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Heads or Tails: An Introduction to Limit For Classroom Heads ot Tails Theorems in Probability Use An JntroduHwn to LlmH Th("ore~ru. Emmanuel Lesigne, Universite Fran~ois Rabelais, Tours, France in ProhahiHt) Some of the most remarkable results in probability are those that are related to limit theorems. In this book, the author explains limit theorems in terms of a game of tossing a coin. This approach makes analyzing results much clearer and helps to cultivate intuition about probability. It is suitable for anyone who would like to learn more about mathematical probability. Student Mathematical Library, Volume 28; 2005; approximately 160 pages; Softcover; ISBN 0-8218-3714-1 ; List $29;AII AMS members $23; Order code STMU28
The Wild World of 4-Manifolds Alexandru Scorpan, University of Florida, Gainesville, FL Dimension four is unlike any other dimension; it is large enough to have room for wild things to happen, but too small to have room to undo them. This book offers a panorama of the topology of simply-connected smooth manifolds of dimension four. The excellent choice of material, over 250 illustrations, and historical notes make this a "must-read" for anyone interested in topology. 2005; approximately 600 pages; Hardcover; ISBN 0-8218-3749-4; List $69;AII AMS members $55; Order code FOURMAN
And don't miss these other important titles. .. Conformally Invariant Processes in the Plane ~ Conlonnally Invariant Gregory F. Lawler, Cornell University, Ithaca, NY Proceases In the Plane Suitable for graduate students and research mathematicians interested in random processes and their applications in theoretical physics, this volume introduces conformally invariant processes that appear as scaling limits. ~ryF.I.owlor Mathematical Surveys and Monographs, Volume 114; 2005; 242 pages; Hardcover; ISBN 0-8218-3677-3; List $59; . :• -- All AMS members $47; Order code SURV/114 Graphs and Patterns in Mathematics and Theoretical Physics Mikhail Lyubich, Stony Brook University, NY, and University of Toronto, ON, Canada, and Leon Takhtajan, Stony Brook University, NY, Editors -, This comprehensive proceedings volume will help researchers and graduate students in mathematics and theoretical physics who encounter graphs in their research to overcome conceptual barriers. · Proceedings of Symposia in Pure Mathematics, Volume 73; 2005; 418 pages; Hardcover; ISBN 0-8218-3666-8; List $89;AII AMS members $71 ; Order code PSPUM/73
-· For many more publications of interest, ~ -2·~~ I !! ~] L'l visit the AMS Bookstore I ::= AMS BOOKSTORE www.ams.org/bookstore AMERICAN MATHEMATICAL SoCIETY 05105 N Discrete and Continuous Dynamical Systems http:// AI MSciences.org ISSN 1078-0947 (print); ISSN 1553-5231 (electronic)
:•:• Editorial Board :•:• I n t r o d u c t i o n Editor in Chief : Published by the American Institute of Mathematical Sciences, Discrete Shouchuan Hu and Continuous Dynamical Systems is one of the world's leading mathematical journals. Of the more than 600 mathematics journals Board of Editors: published worldwide, only about 150 reach the level of Lluis Alseda impact required to be included in the rankings of the Institute for Antonio Ambrosetti Scientific Information. Among these, DCDS is ranked at 7, with an Peter Bates Jerry Bona impact factor of 1.504. The electronic journal is available for Jean Bourgain subscription. Subscription information and the online access at Alberto Bressan http://aimSciences.org. Giuseppe Buttazzo :•:• 0 t h e r l o u r n a I s fro m A I M S Alexandre Chorin Amadeu Delshams Discrete and Continuous Dynamical Systems Series B Mike Field ISSN 1531-3492 (print); ISSN 1553-524X (electronic) Anatole Katok A journal of mathematical modeling, analysis, and computations Yuri Kifer Communications on Pure and Applied Analysis Irena Lasiecka ISSN 1534-0392 (print); ISSN 1553-5258 (electronic) Tatsien Li Yanyan Li Mathematical Biosciences and Engineering Fanghua Lin ISSN 1547-1063 (print); ISSN 1551-0018 (electronic) Carlangelo Liverani Journal of Industrial and Management Optimization , Mikhail Lyubich ISSN 1547-5816 (print); ISSN 1553-166X (electronic) Hiroshi Matano Jean Mawhin :•:• A I M S Boo k S e r i e s Konstantin Mischaikow AlMS Book Series on Differential Equations and Dynamical Systems Wei-Ming Ni Mitsuharu Otani Jerry L. Bona, University of Illinois at Chicago, USA Nikolaos S. Papageorgiou Yacine Chitour, Universite Paris-Sud, France Ricardo Perez-Marco Messoud Efendiev, University of Stuttgart, Germany Peter Polacik Editors Hans G. Othmer, University of Minnesota, USA Mark Pollicott Jeff Webb, University of Glasgow, UK Walter Strauss Jianhong Wu (Editor in Chief), York University, Canada Roger Temam Kok Lay Teo Sebastian van Strien :•:• U p co m i n g Eve n t Marcelo Viana The 6th AIMS international conference on Hans-Otto Walther Glenn F. Webb Dynamical Systems, Differential Equations Lan Wen and Applications, to be held in Poitiers, Jianhong Wu Zhihong Jeff Xi a France, June 25-28, 2006 Eiji Yanagida Keep updated at http://aimSciences.org Jean C. Yoccoz
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Studies in lie Theory The Unity of Mathematics Complex Numbers from A. Joseph Festschrift In Honor of the Ninetieth Birthday of I.M. A to ••• Z JOSEPH BERNSTEIN, Tel Aviv University, Israel; VLADIMIR Gelfand TITU ANDREESCU, University of Texas, Dallas, USA; DORIN HINICH, University of Haifa, Israel; ANNA MELNIKOV, PAVEL ETINGOF, Massachusetts Institute of Technology, ANDRICA, "Babes-Bolyai" University, Cluj-Napoca, Romania University of Haifa, Israel (Eds.) Cambridge, MA; VLADIMIR S RETAKH, Rutgers University, Complex Numbers from A to ... Z is an introduction to this Dedicated to Anthony Joseph in honor of his 60th birth Piscataway, NJ; and l.M. SINGER, Massachusetts Institute of Technology, Cambridge, MA {Eds.} fascinating subject, with a particular emphasis on key day, this volume contains two articles reviewing different concepts and elementary results concerning these aspects of Joseph's impact on the theory of representa Atribute to the vision and legacy of Israel Gelfand, the numbers. The reader learns how complex numbers can tions of Lie algebras and seventeen original contributions invited papers in this volume reflect the unity of be used to solve algebraic equations and to understand from leading specialists in the field. The book contains mathematics as a whole, with particular emphasis on the geometric interpretation of complex numbers and new results on different aspects of Lie theory, including the many connections among the fields of geometry, the operations involving them. Lie superalgebras, quantum groups, crystal bases, physics, and representation theory. Written by leading representations of reductive groups in finite characteris mathematicians, the text is broadly divided into two The theoretical parts of the book are augmented with rich tic, and the geometric Langlands program. sections: the first is devoted to developments at the exercises and problems at various levels of difficulty. A special feature of the book is the last chapter, a selection Contributors: J Alev, A. Beilinson, A Braverman, I. Cherednik, J intersection of geometry and physics, and the second to Dixmier, F. Dumas, P. Etingoff, D. Farkas, D. Gaitsgory, F. lvorra, D. representation theory and algebraic geometry. of outstanding Olympiad and other important mathemati cal contest problems solved by employing methods Joseph, M Kashiwara, D. Kazhdan, A. Kirillov, B. Kostant, S. Contributors: M Atiyah, A. Bei/inson, J Bernstein, A Cannes, P. Kumar, G. Letzter, T. Levasseur, G. Lusztig, L Makar-Limanov, W Deligne, R. Dijkgraaf, D. Gaitsgory, M Gromov, F. Hirzebruch, M presented in the text. McGovern, M Nazarov, K-H. Neeb, L Rybnikov, A. Sergeev, V. Hopkins, D. Kazhdan, F. Kirwan, M Kontsevich, B. Kostant, G. The target audience includes undergraduates, high school Schechtman, T. Stafford, Ya. Varshavsky, N. Wallach, I. Waschkies. Lusztig, D. McDuff, H. Nakajima, S. Novikov, P. Sarnak, A. students, instructors, mathematical contestants (such as 2005/APPROX. 400 PP./HARDCOVER/ISBN O-B176-4342-7 j$99.95 (TENT.) Schwartz, I. Singer, R. Taylor, C. Vafa, A. Vershik, E. Witten. those training for Olympiads or theW L. Putnam PROGRESS IN MATHEMATICS 2005/APPROX. 464 PP., 20 ILLUS./HARDCOVER/ Mathematical Competition) and their coaches. The work ISBN 0-8176-4076-2/$89.95 (TENT.) may serve as an engaging supplemental undergraduate Stucles in Memory of lssai Schur PROGRESS IN MATHEMATICS text for an introductory course on complex numbers or ANTHONY JOSEPH, ANNA MELNIKOV, and, RUDOLF number theory. RENTSCHLER, all, Universite Pierre et Marie Curie, Paris, Fractal Geometry and 2005/344 PP., 821llUS./SOFTCOVER/ISBN 0-8176-4326-5/$49.95 France (Eds.) Number Theory This volume is dedicated to the memory of lssai Schur. It Complex Dimensions of Fractal Strings Time-Fre~uency and features an extended biography of Schur-written in and Zeros of Zeta Functions Time-Scale Methods collaboration with some of his former students-as well Second Edition as survey articles covering many areas inspired by Adaptive Decompositions, Uncertainty Schur's work, as well as more recent developments MICHELL. LAPIDUS, University of California, Riverside, CA Principles, and Sampling involving crystal and canonical bases, Heeke algebras, and and MACHIEL VAN FRANKENHUYSEN, Rutgers University, Piscataway, NJ Jeffrey A Hogan, University ofArkansa s, Fayetteville, AR; Joseph the geometric approach linking orbits to representations. D. Lakey, New Mexico State University, Las Cruces, NM Contributors: Biography: W Ledermann, B. Neumann, P.M "A stimulating guide, well written in a clear and pleasant style." -Mathematical Reviews (Review of First Edition) Developed in this book are several deep connections Neumann, H. Abe/in- Schur,· Review of work: H. Dym, V. between time-frequency (Fourier/Gabor) analysis and Katznelson; Original papers: H. H. Andersen, A. Braverman, S. New to the second edition: Material on Diophantine time-scale (wavelet) analysis, emphasizing the power Donkin, V. Ivanov, D. Kazhdan, B. Kostant, A. Lascoux, N. approximation of complex dimensions of nonlattice Lauritzen, B. Leclerc, P. Littelmann, G. Luzstig, 0. Mathieu, M ful adaptive methods that emerge when separate Nazarov, M Reinek, J-Y. Thibon, G Olshanski, E. Opdam, A. strings· Periodic orbits of self-similar flows · techniques from each area are properly assembled in a Regev, C.S. Seshadri, M Varagnolo, E. Vasserot, A. Vershik Discussion of connections between fractality and self larger context. While researchers at the forefront of similarity in the non-lattice case. 2003/560 PP./HARDCOVER/ISBN O-B176-420B-O/$B9.95 developments in time-frequency and time-scale PROGRESS IN MATHEMATICS, VOL 210 2005/APPROX. 320 PP./HARDCOVER/ISBN 0-8176-3209-3/$64.95 (TENT.) analysis are well aware of the benefits of such a unified approach, there remains a knowledge gap in the larger Marked Point Processes community of practitioners about the precise strengths CompactiRcations of Symmetric and limitations of Fourier/Gabor analysis versus and Locally Symmetric Spaces and Piecewise wavelets. This book fills that gap by presenting the ARMAND BOREl, Institute for Advanced Study, Princeton, NJ; Deterministic Processes interface of time-frequency and time-scale methods as a rich area of work. LIZHEN J~ University of Michigan, Ann Arbor, Ml Theory and Applications 2005/390 PP. 221LLUS./HARDCOVER/ISBN O-B176-4276-5/$74.95 Non-compact Riemannian symmetric spaces and their MARTIN JACOBSEN, University of Copenhagen, Denmark quotients by arithmetically defined groups of isometries APPLIED AND NUMERICAL HARMONIC ANALYSIS occur in many parts of mathematics. For many purposes This text examines the basic theory of marked point it has been necessary to compactify them. This mono processes, developing randomly over time, and shows how this theory may be used to treat piecewise deter graph attempts to give a systematic, and in part, new Looking for more? ministic stochastic processes in continuous time. It exposition of these compactifications as well as new Find it on birkhauser.com ones, their interrelations, and the context out of which details point processes that generate only finitely many they arose. Compactifications of the more general points in finite time intervals, resulting, in particular, in semisimple symmetric spaces are also considered. piecewise deterministic processes with "few jumps." 2005/APPROX. 300 PP./HARDCOVER/ISBN 0-8176-3247-6/$59.95 (TENT.) 2005/APPROX. 350 PP./ HARDCOVER/ ISBN 0-8176-4215-3/$79.95 (TENT.) MATHEMATICS: THEORY &APPLICATIONS PROBABILITY AND ITS APPLICATIONS
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502 Experimental Mathematics: Examples, Methods and Implications David H. Bailey and Jonathan M. Borwein Computing software and hardware advances have made numerical experimentation in mathematics an increasingly important research tool. The authors present examples of such experimentation and consider some of the consequences for mathematicians and mathematics of the availability of such tools.
532 The Importance of MathML to Mathematics
516 WHAT IS ... a Horseshoe? 501 Opinion: Gender and Mathematics Michael Shub Again judith Rottman and Carol Wood 51 8 Picturing the Horseshoe Map Bill Casselman 520 The Works ofArchimedes: Translation and Commentary 526 Mathematics in the Near East: Some Volume 1: The Two Books On the Sphere Personal Observations and the Cylinder- A Book Review David Mumford Reviewed by Alexander ]ones 539 Margulis andNovikovReceive 2005 Wolf Prize 541 2005 JPBM Communications Award 543 MAA Prizes Presented in Atlanta 54 5 AWM Prizes Presented in Atlanta 54{! Happy Birthday, Baley Price! Allyn jackson Notices Departments of the American Mathematical Society Mathematics People ...... 548 Fedkiw Rece ives NAS Award for Initiatives in Research, Hughes EDITOR: Andy Magid Receives AAAS Lifetime Mentor Award, Lander Receives Public ASSOCIATE EDITORS: Understanding Award, Boyd Awarded CRM-Fields Prize, AIM Susanne C. Brenner, B ill Casselman (Graphics Editor), Five-Year Fellow Announced, Packard Fellowships Awarded, Robert J. Daverman, Nathaniel Dean, Rick Durrett, Maz'ya Awarded Celsius Gold Medal. Susan Friedlander, Robion Kirby, Steven G. Krantz, Elliott H. Ueb, Mark Saul, Karen E. Smith, Audrey Terras, Usa Traynor Mathematics Opportunities ...... 550 SENIOR WRITER and DEPUTY EDITOR: Enhancing the Mathematical Sciences Workforce in the Twenty- Allyn Jackson First Century, NSF Postdoctoral Research Fellowships, NSF MANAGING EDITOR: Sandra Frost Graduate Teaching Fellowships in K- 12 Education, SASTRA CONTRIBUTING WRITER: Elaine Kehoe Ramanujan Prize, News from The Fields Institute, News from PRODUCTION ASSISTANT: Muriel Toupin Oberwolfach. PRODUCTION: Kyle Antonevich, Stephen Moye, For Your Erin Murphy, Lori Nero, Karen Ouellette, Donna Information ...... 552 Salter, Deborah Smith , Peter Sykes Comments on Female Math Ability Spark Reaction. ADVERTISING SALES: Anne Newcomb Inside the AMS ...... 553 SUBSCRIPTION INFORMATION: Subscription prices Login Now Required for Online Notices, AMS Holds Workshop for Volume 52 (2005) are $417 list; $334 institutional for Department Chairs, Deaths of AMS Members. member; $250 individual member. (The subscription price for members is included in the annual dues.) A Reference and Book list ...... late charge of 10% of the subscription price will be ...... 554 imposed upon orders received from nonmembers after January 1 of the subscription year. Add for American Mathematical Society- Contributions ...... 558 postage: Surface delivery outside the United States and India- $20; in India- $40; expedited delivery to Mathematics Calendar ...... 565 destinations in North America- $35; elsewhere- $87. Subscriptions and orders for AMS publications should be addressed to t he American Mathematical Society, New Publications Offered by the AMS ...... _...... 571 P.O. Box 845904, Boston, MA 02284-5904 USA. All orders must be prepaid. Classified Advertisement s ...... 577 ADVERTISING: Notices publishes situations wanted and classified advertising, and display advertising for Meetings and Conferences Table of Contents ...... 592 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 not i ces@math. ou. edu, by fax at 405-325-5765, or b y postal mail at Department of Mathematics, 601 Elm, PHSC 423, University of Okla homa, Norman, OK 73019-0001. Email is preferred. Correspondence with the managing editor may be sent From the to noti ces@ams . org. For more information, see the section "Reference and Book List". AMS Secretary NOTICES ON THE AMS WEBSITE: Most of this publi cation is available electronically through the AMS web site, the Society's resource for delivering e lectronic Officers of the products and services. Use the URL http: //www. ams . Society 2004 and 2005 Updates ...... 564 o rg/ noti ces/ to access the Notices on the website.
