, John Hertz, Kresimir Josie, Peter Latham, Tim Lewis, sity), Anna Karlin (University of Washington), Jochen Kannemann Georgi Medvedev, Far zan Nadim, Jonathan Rubin, Michael Rudolph, (University of Waterloo), Kate Larson (University of Waterloo), Maria Victoria Sanchez-Vives, Walter Senn, Brian Smith, Jeffrey C. Yishay Mansour (Tel Aviv University), Peter Marbach (University of Smith, Louis Tao, Mirra Teicher, Misha Tsodyks and ]. Leo van Toronto), SeanMeyn (University of Illinois), Tim Roughgarden (Stan­ Hemmen. ford University), Andreas Schulz (ETH Zentrum), Nahum Shimkin Information: See http: I lwww. crm. esiCMathNeuroscience; email: (Technion), Eva Tardos (Cornell University).''('') To be confirmed [email protected]. Information: http: I lwww. crm. umontreal. ca1Network06l. ,., 4-8 In ternational Seminar on Applied Geometry in Andalusia, '' 21-2 5 Phase Transitions in Physics, Computer Science, Combi­ University of Granada, Granada, Spain. natorics and ProbabilityTheory,AIMResearch Conference Center, Organizers: A. Romero, M. Ortega, C. Ruiz, M. Gutiirrez, M. Palo Alto, Califonia. Angustias Caiiadas-Pinedo, M. Fernandez, A. Carriazo. Organizers: Persi Diaconis, Daniel Fisher, Cris Moore, and Charles Invited Speakers: X. Gual (Univ. Jaume I), L. M. Cruz-Orive (Univ. Radin. Cantabria): "Steorology",]. M. M. Senovilla (UPV /EHU): "Geometry of Workshop Topics: This workshop, sponsored by AlM and the sub manifolds in Lorentzian geometry with applications", 0 . Garay, NSF, will be devoted to the study of phase transitions in several ]. Arroyo (UPV / EHU), R. Lipowsky (Max Planck Institute): "Curvature traditionally separate subjects. We propose to bring together experts energy minimizers. Applications to the physics of elastic and soft in different area to present the various intuitions, motivations, materials", R. Kamien (Univ. Pennsylvania): "Materials Geometry: canonical examples and conceptual techniques of their areas, the An Introduction (Survey on Geometry and Physics; Geometry of hope being to come to agreement on a few key definitions, and Smectics)", Y. S. Cho (Ewha Women's University, Korea): "Group perhaps thereby to bring fresh ideas to bear on open problems. Actions on Gauge Theory", M. Ferri (Univ. Bologna): "Geometrical Application Deadline: May 14, 2006. Methods in Application and Industry in Italy", L. Verstraelen Information: http: I I aimath. orgl ARCCiworkshopsl (Katholieke Univ. Leuven, Belgium): "Understanding Vision through phasetransition.html. Geometry", A. Ferrandez and ]. Pastor (Univ. Murcia): "Geometry applied to DNA", ]. Martinez Aroza (Univ. Granada): "Fractals" '' 30-September 1 Recent Trends in Constructive Approximation M. Cabrerizo (Univ. Granada): "Physics in Action", C. Ruiz (Univ. Theory. Satellite Conference of ICM06, Universidad Carlos III de Granada): "Geometry in the Alhambra." Madrid, Legans, Spain. Information: http: I lgigda. ugr. eslisaga06l; email: isaga06© Topics: Riemann-Hilbert approach to asymptotics of orthogonal ugr . es. polynomials, inverse problems and rational approximation, spec­ tral theory of banded and differential operators, numerical analysis '' 4-9 International Conference on Applied Analysis and Differ­ and rational approximation, integrable non linear dynamical sys­ ential Equations, University "Al.I.Cuza", Faculty of Mathematics, tems, eigenvalues of random matrices, non standard orthogonal Iasi, Romania. polynomials. Purpose: Of the conference is to create a platform for international Main Speakers: P. Deift (Courant Institute, New York University), exchange of ideas and the newest results in the fields of applied B. Simon (California Institute of Technology), F. A. Grunbaum analysis and differential equations. (University of California, Berkeley), S. Khruschev (Atilim University, Topics: Nonsmooth Analysis and Optimization, Ordinary Differ­ Ankara, Turkey), M. E. H. Ismail (University of Central Florida), A. ential Equations, Partial Differential Equations, Control Theory, B. ]. Kuijlaars (Katholieke Universiteit Leuven, Belgium). Stochastic Analysis. Deadlines: Submission of abstracts is May 31, 2006; Registration Speakers: Please see website below. is June 15, 2006. Information: http: I lwww. math. uaic. roricaade; email: icaade© Contact: Francisco Marcellan, Departamento de Matematicas, Uni­ uaic.ro. versidad Carlos III de Madrid, Avenida de la Universidad 30, 28911 Leganes, Spain; email: pacomarc©ing. uc3m . es. '' 1 1-1 6 XV Fall Workshop on Geometry and Physics, Puerto de la Information: Visit the website http: I lwww. uc3m. esluc3ml dptol Cruz (Tenerife, Canary Islands), Spain. MATEMIOrthApproxiiCM06Iuc3m_ICM06.html. Main topics: Continuum Mechanics, Dynamical systems, Geometry Control Theory, Integrable systems, Lie algebroids (groupoids) ' 31-September 5 Advanced Course on Combinatorial and Com­ and its applications, Poisson Geometry, Classical and Quantum putational Geometry: Trends and topics for the future, Centre Field theories, Riemannian and Lorentz Geometry and Relativity, de Recerca Matematica, Barcelona, Spain. Symplectic and Contact Geometry and Topology, String Theory, Speakers: Jimos Pach (City College and Courant Institute, New Supergravity and Supersymmetry. York and Renyi Institute, Budapest, Hungary), Micha Sharir (Tel Programme: Two mini-courses. Invited talks (45 minutes). Con­ Aviv University, Israel). tributed talks (25 minutes). Poster Session. Registration and Payment: Fee with dinner party: 190 ii; Fee Deadline: July 1, 2006. without dinner party: 150 ii, Deadline: May 15, 2006. Information: http : I lwww. gt . matfun . ull . esl15iwgp2006lindex . Grants: The CRM offers a limited number of grants for registration htm. fee and/or accommodation addressed to young researchers. The deadline for application is April14, 2006. * 1 8-20 The 1Oth Workshop on Elliptic Curve Cryptography (ECC Further Information: Visit http: I lwww. crm. esl ACComGeometry; 2006), Fields Institute, Toronto, Canada. email: ACComGeometry©crm. es. Information: http: I lwww. cacr. math. uwaterloo. cal conferencesl2006lecc2006lannouncement.html. September 2006 '' 1 8-22 Hybrid Methods and Branching Rules in Combinatorial " 1-4 Conference on Mathematical Neuroscience, Sant Julia de Optimization, Centre de Recherches Mathematiques, Montreal,

610 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Mathematics Calendar

Canada. Works hop: sponsored by the ESF Programme "Global and Geometric Description: Problems of combinatorial optimization (such as Aspects of Nonlinear Partial Differential Equations". SAT, the problem of recognizing satisfiable boolean formulas in Description: Recent progress in mathematical image processing the conjunctive normal form) have been the subject of intensive shows a surprising success when one applies numerical methods to study by two communities of researchers: Those in mathematical ill-posed partial differential equations. There is hardly any theory programming (often classified under "operations research") and for these equations, it lags far behind their use by engineers, and those in constraint satisfaction programming (often classified the purpose of the workshop is to learn more about the underlying under "artificial intelligence"). Recent years have seen increasing mathematical questions. We shall address for instance issues like interaction between these two initially separate communities. One anisotropic diffusion and Perona-Malik type equations. of the aims of the workshop is to foster this confluence. Organizers: Bernd Kawohl (Cologne), Felix Otto (Bonn). Second Theme: Branching rules are another theme of the workshop. Information: http: I lwww .mi. uni-koeln. de;-jhorakl These rules are an important component of branch-and-bound­ workshop/. based exact algorithms and their choice may have an overwhelming impact on the efficiency of such algorithms. '' 9- l 3 Short-term Cardiovascular-Respiratory Control Mecha­ Organizer: Va'uek Chvatal (Concordia). nisms, AIM Research Conference Center, Palo Alto, California. Information: email: paradis©crm . umontreal. ca. Organizers: Franz Kappel, Vera Novak, Mette Olufsen, and Hien Tran. '' l 8-22 Model Theory of Metric Structures, AIM Research Confer­ Workshop: This workshop, sponsored by AIM and the NSF, will ence Center, Palo Alto, CA. be the first highly focused attempt to tackle complex problems Organizers: C. Ward Henson and Itay Ben-Yaacov. in cardio-respiratory physiology by bringing together researchers Workshop topics: This workshop, sponsored by AIM and the NSF, with expertise in physiology, mathematics, and statistics. The will focus on the use of model theoretic ideas in analysis and metric overall objective of this workshop is to discuss methodologies to geometry, bringing together model theorists and specialists from further develop mathematical models to improve understanding, a few key application areas for a period of intense discussions. diagnosis, and treatment of clinical problems related to short-term A diverse combination of backgrounds will allow the participants cardiovascular-respiratory regulation. to explore from new angles certain examples, applications, and Application deadline: July 9, 2006. theoretical problems that define the frontier of research on the Information: http: I I aimath. orgl ARCCiworkshopsl model theory of metric structures. cardiocontrol.html. Application deadline: June 18, 2006. For more information: http: I I aimath. orgl ARCCiworkshopsl '' l 0-l 3 Data Mining and Mathematical Programming, Centre de continuouslogic.html. Recherches Mathematiques, Montreal, Canada. Description: Data mining is a fast-growing discipline that uses ,., 22-29 Conference on Geometry and Dynamics of Groups and techniques from several subfields of applied mathematics, includ­ Spaces In Memory of Alexander Reznikov, Max-Planck-Institut ing operations research and statistics. This workshop will feature fUr Mathematik, Bonn, Germany. applications of exact or heuristic algorithms for solving mathe­ Organizers: Mil