[Notices of the American Mathematical Society is published monthly except bimonthly in june/July by the American Mathematical Society at 201 Charles Street, Providence, RI 02904-2294 USA, GSTNo.12189 2046 RT****. P~ rio dical s postage paid at Providence, Rl, and additional mailing offices. POSTMASTER: Send address change notices to Notices of the American Mathematical Society, P.O. Box 6248, Providence, Rl 02940-6248 USA.] Publication here of the Society's street address and the other information in brackets above is a tech nical requirement of the U.S. Postal Service. Tel: 401-455-4000, email: noti ces@ams . org. ©Copyright 2005 by the American Mathematical Society. All rights reserved. Printed in the United States of America. The paper used in this journal is acid-free and falls within the guidelines established to ensure permanence and durability. Opinions expressed in signed Notices articles are those of the authors and do not necessarily reflect opinions of the editors or policies of the American Mathematical Society. Opinion
Have you ever heard a grown man described in a pro Gender and fessional context as a little boy? We thought not. Are lit tle girls invited to give hour addresses? Again, we thought not. "Juvenile" is part of the common gender schema for Mathematics-Again women but not for men. So is "not worth listening to". (Read The president of Harvard stands up in a room full of women Valian's book for the evidence; it's extensive.) Many aspects scientists and says that maybe women can't do first-rate of the schema "woman" essentially clash with schemas such science and math for genetic reasons. If this sounds like the as "professional" or "mathematician". Aspects of the start of a bad joke, it is: a bad joke that really happened. schema "man" do not. So many women are doing mathematics and science at Women can lose momentum and heart as a result of the such high levels that we cannot imagine any meaningful accumulation of reactions to them based on these schemas. interpretation of his comment. With about half of all U.S. Gross comments are bad enough, but accumulated mi undergraduate mathematics degrees and about one third croscopic inequities devastate. To persevere despite them of all U.S. mathematics Ph.D.'s going to women, you might takes a tough skin; to avoid being distracted by them-well, think there's no problem anymore. You would be wrong. neither of us has figured that one out yet. Not many peo Recently, one of us got an email which read in its ple have. entirety: "Women and mathematics remind me of Dr. John It is especially insidious that both women and men son's sally about women and preachers." (At Boswell's re hold these schemas. A basic study, replicated often, sends port of hearing a woman preaching, he said, "Sir, a woman's the same vita or the same academic paper to some peo preaching is like a dog's walking on his hinder legs. It is ple under a male name, to other people under a female not done well; but you are surprised to find it done at all.") name. The "woman" is ranked lower when men do the rank Earlier, one of us heard an accomplished postdoc at an ing. And also when women do the ranking. other institution described offhandedly by a colleague as What can be done? We can become more conscious of "the little girl." You do not have to be at Harvard to notice our own gender schemas and the way they influence our that women still are not entirely welcome or completely judgments. Like Dracula, the more pernicious of these accepted. And Harvard is not the only place in the math schemas tend to be destroyed by light. We can consciously ematical community where women faculty are absent or increase the number of women-the talent pool is deep scarce. in our departments, our conferences, the major commit Samuel Johnson's eighteenth-century comment is not . tees of our professional organizations. The more women only smug but unfortunately still relevant. The notion there are, the more normal their presence becomes, the that women are an oddity is destructive to women and to more they become just people, and the easier it is for mathematics. And this notion still exists. Talent needs them to do their work. nurturing-not coddling, but nurturing-and it is diffi We know this is true because there are institutions, cult to nurture people you regard as freaks. both academic and nonacademic, even sub fields of math What about the "little girl" comment? Remarks like this ematics, in which women participate in sufficient numbers may seem innocuous, but they aren't, and potentially hurt so that their presence is no longer remarkable, hence no feelings are only the tip of the iceberg. To explain this, we longer remarked on. There are existence proofs. Next step: need to define the notion of gender schema. A schema is, a universal quantifier. roughly, a cluster of expectations. For example, one part Why bother? of the "driver" schema is "stops for red lights". That is why Because it is right. Because it is just. Because it is dumb we feel safe going when the light turns green. to discourage talented people simply because their geni As anyone who has studied calculus knows, the cumu tals go in instead of out. Because a "mathematician" schema lative effect of small differences can be very large indeed. that does not include women turns off talented students. In her book Why So Slow? (MIT Press) Virginia Valian gives And because the profession is ill served by the ensuing persuasive evidence of how the cumulative effect of small waste of talent if we do not change. differences in treatment, differences resulting from gen As Barbara Grosz of Harvard said, "It's time to put der schemas, has a major impact on women in supposedly solutions in place." male domains. In this op-ed we have addressed the situation only for Consider the following scenario, which has recurred over women. The situation for underrepresented minorities is the thirty-plus years each of us has been in mathematics. We even worse. The schemas they encounter are far more in meet a mathematically talented, enthusiastic young woman sidious. We hope the mathematics community will realize and think to ourselves, "YES! This one can't miss!" Her teach that Grosz's words apply to this situation as well and will ers encourage her and follow her career with interest. But move swiftly to action. a few years down the road something goes awry-maybe a lot, more often just a little, then a little more. Her work is -judith Roitman, University of Kansas fine, but somehow the men have passed her by. Questions -Carol Wood, Wesleyan University and self-doubt appear. Maybe she loses energy. Why continue Former presidents of the Association for Women in to bang her head against this particular wall? Mathematics
MAY 2005 NOTICES OF THE AMS 501 Experimental Mathematics: Examples, Methods and Implications David H. Bailey and Jonathan M. Borwein
The object of mathematical rigor is to limited to a handful of researchers nor to a sanction and legitimize the conquests handful of universities, nor is it limited to one of intuition, and there was never any particular field of mathematics. Instead, it involves other object for it. hundreds of individuals, at many different insti -Jacques Hadamard1 tutions, who have turned to the remarkable new computational tools now available to assist in their If mathematics describes an objective research, whether it be in number theory, algebra, world just like physics, there is no rea analysis, geometry, or even topology. These tools son why inductive methods should not are being used to work out specific examples, be applied in mathematics just the same generate plots, perform various algebraic and as in physics. calculus manipulations, test conjectures, and ex -Kurt Godel2 plore routes to formal proof. Using computer tools to test conjectures is by itself a major timesaver Introduction for mathematicians, as it permits them to quickly Recent years have seen the flowering of "experi rule out false notions. mental" mathematics, namely the utilization of Clearly one of the major factors here is the modern computer technology as an active tool in development of robust symbolic mathematics mathematical research. This development is not software. Leading the way are the Maple and Math ematica products, which in the latest editions are David H. Bailey is at the Lawrence Berkeley National Laboratory, Berkeley, CA 94720. His email address is far more expansive, robust, and user-friendly than [email protected]. This work was supported by the when they first appeared twenty to twenty-five Director, Office of Computational and Technology Re years ago. But numerous other tools, some of which search, Division ofMathematical, Information, and Com emerged only in the past few years, are also play putational Sciences of the U.S. Department of Energy, ing key roles. These include: (1) the Magma com under contract number DE-AC03-76SF00098. putational algebra package, developed at the jonathan M. Borwein is Canada Research Chair in Col University of Sydney in Australia; (2) Neil Sloane's laborative Technology and Professor ofComputer Science and of Mathematics at Dalhousie University, Halifax, NS, online integer sequence recognition tool, available B3H 2W5, Canada. His email address is at http: I /www. research. att. com/ nj as/ j borwei n@cs . da l . ca. This work was supported in part sequences; (3) the inverse symbolic calculator (an by NSERC and the Canada Research Chair Programme. online numeric constant recognition facility), avail 1 Quoted at length in E. Borel, Le <;: ons sur la theorie des able at http: I /www. cecm. sfu. ca/proj ects/ISC; fonctions, 1928. (4) the electronic geometry site at http: I /www. 2 Kurt Godel, Collected Works, Vol. ill, 1951. eg-model s. de; and numerous others. See
502 NOTICES OF THE AMS VOLUME 52, NUMBER 5 http: I jwww. experimental math. info for a more precision in the input vector x to obtain numeri complete list, with links to their respective web sites. cally meaningful results-at least dn-digit precision, We must of course also give credit to the com where d = log10 A. This is the principal reason for puter industry. In 1965 Gordon Moore, before he the interest in very high-precision arithmetic in served as CEO of Intel, observed: experimental mathematics. In one recent integer re lation detection computation, 50,000-digit arith The complexity for minimum compo metic was required to obtain the result [9]. costs has increased at a rate of nent At the present time, the best-known integer roughly a factor of two per year.... Cer relation algorithm is the PSLQ algorithm [26] of over the short term this rate can tainly mathematician-sculptor Helaman Ferguson, who, not to in be expected to continue, if together with his wife, Claire, received the 2002 crease. Over the longer term, the rate of Communications Award of the Joint Policy Board increase is a bit more uncertain, al for Mathematics (AMS-MAA-SIAM). Simple formu though there is no reason to believe it lations of the PSLQ algorithm and several variants will not remain nearly constant for at are given in [10]. The PSLQ algorithm, together least 10 years. [29] with related lattice reduction schemes such as LLL, Nearly forty years later, we observe a record of was recently named one of ten "algorithms of the sustained exponential progress that has no peer in century" by the publication Computing in Science the history of technology. Hardware progress alone and Engineering [4]. PSLQ or a variant is imple has transformed mathematical computations that mented in current releases of most computer al were once impossible into simple operations that gebra systems. can be done on any laptop. Many papers have now been published in the ex Arbitrary Digit Calculation Formulas perimental mathematics arena, and a full-fledged The best-known application of PSLQ in experi journal, appropriately titled Experimental Mathe mental mathematics is the 1995 discovery, by matics, has been in operation for twelve years. means of a PSLQ computation, of the "BBP" formula Even older is the AMS journal Mathematics of Com for rr: putation, which has been publishing articles in the (1) general area of computational mathematics since 00 1960 (since 1943 if you count its predecessor). 1(4 2 1 1) TT = k~O 16k 8k + 1 - 8k + 4 - 8k + 5 - 8k + 6 . Just as significant are the hundreds of other recent articles that mention computations but which oth This formula permits one to directly calculate bi erwise are considered entirely mainstream work. nary or hexadecimal digits beginning at the n-th All of this represents a major shift from when the digit, without needing to calculate any of the first present authors began their research careers, when n - 1 digits [8], using a simple scheme that re the view that "real mathematicians don't compute" quires very little memory and no multiple-precision was widely held in the field. arithmetic software. In this article, we will summarize some of the It is easiest to see how this individual digit discoveries and research results of recent years, by calculating scheme works by illustrating it for a sim ourselves and by others, together with a brief de ilar formula, known at least since Euler, for log 2: scription of some of the key methods employed. 00 1 We will then attempt to ascertain at a more fun log2 =I n2n· damental level what these developments mean for n=l the larger world of mathematical research. Note that the binary expansion of log 2 beginning Integer Relation Detection after the first d binary digits is simply {2d log 2}, One of the key techniques used in experimental where by { ·} we mean fractional part. We can write mathematics is integer relation detection, which in effect searches for linear relationships satisfied by a set of numerical values. To be precise, given a real or complex vector (x1, x2, · · · , Xn), an inte ger relation algorithm is a computational scheme that either finds the n integers (a;), not all zero, such that a 1x1 + a2x2 + · · · anXn = 0 (to within where we insert "mod n" in the numerator of available numerical accuracy) or else establishes the first term of (2), since we are interested only that there is no such integer vector within a ball in the fractional part after division by n. Now the of radius A about the origin, where the metric expression 2d- n mod n may be e valuated very is the Euclidean norm: A = (ai + a ~ + ... + a ~ ) 1 ' 2 • rapidly by means of the binary algorithm for ex Integer relation computations require very high ponentiation, where each multiplication is reduced
MAY 2005 NOTICES OF THE AMS 503 modulo n. The entire scheme "Figure Eight Knot Complement";3 see Figure 1), indicated by formula (2) can be which is given by implemented on a computer using ordinary 64-bit or 128-bit arith oo 1 2n - l 1 metic; high-precision arithmetic v = 2J3 ~ ne:) ~ k software is not required. The re sulting floating-point value, when = 2.029883212819307250042405108549 ... ' expressed in binary format, gives has been identified in terms the first few digits of the of a BBP-type formula by application of Ferguson's binary expansion of log 2 begin own PSLQ algorithm. In particular, British physicist ning at position d + 1. Similar David Broadhurst found in 1998, using calculations applied to each of the a PSLQ program, that four terms in formula (1) yield a J3 "' (- 1)" similar result for rr. The largest V=g L27il n=O computation of this type to date [ 18 18 24 6 2 ] is binary digits of rr beginning at x (6n + 1)2 - (6n + 2)2 - (6n + 3)2 - (6n + 4)2 + (6n + 5)2 · Figure 1. Ferguson's "Figure the quadrillionth (1015 -th) binary Eight Knot Complement" digit, performed by an interna This result is proven in [15, Chap. 2, Prob. 34]. sculpture. tional network of computers organized by Colin Percival. Does Pi Have a Nonbinary BBP Formula? The BBP formula for rr has even found a prac Since the discovery of the BBP formula for rr in tical application: it is now employed in the g95 1995, numerous researchers have investigated, by Fortran compiler as part of transcendental function means of computational searches, whether there evaluation software. is a similar formula for calculating arbitrary digits Since 199 5 numerous other formulas of this of rr in other number bases (such as base 10). Alas, type have been found and proven using a similar these searches have not been fruitful. experimental approach. Several examples include: Recently, one of the present authors (JMB), to (3) gether with David Borwein (Jon's father) and William Galway, established that there is no degree-1 BBP 9 1 8 type formula for rr for rr.f3 = 32 k~ 6~k ( 6k : 1 - 6k + 2 - 6k ~ 4 - 6k ~ 5)' bases other than powers of two (although this does not rule (4) out some other scheme for calculating individual digits). We will 2 1 "' 1 [ 144 216 72 5 4 9 ] sketch this result here. Full details and some related TT = S k~ 64k (6k + 1)2 - (6k + 2)2 - (6k + 3)2 - (6k + 4)2 + (6k + 5)2 ' results can be found in [20]. In the following, ~(z) and D(z) denote the real and imaginary parts of z, respectively. The integer 2 2 "' 1 [ 243 405 81 27 b > 1 is not a proper power if it cannot be written TT = 27 k~O 729k (12k + 1)2 - (1 2k + 2)2 - (1 2k + 4)2 - (12k + 5)2 as em for any integers c and m > 1. We will use the (5) notation ordp(z) to denote the p-adic order of the 72 9 9 5 1 ] rational z E Q.ln particular, ordp(p) = 1 for prime (12k + 6)2 - (1 2k + 7)2 - (1 2k + 8)2 - (1 2k + 10)2 + (1 2k + 11)2 ' p, while ordp(q) = 0 for primes q * p, and ordp(wz) = ordp(w) + ordp(z). The notation vb(P) will mean the order of the integer b in the multi ,[3 1 ( 3 1 ) plicative group of the integers modulo p. We will (6) arctan (7J3 ) = k~"' o 27k 3k + 1 + 3k + 2 ' say that p is a primitive prime factor of bm - 1 if (7) m is the least integer such that p l(bm - 1).,Thus p is a primitive 25 [ 781 ( 57 - 515) -15] "' 1 ( 5 1 ) prime factor of bm - 1 provided z- log 256 57 + 5)5 = k~O 55k 5k + 2 + 5k + 3 . v b(P) = m. Given the Gaussian integer z E Q[i] and the rational prime p =1 (mod 4), let 8p(z) denote Formulas (3) and (4) permit arbitrary-position ordp(z) - ordp(z), where p and p are the two con binary digits to be calculated for rr v'3 and rr2 • jugate Gaussian primes dividing p and where we Formulas (5) and (6) permit the same for ternary require 0 < D(p) < ~(p) to make the definition of (base-3) expansions of rr2 and v'3 arctan(v'3/7). eP unambiguous. Note that Formula (7) permits the same for the base- S ex (8) pansion of the curious constant shown. A com pendium of known BBP- type formulas , with Given K E R, with .2 :::; b E Z and b not a proper references, is available at [5]. power, we say that K has a Z -linear or Q -linear One interesting twist here is that the hyperbolic volume of one of Ferguson's sculptures (the 3 Reproduced by permission of the sculptor.