MAY 2006 NOTICES OF THE AMS 611 Mathematics Calendar

'' 30-November 3 Computational Challenges Arising In Algorith­ Workshop topics and Goals: This workshop, sponsored by AIM and mic Number Theory and Cryptography, Fields, Toronto, Canada. the NSF, will bring together researchers with different perspectives Information: http: I /www. fields . utoronto . cal pr ograms/ in combinatorial representation theory: combinatorial, metric, and scientific/06-07/crypto/number_theory/. algebra-geometric. It has emerged from recently that Bruhat­ Tits buildings play an essential, not yet well-understood role November 2006 in combinatorial representation theory by providing a geometric realization to existing combinatorial models and linking them to '' 1-5 CCA 2006 Third International Conference on Computability the algebra-geometric tools of representation theory. Goals for and Complexity in Analysis, University of Florida, Gainesville, the workshop include examining and comparing the different Florida. approaches to the saturation theorem, with an emphasis on the Scope: The conference is concerned with the theory of computabil­ role of buildings. ity and complexity over real-valued data. Computable Analysis Application Deadline: December 1, 2006. combines concepts from Analysis/Numerical Analysis and Com­ Information: http : I I aimath. org/ ARCC/workshops/buildings. putability/Computational Complexity and studies those functions html. over real-valued data, which can be realized by digital computers. Submissions: Authors are invited to submit a PostScript or PDF version of a paper to cca- submission@FernUni -Hagen. de by July April 2007 2, 2006. '' 2 3-2 7 Problems in Geometric Group Theory, AIM Research Organizing Committee: Gainesville: Paul Brodhead, Douglas Cen­ Conference Center, Palo Alto, California. zer, chair, Rick Smith. Organizers: Mladen Bestvina, Tadeusz Januszkiewicz, and Richard Information: Klaus Weihrauch, email: Klaus. Weihrau ch@FernUni­ Scott. Hagen . de, Douglas Cenzer, email: cenzer@ufl. edu; http : I /cca­ Workshop Topics: This workshop, sponsored by AIM and the NSF , net. de/ cca2006/. will be devoted to compiling a list of unsolved and partially solved problems in geometric group theory. The list will be organized ,., 2 ?- December 1 Cryptography: Underlying Mathematics, Prov­ into various subfields of geometric group theory and other fields ability and Foundations, Fields Institute, Toronto, Canada. that have substantial overlap with geometric group theory. The Information: http: I /www. fields. utoronto. ca/programs/ problems will be annotated with special cases, relationships among scientific/06-07/crypto/crypto_foundations/. the problems, broader implications, and progress to date. Application Deadline: January 12, 2007. December 2006 Information: http: I /aimath. org/ ARCC/workshops/geomgpthy. '' 1 3-1 5 Workshop on "Geometry of vector distributions, differ­ html. ential equations, and variational problems", International School for Advanced Studies (SISSA), Trieste, Italy. Workshop topics: Equivalence problems for various geometric The following new announcements will not be repeated until structures on manifolds, especially nonholonomic distributions, the criteria in the next to the last paragraph at the bottom of sub-Riemannian structures, Cauchy-Riemann (CR) structures with the first page of this section are met. application to control systems, geometry of differential equations and variational problems. May 2007 Expected Participants: Andreas Cap (University of Vienna and ,., 7- 11 Rational Curves on Algebraic Varieties, AIM Research Erwin Schrodinger Institute of Mathematical Physics), Boris Doubrov Conference Center, Palo Alto, California. (Belarus sian State University, Minsk), Svetlana Ignatovich (Kharkov Organizers: Brendan Hassett and Sandor Kovacs. National University, Ukraine), Frederic Jean (ENSTA, Paris), Piotr Workshop Topics and Goal: This workshop, sponsored by AIM Mormul (Warsaw University), Pawel Nurowski (Warsaw University), and the NSF, will be devoted to rationally-connected varieties. The Jean-Baptiste Pomet (INRIA, Sophia Antipolis, France), Jan Slovak workshop will focus on the following tools: deformation theory (Masaryk University in Brno, Czech Republic), Michail Zhitomirskii of curves and combs; constructions of free curves with desired (Technion -Israel Institute of Technology, Haifa, Israel). properties; moduli spaces of stable maps; singularity theory and Organizers: Andrei Agrachev (SISSA) and Igor Zelenka (SISSA). rational-chain connectedness. One main goal will be to present and Information: All who are interested in taking part in this activity discuss state-of-the-art techniques in each of these areas. are invited to contact Igor Zelenka, zelenko@sissa. it; http: I I Application Deadline: January 21, 2007. www.sissa.it/-zelenko/CEIHomepage.html. Information: http: I /aimath. or g/ ARCC/workshops/ '' 1 7-21 Integral Closure, Multiplier Ideals and Cores, AIM Research rationalcurves.html. Conference Center, Palo Alto, California. Organizers: Alberto Corso, Claudia Polini, and Bernd Ulrich. July 2007 Workshop Topics: This workshop, sponsored by AIM and the NSF, '' 2- 6 25th Journees Arithmetiques, University of Edinburgh, Scot­ will be devoted to questions related to the notion of integral closure land, UK. of ideals. Specific aspects of the workshop focus are: computation Information: email: c . smyth@ed . ac. uk. of the integral closure and its complexity; multiplicities and equisingularity theory; cores of ideals and Briancon-Skoda type theorems; multiplier ideals and test ideals; and multiplier ideals and jet schemes. Application Deadline: September 1, 2006. Information: http : I I aimath. org/ ARCC/workshops/ integral cl osure.html.

March 2007 '' 26-30 Buildings and Combinatorial Representation Theory, AIM Research Conference Center, Palo Alto, California. Organizers: Monica Vazirani, Michael Kapovich, and Arun Ram.

612 NOTICES OF THE AMS VOLUME 53, NUMBER 5 New Publications Offered by the AMS

been solved using new and deep mathematical techniques Algebra and Algebraic inspired by physics! Geometry The book begins with an insightful introduction to enumerative geometry. From there, the goal becomes explaining the more advanced elements of enumerative algebraic geometry. Along the way, there are some crash On Higher Frobenius­ courses on intermediate topics which are essential tools for MEJY191RS the student of modern mathematics, such as cohomology and American Mathematical Society Schur Indicators other topics in geometry. l'ol~nl< !8J Nuni0.<8S5 On Higher Yevgenia Kashina, DePaul The physics content assumes nothing beyond a first Frobenius-Schur University, Chicago, IL, Yorck undergraduate course. The focus is on explaining the action Indicators Sommerhauser, Universitdt principle in physics, the idea of string theory, and how these Yevgenia Kashina directly lead to questions in geometry. Once these topics are Yorck Som merha user Miinchen, Munich, Germany, Yongchang Zhu in place, the connection between physics and enumerative and Yongchang Zhu, Hong geometry is made with the introduction of topological Kong University of Science and quantum field theory and quantum cohomology. Technology, Kowloon, Hong Contents: Warming up to enumerative geometry; Enumerative Kong geometry in the projective plane; Stable maps and enumerative geometry; Crash course in topology and Contents: Introduction; The calculus manifolds; Crash course in c oo manifolds and cohomology; of Sweedler powers; Frobenius-Schur indicators; The exponent; Cellular decompositions and line bundles; Enumerative The order; The index; The Drinfel'd double; Examples; geometry of lines; Excess intersection; Rational curves on the Bibliography; Subject index; Symbol index. quintic threefold; Mechanics; Introduction to supersymmetry; Memoirs of the American Mathematical Society, Volume 181, Introduction to string theory; Topological quantum field Number 855 theory; Quantum cohomology and enumerative geometry; Bibliography; Index. April 2006, 65 pages, Softcover, ISBN 0-8218-3886-5, LC 2006040679, 2000 Mathematics Subject Classification: 16W30; Student Mathematical library, Volume 32 17B35, Individual member US$30, List US$ 50, Institutional May 2006, approximately 215 pages, Softcover, ISBN 0-8218- member US$40, Order code MEM0/ 181/ 855 3687-0, 2000 Mathematics Subject Classification: 14N10, All AMS members US$28, List US$35, Order code STML/ 32

.,.._ COURSE ...V ADOPTION

UijO

MAY 2006 NOTICES OF THE AMS 613 New Publications Offered by the AMS provide the foundation for projective varieties. Major topics solutions; The equality case; Nontangentiallirnits; The include the theory of modules over a principal ideal domain, Nevanlinna-Pick boundary problem; A multiple analogue of the and its applications to matrix theory (including the Jordan Caratheodory-Julia theorem; On the solvability of a Stein decomposition), the Galois theory of field extensions, equation; Positive definite solutions of the Stein equation; A transcendence degree, the prime spectrum of an algebra, Caratheodory-Fejer boundary problem; The full matrix localization, and the classical theory of Noetherian and Caratheodory-Fejer boundary problem; The lossless inverse Artinian rings. Later chapters include some algebraic theory of scattering problem; Bibliography. elliptic curves (featuring the Mordell-Weil theorem) and Memoirs of the American Mathematical Socjety, Volume 181, valuation theory, including local fields. Number 856 One feature of the book is an extension of the text through a April 2006, 107 pages, Softcover, ISBN 0-8218-404 7-9, LC series of appendices. This permits the inclusion of more 2006040674, 2000 Mathematics Subject Classification: 30E05, advanced material, such as transcendental field extensions, 47A57, Individual member US$35, List US$ 59, Institutional the discriminant and resultant, the theory of Dedekind member US$47, Order code MEM0/ 181/ 856 domains, and basic theorems of rings of algebraic integers. An extended appendix on derivations includes the Jacobian conjecture and Makar-Limanov's theory of locally nilpotent derivations. Grbbner bases can be found in another appendix. Exercises provide a further extension of the text. The book can Stability of be used both as a textbook and as a reference source. ME1Y[9IRS Alllerlc..,MIIIhemottculSuclety Spherically Contents: Introduction and prerequisites; Exercises-Chapter 0; Part I. Modules: Introduction to modules and their structure Symmetric Wave theory; Finitely generated modules; Simple modules and Stability of composition series; Exercises-Part I; Part II. Affine algebras Spherically Symmetric Maps II. Introduction; Galois theory of Wave Maps and Noetherian rings: Part Joachim Krieger Joachim Krieger, Harvard fields; Algebras and affine fields; Transcendence degree and University, Cambridge, MA the Krull dimension of a ring; Modules and rings satisfying ,-· chain conditions; Localization and the prime spectrum; The Contents: Introduction, controlling Krull dimension theory of commutative Noetherian rings; spherically symmetric wave maps; Exercises-Part II; Part III. Applications to geometry and number Technical preliminaries. Proofs of theory: Part III. Introduction; The algebraic foundations of main theorems; The proof of geometry; Applications to algebraic geometry over the Proposition 2.2; Proof of theorem 2.3; Bibliography. rationals - Diophantine equations and elliptic curves; Memoirs of the American Mathematical Society, Volume 181, Absolute values and valuation rings; Exercises-Part III; Number 853 References; Index. April 2006, 80 pages, Softcover, ISBN 0-8218-3877-6, LC Graduate Studies in Mathematics, Volume 73 2006040673, 2000 Mathematics Subject Classification: 35L05, May 2006, approximately 399 pages, Hardcover, ISBN 0-8218- 35L70, Individual member US$33, List US$ 55, Institutional 0570-3, 2000 Mathematics Subject Classification: 28-XX; 26-XX, member US $44, Order code MEM0/ 181/ 853 31-XX, 42-XX, 46-XX, 49-XX, 81-XX, All AMS members US$52, List US $65, Order code GSM/ 73

Differential Equations Analysis Tangential Boundary ME1Y[9IRS On Boundary /l.lnerlcnn MatbemmtlcalSoclety Stabilization of ME1Y[91RS AmerlcnnMatbewttlculSoclety Tangen tial Boundary Navier-Stokes Interpolation for Stabilization of On Boundary Matrix Valued Schur Navier -Stokes Equations Interpolation Equ ations Viorel Barbu Viorel Barbu and Irena for Matrix Valued Functions Irena L asieck a Robcrlo Trigglani Lasiecka, Un iversity of Schur Functions Vladimir Bolotnikov, The Vladimir Bolotnikov Virginia, Charlottesville, VA, Harry Dym College of William and Mary, and Roberto Triggiani Williamsburg, VA, and Harry AmerlcanMa tbematlcalSoclety Dym, Weiz mann Institute of Contents: Introduction; Main results; Science, Rehovot, Israel Proof of Theorems 2.1 and 2.2 on the linearized system (2.4): d = 3; Boundary feedback uniform Contents: Introduction; Preliminaries; Fundamental matrix stabilization of the linearized system (3 .1.4) via an optimal inequalities; On :H(D) spaces; Parametrizations of all control problem and corresponding Riccati theory. Case d = 3;

614 NOTICES OF THE AMS VOLUME 53, NUMBER 5 New Publications Offered by the AMS

Theorem 2.3(i): Well-posedness of the Navier-Stokes equations problem; The longest lacings; The strongest lacings; The with Riccati-based boundary feedback controL Cased = 3; weakest lacings; Related mathematics; Loose ends; References; Theorem 2.3(ii): Local uniform stability of the Navier-Stokes Index. equations with Riccati-based boundary feedback control; A Mathematical World, Volume 24 PDE-interpretation of the abstract results in Sections 5 and 6; Appendix A. Technical material complementing Section 3.1; June 2006, 125 pages, Softcover, ISBN 0-8218-3933-0, LC Appendix B. Boundary feedback stabilization with arbitrarily 2006040733, 2000 Mathematics Subject Classification: OOA05, small support of the linearized system (3.1.4a) at the 90C27, 05A15, All AMS members US$23, List US$29, Order ~ 1 code MAWRLD/24 (H 2 ~ f (D.) )d n H-level, with LC. yO E (H r ' (D.) )d n H. Cases d = 2, 3. Theorem 2.5 ford= 2; Appendix C. Equivalence between unstable and stable versions of the optimal control problem of Section 4; Appendix D. Proof that FS(·) E L(W;L2 (0, oo; (L2(f))d); Bibliography. Memoirs of the American Mathematical Society, Volume 181, Geometry and Topology Number 852 April2006, 128 pages, Softcover, ISBN 0-8218-3874-1, . ,.COURSE LC 2006040678, 2000 Mathematics Subject Classification: Elements of •; ADOPTION 76D05, 35B40, 35Q30, Individual member US$36, List US$60, Institutional member US$48, Order code MEM0/181/852 Elements of Combinatorial and Combinatorial and Differential Differential Topology Topology V. V. Prasolov, Independent Discrete Mathematics and University of Moscow, Russia ·-~­ Combinatorics Modern topology uses very diverse t5 --- methods. This book is devoted largely to methods of combinatorial topology, which reduce the study of topological The Shoelace Book spaces to investigations of their partitions into elementary sets, and to methods of differential topology, which deal with A Mathematical Guide to smooth manifolds and smooth maps. Many topological the Best (and Worst) problems can be solved by using either of these two kinds of Ways to Lace Your Shoes methods, combinatorial or differentiaL In such cases, both approaches are discussed. Burkard Polster, Monash One of the main goals of this book is to advance as far as University, Clayton, Vic, possible in the study of the properties of topological spaces Australia (especially manifolds) without employing complicated techniques. This distinguishes it from the majority of other Crisscross, zigzag, bowtie, devil, angel, books on topology. or star: which are the longest, the shortest, the strongest, and the The book contains many problems; almost all of them are weakest lacings? Pondering the mathematics of shoelaces, the supplied with hints or complete solutions. author paints a vivid picture of the simple, beautiful, and Contents: Graphs; Topology in Euclidean space; Topological surprising characterizations of the most common shoelace spaces; Two-dimensional surfaces, coverings, bundles, and patterns. The mathematics involved is an attractive mix of homotopy groups; Manifolds; Fundamental groups; Hints and combinatorics and elementary calculus. This book will be solutions; Bibliography; Index. enjoyed by mathematically minded people for as long as there Graduate Studies in Mathematics, Volume 74 are shoes to lace. July 2006, approximately 342 pages, Hardcover, ISBN 0-8218- Burkard Polster is a well-known mathematical juggler, 3809-1, 2000 Mathematics Subject Classification: 57 ~01; magician, origami expert, bubble-master, shoelace charmer, 57Mxx, 57Rxx, All AMS members US$47, List US$ 59, Order and "Count von Count" impersonator. His previous books code GSM/ 74 include A Geometrical Picture Book, The Mathematics of juggling, and QED: Beauty in Mathematical Proof This item will also be of interest to those working in general and interdisciplinary areas. Contents: Setting the stage; One-column lacings; Counting lacings; The shortest lacings; Variations on the shortest lacing

MAY 2006 NOTICES OF THE AMS 615 New Publications Offered by the AMS

theory and integrable spin chains; R. Roiban, M. Spradlin, and Logic and Foundations A. Volovich, Yang-Mills amplitudes from twistor string theory; E. Sharpe, Notes on correlation functions in (0,2) theories. Contemporary Mathematics, Volume 401 The Role of True May 2006, 104 pages, Softcover, ISBN 0-8218-3663-3, LC MEMOIRSof the AmerlcanlllothematicalSoclety 2005058917, 2000 Mathematics Subject Classification: 14D22, Finiteness in the 14F05, 14N10, 14N35, 14Q05, 18E30, 53C80, 81Q30, 81Q60, The Role of True Admissible 81Q70, 81Tl3, 81T45, 81T30, 81T60, All AMS members Finiteness in the US$31, List US$39, Order code CONM/401 Admissible Recursively Recursively Enumerable Degrees Enumerable Degrees Noam Greenberg Noam Greenberg, University of Notre Dame, IN AmerlcRnMathemollcaiSoc!ety Number Theory Contents: Introduction; Coding into the R. E. degrees; Coding effective successor models; A negative result concerning effective successor models; A nonembedding result; Embedding the 1-3- Lectures on Ergodic 1 lattice; Appendix A. Basics; Appendix B. The jump; Appendix LECTURES C. The projectum; Appendix D. The admissible collapse; ON Theory Appendix E. Prompt permission; Appendix. Bibliography. ERGODIC Paul Halmos, Santa Clara Memoirs of the American Mathematical Society, Volume 181, THEORY University, Santa Clara, CA Number 854 This classic book is based on lectures April 2006, 99 pages, Softcover, ISBN 0-8218-3885-7, LC P~UL R. IIALMOS given by the author at the University 2006040677, 2000 Mathematics Subject Classification: 03D60; of Chicago in 1956. The topics A..\iS CHR!.SEA PtJBLISHl:'\'G 03D25, 03D30, Individual member US$35, List US$ 58, covered include, in particular, Institutional member US$46, Order code MEM0/181/854 recurrence, the ergodic theorems, and a general discussion of ergodicity and mixing properties. There is also a general discussion of the relation between conjugacy and equivalence. With minimal prerequisites of some analysis and measure theory, this work Mathematical Physics can be used for a one-semester course in ergodic theory or for self-study. Contents: Introduction; Examples; Recurrence; Mean Snowbird Lectures convergence; Pointwise convergence; Comments on the fa'CONTEMPORARY . ergodic theorem; Ergodicity; Consequences of ergodicity; ft•MATH~!!L~!!~~~ ~ on String Geometry Mixing; Measure algebras; Discrete spectrum; Automorphisms '"' Katrin Becker, Melanie of compact groups; Generalized proper values; Weak topology; Snowbird Lectures Weak approximation; Uniform topology; Uniform on String Geometry Becker, Aaron Bertram, Paul approximation; Category; Invariant measures; Invariant S. Green, and Benjamin measures: the solution; Invariant measures: the problem; Katrin Becker, Melanie Becker, Aaron Bertram, Paul S. Green. McKay, Editors Generalized ergodic theorems; Unsolved problems; References. Benjamin McKay. Editors The interaction and cross-fertilization AMS Chelsea Publishing of mathematics and physics is April 2006, 99 pages, Hardcover, ISBN 0-8218-412 5-4, ubiquitous in the history of both LC 60-8964, 2000 Mathematics Subject Classification: 37-XX, disciplines. In particular, the recent All AMS members US$23, List US$25, Order code CHEL/ 142.H developments of string theory have led to some relatively new areas of common interest among mathematicians and physicists, some of which are explored in the papers in this volume. These papers provide a reasonably comprehensive sampling of the potential for fruitful interaction between mathematicians and physicists that exists as a result of string theory. Contents: P. S. Aspinwall, D-branes, IT-stability and 8- stability; K. Becker, M. Becker, K. Dasgupta, and R. Tatar, Geometric transitions, non-Kahler geometries and string vacua; L.A. Borisov and R. P. Horja, On the K-theory of smooth toric DM stacks; S. Katz, Gromov-Witten, Gopakumar­ Vafa, and Donaldson-Thomas invariants of Calabi-Yau threefolds; A. Krause, Flux compactification geometries and de Sitter vacua in M-theory; R. Roiban, N = 4 super-Yang-Mills

616 NOTICES OF THE AMS VOLUME 53, NUMBER 5 New AMS-Distributed Publications

classical linear groups and from finite groups. The book gives a systematic and detailed introduction to the highlights of New AMS-Distributed that theory. In the beginning, a review of fundamental tools from topology Publications and the elementary theory of topological groups and transformation groups is presented. Completions, Haar integral, applications to linear representations culminating in the Peter-Weyl Theorem are treated. Pontryagin duality for locally compact Abelian groups forms a central topic of the Algebra and Algebraic book. Applications are given, including results about the structure of locally compact Abelian groups, and a structure Geometry theory for locally compact rings leading to the classification of locally compact fields. Topological semigroups are discussed in a separate chapter, with special attention to their relations to groups. The last chapter reviews results related to Hilbert's One Semester of Elliptic Curves Fifth Problem, with the focus on structural results for non­ Abelian connected locally compact groups that can be derived Torsten Ekedahl, University of Stockholm, Sweden using approximation by Lie groups. These lecture notes grew out of a one semester introductory The book is self-contained and is addressed to advanced course on elliptic curves given to an audience of computer undergraduate or graduate students in mathematics or science and mathematics students, and assume only minimal physics. It can be used for one-semester courses on background knowledge. After having covered basic analytic topological groups, on locally compact Abelian groups, or on and algebraic aspects, putting special emphasis on explaining topological algebra. Suggestions on course design are given in the interplay between algebraic and analytic formulas, they go the preface. Each chapter is accompanied by a set of exercises on to some more specialized topics. These include the )­ that have been tested in classes. function from an algebraic and analytic perspective, a A publication of the European Mathematical Society (EMS) . Distributed discussion of elliptic curves over finite fields, derivation of within the Americas by the American Mathematical Society. recursion formulas for the division polynomials, the algebraic Contents: Preliminaries; Topological groups; Topological structure of the torsion points of an elliptic curve, complex transformation groups; The Haar integral; Categories of multiplication, and modular forms. topological groups; Locally compact Abelian groups; Locally In an effort to motivate basic problems the book starts very compact semigroups; Hilbert's fifth problem; Bibliography; slowly, but considers some aspects such as modular forms of Index of symbols; Subject index. higher level which are not usually treated. It presents more EMS Textbooks in Mathematics than 100 exercises and a MathematicaT" notebook that treats a number of calculations involving elliptic curves. February 2006, 312 pages, Hardcover, ISBN 3-03719-016-7, 2000 Mathematics Subject Classification: 22005, 22-01, 20E18, The book is aimed at students of mathematics with a general 22A25, 22B05, 22C05, 22010, 22045, 22F05, 12]10, 43A05, interest in elliptic curves but also at students of computer 54H15, 22A15, All AMS members US$46, List US$ 58, Order science interested in their cryptographic aspects. code EMSTEXT/3 A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. Contents: Elliptic integrals; Elliptic curves; Elliptic functions; A projective interlude; The group structure on an elliptic curve; Equivalence; Formulaire; Finite fields; Division polynomials; Torsion points; Lattice inclusions; Modular forms; Hints to Differential Equations exercises; Solutions to exercises. ESI Lectures in Mathematics and Physics March 2006, 140 pages, Softcover, ISBN 3-03719-015-9, 2000 Dynamics on the Riemann Sphere Mathematics Subject Classification: 14K25, All AMS members US$30, List US$38, Order code EMSSERLEC/2 A Bodil Branner Festschrift Poul Hjorth, Technical University of Denmark, Lyngby, Denmark, and Carsten Lunde Petersen, Roskilde University, Denmark, Editors Dynamics on the Riemann Sphere presents a collection of Locally Compact Groups original research articles by leading experts in the area of Markus Stroppel, University of Stuttgart, holomorphic dynamics. These papers arose from the symposium Dynamics in the Complex Plane, held on the Germany occasion of the 60th birthday of Bodil Branner. Topics covered Locally compact groups play an important role in many areas range from Lattes maps to cubic polynomials over rational of mathematics as well as in physics. The class of locally maps with Sierpinsky Carpets and Gaskets as Julia sets, as compact groups admits a strong structure theory, which well as rational and entire transcendental maps with Herman allows to reduce many problems to groups constructed in rings. various ways from the additive group of real numbers, the This item will also be of interest to those working in analysis.