504 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Machin-type BBP arctangent formula to the base b Thus our assumption that there was' a b-ary Machin if and only if K can be written as a Z -linear or Q type BBP arctangent formula for rr must be false. linear combination (respectively) of generators of the form Normality Implications ofthe BBP Formulas One interesting (and unanticipated) discovery is that arctan ( b1m ) = log ( 1 bim ) ~ + the existence of these computer-discovered BBP
- m oo (-1)k type formulas has implications for the age-old (9) - b I b2mk(2k + 1). question of normality for several basic mathe k ~O matical constants, including rr and log 2. What's We shall also use the following result, first proved more, this line of research has recently led to a full by Bang in 1886: fledged proof of normality for an uncountably in finite class of explicit real numbers. Theorem 1. The only cases where bm - 1 has no Given a positive integer b, we will define a real primitive prime factor(s) are when b = 2, m = 6, number ex to be b-normal if every m-long string of bm- 1 = 32 · 7 or when b = 2N - 1,N E Z, m = 2, base-b digits appears in the base-b expansion of bm- 1 = 2 N+1(2N-1 - 1) . ex with limiting frequency b- m. In spite of the ap parently We can now state the main result: stringent nature of this requirement, it is well known from measure theory that almost all real Theorem 2. Given b > 2 and not a proper power, numbers are b-normal, for all bases b . Nonetheless, there is no Q -linear Machin-type BBP arctangent for there are very few explicit examples of b-normal mula forrr. numbers, other than the likes of Champernowne's constant 0.12 3456789101112131415 .... In par Proof: It follows immediately from the definition ticular, although computations suggest that virtu of a Q -linear Machin-type BBP arctangent formula ally all of the well-known irrational constants of that any such formula has the form mathematics (such as rr, e, ;y, log 2, )2, etc.) are 1 M normal to various number bases, there is not a (10) rr = - I nm~ log(bm - i), single proof- not for any of these constants, not n m ~ l for any number base. where n > 0 E Z, nm E Z, and M :2: 1, nM * 0. This Recently one of the present authors (DHB) and implies that Richard Crandall established the following result. Let p(x) and q(x) be integer-coefficient polyno M mials, with deg p < deg q, and q(x) (11) n (bm _ i)nm E enirrQx = Q x. having no zeroes for positive integer arguments. equidis m ~ l By an tributed sequence in the unit interval we mean a For any b > 2 and not a proper power, it follows sequence (xn) such that for every subinterval (a, b), 4 from Bang's Theorem that b M - 1 has a primitive the fraction #[xn E (a, b)] / n tends to b- a in the prime factor, say p. Furthermore, p must be odd, limit. The result is as follows: since p = 2 can only be a primitive prime factor of bm - 1 when b is odd and m = 1. Since p is a prim Theorem 3. A constant ex satisfying the BBP-type itive prime factor, it does not divide b2M - 1, and formula oo p(n) so p must divide b2M + 1 = (bM + i)(bM - i). We ex = ~~ bnq(n) cannot have both pI bM + i and pI bM - i, since this would give the contradiction that pl(bM + i) is b-normal if and only if the associated sequence = (bM - i) 2 i. It follows that p =1 (mod 4) and that defined by x0 = 0 and, for n :2: 1, Xn = p factors as p = :pp over Z[i], with exactly one of {bxn- 1 + p(n) I q(n)} (where { ·} denotes fractional :p, p dividing bM - i. Referring to the definition of part as before), is equidistributed in the unit in e, we see that we must have ep(bM- i) * 0. Fur terval. thermore, for any m < M, neither :p nor p can di For example, log 2 is 2 -normal if and only if the vide bm - i, since this would imply p I b4m - 1, simple sequence 0 4m < 4M, contradicting the fact that pis a primi defined by x 0 = and {xn = 2Xn - l + 1/ n} isequidistributedintheunitin tive prime factor of b4M - 1. So form < M, we have terval. For rr, the associated sequence is xo = 0 and 8v(bm - i) = 0. Referring to equation (10) and using 120n2 - 89n equation (8) and the fact that nM * 0, we get the { + 16 } X n = 16Xn- l + 5 12n4- 1024n3 + 712n2 - 206n + 2 1 · contradiction Full details of this result are given in [11] [15, 0 '1=- nM8p(bM - i) Section 3.8]. (12) M It is difficult to know at the present time whether = I nmep(bm- o = ep(Qx) = o. this result will lead to a full-fledged proof of nor m ~ l mality for, say, rr or log 2. However, this approach
MAY 2005 NOTICES OF THE AMS 505 has yielded a solid normality proof for another where s(s) = In ~ 1 n-s is the Riemann zeta func class of reals: Given r E [0, 1), let rn be the n-th tion. Au-Yeung had computed the sum in (14) to binary digit of r. Then for each r in the unit inter 500,000 terms, giving an accuracy of five or six dec val, the constant imal digits. Suspecting that his discovery was 00 1 merely a modest numerical coincidence, Borwein (13) OCr = L 3n2 3"+rn sought to compute the sum to a higher level of pre n=1 cision. Using Fourier analysis and Parseval's equa is 2-normal and transcendental [12]. What's more, tion, he wrote it can be shown that whenever r =1= s, then oc, =1= oc5 • Thus (13) defines an uncountably infinite class of (15) distinct 2-normal, transcendental real numbers. A rr oo ("n 1)2 1 r ( 2 1 2 . t d , L.k=1 " similar conclusion applies when 2 and 3 in (13) are 2rr Jo rr - t) og (2 sm 2) t = ~ (n + 1)2 replaced by any pair of relatively prime integers greater than 1. The series on the right of (15) permits one to eval Here we will sketch a proof of normality for one uate (14), while the integral on the left can be com particular instance of these constants, namely puted using the numerical quadrature facility of oco = Lnd 1 I (3n 23"). Its associated sequence can be Mathematica or Maple. When he did this, Borwein seen to be x0 = 0 and Xn = {2Xn-1 + Cn}, where was surprised to find that the conjectured identity Cn = 1/ n if n is a power of 3, and zero otherwise. (14) holds to more than 30 digits. We should add This associated sequence is a very good approxi here that by good fortune, 17/360 = 0.047222 ... mation to the sequence ({2noc 0}) of shifted binary has period one and thus can plausibly be recognized fractions of oc0 • In fact, I{2noc 0} - Xn I < 1/(2n). The from its first six digits, so that Au-Yeung's nu first few terms of the associated sequence are merical discovery was not entirely far-fetched. Borwein was not aware at the time that (14) fol 1 2 121 2 O, o, o, 3' 3' 3' 3' 3' 3' lows directly from a 1991 result due to De Doelder 487 5 1 2 487 5 1 2 4 8 7 5 1 2 and had even arisen in 1952 as a problem in the g' g' g' g' g' g' g' g' g' g' g' g' g' g' g' g' g' g' American Mathematical Monthly. What's more, it 13 26 25 23 19 ll 22 17 7 14 1 2 4 8 16 5 10 20 v ·v ·v ·v·v·v ·v ·v ·v·v·v·v·v·v·v·v·v·v· turns out that Euler considered some related sum l3 26 25 23 19 ll 22 17 7 14 1 2 4 8 16 5 10 20 mations. Perhaps it was just as well that Borwein v· v ·v·v·v·v·v ·v ·v·v·v ·v ·v·v·v ·v ·v·v · was not aware of these earlier results-and indeed v·v·v·13 26 25 v23 ·v·v19 ll ·v·v·v22 17 7 ·v14 ·v1 ·v2 ·v4 ·v8 ·v16· v 5 ·v10 ·v20· of a large, quite deep and varied literature [21] because pursuit of this and similar questions had and so forth. The clear pattern is that of triply re led to a line of research that continues to the pre peated segments, each of length 2 · 3m, where the sent day. numerators range over all integers relatively prime First define the multi-zeta constant to and less than 3m+1. k Note the very even manner in which this se s(s1,s2," . . ,Sk):= L Dn}1511 o71, quence fills the unit interval. Given any subinter n1 >n2> · .. >nk>O }=1 val (c, d) of the unit interval, it can be seen that this sequence visits this subinterval no more than where the s1, s2, .. . , sk are nonzero integers and 3n(d- c) + 3 times, among the first n elements, the O"J := signum(s1). Such constants can be con provided that n > 1/(d- c). It can then be shown sidered as generalizations of the Riemann zeta that the sequence ({21 oc}) visits (c, d) no more than function at integer arguments in higher dimen 8n(d- c) times, among the first n elements of this sions. sequence, so long as n is at least 1/(d- c)2. The 2- The analytic evaluation of such sums has relied normality of oc0 then follows from a result given on fast methods for computing their numerical in [28, p. 77]. Further details on these results are values. One scheme, based on Holder Convolution, given in [15, Sec. 4.3], [6], [12]. is discussed in [22] and implemented in EZFace+, an online tool available at http: I /www. cecm. sfu. Euler's Multi-Zeta Sums ca/projects/ezface+. We will illustrate its ap In April1993, Enrico Au-Yeung, an undergraduate plication to one specific case, namely the analytic at the University of Waterloo, brought to the at identification of the sum tention of one of us (JMB) the curious result (16) I00( 1 + -1 + ... + -1)2 k-2 1 k+11)2 -3 (14) k= 1 2 k 52,3 = L.~( 1 - - + ... + (-1) - (k + 1) . k=1 2 k 1 17rr4 = 4.59987 ... <=:j : ((4) 360 Expanding the squared term in (16), we have
506 NOTICES OF THE AMS VOLUME 52, NUMBER 5 ( -1)i+ J+l Mathematica have some rather effective integration (17) I . ·k3 = -2 s(3, -1, -1) + Q3, 2). facilities, not only for obtaining analytic results O
MAY 2005 NOTICES OF THE AMS 507 where x1 = g(hj) and w1 = g' (hj) are abscissas and where L3(s) = 2.: ~= 1 [1 /(3n - 2)5 - 1/(3n - 1)5 ]. weights that can be precomputed. If g' (t) and its Based on these experimental results, general results derivatives tend to zero sufficiently rapidly for of this type have been conjectured but not yet rig large t, positive and negative, then even in cases orously established. where f(x) has a vertical derivative or an integrable A third example is the following: singularity at one or both endpoints, the resulting integrand f(g(t))g' (t) is, in many cases, a smooth (21) ~ Jrr/2 log I tant + -J? I dt ~ L 7(2) bell-shaped function for which the Euler 7-Jt rr /3 tant - -J7 Maclaurin formula applies. In these cases, the error E(h) in this approximation decreases faster than where any power of h. Three suitable g functions are g 1 (t) = tanh t, "" [ 1 1 1 g2(t) erf t, and g3(t) = tanh(rr I 2 · sinh t). Among L 7(s) = I (7n + l)s + (7n + 2)5 - (7n + 3)5 = n=O these three, g3 (t) appears to be the most effective for typical experimental math applications. For 1 1 1 ] + (7n + 4)5 - (7n + 5)5 - (7n + 6)5 • many integrals, "tanh-sinh" quadrature, as there sulting scheme is known, achieves quadratic con The "identity" (21) has been verified to over 5000 vergence: reducing the interval h in half roughly decimal digit accuracy, but a proof is not yet known. doubles the number of correct digits in the quad It arises from the volume of an ideal tetrahedron rature result. This is another case where we have in hyperbolic space, [15, pp. 90-1]. For algebraic more heuristic than proven knowledge. topology reasons, it is known that the ratio of the As one example, recently the present authors, left-hand to the right-hand side of (21) is rational. together with Greg Fee of Simon Fraser University A related experimental result, verified to 1000 in Canada, were inspired by a recent problem in the digit accuracy, is American Mathematical Monthly [2]. They found by 0 l - 2Jz - 213- 214 + 2Jw + 2J u + 3]Jz + 3] 13 + lJ4 - l!s using a tarili-sinh quadrature program, together - l!6 - l!7 - l!s - ]Jg +ho + h 1 -Jzz -h3 + 2hs, with a PSLQinteger relation detection program, that if C(a) is defined by where In is the integral in (21), with limits nrr /60 and (n + 1)rr /60. The above examples are ordinary one C(a) = arctan( .Jx2 + a2) dx e dimensional integrals. Two-dimensional integrals .Jx2 + a2(x2 + 1) ' Jo are also of interest. Along this line we present a then more recreational example discovered experimen tally by James Klein- and confirmed by Monte C(O) = rrlog2/ 8 + G/ 2, Carlo simulation. It is that the expected distance C(1) = rr/4 - rrJZ/ 2 + 3arctan(J2); )2, between two random points on different sides of a unit square is C(J2) = 5rr2 / 96. 1 ~ e e~ x 2 + y 2 dx dy + ! r e ~ 1 + (u- v)2 du dv Here G = 2.: k?:o (-1)k /(2k + 1)2 is Catalan's con 3 Jo Jo 3 Jo Jo stant-the simplest number whose irrationality is 1 ,-;:;- 5 ,-;:;- 2 = -v 2 + log(-v 2 + 1) + g' not established but for which abundant numerical 9 9 evidence exists. These experimental results then led the expected distance between two random to the following general result, rigorously estab and points on different sides of a unit cube is lished, among others:
( "" arctan(.JXZ"+/12 ) dx -4 1111 1111 ~x2 + y 2 + (z - w)2 dw dx dy dz Jo .JXZ"+/1Z(x 2 + 1) 5 0 0 0 0 1 r1 e r1 r1 = ~ [2 arctan(.JCI2=1) - arctan(.Ja4 - 1)]. +S Jo Jo Jo + (y - u)2 + (z- w)2 dudwdydz 2 a - 1 Jo ~ 1 4 17 2 7 =- + - F2 - -J3 - - rr As a second example, recently the present au 75 75 25 75 thors empirically determined that + ; 5 log ( 1 + F2) + ; 5 log ( 7 + 4-vS) . _I_ f1 log6(x) arctan[xJ3/(x - 2 )] d = _ 1 _ [- 229635L ( 8) J3 Jo X + 1 X 81648 3 + 29852550L3(7)log 3- l 63 2960L3(6)rr2 + 27760320L3(5)((3) See [7] for details and some additional examples. - 275184L3(4)rr4 + 36288000L3(3)((5) - 30008L3(2)rr6 It is not known whether similar closed forms exist - 57030120L3(1)((7)], for higher-dimensional cubes.
508 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Ramanujan's AGM Continued Fraction Given a, b, 17 > 0, define a RTJ(a, b) = b2 11 + ---4.,--a.....,2.----- 17 + 9b2 1]+-- 1]+
This continued fraction arises in Ramanujan's Note books. He discovered the beautiful fact that
RTJ(a,b)+RTJ(b,a) = R (a+b ~b) 2 TJ 2 , vao .
The authors wished to record this in [15] and wished to computationally check the identity. A first attempt to numerically compute R1 (1, 1) directly failed miserably, and with some effort only three .. ~. reliable digits were obtained: 0.693 .... With hind .e·;. 15 ..... sight, the slowest convergence of the fraction oc .... curs in the mathematically simplest case, namely 10 when a = b. Indeed R1 (1, 1) = log 2, as the first primitive numerics had tantalizingly suggested. (...{I . Attempting a direct computation of R1 (2, 2) using a depth of 20000 gives us two digits. Thus we must seek more sophisticated methods. From -15 -10 10 15 formula(l.11.70)of[16]weseethatfor0 < b < a, -5'
(~ii. -1 0 ·-~ where k = b j a = e ~ ;e ~ ,k' = Jl=k2. Here e2,e3 ·Jo& ·. ·.: · are jacobian theta functions and K is a complete .. elliptic integral of the first kind. Writing the previous equation as a Riemann Figure sum, we have 2. Dynamics and attractors of various iterations. /f)( ) ·= 'R ( ) = J"" sech(rrx/(2a))d .£\.,a . 1 a, a 1 2 x 0 + X (23) 'R(2) = n [rr / 2 - log(1 + n )]. (- 1)k+1 Indeed, it follows (see [19]) that = 2a ~ 1 + (2k - 1)a' r1 t11a where the final equality follows from the Cauchy 'R(a) = 2 Jo 1 + t2 dt. Lindelof Theorem. This sum may also be written 2 Note that 'R(1) = log 2. No nontrivial closed form as 'R(a) = 1:aF (-fa +!.1; fa + ~ ; - 1). The latter is known for 'R(a ; b) a form can be used in Maple or Mathematica to with * b, although determine R (.l... /3 (.!. .!. ) J2 /3 (.!. .!. )) = .!_ L sech(mr) 1 R(2) = 0.974990988798722096719900334529 ... . 4rr 4 ' 4 ' 8rr 4' 4 2 nEZ 1 + n2
This constant, as written, is a bit difficult to is close to closed. Here f3 denotes the classical Beta recognize, but if one first divides by v?., one can function. It would be pleasant to find a direct proof obtain, using the Inverse Symbolic Calculator, an of (23). Further details are to be found in [19], [17], online tool available at the URL http: I jwww. [16]. cecm.sfu.ca/projects/ISC/ISCmain.html,that Study of these Ramanujan continued fractions the quotient is rr / 2 - log(l + v?_). Thus we con has been facilitated by examining the closely related clude, experimentally, that dynamical system t0 = 1, t 1 = 1, and
MAY 2005 NOTICES OF THE AMS 509 - even - odd There are some exceptional ]acobsen 7 - even cases. - odd Masson theory [17), [18) shows that the even/odd 6 fractions for 'R 1 (i, i) behave "chaotically"; neither converge. Indeed, when a = b = i, (tnU, i)) exhibit a 5 fourfold quasi-oscillation, as n runs through val ues mod 4. Plotted versus n, the (real) sequence tn(i) ~ 4 :.:=..0 exhibits the serpentine oscillation of four sepa o)3 rate "necklaces". The detailed asymptotic .Q is 2 tnU, i) =)!cosh; }n (1 + 0 ( ~)) (-l)n /2 cos(e -log(2n)/2) n is even 0 { -1 -2 x (-1)(n+l)/2 sin(e - log(2n)/ 2) n odd
real where e := argf((1 + i) / 2). imaginary 1 2 Analysis is easy given the following striking hy· pergeometric parametrization of (24) when Figure 3. The subtle fourfold serpent. a = b * 0 (see [18)), which was both experimen tally discovered and is computer provable: 1 1 (25) tn(a, a) = 2 Fn(a) + 2Fn( - a),
15000 where 510000 anz!-w ( .., Fn(a) := - fl( ) zh w, w; n + 1 + w; -1) . tU Wp n + w,-w 2 ~5000 -~ 0 Here 6 6 r (n + 1) {3(n + 1 + w, -w) := f(n + 1 + w)f(-w)' and 1- 1/ a w:= 2 Indeed, once (2 5) was discovered by a combination imaginary -6 - 6 real of insight and methodical computer experiment, its proof became highly representative of the chang ing paradigm: both sides satisfy the same recursion Figure 4. A period three dynamical system (odd and even and the same initial conditions. This can be checked iterates). in Maple, and if one looks inside the computation, one learns which conf7uent hypergeometric identi ties are needed for an explicit human proof. As noted, study of 'R devolved to hard but com· pelling conjectures on complex where Wn = a2 or b2 (from the Ramanujan con dynamics, with many tinued fraction definition), depending on whether interesting proven and unproven general n is even or odd. izations. In [23) consideration is made of contin ued fractions If one studies this based only on numerical val like ues, nothing is evident; one only sees that tn ~ 0 fairly slowly. However, if we look at this iteration pictorially, we learn significantly more. In particu lar, if we plot these iterates in the complex plane and then scale by fo and color the iterations blue or red depending on odd or even n, then some re markable fine structures appear; see Figure 2. With for any sequence a =(an) ;;'~ 1 and convergence assistance of such plots, the behavior of these it properties obtained for deterministic and random erates (and the Ramanujan continued fractions) is sequences (an). For the deterministic case the best now quite well understood. These studies have results obtained are for periodic sequences, satis ventured into matrix theory, real analysis, and even fying a J = aJ+c for all j and some finite c. The dy the theory of martingales from probability theory namics are considerably more varied, as illustrated [19), [17), [18), [2 3). in Figure 4.