MAY 2006 NOTICES OF THE AMS 617 New AMS-Distributed Publications

A publication of the European Mathematical Society (EMS). Distributed This item will also be of interest to those working in within the Americas by the American Mathematical Society. applications. Contents: J. Milnor, On Lattes maps; C. L. Petersen and T. Lei, A publication of the European Mathematical Society (EMS) . Distributed Branner-Hubbard motions and attracting dynamics; A. Avila within the Americas by the American Mathematical Society. and M. Lyubich, Examples of Feigenbaum Julia sets with small Contents: I. Analysis of differential-algebraic equations: Hausdorff dimension; A. Che'ritat, Parabolic explosion and the Introduction; Linear differential-algebraic equations with size of Siegel disks in the quadratic family; P. Blanchard, constant coefficients; Linear differential-algebraic equations R. L. Devaney, D. M. Look, M. Rocha, P. Seal, S. Siegmund, with variable coefficients; Nonlinear differential-algebraic and D. Uminsky, Sierpinski carpets and gaskets as Julia sets equations; II. Numerical solution of differential-algebraic of rational maps; P. Roesch, On capture zones for the family equations: Numerical methods for strangeness-free problems; j(z) = z2 + ?l(z2 ; T. Kawahira, Semiconjugacies between the Numerical methods for index reduction; Boundary value Julia sets of geometrically finite rational maps II; W. Jung, problems; Software for the numerical solution of differential­ Homeomorphisms of the Mandelbrot set; N. Fagella and algebraic equations; Final remarks; Bibliography; Index. C. Henriksen, Arnold disks and the moduli of Herman rings of the complex standard family; T. Lei, Stretching rays and their EMS Textbooks in Mathematics accumulations, following Pia Willumsen; A. Douady, February 2006, 392 pages, Hardcover, ISBN 3-03719-017-5, Conjectures about the Branner-Hubbard motion of Cantor sets 2000 Mathematics Subject Classification: 34A09, 65L80, All in C. AMS members US$54, List US$68, Order code EMSTEXT/ 2 January 2006, 240 pages, Hardcover, ISBN 3-03719-011-6, 2000 Mathematics Subject Classification: 37F45, All AMS members US$63, List US$79, Order code EMSDRS

Differential-Algebraic Equations Analysis and Numerical Solution Peter Kunkel, University of Leipzig, Germany, and Volker Mehrmann, Technical University of Berlin, Germany Differential-algebraic equations are a widely accepted tool for the modeling and simulation of constrained dynamical systems in numerous applications, such as mechanical multibody systems, electrical circuit simulation, chemical engineering, control theory, fluid dynamics and many others. This is the first comprehensive textbook that provides a systematic and detailed analysis of initial and boundary value problems for differential-algebraic equations. The analysis is developed from the theory of linear constant coefficient systems via linear variable coefficient systems to general nonlinear systems. Further sections on control problems, generalized inverses of differential-algebraic operators, generalized solutions, and differential equations on manifolds complement the theoretical treatment of initial value problems. Two major classes of numerical methods for differential-algebraic equations (Runge-Kutta and BDF methods) are discussed and analyzed with respect to convergence and order. A chapter is devoted to index reduction methods that allow the numerical treatment of general differential-algebraic equations. The analysis and numerical solution of boundary value problems for differential-algebraic equations is presented, including multiple shooting and collocation methods. A survey of current software packages for differential-algebraic equations completes the text. The book is addressed to graduate students and researchers in mathematics, engineering and sciences, as well as practitioners in industry. A prerequisite is a standard course on the numerical solution of ordinary differential equations. Numerous examples and exercises make the book suitable as a course textbook or for self-study.

618 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Meetings & Conferences oftheAMS

IMPORT ANT INFORMATION REGARDING MEETINGS PROGRAMS: AMS Sectional Meeting programs do not appear in the print version of the Notices. However, comprehensive and continually updated meeting and program information with links to the abstract for each talk can be found on the AMS website. See http: I /www . ams. org/meeti ngs/. Final programs for Sectional Meetings will be archived on the AMS website accessible from the stated URL and in an electronic issue of the Notices as noted below for each meeting.

Special Sessions Miami, Florida Approximation Theory and Orthogonal Polynomials, Doron Florida International University S. Lubinsky, Georgia Institute of Technology, and Edward B. Saff, Vanderbilt University. April l-2, 2006 Commutative Algebra and Algebraic Geometry, Laura Saturday - Sunday Ghezzi, Florida International University, Huy T

History of Mathematics, Karen H. Parshall, University of Karen V. H. Parshall, University of Virginia, The British de­ Virginia. velopment of the theory of invariants, 1841-1895. Imaging, Homogenization, and Shape Optimization, Michael Michael S. Vogelius, Rutgers University, Electromagnetic S. Vogelius, Rutgers University, and Shari Moskow, Uni­ imaging- An applied analyst's perspective. versity of Florida.

MAY 2006 NOTICES OF THE AMS 619 Meetings & Conferences

Interpolation Theory and Applications, Michael Cwikel, For consideration of contributed papers in Special Sessions: Technion, Laura De Carli, Florida International Univer­ Expired sity, and Mario Milman, Florida Atlantic University. For abstracts: Expired Invariants of Low-Dimensional Manifolds, Thomas G. Invited Addresses Leness, Florida International University, and Nikolai N. Saveliev, University of Miami, Coral Gables. Douglas N. Arnold, Institute for Math and Applications, Finite element exterior calculus and Mathematical Models in Image and High-Dimensional Data University of Minnesota, its applications. Analysis, Hanna E. Makaruk and Robert M. Owczarek, Los Alamos National Laboratory, and Nikita Sakhaneko, Uni­ Bela Bollobas, University of Memphis and Cambridge Uni­ versity of New Mexico and Los Alamos National Labora­ versity, Inhomogeneous random graphs (Erdos Memorial tory. Lecture). Monomials and Resolutions, Joseph P. Brennan, North Steve C. Hofmann, University of Missouri, Columbia, Local Dakota State University, and Heath M. Martin, University Tb theorems and applications in PDE. of Central Florida. Michael J. Larsen, Indiana University, Representation zeta Nonlinear Waves, Andrea R. Nahmod, University of Mass­ functions. achusetts, Amherst, and Sijue Wu, University of Michigan Christopher M. Skinner, University of Michigan, Modular at Ann Arbor. forms and special values of L-functions. Partial Differential Equations and Several Complex Vari­ Special Sessions ables, Shiferaw Berhanu, Temple University, and Hamid Meziani, Florida International University. Algebraic Structures of Exactly Solvable Models, Michael Qualitative Analysis of Partial Differential Equations, Con­ Gekhtman, University of Notre Dame, Mikhail Shapiro, gming Li, University of Colorado, and Wenxiong Chen, Michigan State University, and Alexander Stolin, Univer­ sity of Gothenburg. Yeshiva University. Analysis and Geometry of Non-linear Evolution Equations, Recent Developments on Fluid and Geophysical Fluid Dy­ Alexandrou A. Himonas and Gerard K. Misiolek, Univer­ namics, C. Cao and T. Tachim Medjo, Florida Interna­ sity of Notre Dame. tional University, and X. Wang, Florida State University. Combinatorial Algebraic Geometry, juan C. Migliore, Uni- · Singular Integrals, Geometric Analysis, and Free Boundary versity of Notre Dame, and Uwe R. Nagel, University of Ken­ Problems, Marianne Korten and Charles N. Moore, Kansas tucky. State University, and Laura DeCarli, Florida International University. Commutative Algebra and Algebraic Geometry, Alberto Corso, University of Kentucky, Claudia Polini, University Spectral Geometry ofManifolds with Boundary and Singular of Notre Dame, and Bernd Ulrich, Purdue University. Spaces, Juan B. Gil, Pennsylvania State University, Altoona, and Patrick T. McDonald, New College, University of South Developments and Applications in Differential Geometry, Florida. Jianguo Cao, Xiaobo Liu, and Brian Smyth, University of Notre Dame. Structure of Function Spaces and Applications, Jan Lang, The Ohio State University, and Osvaldo Mendez, Univer­ Dynamical Systems, Francois Ledrappier, University of Notre Dame, and Arnie Wilkinson, Northwestern Univer­ sity of Texas at El Paso. sity. Harmonic Analysis, PDE and Geometric Function Theory, John L. Lewis, University of Kentucky, and Steve C. Hof­ Notre Dame, Indiana mann, University of Missouri, Columbia. University of Notre Dame Holomorphic Methods and Heat Kernels in Harmonic Analy­ sis and Quantization Theory, Brian Hall and William Kir­ April8-9, 2006 win, University of Notre Dame. Saturday - Sunday Mathematical Biology, Mark Alber and Bei Hu, University of Notre Dame. Meeting #1 016 Model Theory and Computability, Steven Allen Buechler Central Section and Julia Knight, University of Notre Dame, Steffen Lempp, Associate secretary: Susan ]. Friedlander University of Wisconsin, and Sergei Starchenko, Univer­ Announcement issue of Notices: January 2006 sity of Notre Dame. Program first available on AMS website: February 23, 2006 Program issue of electronic Notices: April 2006 New Developments in Optimization, Leonid Faybusovich, issue of Abstracts: Volume 27, Issue 2 University of Notre Dame. Nonlinear Waves, Mark S. Alber and Pavel Lushnikov, Deadlines University of Notre Dame, and Ildar Gabotiv and Vladimir For organizers: Expired E. Zakharov, University of Arizona.

620 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Meetings & Conferences

Number Theory, Scott T. Parsell and jonathan P. Soren­ Special Sessions son, Butler University. Algebraic Groups, George j. McNinch, Tufts University, and Numerical Solution of Polynomial Systems, Christopher S. Eric Sommers, University of Massachusetts, Amherst. Peterson, Colorado State University, and Andrew j. Arithmetic Geometry and Modular Forms, Paul E. Gun­ Sommese, University of Notre Dame. nells and Farshid Hajir, University of Massachusetts, PDEs and Geometric Analysis, Matt Gursky and Qing Han, Amherst. University of Notre Dame. Arrangements and Configuration Spaces, Graham C. Den­ Several Complex Variables, Nancy K. Stanton and Jeffrey ham, University of Western Ontario, and Alexander I. A. Diller, University of Notre Dame. Suciu, Northeastern University. Special Functions and Orthogonal Polynomials, Diego Do­ minici, State University of New York at New Paltz. Banach Lattices, Regular Operators, and Applications, A. K. Kitover, Community College of Philadelphia, M. Topics in Representation Theory, Sam Evens, University of Orhon, University of New Hampshire, and A. W. Wickstead, Notre Dame, and Jiu-Kang Yu, Purdue University. Queen's University of Belfast. Topology and Physics, Stephan A. Stolz and Bruce Williams, Functions, Rita A. Hibschweiler, University of Notre Dame. Banach Spaces ofAnalytic University of New Hampshire, and Thomas H. MacGregor, Undergraduate Mathematical Research, Francis X. Con­ SUNY Albany and Bowdoin College. nolly, University of Notre Dame, and Zsuzsanna Szanis­ zlo, Valparaiso University. Discrete and Convex Geometry, Daniel A. Klain, University Lowell, Barry R. Monson, University of Water Waves, David Nicholls, University of Illinois at of Massachusetts, Univer­ Chicago. New Brunswick, and Egon Schulte, Northeastern sity. Galois Theory in Arithmetic and Geometry, Florian Pop and Durham, New David Harbater, University of Pennsylvania, and Rachel j. Pries, Colorado State University. Hampshire Geometric Methods in Group Theory and Topology, Kim Taback, Bowdoin Col­ University of New Hampshire Ruane, Tufts University, jennifer lege, and Peter N. Wong, Bates College. April22-23, 2006 Global Perspectives on the Geometry of Riemann Surfaces, Saturday - Sunday Eran Makover and Jeffrey K. McGowan, Central Con­ necticut State University. Meeting #l 017 Hopf Algebras and Galois Module Theory, Timothy Kohl, Eastern Section Boston University, and Robert G. Underwood, Auburn Associate secretary: Lesley M. Sibner University Montgomery. Announcement issue of Notices: February 2006 Program first available on AMS website: March 9, 2006 Mathematical Challenges in Physical and Engineering Sci­ Program issue of electronic Notices: April 2006 ences, Marianna A. Shubov, University of New Hampshire. Issue of Abstracts: Volume 27, Issue 2 Quantum Invariants of Knots and 3-Manifolds, Charles D. Frohman, University of Iowa, and Razvan Gelca, Texas Tech Deadlines University. For organizers: Expired For consideration of contributed papers in Special Sessions: Symplectic and Contact Topology, Weimin Chen, Michael Expired G. Sullivan, and Hao Wu, University of Massachusetts, For abstracts: Expired Amherst. Topological Algebras and Applications, Alexander A. Katz, Invited Addresses St. John's University, and Genady Y. Grabarnik, IBM T . ]. Ailana M. Fraser, University of British Columbia, Analytic Watson Research Center. methods in studying curvature and topology. Dmitri Nikshych, University of New Hampshire, Algebraic theory of tensor categories. Florian Pop, University of Pennsylvania, From topological covers to algebraic numbers.

Konsta~tina Trivisa, University of Maryland, College Park, On the dynamics of binary fluid mixtures.

MAY 2006 NOTICES OF THE AMS 621 Meetings & Conferences

Geometry of Grabner Bases, Bernd Sturmfels, University San Francisco, of California Berkeley, and Alexander Yong, University of Minnesota and Fields Institute. California Hilbert Functions and Resolutions, Benjamin Richert, Cal­ ifornia Polytechnic State University, and Sean Sather­ San Francisco State University Wagstaff, California State University, Dominguez Hills. History and Philosophy of Mathematics, Shawnee April29-30, 2006 L. Mc­ Murran, California State University, San Bernardino, and Saturday - Sunday James j. Tattersall, Providence College. Meeting #1 018 Homological and K-theoretical Trends in Algebraic Com­ binatorics, joseph Gubeladze and Serkan Hosten, San Western Section Francisco State University. Associate secretary: Michel L. Lapidus Announcement issue of Notices: February 2006 Liapunov Exponents and Nonuniform Hyperbolicity, Anton Gorodetski and Vadim Kaloshin, California Program first available on AMS website: March 16, 2006 Institute of Technology. · Program issue of electronic Notices: April 2006 Issue of Abstracts: Volume 27, Issue 2 Lie Algebras and Applications, Dimitar Grantcharov, San Jose State University, Vera Serganova, University of Cali­ Deadlines fornia Berkeley, and Arturo Pianzola, University of Al­ For organizers: Expired berta. For consideration of contributed papers in Special Sessions: Partial Differential Equations and Their Applications, Steve Expired Shkoller, University of California Davis. For abstracts: Expired Probability and Statistical Physics, Marek Biskup, Univer­ sity of California Los Angeles, Noam Berger, California In­ Invited Addresses stitute of Technology and University of California Los An­ Lincoln Chayes, University of California Los Angeles, geles, and Balint Virag, University of Toronto. Mean-field approach to the problem of phase transitions in Q-series and Partitions, Neville Robbins, San Francisco physically realistic systems. State University. C. Robin Graham, University of Washington, Ambient met­ rics, jet isomorphism and parabolic invariant theory in conformal geometry. Salt Lake City, Utah Vadim Kaloshin, California Institute of Technology, Non­ University of Utah local instabilities of the planar three body problem. Benoit B. Mandelbrot, Yale University, The nature of rough­ October 7-8,2006 ness in mathematics, science, and art (Einstein Public Lec­ Saturday - Sunday ture in Mathematics). Meeting#1019 Yuval Peres, University of California Berkeley, Hex, ran­ Western Section dom-turn games, and the infinity Laplacian. Associate secretary: Michel L. Lapidus Announcement issue of Notices: August 2006 Special Sessions Program first available on AMS website: August 24, 2006 Computational Arithmetic Geometry, Kenneth A. Ribet, Uni­ Program issue of electronic Notices: October 2006 versity of California Berkeley, and Kristin Estrella Lauter, Issue of Abstracts: Volume 27, Issue 3 Microsoft Corporation. Deadlines Elliptic Methods in Geometry, C. Robin Graham, University For organizers: Expired of Washington, and Rafe Mazzeo, Stanford University. 1 For consideration of contributed papers in SpecialJSessi~ns: Enumerative Aspects of Polytopes, Federico Ardila and June 20, 2006 Matthias Beck, San Francisco State University. For abstracts: August 15, 2006 Fractal Geometry: Connections to Dynamics, Geometric · Invited Addresses Measure Theory, Mathematical Physics and Number The­ ory, MichelL. Lapidus and Erin P. Pearse, University of William Arveson, University of California Berkeley, Title California Riverside, and Machiel van Frankenhuijsen, to be announced. Utah Valley State College. Alexei Borodin, California Institute of Technology, Title Geometric Dynamics and Ergodic Theory, Yitwah Cheung to be announced. · and Arek Goetz, San Francisco State University, and Slo­ Izabella joanna Laba, Un)versity of British Columbia, Title bodan Simic, San Jose State University. to be announced.

622 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Meetings & Conferences

Darren Long, University of California Santa Barbara, Title to be announced. Cincinnati, Ohio Special Sessions University of Cincinnati

Commutative Algebra (Code: SS 3A), Paul Roberts, Anurag October 21 - 22, 2006 K. Singh, and Oana Veliche, University of Utah. Saturday - Sunday Complex Geometry, Kaehler Groups, and Related Topics (Code: SS 9A), Terrence Napier, Lehigh University, Mohan Meeting # l 020 Ramachandran, State University of New York at Buffalo, Central Section and Domingo Toledo, University of Utah. Associate secretary: Susan]. Friedlander Floer Methods in Low-dimensional Topology (Code: SS SA), Announcement issue of Notices: August 2006 Alexander Felshtyn and Uwe Kaiser, Boise State Univer­ Program first available on AMS website: September 7, 2006 sity. Program issue of electronic Notices: October 2006 Harmonic Analysis: Trends and Perspectives (Code: SS lA), Issue of Abstracts: Volume 27, Issue 3 Alex Iosevich, University of Missouri, and Michael T. Deadlines Lacey, Georgia Institute of Technology. For organizers: Expired Interface of Stochastic Partial Differential Equations and For consideration of contributed papers in Special Sessions: Gaussian Analysis (Code: SS 7A), Davar Khoshnevisan, Uni­ July 5, 2006 versity of Utah, and Eulalia Nualart, University of Paris XIII. For abstracts: August 29, 2006 Low Dimensional Topology and Geometry (Code: SS 4A), Mladen Bestvina and Kenneth W. Bromberg, University Invited Addresses of Utah. Suncica Carrie, University of Houston, Title to be an­ Mathematics Motivated by Physics (Code: SS SA), Aaron J. nounced. Bertram, Yuan-Pin Lee, and Eric R. Sharpe, University of Bryna R. Kra, Northwestern University, Title to be an­ Utah. nounced. Multi-variable Operator Theory (Code: SS 13A), William B. Ezra N. Miller, University of Minnesota, Title to be an­ Arveson, University of California Berkeley, Scott A. Mc­ nounced. Cullough, University of Florida, and Geoffrey L. Price, Jon G. Wolfson, Michigan State University, Title to be an­ U.S. Naval Academy. nounced. Noncommutative Dynamical Systems (Code: SS 12A), William B. Arveson, University of California Berkeley, Special Sessions Scott A. McCullough, University of Florida, and Geoffrey Algebraic Coding Theory-Honoring the Retirement of L. Price, U.S. Naval Academy. Vera Pless (Code: SS SA), William Cary Huffman, Loyola Nonconvex Variational Problems: Recent Advances and Ap­ University, and Jon-Lark Kim, University of Louisville. plications (Code: SS lOA), Marian Bocea, North Dakota Analysis and Potential Theory on Metric Spaces (Code: SS State University, and Andrej Cherkaev, University of Utah. 4A), Thomas Bieske, University of South Florida, and Zair Nonlinear Differential Equations: Methods and Applica­ Ibragimov and Nageswari Shanmugalingam, University tions (Code: SS 2A), David G. Costa, University of Nevada, of Cincinnati. and Zhi-Qiang Wang, Utah State University. Applied Algebraic Geometry and Cryptography (Code: SS Random Motion in Random Media (Code: SS llA), Fir as Ras­ 3A), Jintai Ding, Jason Eric Gower, and Timothy J. Hodges, soul-Agha, University of Utah, and Tom Schmitz, Swiss Fed­ University of Cincinnati, Lei Hu, Chinese Academy of Sci­ eral Institute of Technology. ences, and Dieter S. Schmidt, University of Cincinnati. Theory and Applications of Infinite Dimensional Dynami­ Birational Geometry(Code: SS 2A), Mirel Constantin Caibar cal Systems (Code: SS 6A), Peter W. Bates, Michigan State and Gary P. Kennedy, Ohio State University. University, and Kening Lu, Brigham Young University. Boundary Value Problems for Differential Equations with Applications (Code: SS llA), Xiaojie Hou, Philip L. Korman, and Bingyu Zhang, University of Cincinnati. Ergodic Theory (Code: SS lA), Nikos Frantzikinakis, Penn­ sylvania State University, Bryna R. Kra, Northwestern Uni­ versity, and Mate Wierdl, University of Memphis. Geometric Combinatorics (Code: SS 6A), Ezra N. Miller, University of Minnesota, and Igor Pak, Massachusetts In­ stitute of Technology.