510 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Coincidence and Fraud C(x) Coincidences do occur, and such examples drive home the need for reasonable caution in this en terprise. For example, the approximations 3 rr "" .Jf63 log(640320), 1T "" .J2 9801 4412 occur for deep number theoretic reasons: the first good to fifteen places, the second to eight. By con trast
err- rr = 19.999099979189475768 ... , most probably for no good reason. This seemed more bizarre on an eight-digit calculator. Likewise, as spotted by Pierre Lanchon recently,
e = 10.10110111111000010101000101100 . . . Figure 5. First few terms of Tin ~ l cos(x/k). while a few correct digits. Thus it is necessary to rewrite 7T = 11.0010010000111111011010101000 ... the integrand function in a form more suitable for have 19 bits agreeing in base two-with one reading computation. This can be done by writing right to left. More extended coincidences are almost always contrived, as illustrated by the following: (27) f(x) = cos(2x) [ Qcos(x/k)] exp(fm(x)),
~ [ntanh(rr/2)] "'_.!._ ~ [ntanh(rr)] "' _.!.__ L.... 10n 81' L.... 10n 81 where we choose m > x, and where n~l n ~ l (28) fm(x) = log cos The first holds to 12 decimal places, while the sec I G). k~m+l ond holds to 268 places. This phenomenon can be understood by examining the continued fraction ex The log cos evaluation can be expanded in a Tay pansion of the constants tanh(rr /2) and tanh(rr): lor series [1, p. 75], as follows: the integer 11 appears as the third entry of the first, while 267 appears as the third entry of the second. ( ~) - ~ (-1)1221-1(221 -1)B2J (~)2J Bill Gasper, commenting on the extraordinary ef log cos k - 1~1 j(2j)! k , fectiveness of continued-fraction expansions to "see" what is happening in such problems, de where B21 are Bernoulli numbers. Note that since clared, "It looks like you are cheating God some k > m > x in (28), this series converges. We can now how." write A fine illustration is the unremarkable decimal (-1)J22J-1(22J- 1)B2J (~)2J fm(X) = I I cx = 1.4331274267223117583 .. . whose contin k ~ m+l J~l j(2j)! k ued fraction begins [1, 2, 3, 4, 5, 6, 7, 8, 9 ... ] and so most probably is a ratio of Bessel functions. Indeed, - ~ (2 2) - 1)((2j) [ ~ _1 ] 2J L.... ·rr2J L.... k2J x 10(2) I h (2) was what generated the decimal. Simi J~l 1 k~m+l larly, rr and e are quite different as continued frac _ ~ (2 21 - 1)((2j) [.,..(2 .) _ ~ ...!._] 21 tions, less so as decimals. L.... . 2) "' J L.... k2J X • A more sobering example of high-precision J~l )IT k~l "fraud" is the integral This can now be written in a compact form for com putation as (26) rr2 := f"" cos(2x) TI cos(~) dx. Jo n~l (29) The computation of a high-precision numerical value for this integral is rather challenging, due in where part to the oscillatory behavior of Tin ~ l cos(x/n) (see Figure 2), but mostly due to the difficulty of computing high-precision evaluations of the inte (30) m grand function. Note that evaluating thousands of bj,m = 1;(2))- I 1/k21. terms of the infinite product would produce only k~l
MAY 2005 NOTICES OF THE AMS 511 However,
I 8 : = fa"' sinc(x)sinc ( ~) · · · sine cxs) dx 467807924713440738696537864469 = 935615849440640907310521750000 TT ::::: 0.499999999992646TT. When this was first found by a researcher using a well-known computer algebra package, both h e and the software vendor concluded there was a "bug" in the software. Not so! It is easy to see that the limit of these integrals is 2 TT1, where (31) TT1 :=fa''" cos(x) lJ cos(~) dx.
This can be seen via Parseval 's theorem, which links the integral Figure 6. Advanced Collaborative Environment in Vancouver. IN := sinc(alx)sinc (a2x) · · ·sine (aNx) Computation of these b coefficients must be done fa"' dx to a much higher precision than that desired for the quadrature result, since two very nearly equal with the volume of the polyhedron PN given by quantities are subtracted here. N The integral can now be computed using, for ex PN := {x: I I akxkl :5 a1, lxkl :5 1,2 :5 k :5 N}, ample, the tanh-sinh quadrature scheme. The first k=2 60 digits of the result are the following: where x := (x 2 ,x3 , · • • ,xN). If we let
0.3926990816987241548078304229099 eN := {(x2,x3, ... ,XN): -1:5 Xk :51,2:5 k :5 N}, 37860524645434187231595926812 .... then At first glance, this appears to be TT I 8. But a care ful comparison with a high-precision value of TT I 8, namely Thus, the value drops precisely 0.3926990816987241548078304229099 when the con straint L: ~= 2 akxk :5 a1 becomes active and bites 37860524646174921888227621868 ... , the hypercube eN. That occurs when L:~= 2 ak > a1. reveals that they are not equal: the two values dif In the above, ~ + ~ + · · · + f3 < 1, bu t on addition fer by approximately 7.407 x 10- 43 . Indeed, these of the term -fs, the sum exceeds 1, the volume two values are provably distinct. The reason is drops, and IN = .g. no longer holds. A similar analy governed by the fact that L: ~~ 1 1/(2n + 1) > 2 > sis applies to TT2. Moreover, it is fortunate that we L:~~ 1 1/(2n + 1). See [16, Chap. 2] for additional began with TT1 or the falsehood of the identity anal details. ogous to that displayed above would have been A related example is the following. Recall the sine much harder to see. function . sinx smc(x) := --. Fu rther Directions and Implications X In spite of the examples of the previous section, Consider the seven highly oscillatory integrals it must be acknowledged below. that computations can in many cases provide very compelling evidence for h := Jo''" sine(x)dx ~ ~· mathematical assertions. As a single example, re cently Yasumasa Kanada of Japan calculated TT to h := fo"' sine(x)sine G) dx = ~, over one trillion decimal digits (and also to over one trillion hexadecimal digits). Given that such com h := fa"' sine(x)sine G) sine ( ~) dx = ~, putations- which take many hours on large, state of-the-art supercomputers-are prone to many types of error, including hardware failures, system software problems, and especially programming 16 := roo sine(x)sine (~) . .. sine( ~ ) dx = !:!:., bugs, how can one be confident in such results? Jo 3 · 11 2 In Kanada's case, he first used two different arctangent-based .formulas TT f? := Jo''" sine(x)sine G) ·· ·sine ( dx = ~ · to evaluate to over tJ one trillion hexadecimal digits. Both calculations
512 NOTICES OF THE AMS VOLUME 52, N UMBER 5 agreed that the hex expansion beginning at position 1,000,000,000,001 is B4466E8D21 5388C4E014. He then applied a variant of the BBP formula for rr, mentioned in Section 3, to calculate these hex digits directly. The result agreed exactly. Needless to say, it is exceedingly unlikely that three different computations, each using a com pletely distinct computational approach, would all perfectly agree on these digits unless all three are correct. Another, much more common, example is the usage of probabilistic primality testing schemes. Damgard, Landrock, and Pomerance showed in 1993 that if an integer n has k bits, then the prob ability that it is prime, provided it passes the most commonly used probabilistic test, is greater than Figure 7. Polyhedra in an immersive environment. 1- k242- .Jk, and for certain k is even higher [25]. For instance, if n has 500 bits, then this probabil ity is greater than 1 - 1/428m . Thus a 500-bit is any error among the remaining digits after the integer that passes this test even once is prime first million is less than 1.465 x 10- 8 [15, Section with prohibitively safe odds: the chance of a false 4.3]. Additional computations could be used to than one part in Avo declaration of primality is less lower this probability even more. gadro's number (6 x 1023 ). If it passes the test for Although few would bet against such odds, these four pseudorandomly chosen integers a, then the computations do not constitute a rigorous proof chance of false declaration of primality is less than that the sequence (yn) is identical to the hexadec one part in a googol (10100). Such probabilities are many orders of magnitude more remote than the imal expansion of rr. Perhaps someday someone chance that an undetected hardware or software will be able to prove this observation rigorously. error has occurred in the computation. Such meth On the other hand, maybe not-maybe this ods thus draw into question the distinction be observation is in some sense an "accident" of tween a probabilistic test and a "provable" test. mathematics, for which no proof will ever be Another interesting question is whether these found. Perhaps numerical validation is all we can experimental methods may be capable of discov ever achieve here. ering facts that are fundamentally beyond the reach of formal proof methods, which, due to Godel's re Conclusion sult, we know must exist; see also [24]. We are only now beginning to digest some very old One interesting example, which has arisen in our ideas: work, is the following. We mentioned in Section 3 the fact that the question of the 2 -normality of rr Leibniz's idea is very simple and very reduces to the question of whether the chaotic it profound. It's in section VI of the Dis e]. eration x0 = 0 and cours [de metaphysiqu It's the obser vation that the concept of law becomes x = { 16x + 120n2 - 89n + 16 } vacuous if arbitrarily high mathemati- n n -l 512n4- 1024n3 + 712n2 - 206n + 21 ' . cal complexity is permitted, for then where { ·} denotes fractional part, are equidistrib there is always a law. Conversely, if the uted in the unit interval. law has to be extremely complicated, It turns out that if one defines the sequence then the data is irregular, lawless, ran Yn = L16xn J (in other words, one records which of dom, unstructured, patternless, and the 16 subintervals of (0, 1), numbered 0 through also incompressible and irreducible. A 15, Xn lies in), that the sequence (Yn), when inter theory has to be simpler than the data preted as a hexadecimal string, appears to pre that it explains, otherwise it doesn't ex cisely generate the hexadecimal digit expansion of plain anything. - Gregory Chaitin [24] rr. We have checked this to 1,000,000 hex digits and have found no discrepancies. It is known that (yn) Chaitin argues convincingly that there are many is a very good approximation to the hex digits of mathematical truths which are logically and com rr, in the sense that the expected value of the num putationally irreducible- they have no good reason ber of errors is finite [1 5, Section 4.3] [11]. Thus in the traditional rationalist sense. This in turn one can argue, by the second Borel-Cantelli lemma, adds force to the desire for evidence even when that in a heuristic sense the probability that there proof may not be possible. Computer experiments
MAY 2005 NOTICES OF THE AMS 513 can provide precisely the sort of evidence that is [15) jONATHAN BORWEIN and DAVID BAILEY, Mathematics by required. Experiment, A K Peters Ltd., Natick, MA, 2004. Although computer technology had its roots in [16) jONATHAN BORWEIN, DAVID BAILEY, and ROLAND mathematics, the field is a relative latecomer to the GIRGENSOHN, Experimentation in Mathematics: Compu application of tational Paths to Discovery, A K Peters Ltd., Natick, MA, computer technology, compared, 2004. say, with physics and chemistry. But now this is [17) jONATHAN BORWEIN and RICHARD CRANDALL, On the Ra changing, as an army of young mathematicians, manujanAGM fraction. Part II: The complex-parameter many of whom have been trained in the usage of case, Experiment. Math. 13 (2004), 287-96. sophisticated computer math tools from their high [18) JONATHAN BORWEIN, RICHARD CRANDALL, DAVID BORWEIN, school years, begin their research careers. Further and RAYMOND MAYER, On the dynamics of certain re advances in software, including compelling new currence relations, Ramanujan]. (2005). mathematical visualization environments (see Fig [19) jONATHAN BORWEIN, RICHARD CRANDALL, and GREG FEE, On ures 6 and 7), will have their impact. And the re the Ramanujan AGM fraction. Part 1: The real-parameter case, Experiment. markable Math. 13 (2004), 275-86. trend towards greater miniaturization [20) jONATHAN M. BORWEIN, DAVID BORWEIN, and WIWAM F. (and corresponding higher power and lower cost) GALWAY, Finding and excluding b-ary Machin-type BBP in computer technology, as tracked by Moore's formulae, Canadian]. Math. 56 (2004), 897-925. Law, is pretty well assured to continue for at least [21) JONATHAN M. BORWEIN and DAVID M. BRADLEY, On two another ten years, according to Gordon Moore him fundamental identities for Euler sums, http: I lwww. self and other industry analysts. As Richard Feyn cs. da l . cal-jborwei nlz21. pdf (2005). man noted back in 1959, "There's plenty of room [22) JONATHAN M. BORWEIN, DAVID M. BRADLEY, at the bottom" [2 7]. It will be interesting to see what DAVID ]. BROADHURST, and PETR L!SONEK, Special values the future will bring. of multiple polylogarithms, Trans. Amer. Math. Soc. 353 (2001), 907-41. [23) jONATHAN M. References BORWEIN and D. RUSSELL LUKE, Dynamics of generalizations of the AGM continued [1) MILTON ABRAMOWITZ and IRENE A. STEGUN, Handbook of fraction of Ramanujan. Part 1: Divergence, Mathematical Functions, New York, 1970. http:llwww.cs.dal .cal-jborweiniBLuke.pdf [2) ZAFAR AHMED, Definitely an integral, Amer. Math. (2004). Monthly 109 (2002), 670-1. [24) GREGORY CHAITIN, Irreducible complexity in pure [3) KENDALL E. ATKINSON, An Introduction to Numerical mathematics,http:llarxiv.orglmath.HOI0411091 Analysis, Wiley and Sons, New York, 1989. (2004). [4) DAVID H. BAILEY, Integer relation detection, Comput. Sci. [25)1. DAMGARD, P. LANDROCK, and C. POMERANCE, Engineering 2 (2000), 24-8. Average case error estimates for the strong probable [5) __ , A compendium of BBP-type formulas for prime test, Math. of Comp. 61 (1993), 177-94. mathematical constants, http: I I c rd. l b l . gov I [26) HELAMAN R. P. FERGUSON, DAVID H. BAILEY, and STEPHEN -dhbai leyldhbpaperslbbp-formul as. pdf (2003). ARNo, Analysis of PSLQ, an integer relation finding [6) __ , A hot spot proof of normality for the alpha algorithm, Math. ofComp. 68 (1999), 351-69. constants, http:llcrd.lbl.govl-dhbaileyl [27) RICHARD FEYNMAN, There's plenty of room at the dhbpaperslalpha-normal.pdf~005~ bottom, http: I I eng r. smu. edul eel smuphotoni csl [7) DAVID H. BAILEY, jONATHAN M. BORWEIN, VISHAA KAPOOR, NanoiFeynmanPlentyofRoom.pdf(1959). and ERIC WEISSTEIN, Ten problems of experimental [28) L. KUIPERS and H. NIEDERREITER, Uniform Distribution of mathematics, http: I lcrd. l bl. gov 1-dhbail ey I Sequences, Wiley-lnterscience, Boston, 1974. dhbpapersltenp rob l ems. pdf (2004). [29) GORDON E. MooRE, Cramming more components onto [8) DAVID H. BAILEY, PETER B. BORWEIN, and SIMON PLOUFFE, On integrated circuits, Electronics 38 (1965), ll4-7. the rapid computation of various polylogarithmic con stants, Math. ofComp. 66 (1997), 903-13. [9) DAVID H. BAILEY and DAVID]. BROADHURST, A seventeenth order polylogarithm ladder, http: I I c rd. l b l . govI -dhbai leyldhbpapersll adder. pdf (1999). [10) __ , Parallel integer relation detection: Techniques and applications, Math. of Comp. 70 (2000), 1719-36. [ll) DAVID H. BAILEY and RICHARD E. CRANDALL, Random generators and normal numbers, Experiment. Math. 10 (2001), 175-90. [12) __ , Random generators and normal numbers, Experiment. Math. 11 (2004), 527-46. [13) DAVID H. BAILEY and XIAOYE S. LI, A comparison of three high-precision quadrature schemes, http: I I c rd. l b l . gov1-dhbai l eyldhbpaperslquadrature. pdf (2004). [14) DAVID H. BAILEY and SINAI ROBINS, Highly parallel, high precision numerical quadrature, http:llcrd. lbl.govl-dhbaileyldhbpaperslquadparallel.pdf (2004).