MAY 2006 NOTICES OF THE AMS 623 Meetings & Conferences

Limit Theorems of Probability Theory (Code: SS 9A), Wlodek Harmonic Analysis and Integral Geometry (Code: SS 7 A), Bryc and Magda Peligrad, University of Cincinnati. William 0. Bray, University of Maine, and Wolodymyr R. Nonlinear Functional Analysis and Applications (Code: SS Madych, University of Connecticut, Storrs. SA), S. P. Singh and Bruce Watson, Memorial University Nonlinear Elliptic and Parabolic Equations (Code: SS SA), of Newfoundland. Yung-Sze Choi, Changfeng Gui, and Joseph McKenna, Nonlinear Partial Differential and Its Applications (Code: University of Connecticut, Storrs. SS 7A), Changyou Wang, University of Kentucky, and Guan Number Theory (Code: SS 2A), Keith Conrad, University Bo, Ohio State University. of Connecticut, Storrs, David Pollack, Wesleyan University, and Thomas A. Weston, University of Massachusetts, Recent Results on Operator Algebras (Code: SS lOA), Her­ Amherst. bert Halpern, Gary Weiss, Costel Peligrad, Shuang Zhang, and Victor G. Kaftal, University of Cincinnati. Topology and Computing (Code: SS 6A), Thomas J. Pe­ ters, University of Connecticut, Storrs. Undergraduate Mathematics Education (Code: SS SA), Tom Storrs, Connecticut Roby, University of Connecticut, Storrs. University of Connecticut Fayetteville, October 28-29, 2006 Arkansas Saturday - Sunday University of Arkansas Meeting #1 021 Eastern Section November 3-4, 2006 Associate secretary: Lesley M. Sibner Friday - Saturday Announcement issue of Notices: August 2006 Program first available on AMS website: September 14, Meeting #1 022 2006 Southeastern Section Program issue of electronic Notices: October 2006 Associate secretary: Matthew Miller Issue of Abstracts: Volume 27, Issue 4 Announcement issue of Notices: September 2006 Program first available on AMS website: September 21, Deadlines 2006 For organizers: Expired Program issue of electronic Notices: November 2006 For consideration of contributed papers in Special Sessions: Issue of Abstracts: Volume 27, Issue 4 July 11, 2006 Deadlines For abstracts: September 6, 2006 For organizers: Expired Invited Addresses For consideration of contributed papers in Special Sessions: July 18, 2006 Changfeng Gui, University of Connecticut, Storrs, Title to For abstracts: September 12, 2006 be announced. Niranjan Ramachandran, University of Maryland, College Invited Addresses Park, Title to be announced. Richard P. Anstee, University of British Columbia, Title to Karman Soundararajan, University of Michigan, Title to be be announced. announced. Arun Ram, University of Wisconsin, Title to be announced. Katrin Wehrheim, Institute for Advanced Study, Title to Donald G. Saari, University of California Irvine, Title to be be announced. announced. Special Sessions Andras Vasy, Massachusetts Institute of Technology, Title to be announced. Analysis and Probability on Fractals (Code: SS 3A), Robert S. Strichartz, Cornell University, and Alexander Teplyaev, Special Sessions University of Connecticut, Storrs. Analytic Number Theory and Modular Forms (Code: SS Combinatorial Methods in Equivariant Topology (Code: SS 2A), Matthew Boylan, University of Illinois, Urbana-Cham­ lA), Tara Holm, University of Connecticut, Storrs, and paign, and Gang Yu, University of South Carolina. Tom C. Braden, University of Massachusetts, Amherst. Boundary Operators in Real and Complex Domains (Code: Computability Theory in Honor of Manuel Lerman's Re­ SS 3A), Loredana Lanzani, University of Arkansas, Fayet­ tirement (Code: SS 4A) , Joseph S. Miller and David Reed teville, and David E. Barrett, University of Michigan, Ann Solomon, University of Connecticut, Storrs. Arbor.

624 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Meetings & Conferences

Dirac Operators in Analysis and Geometry (Code: SS lA), Deadlines John Ryan, University of Arkansas, Marius Mitrea, Uni­ For organizers: August 3, 2006 versity of Missouri, and Mircea Martin, Baker University. For consideration of contributed papers in Special Sessions: Evolution Equations in Physics and Mechanics (Code: SS 4A), To be announced John P. Albert, University of Oklahoma, Jerry L. Bona, Uni­ For abstracts: To be announced versity of Illinois at Chicago, and Jiahong Wu, Oklahoma State University. Oxford, Ohio New Orleans, Miami University March 16-1 7, 2007 Louisiana Friday - Saturday

New Orleans Marriott and Sheraton New Meeting #1 025 Orleans Hotel Central Section January 5-8,2007 Associate secretary: Susan]. Friedlander Announcement issue of Notices: To be announced Friday - Monday Program first available on AMS website: To be announced Meeting # 1023 Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced ]oint Mathematics Meetings, including the 113th Annual Meeting of the AMS, 90th Annual Meeting of the Mathe­ Deadlines matical Association of America (MAA), annual meetings of For organizers: To be announced 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 with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Special Sessions Associate secretary: Susan]. Friedlander Announcement issue of Notices: October 2006 Finite Geometry and Combinatorics (Code: SS 3A), Mark A. Program first available on AMS website: November l, 2006 Miller, Marietta College. Program issue of electronic Notices: January 2007 Geometric Topology (Code: SS 2A), Jean-Francois LaFont, Issue of Abstracts: Volume 28, Issue l SUNY Binghamton and Ohio State University, and Ivonne ]. Ortiz, Miami University. Deadlines Large Cardinals in Set Theory (Code: SS lA), Paul B. Lar­ For organizers: Expired son, Miami University, Justin Tatch Moore, Boise State Uni­ For consideration of contributed papers in Special Sessions: versity, and Ernest Schimmerling, Carnegie Mellon Uni­ August l, 2006 versity. For abstracts: September 26, 2006 Davidson, North Hoboken, New Jersey Stevens Institute of Technology Carolina April14-1 5, 2007 Davidson College Saturday - Sunday March 3-4, 2007 Meeting #1 026 Saturday - Sunday Eastern Section Associate secretary: Lesley M. Sibner Meeting # 1024 Announcement issue of Notices: To be announced Southeastern Section Program first available on AMS website: To be announced Associate secretary: Matthew Miller Program issue of electronic Notices: To be announced Announcement issue of Notices: To be announced Issue of Abstracts: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Deadlines Issue of Abstracts: To be announced For organizers: September 14, 2006

MAY 2006 NOTICES OF THE AMS 625 Meetings & Conferences

For consideration of contributed papers in Special Sessions: To be announced Albuquerque, New For abstracts: To be announced Mexico Tucson, Arizona University of New Mexico University of Arizona October 1 3-14, 2007 Saturday - Sunday April21 -22, 2007 Western Section Saturday - Sunday Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced Meeting #1 027 Program first available on AMS website: To be announced Western Section Program issue of electronic Notices: To be announced Associate secretary: Michel L. Lapidus Issue of Abstracts: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Deadlines Program issue of electronic Notices: To be announced For organizers: To be announced Issue of Abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced Deadlines For abstracts: To be announced For organizers: September 21, 2006 For consideration of contributed papers in Special Sessions: To be announced For abstracts: To be announced Murfreesboro, Tennessee Warsaw, Poland Middle Tennessee State University

University of Warsaw November 3-4, 2007 July 31 -August 3, 2007 Saturday - Sunday Tuesday - Friday Southeastern Section Associate secretary: Matthew Miller Meeting #1 028 Announcement issue of Notices: To be announced First joint International Meeting between the AMS and the Program first available on AMS website: To be announced Polish Mathematical Society Program issue of electronic Notices: To be announced Associate secretary: Susan]. Friedlander Issue of Abstracts: To be announced Announcement issue of Notices: To be announced Deadlines Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced For organizers: April4, 2007 Issue of Abstracts: To be announced For consideration of contributed papers in Special Sessions: To be announced Deadlines For abstracts: To be announced For organizers: To be announced For consideration of contributed papers in Special Sessions: To be announced San Diego, California For abstracts: To be announced San Diego Convention Center Invited Addresses January 6-9, 2008 Henryk Iwaniec, To be announced. Sunday - Wednesday Tomasz Mrowka, To be announced. joint Mathematics Meetings, including the 114th Annual Madhu Sudan, To be announced. Meeting of the AMS, 9lst Annual Meeting of the Mathe­ matical Association ofAmerica (MAA), annual meetings of the Association for Women in Mathematics (A WM) 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 Applied Mathematics (SIAM).

626 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Meetings & Conferences

Associate secretary: Michel L. Lapidus Announcement issue of Notices: October 2007 Bloomington, Indiana Program first available on AMS website: November 1, 2007 Program issue of electronic Notices: January 2008 Indiana University Issue of Abstracts: Volume 29, Issue 1 April4-6, 2008 Deadlines Friday - Sunday For organizers: April1, 2007 Central Section For consideration of contributed papers in Special Sessions: Associate secretary: Susan]. Friedlander To be announced Announcement issue of Notices: To be announced For abstracts: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced New York, New York Issue of Abstracts: To be announced Deadlines Courant Institute of New York University For organizers: September 4, 2007 March 22-23, 2008 For consideration of contributed papers in Special Sessions: To be announced Saturday - Sunday For abstracts: To be announced Eastern Section Associate secretary: Lesley M. Sibner Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Claremont, California Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Claremont McKenna College Deadlines May 3- 4, 2008 For organizers: August 22, 2007 Saturday - Sunday For consideration of contributed papers in Special Sessions: Southeastern Section To be announced Associate secretary: Michel L. Lapidus For abstracts: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Baton Rouge, Issue of Abstracts: To be announced Deadlines Louisiana For organizers: October 4, 2007 Louisiana State University, Baton Rouge For consideration of contributed papers in Special Sessions: To be announced March 28- 30,2008 For abstracts: To be announced Friday - Sunday Southeastern Section Associate secretary: Matthew Miller Rio de Janeiro, Brazil Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Instituto Nacional de Matematica Pura e Program issue of electronic Notices: To be announced Aplicada (IMPA) Issue of Abstracts: To be announced June 4- 7,2008 Deadlines Wednesday - Saturday For organizers: August 28, 2007 First ]oint International Meeting between the AMS and the For consideration of contributed papers in Special Sessions: Sociedade Brasileira de Matematica. To be announced Associate secretary: Lesley M. Sibner For abstracts: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: Not applicable Program issue of electronic Notices: Not applicable Issue of Abstracts: Not applicable