514 NOTICES OF THE AMS VOLUME 52, NUMBER 5 AMERICAN MATHEMATICAL SOCIETY
._ t (:.'Xt•W zl ~t"' G~ometry
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The Smale horseshoe is the hallmark of chaos. With n ~ co; while if y lies on the vertical through p, then striking geometric and analytic clarity it robustly the inverse iterates off squeeze it top. With respect describes the homoclinic dynamics encountered to linear coordinates centered at p, f(x, y) = (kx, my) by Poincare and studied by Birkhoff, Cartwright where (x, y) E B and 0 < k < 1 < m. Similarly, Littlewood, and Levinson. We give the example first and the definitions later. f(x, y) = (-kx, -my) with respect to linear coordi Consider the embedding f of the disc ~ into nates on D at s. itself exhibited in the figure. It contracts the The sets semidiscs A, E to the semidiscs f(A), f(E) in A and it sends the rectangles B, D linearly to the rectan ws = {z : fn(z) ~ p as n ~ co}, gles{ (B), f (D), stretching them vertically and shrink wu = {z: fn(z) ~pas n ~-co} ing them horizontally. In the case of D, it also rotates by 180 degrees. We don't really care what are the stable and unstable manifolds of p. They the image f(C) of Cis as long as it does not intersect intersect at r, which is what Poincare called a the rectangle B u C u D. In the figure it is placed so homoclinic point. The figure shows these invariant that the total image resembles a horseshoe, hence the manifolds only locally. Iteration extends them name. globally. It is easy to see that f extends to a diffeomor The key part of the dynamics of f happens on phism of the 2 -sphere to itself. We also refer to the horseshoe extension as f and work out its dynamics in~. i.e., the its iterates fn for n E 7L. A= {z: fn(z) E BuD for all n E ~}. Necessarily there are three fixed points p, q, s. The point q is a sink in the sense that all points Everything there is explained as the "full shift on z E A u E u C converge to q under forward itera the space of two symbols". Take two symbols, 0 and tion, fn(z) ~ q as n ~ co. 1, and look at the set~ of all hi-infinite sequences The points p, s are saddle points. If x lies on the a = (an) where n E ~ and for each n, an is 0 or an horizontal through p, then squeezes it to p as rn is 1. Thus~= {0, 1}., is homeomorphic to the Can Michael Shub has been a Research Staff Member at IBM's tor set. The map u : ~ ~ ~ that sends a = (an) to T] Watson Research Center for twenty years. He is currently u(a) = (an+l) is a homeomorphism called the shift professor of mathematics at the University of Toronto. His map. It shifts the decimal point one slot rightward. email address is mi chae 1 . shub@uto ron to. ca. Every dynamical property of the shift map is pos The author would like to thank Charles Pugh, who sug gested changes to an early version of this article and who sessed equally by fiA, because there is a homeo produced the original version of the figure. morphism h : ~ ~ A such that the diagram
516 NOTICES OF THE AMS VOLUME 52, NUMBER 5 dimension is expressed in terms of expansion and contraction of the derivative on subbundles of the tangent bundle. Smale unified such examples as the horseshoe and the geodesic flow on manifolds of negative curvature, defining what is now called uniformly hyperbolic dynamical systems. The study of these systems has led to many fruitful discov eries in modern dynamical systems theory. David Ruelle has called Smale's 1967 article [3) "a masterpiece of mathematical literature". It is still worth reading today. Hyperbolic dynamics flourished in the 1960s and 1970s. Anosov proved the stability and ergodicity of the globally hyper bolic systems that now bear his name. Sinai initi ated the more general investigation of the ergodic theory of hyperbolic dynamical systems, and in particular showed that the Markov partitions of Adler and Weiss could be constructed for all hy perbolic invariant sets, thus giving a coding simi lar to the two-shift coding for the horseshoe. This f work was carried forward by Ruelle and Bowen. The A A invariant measures they found, now called Sinai Ruelle-Bowen (SRB) measures, describe the as commutes. The conjugacy h is easy to describe. ymptotic dynamics of most Lebesgue points in the Given any a E I, there is a unique z E A such that manifold, even for dissipative sytems. Uniformly fn(z) E B whenever an = 1, while fn(z) ED when hyperbolic dynamical systems are remarkable. They ever an = 0. Thus u codes the horseshoe dynamics. exhibit chaotic behaviour. By the work of Anosov, For instance, ( · · · 11.111 · · · ) codes the point p, Smale, Palis, and Robbin, they are structurally sta ( · · · 00.000 · · ·) codes s, while ( · · · 111.0111 · · ·) ble; that is, the dynamics of a perturbation of a uni codes r. formly hyperbolic system is topologically conjugate u has 2n periodic orbits of period n, and so to the original. By the work of Sinai, Ruelle, and must fiA· The set of periodic orbits of u is dense Bowen, they are described statistically. in I and so must be the set of periodic orbits of In the early days of the 1960s it was hoped that f IA. Small changes of initial conditions in I can pro uniformly hyperbolic dynamical systems might be duce large changes of a u-orbit, so the same must in some sense typical. While they form a large open be true of fiA· In short, due to conjugacy, the chaos set on all manifolds, they are not dense. The search of u is reproduced exactly in the horseshoe. for typical dynamical systems continues to be a The utility of Smale's analysis is this: every great problem. For progress see the survey [1). dynamical system having a transverse homoclinic Hyperbolic periodic points, their global stable and point, such as r, also has a horseshoe containing unstable manifolds, and homoclinic points remain r and thus has the shift chaos. Nowadays, this some of the principal features of, and tools for, fact is not hard to see, even in higher dimensions. understanding the dynamics of chaotic systems. The mere e xistence of a transverse intersection Indeed, transversal homoclinic points are proven between the stable and unstable manifolds of a to exist in many of the dynamical systems en periodic orbit implies a horseshoe. In the case of countered in science and engineering from celes flows, the corresponding assertion holds for the tial mechanics, where Poincare first observed them, Poincare map. To recapitulate, to ecology and beyond. transverse homoclinicity => horseshoe => chaos. References Since transversality persists under perturbation, it [1] CHRISTIAN BONA TTl, LORENZO]. DIAZ, and MARCELO VIANA, follows that so does the horseshoe and so does its Dynamics beyond Uniform Hyperbolicity: A G lobal chaos. Geometric and Probabilistic Approach, Encyclopedia The analytical feature of the horseshoe is Math. Sci., Springer, 2004. hyperbolicity, the squeeze/stretch phenomenon ex [2] MICHAEL SHUB, Global Stability of Dynamical Systems, pressed via the derivative. The derivative of f Springer, 1986. stretches tangent vectors that are parallel to the [3] STEPHEN SMALE, Differentiable dyn amical system s, vertical and contracts vectors parallel to the Bull. A mer. Math. Soc. 73 (1 967), 747-817. [4] _ _ , F horizontal, not only at the saddle points, but inding a horseshoe on the beaches of Rio, Math. Intelligencer 20 (1998), 39-44. uniformly throughout A. In general, hyperbolicity of a compact invariant set such as A in any
MAY 2005 NOTICES OF THE AMS 517 Picturing the Horseshoe Map
Smale's horseshoe map takes a disk into itself. Oo,o =BuD f(Oo,o) Points in A and E are mapped into A, points in C into E. ~ I •I II I I 111111111 . - Oo,l
Upon iteration, any point that finds itself in A, C, orE eventually winds up in A, converging to an attracting fixed point in that region. The interest ing dynamics happens to points that never arrive in A, C, E. This is the intersection of all the sets
O o,n = {xI r(x) E Bu D for 0 ~ k ~ n},
defined recursively by
Oo,o =BuD Oo,n+l = Oo,n n f - 518 NOTICES OF THE AMS VOLUME 52, NUMBER 5 is the set ~Loo ,oo of points x for which all fn(x) lie than B itself are homoclinic points, attracted to B in B u D. The condition that r - n(x) lie in B u D is 1 upon iteration by both f and r- 0 equivalent to the condition that x lie in fn(B u D). This set is approximated by sets fLm,n with m and n large, and amounts to a two-dimensional Cantor dust. Hyperbolic fixed points and a homoclinic point The simplest homoclinic point is ... BBD.BB ... , which lies on both the horizontal line through the hyperbolic point P = ... BBB.BB ... and the image Because the map f is an affine transformation under iterations off applied to a small vertical seg on B and D, these sets are easy to calculate, and it ment through P. is easy to see that the connected components con verge to isolated points. Each one of the connected sets fLm,n can be labeled by a finite string S-m ... S- 1.sos1 .. . Sn of characters B and D where sk = B if fk(Q) ~ B, otherwise D. Thus B.D is the set of points x with x in D, f - 1(x) in B. The points to which the sets converge are indexed by infinite strings (sk) fork in Z. The limit set is stable under f, which acts as a left shift. The image of BB.DB under fis BBD.B. The coordinate succession is semi-inverted: More homoclinic points But the homoclinic points are in fact dense in 0.- oo, oo , as the picture above barely suggests. In fact, as it I hope it also suggests, the horseshoe map and the network of homoclinic points in very gen eral dynamical systems are intimately related. This was Smale's original insight on the beach at Rio, mentioned in Shub's article in this issue. Coordinates x.y - Bill Casselman There are exactly two points fixed by f, namely B = ... BB.BB . .. and D = .. . DD.DD .. . , which are both hyperbolic- attracting horizontally and re pelling vertically. The points with an infinite string ... BBB ... at the right end are those attracted to the first, those with an infinite terminal string ... ODD ... to the second. Points ... BB ... BB ... with an infinite string. of B at both right and left other MAY 2005 NOTICES OF THE AMS 519 Book Review The Works of Archimedes: Translation and Commentary Volume 1: The Two Books On the Sphere and the Cylinder Reviewed by Alexander ]ones The Works of Archimedes: Translation and He was the sub Commentary. Volume 1: The Two Books On the ject of a biogra Sphere and the Cylinder phy- now lost · Edited and translated by Reviel Netz alas!-and stories Cambridge University Press, 2004 about him are told Hardcoverx + 376 pp., $130.00 by ancient histo ISBN 0-521-66160-9 rians and other writers who gen Ancient Greek mathematics is associated in erally took little most people's minds with two names: Euclid and interest in scien Archimedes. The lasting fame of these men does tific matters. The not rest on the same basis. We remember Euclid as stories of Archi the author of a famous book, the Elements, which medes' inven for more than two millennia served as the funda tions; his solution mental introduction to ruler-and-compass geome of the "crown try and number theory. About Euclid the man we problem"; the ma know practically nothing, except that he lived b e chines by which, fore about 200 B.C. and may have worked in Alexan as an old man, he dria. He wrote works on more advanced mathe defended his native city, Syracuse, from the be matics than the Elements, but none of these have sieging Roman fleet in 212 B.C.; and his death-still survived, though we have several fairly basic books doing geometry-at the hands of a Roman soldier on mathematical sciences (optics, astronomy, har when Syracuse at last fell have never lost their ap monic theory) under his name. All his writings dive peal. Archimedes became paradoxically' emblem straight into the mathematics with no introduc atic of two stereotypes of the mathematician: a tions. There are hardly even any unreliable anec man who could harness reasoning to the seemingly dotes about Euclid. superhuman performance of practical tasks, yet Archimedes, by contrast, is not just an author whose preoccupation with abstract problems could to us but a personality. He was famous in his time, make him fatally oblivious to his surroundings. not only among mathematicians and intellectuals. Alongside the public Archimedes of the stories, we also have the private Archimedes of the writ Alexander ]ones is professor of classics at the University ings. In the manner of his time, Archimedes wrote . of Toronto. His email address is a l exander.jones@ his mathematics in the form of substantial books utoront o . ca. built up of theorems that cumulatively lead to the 520 NOTICES OF THE AMS VOLUME 52, NUMBER 5 proof of a series of major results. Copies of these parchment sheets bound like modern books, and books were sent, in the form of papyrus rolls, to they could hold the equivalent of many of the old other mathematicians to be read, copied, and ap rolls. The parchment codices were exceedingly preciated. The third century B.C., when Archimedes costly, both because of the materials (good parch lived, was the heyday of the Greek mathematical ment was always expensive) and because of the cal sciences. Mathematicians were scattered about the ligraphy, to say nothing of special skills such as Greek-speaking world, but there was a particular copying geometrical diagrams. Nevertheless, at concentration of them, as of other intellectuals, least three codex collections of Archimedes' works in Alexandria. Archimedes sent his books from were made, each containing a different selection. Syracuse to three Alexandrian mathematicians: Yet it seems as if no one in the Byzantine Em Eratosthenes, remembered for his measurement of pire read them. Archimedes' name continues to the Earth and his scientifically based world map; crop up in Byzantine literature, but he is the Corron, whose identification of a new constella Archimedes of the old anecdotes, not the mathe tion in honor of Berenice, the queen of Egypt, was matical writer. No further copies of the Archimedes immortalized by the poets Callimachus and Cat codices were made. In fact, by about A.D. 1300 all ullus; and one Dositheus. three codices were in situations where Byzantine There was a perverse, teasing streak in Archimedes' scholars could no longer read them. Two of them relations with his Alexandrian correspondents. At a had somehow made their way to western Europe, time when, reflecting the cosmopolitan nature of where Greek learning was still rather scarce. Per Greek culture after the conquests of Alexander the haps they were part of a royal gift, like the manu Great, writers had embraced a standard form of Greek script of Ptolemy's Almagest that the Byzantine in place of the many local dialects, Archimedes per emperor Manuel Comnenus gave to the Norman sisted in writing his mathematics in the provincial king William I of Sicily about 1160. Whatever the Doric dialect of his native city. In the letters that he story, by 1300 these manuscripts had become part prefaced to his books, he speaks in less-than of the small collection of Greek manuscripts in flattering terms of the mathematicians to whom he the papal library (one of them was in pretty bad con is sending them, and on one occasion he sent them a dition). After the papacy was moved to Avignon in list of theorems without proofs, including two false 1309, the papal manuscripts seem to have been ones that were laid as a trap to catch anyone claim dispersed, and only one of the Archimedes manu ing priority of discovery. In spite of his best efforts scripts eventually resurfaced in the fifteenth to get their backs up, Archimedes' contemporaries century. Greek humanism was now in full flower evidently thought well of his work, and later people in Italy, and good copies were made of this survivor not only preserved them but wrote commentaries on before it again vanished, this time for good. Ap some of them, in some instances also stripping the parently the other manuscript had never been texts of their dialect for easier reading. We can still copied, though a painstaking Latin translation of read the commentaries of a very late Alexandrian the contents of both manuscripts had been made philosopher-mathematician, Eutocius, who lived at by the Dominican scholar William of Moerbeke in the time of the emperor Justinian, in the sixth cen 1269, and this Latin version has survived. tury of our era. The third codex met a different fate: around It was about Eutocius' time that Constantinople 1300 it was dismantled, its parchment leaves were began supplanting Alexandria as the focus of Greek cut in two, to some extent cleaned, and rewritten learning-the Islamic conquest of Egypt was just with a Greek prayer book. In this new guise, as a century away-and the selection of ancient Greek a palimpsest (recycled manuscript) it passed literature that was to survive through the Middle through one or more monastic libraries. By the Ages was to a large extent determined by which end of the nineteenth century it was in the library books were brought to Constantinople. There was of a monastery in Constantinople, and a scholar a very high risk of loss, especially in the seventh writing a catalogue of Greek manuscripts in that and eighth centuries, when classical learning, es city noticed and copied a bit of the partially pecially in the sciences, was at a low ebb, but some erased text. The Danish classicist Johan Ludvig how a dozen or so papyrus rolls of works of Heiberg, who had recently published an edition Archimedes were still around and in copyable of Archimedes' works based on the manuscripts condition by the time intellectual conditions then known, recognized that the lines copied from were improving, about A.D. 800. (A few works of the palimpsest were part of Archimedes' On the Archimedes, and apparently some others falsely at Sphere and the Cylinder, and he made haste to tributed to him, were translated into Arabic about get access to the manuscript. Heiberg succeeded this time.) During the next two hundred years or in transcribing a substantial part of the faded so, a large number of manuscripts of ancient philo Archimedean texts written crossways underneath sophical, scientific, and mathematical works were the prayers; they turned out to include two works produced. These were codices, manuscripts of that were otherwise entirely unknown and a third MAY 2005 NOTICES OF THE AMS 521 Uncovering New Views on Archimedes In 1996 when as a postdoc Reviel Netz launched his project of translating the works of Archimedes, his historian colleagues were not encouraging. "They said that obviously it would be a misguided project," he recalled. "In scholarly terms it would be incomplete because I would not have access to the palimpsest." That Netz forged ahead anyway proved to be fortuitous. When the long-lost Archimedes palimpsest resurfaced in 1998, his work on translating the first volume, On the Sphere and Cylinder, was ideal preparation for working on the palimpsest. "I was incredibly lucky," he says. The program of preserving, imaging, and transcribing the Archimedes palimpsest is under way at the Walters Art Museum in Baltimore. Netz, now at Stanford University, and classics scholar Nigel Wilson of Oxford University are leading an international team analyzing the content of the palimpsest. The museum's work on Sphere and Cylinder has been completed. The imaging of The Method is also finished, and the Stanford Linear Accelerator is now helping to uncover additional information. Transcription of The Method is about one-third fin ished. The palimpsest is the only extant source for The Method, and Netz lost no time in setting about studying it. He has formed some intriguing new in sights. For example, in The Method Archimedes constructs in the course of a proof a one-to-one correspondence between two infinite (in fact, un countable) sets. "This was completely unknown," Netz states. "The as sumption was that Greek mathematicians never made this kind of claim" about two infinite sets being the same size. Another work unique to the palimpsest is The Stomachion. This mere half Reviel Netz page fragment has puzzled scholars. "There wasn't an interpretation of it," Netz says. "No one really ventured to say what it was." He has now conjec tured that the purpose of The Stomachion was to treat a combinatorics problem, a surprising idea, for it was not thought that combinatorics existed in ancient mathematics. In fact Netz is changing many of the standard views on Archimedes and his work. "The original pic ture historians had of Archimedes is as a practical engineer," Netz remarks. "I don't find any evidence for this ....He was strictly a mathematician." Even in Archimedes' works on physics Netz sees mathe matics as the ultimate goal. For example, Netz believes that Archimedes invented statics as a way of deriving results in geometry. The strategy was to use imaginary situations involving bodies in equi librium to derive proportions that lead to measurements of geometric figures. Netz sees in Archimedes' work a personality that is "very playful, cunningly and even maliciously playful." Hellenistic mathematicians produced works that juxtaposed different things in surprising ways and that set challenges and puzzles for readers. Netz says this style of presentation has paral lels in the larger cultural tendencies of Alexandrian and Hellenistic society. His next planned book, Lu dic Proof: Greek Mathematics and the Alexandrian Aesthetic, will examine the playful strands running through Hellenistic mathematics. "Archimedes, among the truly great, is relatively neglected," Netz comments. "There is a Newton in dustry and an Einstein industry, but there isn't an Archimedes industry, and there ought to be one." He believes one barrier to the study of Archimedes has been the lack of a complete and faithful translation of his works into English. Netz's translation is so faithful, he says, that one could use it for serious his torical studies. It is not an easy read, but then it is not an easy thing to plunge into the mind of a writer from a completely different culture and time. The translation is "very Greek-it's a Greek book," Netz says. He has not transcribed the mathematics into modern notation, preferring instead to let the reader puzzle through the mathematics just as Archimedes' contemporaries did. Says Netz, "If you are really interested in Archimedes, invest the extra effort to see what he did." -Allyn jackson 522 NOTICES OF THE AMS VOLUME 52, NUMBER 5 that had hitherto only been available in Moerbeke's As well as the intrinsic interest of Archimedes' Latin version. After a rather obscure history in the mathematics in On the Sphere and the Cylinder, twentieth century, the Archimedes palimpsest is there are two "loose ends" that provoked Eutocius now in private hands and the subject of an ambi to collect for us some remarkable specimens of tious project of conservation and research intended Greek geometry that we would not otherwise know to recover text that Heiberg was unable to read. In about. These are the solutions by several geome the meantime, Heiberg's second edition, which is ters of the problem of finding two mean propor very good, remains after a century the basis for tionals between given magnitudes (i.e., given A and studying what Archimedes wrote. D, to find Band C such that A: B = B: C = C: D) Astonishingly, given Archimedes' fame and the and another problem mathematically equivalent to importance of the works, Archimedes' books have solving a cubic equation. Both problems fall into never really been translated into English. Thomas the class that Greek mathematicians sensed from Heath, who made an excellent translation of Euclid's experience, though they could not prove, to be Elements, published The Works of Archimedes in insoluble using only the postulates of Euclid's 1897 (before Heiberg's rediscovery of the palimpsest), Elements, and the solutions that Eutocius preserves but except for the prefatory letters this was not a illustrate how they extended their "toolbox" by translation so much as a mathematical paraphrase allowing certain mechanical constructions, inter that not only uses modern notation but liberally sections of conic sections, or special curves. reorganizes parts of the proofs. E.]