Deadlines For organizers: To be announced

MAY 2006 NOTICES OF THE AMS 627 Meetings & Conferences

For consideration of contributed papers in Special Sessions: Issue of Abstracts: Not applicable To be announced For abstracts: To be announced Deadlines For organizers: To be announced For consideration of contributed papers in Special Sessions: Vancouver, Canada To be announced For abstracts: To be announced University of British Columbia October 4-5,2008 Washington, District Saturday - Sunday Western Section Associate secretary: Michel L. Lapidus of Columbia Announcement issue of Notices: To be announced Marriott Wardman Park Hotel and Omni Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Shoreham Hotel Issue of Abstracts: To be announced january 7-1 0, 2009 Deadlines Wednesday - Saturday For organizers: March 9, 2008 ]oint Mathematics Meetings, including the 115th Annual Meeting of the AMS, 92nd Annual Meeting of the Mathe­ For consideration of contributed papers in Special Sessions: To be announced matical Association ofAmerica (MAA), annual meetings of For abstracts: To be announced the Association for Women in Mathematics (A WM) 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 Huntsville, Alabama Applied Mathematics (SIAM). University of Alabama, Huntsville Associate secretary: Lesley M. Sibner Announcement issue of Notices: October 2008 October 24-26, 2008 Program first available on AMS website: November 1, 2008 Friday - Sunday Program issue of electronic Notices: January 2009 Southeastern Section Issue of Abstracts: Volume 30, Issue 1 Associate secretary: Matthew Miller Deadlines Announcement issue of Notices: To be announced Program first available on AMS website: To be announced For organizers: April1, 2008 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: March 24, 2008 For consideration of contributed papers in Special Sessions: San Francisco, To be announced For abstracts: To be announced California Moscone Center West and the San Fran­ Shanghai, People's cisco Marriott Republic of China january 6-9,2010 Wednesday - Saturday Fudan University ]oint Mathematics Meetings, including the 116th Annual Meeting of the AMS, 93rd Annual Meeting of the Mathe­ December 17-21,2008 matical Association of America (MAA), annual meetings of Wednesday - Sunday the Association for Women in Mathematics (A WM) and the First ]oint International Meeting between the AMS and the National Association of Mathematicians (NAM), and the Shanghai Mathematical Society winter meeting of the Association for Symbolic Logic (ASL), Associate secretary: Susan]. Friedlander with sessions contributed by the Society of Industrial and Announcement issue of Notices: To be announced Applied Mathematics (SIAM). Program first available on AMS website: Not applicable Associate secretary: Matthew Miller Program issue of electronic Notices: Not applicable Announcement issue of Notices: October 2009

628 NOTICES OF THE AMS VOLUME 53, NUMBER 5 Meetings & Conferences

Program first available on AMS website: November 1, 2009 contributed by the Society for Industrial and Applied Math­ Program issue of electronic Notices: January 2010 ematics (SIAM). Issue of Abstracts: Volume 31, Issue 1 Associate secretary: MichelL. Lapidus Announcement issue of Notices: October 2011 Deadlines . Program first available on AMS website: November 1, 2011 For organizers: April1, 2009 Program issue of electronic Notices: January 2012 For consideration of contributed papers in Special Sessions: Issue of Abstracts: Volume 33, Issue 1 To be announced For abstracts: To be announced Deadlines For organizers: April1, 2011 For consideration of contributed papers in Special Sessions: To be announced New Orleans, For abstracts: To be announced Louisiana New Orleans Marriott and Sheraton New San Diego, California Orleans Hotel San Diego Convention Center and San Diego Marriott Hotel and Marina January 5- 8,2011 Wednesday - Saturday January 9- 12,2013 ]oint Mathematics Meetings, including the 117th Annual Wednesday - Saturday Meeting of the AMS, 94th Annual Meeting of the Mathe­ ]oint Mathematics Meetings, including the 119th Annual matical Association ofAmerica, annual meetings of the As­ Meeting of the AMS, 96th Annual meeting of the Mathe­ sociation for Women in Mathematics (A VVM) and the National matical Association ofAmerica, annual meetings of the As­ Association ofMath ematicians (NAM), and the winter meet­ sociation for Women in Mathematics (A VVM) and the National ing of the Association for Symbolic Logic (ASL), with sessions Association of Mathematicians (NAM), and the winter meet­ contributed by the Society for Industrial and Applied Math­ ing of the Association for Symbolic Logic (ASL), with sessions ematics (SIAM). contributed by the Society for Industrial and Applied Math­ Associate secretary: Susan ]. Friedlander ematics (SIAM). Announcement issue of Notices: October 2010 Associate secretary: Lesley M. Sibner Program first available on AMS website: November 1, 2010 Announcement issue of Notices: To be announced Program issue of electronic Notices: January 2011 Program first available on AMS website: To be announced Issue of Abstracts: Volume 32, Issue 1 Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Deadlines Deadlines For organizers: April1, 2010 For consideration of contributed papers in Special Sessions: For organizers: April1, 2012 To be announced For consideration of contributed papers in Special Sessions: For abstracts: To be announced To be announced For abstracts: To be announced Boston, Massachusetts john B. Hynes Veterans Memorial Conven­ tion Center, Boston Marriott Hotel, and Boston Sheraton Hotel January4- 7, 2012 Wednesday - Saturday ]oint Mathematics Meetings, including the 118th Annual Meeting of the AMS, 95th Annual Meeting of the Mathe­ matical Association ofAmerica, annual meetings of the As­ sociation for Women in Mathematics (A VVM) and the National Association ofMathematicians (NAM), and the winter meet­ ing of the Association for Symbolic Logic (ASL), with sessions

MAY 2006 NOTICES OF THE AMS 629 ~ Whether they become Please give generously. t.!!!l scientists, engineers, or entrepreneurs, young people Learn about giving opportunities and estate planning www.ams.org/giving-to-ams with mathematical talent need to be nurtured. Income from this Contact the AMS Development Office fund supports the Young Scholars 1.800.321.4267 Program, which provides grants to (U.S. and Canada) or 1.401.455.4000 summer programs for talented high (worldwide) www.ams.org school students. email: [email protected] 09/04 Meetings and Conferences of the AMS

Associate Secretaries of the AMS 249), Chicago, lL 60607-7045; e-mail: susan@math. nwu. edu; tele­ Western Section: MichelL. Lapidus, Department of Math­ phone: 312-996-304 1. ematics, University of California, Sproul Hall, Riverside, CA Eastern Section: Lesley M. Sibner, Department of Mathe­ 92521 -0135; e-mail: l api dus@math. ucr. edu; telephone: 951 - matics, Polytechnic University, Brooklyn, NY 11201-2990; 827-5910. e-mail: lsi bner@duke. pol y. edu; telephone: 718-260-3505. Central Section: Susan J. Friedlander, Department of Math­ Southeastern Section: Matthew Miller, Department of Math­ ematics, University of Illinois at Chicago, 851 S. Morgan (M/C ematics, University of South Carolina, Columbia, SC 29208- 0001, e-mail: mill er@math. sc . edu; telephone: 803-777-3690.

The Meetings and Conferences section of the Notices April4-6 Bloomington, Indiana p.627 gives information on all AMS meetings and conferences May 3-4 Claremont, California p.627 approved by press time for this issue. Please refer to the page June 4-7 Rio de Janeiro, Brazil p.627 numbers cited in the table of contents on this page for more October 4-5 Vancouver, Canada p.628 detailed information on each event. Invited Speakers and October 24-26 Huntsville, Alabama p.628 Special Sessions are listed as soon as they are approved by December 17- 21 Shanghai, People's the cognizant program committee; the codes listed are needed Republic of China p.628 for electronic abstract submission. For some meetings the list 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. 628 found at www. ams. org/meeti ngs/. Annual Meeting 2010 Meetings: January 6- 9 San Franciso, California p.628 2006 Annual Meeting April1-2 Miami, Florida p . 619 2011 April8- 9 Notre Dame, Indiana p.620 January 5- 8 New Orleans, Louisiana p.629 April 22- 23 Durham, New Hampshire p.621 Annual Meeting April29- 30 San Francisco, California p. 622 2012 October 7-8 Salt Lake City, Utah p. 622 January 4-7 Boston, Massachusetts p. 629 October 21-22 Cincinnati, Ohio p . 623 Annual Meeting October 28- 29 Storrs, Connecticut p. 624 2013 November 3-4 Fayetteville, Arkansas p.624 January 9-12 San Diego, California p. 629 Annual Meeting 2007 Important Information Regarding AMS Meetings January 5-8 New Orleans, Louisiana p.625 Potential organizers, speakers, and hosts should refer to Annual Meeting page 296 in the February 2006 issue of the Notices for gen­ March 3- 4 Davidson, North Carolina p . 625 eral information regarding participation in AMS meetings and March 16-17 Oxford, Ohio p . 625 conferences. April14-15 Hoboken, New Jersey p.625 Abstracts April21- 22 Tuscan, Arizona p . 626 Speakers should submit abstracts on the easy-to-use interactive July 31-August 3 Warsaw, Poland p.626 Web form. No knowledge of 1:\TfX is necessary to submit an October 13-14 Albuqu erque, New Mexico p.626 electronic form, although those who use LAJ'fX may submit November 3-4 Murfreesboro, Tennessee p.626 abstracts with such coding, and all math displays and simi­ larily coded material (such as accent marks in text) must 2008 be typeset in 1:\TfX. Visit http: //www.ams. org/cgi-bin/ January 6-9 San Diego, California p. 626 abstracts/abstract . pl . Questions about abstracts andre­ Annual Meeting quests for paper forms may be sent to abs-i nfo@ams. org. Close attention should be paid to specified deadlines in this March 22-23 New York, NY p.627 issue. Unfortunately, late abstracts cannot be accommodated. March 28-30 Baton Rouge, Louisiana p.627 Conferences: (see http : I jwww. ams. org/meeti ngs/ for the most up-to-date information on these conferences.) June 4-June 29, 2006: Joint Summer Research Conferences, Snowbird, Utah (see November 2005 No tices, page 1296). Co-sponsored conferences: 22nd Annual Workshop on Mathematical Problems in Industry, June 12-16, 2006, Olin College, Need­ ham, MA. For details see http: I /projects. olin. edu /mpi 2006/. Recent advances in nonlinear partial differential equations and applications: A conference in honor of Peter D. Lax and Louis Niren­ berg, June 7-10, 2006, Toledo, Spain. For more details see http: I jwww. mat. ucm. es/- 1 n06/. Poisson 2006: Poisson Geometry in Mathematics and Physics, June 5-9, 2006, Tokyo, Japan. For more details see http: //www.ams. org/mathcal / info/2006_jun5-9_tokyo.html .