. Dijksterhuis's In due course Netz will follow this volume with two Archimedes, first published in 1956, contains math more. Of the books awaiting translation, some are, ematical paraphrases of the proofs that are closer to like On the Sphere and the Cylinder, works of pure the reasoning of the original than Heath's, and his mathematics, while others give a mathematical treat book, which has been reprinted by Princeton Uni ment of problems in statics such as centers of grav versity Press, is still the best general introduction ity of figures and conditions of stability of floating to Archimedes' thought. But neither Heath nor solids. Netz does not intend to include in his scope Dijksterhuis gives a complete restatement of all the medieval Arabic tradition of Archimedes' works. the proofs, and the "feel" of Archimedes' writing is In fact, it is likely that none of the works that pass un entirely lost. der Archimedes' name in Arabic, aside from those Now Reviel Netz offers us the first installment that we also have in Greek, are authentic, although of a translation of all the works of Archimedes several contain interesting mathematics. An edition that survive in Greek. The first volume contains a and translation collecting these would be a worth single large work, On the Sphere and the Cylinder, while project. together with the commentary on that work by Netz's English Archimedes could hardly be more Eutocius. On the Sphere and the Cylinder is the book different from Heath's. To begin with, it is ruthlessly in which Archimedes proved several relations literal. Archimedes, like all Greek geometers, wrote between the volumes and surface areas of spheres, his mathematics in continuous prose, using words segments of spheres, and cylinders, including of to represent concepts and relations, and letters of course the proof that the volume of a sphere is two the Greek alphabet to name them. The vocabulary thirds the volume of the smallest containing cylin and sentence structures of Greek geometrical writ der; a diagram illustrating this relation is reputed ing were highly standardized and formulaic by the to have been inscribed on Archimedes' tomb. The third century B.C., a quality that gives the arguments central proofs employ what are traditionally called something of the same clarity and freedom from "exhaustion methods", which are rigorous limit ambiguity that notation provides in modern math arguments founded on inscribed or circumscribed ematics. But a mathematical argument written out solids (such as the solids of rotation of polygons). as prose (sometimes referred to as "rhetorical" Because exhaustion arguments are not easy to mathematics) has two characteristics in contrast to transfer from one geometrical figure to another, notation: the relation to the spoken word is much the proofs have a certain virtuosic character, very more immedate, so that one can read the argument different from the more readily generalizable aloud correctly even if one does not understand the approaches of European mathematicians to such mathematics at all, but it takes much more space problems leading up to the calculus. In another on the page to write- and more time for the eye book, The Method (to appear in a subsequent vol to take in- each step of a proof. Mathematicians ume), Archimedes presented a more heuristic and are often more comfortable with translations of less rigorous strategy for demonstrating volume early mathematics that employ notations to com and area relations by comparison of infinitesimal press the argument, whereas nonmathematicians slices of figures. This work, however, was unknown interested in the history of science may find the to modern mathematicians until Heiberg discovered rhetorical style more approachable, though in it in the palimpsest. the case of Archimedes they soon find that the MAY 2005 NOTICES OF THE AMS 523 The first of three views (see next page) of a page f rom t he Archimedes Palimpsest (fols. 93v-92r). The photograph at right is taken in regular light. Photographs taken by t he Rochester Institute of Technology and Johns opkins University. Copyright: The owner of the Archimedes Palimpsest. easy flow of words expresses some very difficult has not provided such a step-by-step synopsis to mathematics. accompany the translation. Netz does not hold that it is the translator's For Archimedes' text Netz has relied on Heiberg's business to cater to the comfort of the reader; he edition. (In subsequent volumes he intends to take writes (p. 3) that "the purpose of a scholarly trans account of new readings arising from study of the lation as I understand it is to remove all barriers palimpsest.) The figures are a different matter. having to do with the foreign language itself, leav Heiberg, like most scholars editing works of Greek ing all other barriers intact." And whatever one mathematics, preferred to redraw the figures that may say of Netz's Archimedes, it cannot in any way accompany each theorem according to the sense be charged with looseness. He maintains a close of the text rather than reproduce with necessary and consistent correspondence between the Greek corrections the drawings that appear in the man uscripts. In many cases the res terminology and his English equivalents. The for ulting reconstructed figures look quite different from the transmitted mulaic character of Greek geometrical prose makes ones, though they are usually mathematically equiv it often possible to omit certain common words alent. Netz has made a close study of the role of without ambiguity, and the geometers took full figures in Greek mathematics, and in his earlier advantage of this means of shortening their sen book, The Shaping of Deduction in Greek Mathe tences: for example, the conventional Greek way of matics (1999), he showed how the figures were saying "the angle contained by AB and BC" would intended as an integral and indispensible part of translate word-for-word as "the contained by ABC." the proofs, not mere adjuncts to aid visualization. Netz carefully marks all the implied words by So here Netz has gone back to the manuscripts and enclosing them in angle brackets, as in this case "the produced the first "edition" of the figures, report 524 NOTICES OF THE AMS VOLUME 52, NUMBER 5 The photograph at left is taken in ultraviolet light, showing some of the underlying text of Archimedes' Sphere and Cylinder. The photograph below is a proces sed image, making the Archimedes undertext and drawing appear in red. commentary is that there may be places where he has not detected that the text as transmitted by the manuscripts and edited by Heiberg is incorrect. I have noticed one place where this seems to be the case: in the fourth theorem of Book 2 (p. 203, step 24) a statement about ra tios in Heiberg's text is both false and different from what Eutocius seems to have read in his copy of the same passage. In general the particular interest to people comparing the trans translation has been made consistently and with lation to Heiberg's edition of the Greek), the other care, and I have noticed very few misprints, except offering more general remarks. The general com in the bibliography, where the.typesetters seem to ments are not, for the most part, mathematical in have been uncharacteristically creative (and, oddly, scope, but discuss linguistic and stylistic aspects Lewis Carroll appears disguised as Carol wherever of the text. For mathematical commentary, Netz he turns up). The typography, layout, and drafts refers us to Eutocius and Dijksterhuis, who are manship of the diagrams are of a standard that one certainly worthy and sensible guides, though Dijk can unfortunately no longer take for granted in sterhuis is difficult to use in close comparison scholarly books. A regrettable corollary is the high with Netz's translation, because the lettering of price; individuals planning to own all three volumes the diagrams is different, and Eutocius is writing will need deep pockets. for a reader more conversant in the idioms of an cient mathematics than most modern readers are likely to be. A possible consequence of Netz's de cision not to provide his own mathematical MAY 2005 NOTICES OF THE AMS 525 Mathematics in the Near East: Some Personal Observations David Mumford In October 2003 I visited Turkey, and in June 2004 most of my life), I felt a strong urge to get a first I made a circuit through Lebanon, Israel, and the hand impression of what the Near East is like and Palestinian West Bank. There is a lot of mathe how people are living. As a mathematician, I wanted matics going on in the Near East, 1 some in places to meet and talk with mathematicians and see fairly well known to the West and others in places what conditions were like for them. ill particular, that are less well known. I would like to share with during these visits I promised to help some of my colleagues here in the U.S. something about the those I met by publicizing a few of their needs and conditions in the places I visited as well as some initiatives. We in the West can help and thereby of the things the mathematicians in these univer further the development of mathematics in this sities would like from their colleagues in the part of the world. West. I had been in Israel twice before, in Septem As these were my first visits to largely Muslim · ber 1967 and in June 1995, but had never before countries, I was curious to see if there was a dif visited any of the largely Muslim countries there. ferent "feel" to Islam. In fact, what I found was that As the newspapers have been filled with articles the campuses were extremely lively and full of en all on the wars there (not merely now, but in fact for ergy and not that different from large state uni versities in the U.S. For instance, on all the campuses I visited half or so of the women did not cover their heads at all, and I saw no one in a burka. In the coun try the women were more conservative, and al most everywhere-in cities and small towns-you hear the call to prayer at dawn, something I loved, as I like to wake up early. But only on Friday at noon did there seem to be large crowds going to the mosques. I loved visiting the mosques: their aus tere beauty is very peaceful and moving (though sadly I have been denied entry to the al-Aqsa mosque on each visit to Jerusalem by one or the other party). Turkey, the Middle East Technical University (METU), and Bogazi~i University Though I had heard of Mustafa Kemal Ataturk, it was not until my trip to Turkey that I appreciated David Mumford is professor of applied mathematics at Brown University. His email address is davi d_mumford@ brown .edu. 1 Following what I believe is the most widespread usage, The METU mathematics department. Safak AI pay is in the the term "Near East" is used here to refer to the land in front row standing, with the white mustache. Ersan Akyildiz is the convex hull of Turkey, Iran, and Egypt. Th e "Middle to his right in the blue shirt. East" is somewhat more restricted. 526 NOTICES OF THE AMS VOLUME 52, NUMBER 5 The Bogazi<;:i Center for Mathematical Sciences and its view over the Bosporus. the amazing transformation he brought about in impossible at first to even remember names. As in his country, transforming the disintegrating Ot almost all places outside the U.S.-European loop, toman Empire into a modern country now on the one of their greatest needs is to have more visitors verge of joining the EU. I had not realized that he who can help to keep them in touch with the de was a champion of women's rights and universal velopments in the West. Their website is http: I I education and embraced an integration of Western www. math. metu. tr. I was astonished to learn that ideas with Islam. The universities reflect his phi Bob Langlands is fluent in Turkish and has visited losophy. several times. METU is in Ankara and was founded in 1956 to Going there for a sabbatical or as a visitor for a train Turks and, more widely, people from all over year is very possible, and I believe many mathe the Middle East in the sciences and engineering. The language of instruction is English, and today it has maticians would find it fascinating (though salaries over 20,000 students, of whom about 1,000 come from outside Turkey. It is a huge and bustling cam pus, with much green still-undeveloped space. A strong mathematics program was started at the outset through the guidance of the distinguished Turkish mathematician Cahit Arf and through bringing Masatoshi Ikeda to METU. They created an environment in which research was held to an in ternational standard. They have a large and active Ph.D. program-sixty-six students are currently working for Ph.D.'s-and have forty faculty mem bers covering all areas of mathematics. Many other students from there go to the West to complete their Ph.D.'s; we have had a stream of strong METU students coming to Brown. Their quandary is that they want to maintain their successful local Ph.D. program, yet if their students go to the West, they are exposed to more ideas, because the mathe matical community is so much larger. I had a very warm welcome from Safak Alpay, a functional analyst and the chair; Ayse Berkman, a group theorist and the vice chair; and Ersan Aky ildiz, an algebraic geometer. My lectures were about vision and applied topics, and I met some excellent people from the Computer Engineering Depart ment as well, in particular Sibel Tari, who has started up a strong modern program in vision (hard because the area often falls between departmen tal lines). Ayse is married to a colleague and model theorist, pavid Pierce, who proved to me that it is possible to learn Turkish, though it seemed AUB guides (left) with the author;s son Jeremy. MAY 2005 NOTICES OF THE AMS 527 in Turkey are a prob organized by Rahmi Guven was held at Bogazi<;:i. lem, as I discuss Because of its location exactly on the boundary below). Ankara has between Europe and Asia (traditionally drawn along one of the best arche the Bosporus) and, more importantly, because of ological museums in the political neutrality of Turkey, to which every the world, a reflec one can get a visa, it seems an ideal location for tion of the fact that bringing together the Western and Near Eastern so many civilizations mathematical communities. have flourished there I should mention one problem which seems en in the last ten mil demic to academic institutions in many countries lennia, and it is close but is especially bad today in Turkey: salaries. The to Cappadocia, amaz situation in Turkey, which was brought up by col ing for its physical leagues both in Ankara and Istanbul, is that the pub beauty and for its un lic universities, such as METU and Bogazi<;:i, have derground churches traditionally been the best in the country, but they and whole subter pay very low salaries. Today many private univer ranean villages. All of sities have been started, especially in science and Turkey, in fact, is engineering, that are tempting professors away by covered with inter offering more reasonable salaries. Thus profes esting places to visit, sors are split between those maintaining the high The "Wall" with (left to right) lyad although Ankara it standards of the older public universities and those Suwan, Suwan's father, and the author. self is not very beau who feel the future is with the new private uni tiful. What Ankara versities. I had a good talk about this with a Brown lacks in charm, Is Ph.D. in computer vision, Aytul Ercil, who has tanbul makes up for in spades. Every era has left shifted to Sabanci University, where she can offer incredible structures, from the early Roman cistern a strong program in vision. There is very little a for through the Topkapi palace and the immense "cov eigner can do to help this situation, except to be ered market" of the Ottoman era. aware and sympathetic, unless you are on an ex The primary center of mathematics in Istanbul ternal advisory committee or the like. is Bogazi<;:i University (pronounced 'bowazichi', which means 'Bosporus' in Turkish). Bogazi<;:i was Lebanon and the American University of formerly Robert College, founded in 1869 by two Beirut (AUB) Americans, Christopher Robert and Cyrus Ham Lebanon is arguably the most complex country" in lin, with a decree from the Ottoman sultan. It sits the Middle East from a sociopolitical point of view. on a spectacular site overlooking the Bosporus and It has deep divisions going back over a millennium flourished for a long time with American philan between Christians, Muslims, and, until recently, thropy and trustees, teaching an American-style ed Jews. But it is much more complex: there are ucation in English to Turks. In 1971 the college Maronite Christians, other Christians, Druze, was taken over by the Turkish government and re Muslims, Palestinian refugees, and many other named Bogazi<;:i University, though the medium subgroups of the population that have alternately of instruction remains English. It has about 10,000 fought and made alliances with each other and students and is wonderful and beautiful campus a with Israelis. A Ph.D. in Lebanese history is re just to the north of the city. Like METU, they have quired to read the newspaper intelligently! The vigorous master's and Ph.D. programs and offer a economy that sustains the country is even murkier. full range of advanced mathematics courses. How While my son, Jeremy, and I were there, there was ever, also like METU, they face a dilemma about an incident in which some people in a mob were whether to send their students abroad for their shot by the police. We heard a dozen theories about Ph.D. or not. Their website is http: I jwww. math. which group had instigated the mob, the boun. edu. tr. con frontation, the police response, and why. On the I was welcomed by the chair, Betul Tanbay, a very positive side, we were taken on a fascinating tour dynamic woman working in operator algebras. She through some of the remaining old districts of has created a new international mathematical in Beirut by AUB professor Jihad Touma. Here many stitute, called the Bogazi<;:i Center for Mathemati of these groups were living in close proximity, and cal Sciences (BMBM in Turkish), which is expected Jihad strongly recommended returning at 4:00 a.m. to open in 2005 and already has a new building on to sample the bread fresh from the ovens of an the campus of Bogazi<;:i University, where work ancient bakery (something he assured us would shops and semester-long programs are planned be quite safe to do). in the style of the other major mathematics In the midst of all this confusion, the American institutes. Last summer a workshop on M-theory University in Beirut plays a remarkable role: like 528 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Bogazi<;:i University, it was founded by an Ameri can, Daniel Bliss, in 1866 and has been a neutral force for higher education and medical training. To this day it is governed by an independent board of trustees, now fully international, which gives it insulation from the byzantine politics of Lebanon. During the ghastly twenty-year civil war, it re mained an island of peace, receiving only one hit from the bombardment while the center of Beirut was leveled, as it was considered an indispensable part of the city by all communities. AUB has about 7,000 students and offers many master's programs but no Ph.D.'s. The student body is very mixed: Christian and Muslim, Lebanese and foreign (from sixty-seven countries!). It sits on the corniche of Beirut overlooking the Mediter ranean, about half an hour's walk west of Beirut's center. For the visiting mathematician, the most ex Birzeit University, Jeremy Mumford on left, lyad Suwan on the citing part of AUB is their new Center for Advanced right. Mathematical Sciences (CAMS), which was started with the help of Sir Michael Atiyah and Nicola Khuri (a physicist at Rockefeller University). Though still small, they are hosting a large number of work shops and summer schools under the leadership of the director, Wafic Sabra. This is a great place to visit, and I strongly recommend it. Their web site is http: I jwww- lb. cams. aub. edu. lb. There is only one hitch! You cannot enter Lebanon if there are any Israeli stamps on your passport or if your ticket shows you have been or are going to Israel. Lebanon and Israel are still technically at war, though Israel has withdrawn from all but contested parts of Lebanon and there are only the usual skirmishes. To make our trip, Jeremy and I bought a round-trip ticket to Amman, stopping in Beirut, and made separate arrange ments to get from Amman to Israel and back. This makes it impossible for Israelis to come to Beirut and is a major obstacle to CAMS's desire to be The Weizmann Institute. open to everyone. A peace treaty between Lebanon and Israel is conceivable sometime in the future, took us six hours, mainly because it took three but right now no one is hoding their breath. hours for the Israeli soldiers at the border to allow us to enter. We met Iyad in East jerusalem and went The West Bank and Birzeit University to dinner that night at his house, which is fifty feet My son and I were very fortunate in having a guide from "the Wall" (see photo). It has to be seen to be to help us travel in the West Bank, without whom believed. Iyad's grandfather built the house in the this leg of our trip would certainly not have been 1940s, and the extended family has lived there possible. Iyad Suwan is a graduate student work ever since, paying taxes to whoever runs the city ing with Achi Brandt at the Weizmann Institute. He of Jerusalem. But now Iyad is moving because his is Palestinian but is fortunate in that his family has children are 45 minutes and a checkpoint away lived in East Jerusalem for three generations, so his from the hospital, which is less than a mile away Israeli identity card, identifying him as a Palestin across the Wall. ian from Jerusalem, allows him to enter Israel and Going to Birzeit the next day, the Wall was mas work at the Weizmann. sively visible from many points. Nor did there seem We traveled to the West Bank by flying to Amman, to be any spots in the West Bank where there were taking a taxi to the Allenby (or King Hussein) Bridge, not Israeli settlements as well as Palestinian towns crossing the border there, and taking a shared in view. My impression was of a fractal-like inter minivan to East Jerusalem, where Iyad met us. It is penetration which one cannot imagine undoing. 44 miles from Amman to Jerusalem, but the trip There were stark contrasts: the huge settlement MAY 2005 NOTICES OF THE AMS 529 Ma'aleh Adumim, as posh as a First World resort, his Ph.D. there with Achi Brandt, and is now a post in the middle of the rocky, uncultivatible desert that doc with me at Brown. Achi is also the advisor of constitutes much of the West Bank. We took a "ser Iyad, and at dinner in his house we learned a lot vice taxi", the ubiquitous minibus, to Birzeit to about the history of Israel. The Weizmann Institute avoid bringing a car through the various check is a combination of a research laboratory and a points. Besides the delays of waiting in line, the small and select graduate school (700 students problem of living with the checkpoints seems to be and a comparable number of researchers) covering that the rules are never clear, rather like Kafka's most of the sciences. The closest analog in the story The Castle. And even if the rules are clear, U.S. might be Rockefeller University, if it covered some of them are enforced only rarely but are al more fields, or Caltech. It was started in 1934 ways a threat (for instance, the rule prohibiting West as a chemistry research institute under Chaim Bank Palestinians from working in Israel, which, Weizmann. It has a gorgeous campus where the when enforced, financially ruins a worker). This was faculty live in a communal atmosphere, almost like quite a surprise to us when we encountered it at a kibbutz. Scientifically, it is far and away the most the Allenby Bridge, and Iyad counseled us later to exciting place I visited. I am good friends with and be patient. admire tremendously the work of people in several In the minibus I sat next to a young Arab woman departments there. in head scarf studying reinforced concrete. She Most of us in the U.S. know quite a few people was deep in her math but told me they make their in the seven Israeli universities and do not need my own concrete in the West Bank, though they must fairly random comments. The stimulus to my trip import some key components for construction. was an invitation to the University of Haifa, where Birzeit amazed me: the students looked as well I was hosted by Larry Manevitz. Overlooking Haifa dressed and happy as their counterparts in the from a hilltop, the university is moving ahead U.S. Though there was a meeting at noon to discuss strongly by virtue of a major grant from the Roth Palestinian students jailed by Israel, the campus did schild Foundation, which gives it funds for a diverse not feel as angry as I had expected. But clearly the array of new programs. I then visited my old friend university operates under great stress. In the last Mina Teicher, now vice president for research at few years a new checkpoint has effectively shut Bar-Ilan University. Bar Ilan is a religious institu them down intermittently. I have described the tion, but, like the historically Christian AUB and like above not to make a political point-! am aware of Birzeit, the scientists share a nonsectarian culture the Israeli reasons for both the Wall and the check that keeps religious politics at arm's length. They points- but to explain the conditions under which have just opened the interdisciplinary Gonda Brain the Palestinians work and do math. Research Center, bringing in Moshe Abeles, the Birzeit has been operating as a university since best brain researcher in Israel (in my view), as 19 74 and today has about 5,000 students, all from director. Having been part of a group that sought the West Bank and Gaza, mixed Muslim and Chris to create something very much like this at Brown tians. They offer a master's program in some areas but with only partial success due to lack of funds, but not in mathematics. Their greatest need is to I can appreciate how much Mina and Moshe have have more contacts with universities in the West accomplished at Bar Ilan. Economically, the Israeli to which they can send their best students. Walk universities and their science programs seem as well ing with Alaeddin Elayyan, the chair of the math endowed, if not better, as comparable universities ematics department, we happened to meet two and programs in the U.S. students working on Cauchy's theorem; their best For me, three things stand out after these trips. students seemed well prepared to make the jump One is the desirability of U.S. mathematicians to a regular graduate program. Birzeit has had sup engaging productively with all parts of this fasci port and collaboration with the EU and especially nating and troubled region through the universal with France, where they have an interdisciplinary language of mathematics. The second is the hope mathematics and economics program jointly with that the mathematical community in the Near East, the University of Paris and hope to start a master's on both sides of the religious divide, can bypass program jointly with the University of Tours. Their the political stalemate and find ways to enrich website is http: I /home. bi rzei t. edu/math/. their mathematical collaboration. The third is the great potential for fruitful interchange between Israel, the Weizmann Institute, Haifa the isolated but scientifically motivated Turkish and University, and Bar llan University Arab universities and American mathematicians. After our visit to Birzeit, Iyad drove us to the Weiz mann Institute, which is a truly unique and extra ordinary place. Our main hosts at the Weizmann were Ronen Basri, from computer science, and Eitan Sharon, who was born there, went on to get 530 NOTICES OF THE AMS VOLUME 52, NUMBER 5 CAMBRIDGE NEW AND NOTEWORTHY FROM CAMBRIDGE New from the Quantum Theory as an Mathematical Sciences Research Institute Publications Emergent Phenomenon The Statistical Dynamics of Global Unitary Invariant Matrix Models as the Precursor Perspectives in Riemann-Finsler Geometry of Quantum Field Theory D. Bao, R.L. Bryant, S. Chern, and Z. 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Welzl theory is not a complete, final theory, Thi s volume contains 32 articles explorin g geometric arrangements, polytopes, but an emergent phenomenon arising packing, covering, discrete convexity, geometric algorithms and their complexity, and from a more profound leve l of dynamics. the combinatorial complexity of geometric objects, particularly in low dimension. $50.00: Hardback: 0-521 -83194-6: 238pp $75.00*: Hardback: 0-521-84862-8: c.528pp Introduction to Partial Inside Out Differential Equations Inverse Problems and Applications Yehuda Pinchover and Jacob Rubinstein Gunther Uhlmann A complete introduction to partial differ "It's safe to say that Uhlmann's book is a fingerpost in mathematical imaging for ential equations, this textbook provides a some time to come." rigorous yet accessible guide to students -Bulletin of the American Mathematical Society in mathematics, physics and engineering. $80.00: Hardback: 0-521 -82469-9: 412pp $120.00: Hardback: 0-521-84886-5: 370pp $45.00: Paperback: 0-521-61323-X Lectures in Contemporary Commutative Algebra L.L. 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This vo lume covers the classical theory of finding roots of a univariate polynomial, Hydrodynamic Stability emphasizing computational aspects, especially the representation and manipulation of algebraic numbers, enlarged by more recent representations like the Duval Model P.G. Drazin and W.H. Reid and the Thom Codification. "The work is undeniably of high $.100.00: Hardback: 0-521-81154-6: 438pp scholarship, consummate accuracy and penetrating insight. .. All specialists in stability theory will be happy that two Solving Polynomial Equation Systems II such authorities have found the time, Macaulay's Paradigm and Grabner Technology and spared so few pains, to produce a Teo Mora work of such excellence." A complete survey of Grabner bases and their applications, this volume focuses on -Nature Buchberger theory and its application to the algorithmic view of commutative algebra. $45.00: Paperback: 0-521-52541-1: 626pp $130.00*: Hardback: 0-521-81156-2: 785pp CAMBRIDGE 800-872-7 423 • www.cambridge.org UNIVERSITY PRESS *Prices subject to change. The Importance of MathMLto Mathematics Communication Robert Miner ompared to plain text, mathematical MathML is an XML-based encoding for mathe notation is hard to represent, edit, dis matics. XML (short for eXtensible Markup Lan play, and process with computers. On guage) has emerged as the dominant data format C underlying the information infrastructure of the the surface the formalism and rigor of mathematics seem as if they should lend World Wide Web. XML defines a method of repre themselves to computing. However, advances in senting structured data types, essentially the now user interfaces and electronic communication have familiar pointy-bracket tagging of HTML. XML itself led us to expect that information should flow merely defines the common syntax and leaves it to searnlessly from application to application, intel specific areas of application to define appropriate ligently doing the right thing for the given context. data types. MathML is one such data type. XML is Double-click on a picture and the expectation is that significant because it is becoming deeply embed it will open in a photo editor or viewer. Drag that ded in the software systems and workflows that will same image into your word processor and it should shape the information landscape for years to come. Because of MathML, mathematics is a full-fledged be imported automatically into the document. For part of that information landscape, and this bodes mathematics notation, these expectations are well for the scientific community. particularly demanding, since math is used in a In this article we will look closer at MathML and wide variety of applications with very different examine some of the implications it has for scien requirements. Not only should double-clicking tific communication. We will describe some of the on an equation open it in an equation editor, but ways it is currently being put to use in publishing, dragging it onto a computer algebra system or e-learning, searching, knowledge management, and graphing program should be supported as well. accessibility for the visually impaired and learning Similarly, math should "work" with screen readers disabled. We will also describe areas of active re for the visually impaired, page composition search and suggest some of the possibilities that engines in publishing workflows, search engines, may lie ahead. online testing systems, and more. Devising a world wide software infrastructure that facilitates the The Mathematical Software Landscape use of mathematics in all these contexts is a chal The challenges of dealing with math notation have lenging problem with a long history. Recent years, long had a polarizing effect on mathematical soft however, have witnessed a significant step for ware development. On one hand, because sup ward in the form of a standardized encoding for porting math is difficult and expensive, generic mathematical notation called Mathematical Markup applications such as word processors, databases Language, or MathML. and web browsers have not usually supported math directly on economic grounds. Instead, math func Robert Miner is director of New Product Development tionality is delegated to third-party math add-ons. at Design Science, Inc. His email address is robertm@ On the other hand, within particular communities des sci . com. with a strong need for math, software packages 532 NOTICES OF THE AMS VOLUME 52, NUMBER 5 such as Tp{, Maple, and Mathematica have arisen, The technological key to information exchange in with integrated, specialized support for math. the context of the Web has been standards. Standard The appeal of specialized software is that it is protocols and formats for exchanging structured less constrained by the requirements of a diverse data, such as HTTP, HTML, and XML are fundamen user group and can therefore focus on the math pro tal to the software architecture of the Web. In the cessing, adding value through ease of authoring, case of math, MathML has become established as quality of output, computational power, and so the standard format formathinXML, and this stands on. But the downside of specialized software to benefit mathematical software development in a is that to the extent its functionality overlaps that number of ways. Just by existing, MathML draws of generic applications, it tends to get left behind, attention to the requirements of mathematics inXML struggling to keep up with desirable new features and Web contexts, which is significant. Having a as they evolve in mainstream generic applications. standard format also provides a clear direction for More importantly, specialized systems tend to be developing conversion and import and export difficult to interface with other software, a key capabilities in specialized math applications. A consideration in networked information environ standard also encourages better, more uniform ments. By contrast, adding math functionality to extension architectures in generic applications, generic applications via third-party add-ons allows which benefit add-on math components as well. math components to benefit from the functional Finally, and not least significantly, a standard ity of the highly engineered host application, and decreases the.risk of investment and increases the component architectures lend themselves well to potential for return, which stimulates software reuse in networked environments. However, the development. As a measure of the effectiveness of add-on component approach has pitfalls of its own, MathML in this regard, the World Wide Web Consor since extension mechanisms in generic applica tium (W3C) maintains a list of over fifty MathML-aware tions have tended to be inadequate for math and software packages, about half of which have appeared lack of standards has hindered interoperability. in the last year. As a result, prior to the rise of the Web, the math software landscape was balkanized, consist About MathML ing of a collection of specialized applications and To better understand how MathML is being used, math add-on components that couldn't easily share it is useful to know a bit about MathML itself. data or interoperate to any great degree. A number MathML is an encoding of the visual presentation of these software packages achieved success within and semantic content of mathematical expressions. particular communities, such as commercial pub It was developed under the auspices of the W3C, lishing, mathematical computation, and research the body that has responsibility for most Web mathematics. An obvious example is Tp(, which related standards such as HTML, XML, and many came closer to providing a standard electronic others. The initial impetus behind the Math Activ format for math than anything before it. To the ity at W3C was to provide a better way of display limited extent that math applications of this era ing equations in Web pages than as graphics. talked to each other, they probably exchanged Tp( However, MathML rapidly evolved into a general code. However, in nonnetworked desktop com means of representing and communicating math puting environments before the Internet and the ematical expressions in XML. Web became ubiquitous, there was more emphasis MathML was first published as a W3C Recom on self-contained end-to-end solutions for partic mendation in 1998, which means it is comparatively ular tasks and less demand for interoperability. old and mature for a Web technology. MathML The success of the Web, however, dramatically appeared within a few weeks of the initial release changes the value of information exchange. The of the XML specification itself and was the first process of taking information and moving it around major application of XML. Because of this it has had via email, publishing it in print and in electronic for a high profile within the XML community and has mat, and making it almost instantly accessible using been an important test case for many subsequent search engines and web browsers has tremendous Web technologies. In particular, math was a moti value and offers thought-provoking opportunities. vating example when the namespace mechanism By enabling this kind of information exchange, the for mixing XML markup languages in a single doc Web has had a profound impact on society in general ument was devised. and the way scholarship is conducted in particular. In order to support the diverse demands As a measure of its success, we now take it for granted placed on math in different contexts, MathML is an that by typing "soliton" and "Cameroon" into Google, information-rich encoding. This reflects a funda within seconds one can discover that there is a physi mental design decision to make low-level informa cist at the Universite Yaounde I who is interested in tion explicitly available in the markup, information the topic and whose latest paper can be read online that is usually inferred from context by human for a fee. readers. While this is usually advantageous and MAY 2005 NOTICES OF THE AMS 533 sometimes essential for machine processing, it conventions to tokens if it weren't for the low means MathML is not suitable for writing by hand. level token tagging in MathML. Operator tokens