MAY 2006 NOTICES OF THE AMS 631 CAMBRIDGE NJEW ANJD NJOTEWORTHY

Free Ideal Rings and Central Simple Algebras and Extending Mechanics to Minds Localization in General Rings Galois Cohomology The Mechanical Foundations of Paul Cohn Philippe Gille and Tamas Szamuely Psychology and Economics This book presents the theory of free ideal Suitable as a graduate textbook and as a Jon Doyle rings (firs) in detail. There is also a full account reference for researchers, the book covers the This book deploys the mathematical axioms of of localization, which is treated for general rings, basic theory of central simple algebras, methods modern rational mechanics to understand minds but the features arising in firs are given special of Galois descent and Galois cohomology, as mechanical systems that exhibit forces, inertia, attention. Severi-Brauer varieties, residue maps and, and motion. Using precise mental models devel­ New Mathematical Monographs finally, Milnor K-theory and K-cohomology. oped in artificial intelligence, the author analyzes $140 .00: Hardback: 0-52 1-85337-0: 704pp Cambridge Studies in Advanced Mathematics motivation, attention, reasoning, learning, and $75.00*: Hardback: 0-52 1-86103-9: c.356pp communication in these mechanical terms. $95.00: Hardback: 0-52 1-86197-7: c.48bpp Heights in Diophantine Geometry Classical and Enrico Bombieri and Walter Gubler Quantum Orthogonal Classical Mechanics An Undergraduate Text Diophantine geometry has been studied by Polynomials in One Variable number theorists since the time of Pythagoras Mourad E. H. Ismail R. Douglas Gregory and is a rich area of ideas, including Fermat's This is the first modern treatment of orthogonal Topics covered in this complete guide to Last Theorem and the ABC conjecture. This polynomials from the viewpoint of special func­ mechanics include Newton's, Lagrange's, and monograph is a bridge between the classical tions. The coverage is encyclopedic and includes Hamilton's equations of motion, kinematics, theory and modern approach via arithmetic classical topics such as Jacobi, Hermite, Laguerre, oscillation, particle mechanics, and rigid body geometry. Hahn, Charlier, and Meixner polynomials as well motion. A thorough understanding of the theory New Mathematical Monographs as those (e.g., Askey-Wilson and AI-Salam-Chihara is provided by illustrating the classical laws of $130.00: Hardback: 0-521-84615-3: 668pp polynomial systems) discovered over the last motion with many relevant examples, Matlab 50 years. code, and case studies. Encyclopedia ofMathematics and its Applications $11 0.00*: Hardback: 0-521-82678-0: 608pp $50.00*: Paperback: 0-521-53409-7 Foundations of $140.00: Hardback: 0-521 -78201-5: 724pp Computational Mathematics, Santander 2005 Computational The New Physics Edited by Luis Pardo, Allan Pinkus, Endre Suli, For the Twenty··First Century and Mike Todd Oriented Matroids Second Edition This volume is a collection of articles based on Juergen Bakowski Edited by Gordon Fraser the plenary talks presented at the 2005 meeting Oriented matroids play the role of matrices in Written by leading international experts, in Santander of the Society for the Foundations discrete geometry, when metrical properties, each of the nineteen chapters will fascinate of Computational Mathematics. The topics such as angles or distances, are neither required scientists in all disciplines, as well as anyone covered reflect the breadth of research within nor available. The combination of concrete wanting to know more about modern physics. the area as well as the richness and fertility of applications and computation, the profusion $60.00: Hardback: 0-521-81600-9: 556pp interactions between seemingly unrelated of illustrations, and a large number of examples branches of pure and applied mathematics. and exercises make this an ideal introductory London Mathematical Society Lecture Note Series text on the subject. $75.00*: Paperback: 0-521-68161-8 $75.00: Hardback: 0-521-84930-6: 450pp *Prices subject to change.

Fundamentals of Hyperbolic Manifolds Selected Expositions Edited by R. D. Canary, A. Marden, and D. B. A. Epstein Graduate students and researchers will welcome this rigorous introduction to the modern theory of hyperbolic manifolds. London Mathematical Society Lecture Note Series $70.00*: Paperback: 0-521-61558-5: 350pp

For more information, please visit us at www.cambridge.org/us or call toll-free at 1-800-872-7423 a CAMBRIDGE ~ UNIVERSITY PRESS AUSTRALIAN MATHEMATICAL SOCIETY INC. AUSTRALIAN & NEW ZEALAND INDUSTRIAL AND APPLIED MATHEMATICS www.austms.org.au/

JOURNAL OF THE AUSTRALIAN BULLETIN OF THE AUSTRALIAN MATHEMATICAL SOCIETY MATHEMATICAL SOCIETY

The weak- type (1 ,1) of JOURNAl On the injectivity of the Fourier integral operators ...... Vassiliev homomorphism of AlJSl'lALIAN MATHEMATICAL SOCim of order -{n-1)12 - Terence BULLETIN singular Artin monoids - N. Tao ; Gradings of non­ -- .-----.. - graded Hamiltonian Lie Antony; Properly embedded .... - -~--- :: and immersed minimal --_..__ ---·-·-"'----~--­...- ___ ...... "' algebras - A. Caranti and ·:=--==..f.-~ surfaces in the Heisenberg S. Mattarei; Modular Lie ":!="" ----- group - 1-H. Cheng and representations of finite 1-F. Hwang; On the Ky groups - R .M . Bryant; Fan inequality and related Operator algebras related to inequalities II- E . N euman Thompson's group F - Paul and]. Sandor ...... ----·-- - k 1olissaint Print and online Print and online in the future 6 issues/880 pages per annum 6 issues/1090 pages per annum

GAZETTE OF THE AUSTRALIAN MATHEMATICAL SOCIETY THE ANZIAM JOURNAL

Direction of vorticity and a On the twin prime new reg ularity criterion for conjecture - A. van der the Navier-Stokes equations Poorten and T. Ah-Mah; - Y. Zhou; Hyperelliptic THE Modular equations and eta •. 4-:: parametrisation of the ~,.ff ANZIAM evaluations - W.B. Hart; generalised order parameter JOURNAL Ma thematics & bodysurfing of the N=3 chiral Potts - N. de M estre; Markov (fit,· Au"-fr,iHrm ~· .\·,.,,, /,·,fi,,,hf model- RJ. Baxter; Some process modelling of gene t l llrlu ~ tri, l l

AUSTRALIAN MATHEMATICAL ANZIAM JOURNAL­ SOCIETY LECTURE SERIES ELECTRONIC SUPPLEMENT

Chaos: A Mathematical Introduction -john Banks, Sigmoidal-trapezoidal quadrature for ordinary and Valentina Dragan, Arthur ]ones; Founda tions of Cauchy principal value integ rals - David Elliott; Convex Geometry - W.A. Coppel; A Course in Modern Simulation of monthly rainfall totals - K. Rosenberg, Analysis and its Applications - Graeme L. Cohen 1. Boland and P.G. Howlett

Online versions of print journals are accessible for FREE for 6 months from date of publication

FOR FURTHER DETAILS ON JOURNALS AND MEMBERSHIP CONTACT:

61 2 6125 8923 . ' f:l Springer springer.com the language of science

Mathematical Numerical Introduction to Numerical Problems Optimization Methods in Differential in Image Processing Jorge Nocedal, Northwestern Equations Partial Differential Equations University, Illinois and Mark H. Holmes, Rensselaer Polytechnic Stephen Wright, University of Wisconsin and the Calculus of Variations Institute, New York This well illustrated textbook is aimed at Gilles Aubert, University of Nice-Sophia "[A] well-written treatment of continuous se nior undergraduate and beginning Antipolis, Nice, France and nonlinear optimization and I would graduate students in science and engi­ Pierre Kornprobst, INRIA, Odyssee Lab, recommend its use for upper level neering. It teaches readers to derive, test Sophia Antipolis, France undergraduate or graduate level courses and analyze numerical methods for solv­ in nonlinear optimization ... The book ing both ordinary and partial differential "The authors have substantially contrib­ is sufficiently detailed to be useful to equations. An essential component of uted to many aspects of the subject ... " re-searchers, but its real merit is as an this is the numerous exercises which "' Mathematical Reviews (on the previous educational resource." "' Mathematical develop both the analytical and computa­ edition) Reviews (on the previous edition) tional aspects of the material. Updates for the new edition include: new During the four years since the publica­ 2006. approx. 230 pp., (Texts in Applied topics such as nonlinear interior methods tion of the first edition much progress Mathematics, Vol. 52), Hardcover and filter methods; new chapter on has been made in the range of image ISBN 0-387-30891-1 "'approx. $49.95 processing applications covered by the derivative-free method; extensive revision PDE framework. Th e new edition aims to of unconstrained optimization chapter; give a coherent account of some of the and many new exercises. Techniques of Constructive recent challenging applications. It also 2nd Edition, 2006. approx. 680 pp. 85 Analysis provides the opportunity for readers to ill us., (Springer Seri es in Operations DouglasS. Bridges and Luminita Simona make their own simulations with minimal Research and Financial Engineering) both at the University of Canterbury, effort, in particular providing program­ Vita, Hardcover Christchurch, New Zealand ming tools to allow readers to test easily ISBN 0-387-30303-0 ... $79.95 some classical approaches. This text provides a rigorous, wide-rang­ ing introduction to modern constructive 2nd Edition, 2006. approx. 400 pp. 100 Fractal Geometry, Complex analysis intended for anyone with a illus., (Applied Mathematical Sciences, Vol. Dimensions and Zeta strong mathematical background who is 147), Hardcover interested in the challenge of developing ISBN 0-387-32200-0 "'approx. $69.95 Functions mathematics algorithmically. The authors Geometry and Spectra of Fractal begin by outlining the history of construc­ Strings tive mathematics, and the logic and set Bounded Analytic theory that are used throughout the Michel L. Lapidus, University of California, Functions book. They then present a new construc­ Riverside and Machiel Van Franken­ tion of the real numbers, followed by the John Garnett, University of California, LA huysen, Utah Vall ey State College fundamentals of the constructive theory Number theory, spectral geometry, and of metric and normed spaces, the A-tech­ "Th e book, which covers a wide range of fractal geometry are interlinked in this in­ nique as applied to finite-dimensional beautiful topics in analysis, is extremely depth study of the vibrations of fractal and Hilbert spaces, convexity, and major well organized and well written, with strings, that is, one-dimensional drums results in functional analysis. elegant, detailed proofs." with fractal boundary. "' From the citation for the AMS Steele 2006. 250 pp. 10 ill us, (Universitext), Key features include a natural geometric Prize for Exposition Softcover reformulation of the Riemann hypothesis ISBN 0-387-33646-X ... approx. $49.95 "The motif of the theory of Hardy spaces is in the context of vibrating fractal strings; the interplay between real, complex, and a detailed study of complex dimensions of abstract analysis. While paying proper a fractal string; explicit formulas extended The Square Root of 2 ... attention to each of the three aspects, the to apply to the geometric, spectral, and ... dynamic zeta functions associated with a A Dialogue Concerning ==N--- author has underscored the effectiveness of a Number and a Sequence the methods coming from real analysis ... " fractal; and the examination of new ... --- "'!!!!!!!!!!!!!~= resu lts throughout. David Flannery, Cork Institute of "' MathSciNet N=== Technology, Cork, Ireland =­== 2006. approx. 490 pp. 31 ill us., (Graduate 2006. approx. 452 pp. 53 ill us., (Springer =--- Texts in Mathematics, Vol. 237), Hardcover Monographs in Mathematics), Hardcover 2006. 260 p. 31 ill us., Hardcover U1==""-= U1--- ISBN 0-387-33621-4 "'approx. $69.95 ISBN 0-387-33285-5 "'approx. $54.95 ISBN 0-387-20220-X ... $25.00 --"'= ~- l.- Easy Ways to Order for the Americas 11> Write: Sp ringer Ord er Department, PO Box 2485, Seca ucus, NJ 07096-2485, USA 11> Call: (toll free) 1-800-SPRINGER - 11> Fax: + 1[ 201 )348-4505 11> Email: [email protected] or for outside the Americas 11> Write: Springer Distribution Ce nter GmbH, Haberstrasse 7, 69 126 Hei delberg, Germany 11> Call: +49 [0) 622 1-345-430 1 11> Fax: +49 [0) 622 1-345-4229 11> Email: SDC-bookorder@s pringer.com.,.. Prices are subject to - change without notice. All prices are net prices. 012378x