It is a text -based format, however, so it can be read are quite general in Presentation MathML. They by persistent humans. In this regard, it is somewhat include "grouping operators" such as parentheses analogous to other low-level structured formats as well as more traditional operators like "+". such as PostScript, for example. Operators can be "stretchy" in the case of notations One particularly notable feature of MathML that such as arrows, bars, and parentheses. distinguishes it from most other math encodings A second notable feature of presentation markup is that is contains two separate vocabularies, which is that it goes to some lengths to make the hierarchi can either be used alone or in conjunction. One cal markup structure reflect the underlying mathe vocabulary, termed presentation markup, describes matical structure. Thus, for example, the 534 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Science, Technical, and Medical (STM) ican Physical Society and American Institute of Publishing Physics have already begun using MathML directly As many commentators have noted, mathematics for production purposes. The oldest and probably research literature has a long lifespan. An article largest volume MathML-based workflow is that of can remain relevant for many decades. Conse the U.S. Patent Office, operated under contract by quently, making more effective use of the breadth Reed-Elsevier. That workflow handles thousands of and depth of the literature is an enticing possibil equations per week. Other projects are currently ity. By connecting ideas and thinkers through time gearing up, and the next year or two promise a and space, information technology has enormous substantial increase in MathML use in production potential for mathematics. Organizations such as workflows. MathSciNet, JSTOR, the arXiv, and many others Gaps in software support remain. Ironically, have begun to suggest the outlines of what is pos MathML support in web browsers continues to sible. A large amount of material has already been present challenges. The newer Netscape and made available in electronic format, and retro Mozilla-based browsers such as Firefox now have digitization projects continue to push the frontier built-in MathML support, and the free MathPlayer of electronically accessible documents back into the extension from Design Science adds native-quality past. Schemes for durable references to electronic MathML support to Internet Explorer for Windows. publications have been established by STM pub But the Safari browser for Macintosh does not lishers, and articles and bibliographic databases support MathML yet. Support for MathML in page have been cross-linked. To make this substantial layout programs has yet to be developed, and T£)1: achievement happen, STM publishers have devel conversion is another area requiring further work. oped cross-media publishing workflows, which The Hermes W£)1:-to-MathML translator, being involve creating and managing content in print, over developed as part of the MowGLI project, seems the Web, in databases, and so on. promising in this regard. Across the publishing industry as a whole, XML centric workflows have become the strategy of Math-Aware Searching choice to meet the challenges of cross-media One of the most interesting possibilities of MathML publishing. In the XML-centric model, articles are is the potential for enhancing searching of technical stored in XML format in a central repository and literature and educational material. By integrating formatted for various media by stylesheet-driven mathematics with the surrounding document in a composition engines. Fueled by public and private highly structured way, MathML opens the door to investment throughout the dot-com boom, there is mathematical keyword searching: type an equation widespread support for XML in current publishing into a search engine and get back a list of papers in software systems. But in order for STM publishers which it occurs. MathML could also play a role in to take advantage of these XML-centric publishing automated or semiautomated creation ofmetadata, systems, they must be able to deal with math, and where the content of a document is analyzed by a that's where MathML comes in. software agent to suggest keywords from subject In the last several years there has been a pro taxonomies or other metadata ontologies. In this liferation of MathML software targeting STM way, MathML may have a role in enhancing existing publishing, reflecting a significant shift toward search systems geared toward bibliographic meta XML workflows amongst major content providers. data. STM composition software, XML editors, and con In most cases, current searching of online STM version software have added MathML support. This content is limited to keyword searches on text. As a has been greatly facilitated by the fact that MathML result, a researcher typically needs to know appro is a completely integrated XML data type that can priate keywords in advance to search for the desired be accessed and manipulated through standard mathematical subject matter. This is limiting in APis in XML software systems. This enables math three ways. First, searching is frequently restricted add-ons to do a better job more easily and in an to abstracts and bibliographic metadata, and appro interoperable way. priate keywords may not appear there, even though At the same time, MathML has been incorporated the full document may contain the desired informa into a number of XML document types used in STM tion. This is especially likely for material in secondary publishing such as DocBook and the Journal Archiv topics, background information, and introductory ing and Interchange format used by the National material. This is the classic metadata problem of Library of Medicine and its PubMed system. As a insuring appropriate keywords are accessible and result, a number of publishers are now running pilot appear in close proximity to the resources that they projects using MathML, and a few have embarked describe. on plans to shift major workflows to an XML Secondly, keywords describing mathematical centric model using MathML. John Wiley & Sons is objects are typically overgeneral. One can search conducting pilot projects using MathML. The Amer- for "quadratic polynomial", but there is no effec- MAY 2005 NOTICES OF THE AMS 535 tive way to narrow the search to a particular poly could at least be replicated for many categories of nomial or class of polynomials. This is particularly content, if not in full generality. limiting for educational resources, where the same However, to scale up mathematical searching and generic label applies to many different treatments integrate it with text searching to any appreciable of the same material. Searching for "rate of work" degree, the ability to automatically identify and with Google produces some 20,000 references. normalize large classes of mathematical expressions Finding out which of these documents might shed is essential. Here MathML plays an important role in light on the particular rate-of-work problem in two ways. First, since it takes pains to insure markup your child's homework assignment is a laborious, structure generally reflects mathematical structure, and likely fruitless, task. it significantly simplifies the computational com Finally, it is commonplace in technical subjects plexity of recognizing and manipulating expressions. to be confronted with mathematical problems of This facilitates the creation of specialized, math a type beyond one's experience for which one does aware search engines. Second, by integrating math not know appropriate keywords at all. A variant of with text in a common XML-based format, MathML this problem arises when different fields of study makes math accessible to generic XML search tech have different terminology for identical mathe nologies. The XQuery standard currently under matical objects. In such instances, a problem may development, for example, is expected to make a in fact be .well understood, but the researcher substantial impact in this area. Because of their has no way to discover what keywords will find the obvious commercial potential, generic XML search answer. While the problem of differing nomencla technologies are attracting widespread investment ture also affects mathematical notation, at least in and support, so the potential benefit to STM search some cases the problem is more tractable. ing is highly leveraged. At the Future of Math Communication II work The potential benefit of math-aware searching shop [3], held at MSRl in 1999, Rob Corless of the has been recognized in many quarters. The National Ontario Research Centre for Computer Algebra Science Foundation awarded Design Science a grant related an incident that makes this point well. In through the National Science Digital Library (NSDL) program the course of working on a nonlinear initial value to investigate ways of enhancing math searching. The MowGLI project is that arose in conjunction with a dynamical system also investigat ing applications of MathML to math searching. A he was studying, he needed to understand the number of other projects, both commercial and behavior of a certain power series. He knew that academic, are under way. For example, a workshop Neil Sloane of AT&T Research had recently set up on the topic, funded by Design Science, was held a website where one could search the Handbook of at the Institute for Math and its Applications at Integer Sequences and Series [4], so he entered the the University of Minnesota in April 2004 [5]. coefficients of his power series. It turned out that Searching and related topics also featured promi the series was known, and the initial value prob nently at a conference on Mathematical Knowledge lem had been solved in generality by Gilbert Labelle Management, held in conjunction with the Joint in a paper in the European journal of Combinatorics. Mathematics Meetings in January 2004. Not being a combinatorialist, Corless thought it unlikely he would have found this information in E-learning any other way. In particular, Corless would not £-learning is another area where MathML has nat have found it using text-based keyword searches, ural applications. In many ways, MathML has its roots since he did not know in advance that the solution in online learning, as much of the original motivation had anything to do with combinatorics. for MathML was to provide a better means of incor Of course, a key point in this anecdote is that porating mathematics into Web pages for educa integer sequences and series have an obvious, tional purposes. However, in practice, MathML has unique, easy-to-type canonical form that make probably had a larger impact to date as a backend mathematical keyword searches particularly easy technology used to add math support in course in this very narrow area. However, it is important management systems, online assessment systems, to note that many other kinds of mathematical and the like. objects also have easily computed canonical forms. From the outset, the Web has had a strong Similarly, other techniques involving normalizing appeal as a medium where educational content can expressions and pattern matching can be quite be dynamic and interactive and where concepts effective for suitably restricted kinds of mathe can be presented in multiple educational modali matical content, especially when augmented by ties: text, images, sound, animation, and even metadata. Consequently, it is reasonable to suppose manipulatives and simulations. Much effort has that mathematical search technology can be ex been invested in the creation of such content, in tended without undue effort to the point where many cases with highly engaging and effective search success stories of the kind Corless describes results. Different projects have used a wide variety 536 NOTICES OF THE AMS VOLUME 52, NUMBER 5 of Web technologies, including some that make to target student skill deficits. In a similar vein, use of MathML. One of the larger projects making Integre Technical Publishing is investigating ways extensive use of MathML to create dynamic educa of using standard XML tools to take advantage of tional content is the Homework Help feature of the structured nature of MathML to analyze student Microsoft's MSN Premium service. work through an NSF-funded NSDL grant. However, highly dynamic educational Web con tent is problematic in several ways. First, it is quite Accessibility difficult to create. It requires not only skill in educa One last area where MathML is making a notewor tional design but also substantial technical skill. thy contribution is accessibility for the visually Dynamic Web content also places great demands on impaired and learning disabled. As the work force bandwidth and browser technology. But education has aged, disabilities and impairments of all sorts is precisely where one finds the broadest diversity of have grown to affect nearly two-thirds of adults. browsers and platforms and where the requirement Most are mild, but, according to a survey by Forster of universal access is strongest. As a consequence, Research commissioned by Microsoft, 17 percent efforts to use the Web as a means of integrating rich of computer users have a mild visual difficulty or media into math curricula have generally had mixed impairment and 9 percent have a severe visual success. difficulty or impairment. At the same time, use of the Web to provide a The needs of users vary substantially depend learning environment where students and teachers ing on the nature and degree of the impairment. can interact via various modes of electronic com Individuals with severe visual impairments often munication has grown to the point where it is now rely on tactile feedback through braille displays and commonplace. Such uses range from email and embossers, as well as audio renderings of the math. simple course homepages, where syllabi, assign Individuals with low vision typically require audio ments, and office hours are posted, to sophisticated renderings, in conjunction with conventional type learning management systems (LMS) such as set representations using large font sizes or high WebCT, Blackboard, eCollege, and others. Because contrast colors. Those with learning disabilities electronic communication has an entirely different such as dyslexia benefit most from synchronized dynamic than face-to-face interactions, it can lower highlighting of visual display along with an audio barriers to participation for students who otherwise rendering. might sit silent and unnoticed in the back row. In the United States, the American Disabilities Act Some students may be more comfortable inter (ADA), the Individuals with Disabilities Education Act acting with teachers and their peers via electronic (IDEA), and other federal legislation require schools means, where there is a slight element of anonymity and publishers to provide accessible versions of texts and the ability to consider questions and responses in many circumstances. A majority of states have without the real-time pressure of face-to-face similar laws mandating accessibility of content for interaction. organizations that receive state funding, and a MathML is now widely used behind the scenes majority of states require that textbook publishers for adding math support to online collaboration provide versions of their textbooks for the blind. tools. To a large extent, this has been a conse Similar laws also apply in Europe and elsewhere. quence of the fact that MathML fostered the Currently, in the case of mathematics, these development of a number of interoperable requirements are typically met by providing a text math-aware components that LMS vendors have equivalent for equations. For example, in an HTML been able to utilize. WebEQ, webMathematica, page, this typically means equations are displayed techexplorer, and MathiWYG are among the most using images, with a textual ALT description of an common. In particular, a number of LMS vendors image. now provide math support in their whiteboard and Unfortunately, text descriptions of mathemati message board systems using these components. cal expressions only meet the letter of the law and Typically such systems use a MathML equation do not really address user needs. At a practical level, editor component that can be embedded in a Web the preparation of text descriptions is labor in page for authoring, together with server-side com tensive and error prone. At a deeper level, for audio ponents for processing MathML for display. rendering of mathematics, the ability to "navigate" Another area where MathML is being used is around a long expression is critical to compre online assessment. For lower-level mathematics, hension. Moreover, static text cannot take advan where MathML is most successful at capturing tage of locale or user preference information to mathematical semantics, several systems utilize choose the language or customize the vocabulary. computer algebra systems and other techniques to MathML facilitates solutions to all of these prob perform automatic scoring. Some systems, such as lems. Most assistive technology utilizes standard MapleTA, are even able to analyze student errors software APis developed for HTML and XML. By to provide adaptive hints and customize tutorials making the math notation available through these MAY 2005 NOTICES OF THE AMS 537 standard methods, MathML enables screen readers and other assistive technologies to properly han dle the math with minimal additional effort. And Bathsheba Grossman since MathML tries to insure that markup structure sculpting geometry reflects the underlying semantics, navigation is also greatly facilitated. Finally, since the audio ren dering of a MathML expression is generated on the client machine, it can take full advantage of locale and user preference information. Design Science has received an Small Business Innovation Research grant from the National Sci ence Foundation to add accessibility functionality to its MathPlayer extension for Internet Explorer. MathPlayer 2.0 has a demo "speak expression" fea ture that works with leading screen readers, and support for expression navigation and synchro nized highlighting is under way. A number of other groups are also exploring ways of using MathML for accessibility, and it is anticipated MathML will soon be incorporated into standard XML formats for accessibility currently under development. Conclusion Art for both sides of the brain: symmetrical sculpture and mathematical models realized From a certain point of view, MathML is merely in metal and glass. Commissions welcome! another data format for math notation-not the first, and assuredly not the last. However, as has www. bathshe ba. com so often been the case in the history of technology, the larger significance of MathML depends only tangentially on its particular strengths and weak nesses as a technology. Instead, MathML is signif icant because of the opportunity it represents for math and science to participate more fully in the information revolution that is one of the great intellectual movements of our time. MathML reserves a place at the XML technology table for the interests of math and science. Aworld leading journal Many groups have already seized upon the opportunity presented by MathML and are using it -now available online and in print in innovative ways in STM publishing, searching and knowledge management, e-learning, and accessi bility. Moreover, momentum is building, as devel ASYMPTOTIC opment efforts cross-pollinate and reinforce one ANALYSIS another. While many of these projects are still in their early stages, the future generally looks bright Editor-in-Chief Alain Bensoussan University of Texas at Dallas for electronic communication in the sciences. In a world where math phobia is pervasive and the cold Critical insights for the analysis calculus of the marketplace rarely favors academia, of asymptotic problems that is no small achievement. • Original mathematical results in the asymptotic theory of prob lems affected by the presence of small or large parameters References • Possible applications to different fields of natural sciences [1] The Connexions Project, http: I I cnx. rice. edu. 2005: Volume 41-45; 20 issues [2] MowGLI, http: I lmowgl i . cs. uni bo. it. € 1355 I US$1560 [3] The Future of Mathematical Communication II, (includes electronic access and print) http:llwww.msri .orglactivitiesleventsi9900I ISSN 0921-7134 fmc99l fmc_ABS.html. Recommend Asymptotic Analysis /OS [4] NEIL J. A. SLOANE, The On-Line Encyclopedia of Integer to your librarian Sequences,http:llwww.research.att.coml-njasl Press sequences. Visit our website at www.iospress.nl for additional information [5] Enhancing the Searching of Mathematics, http: I lwww. and to download a free sample copy ima.umn.edulcomplexlspringlsearching.html. 538 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Margulis and Novikov Receive 2005 Wolf Prize On January 18, 2005, the Wolf Foundation announced thatthe 2005 WolfPrizeinMath ematics will be awarded to GREGORY A. MAR Guus of Yale University "for his monumental contributions to algebra, in particular to the theory of lattices in semi-simple Lie groups, and striking applications of this to ergodic theory, representation theory, number theory, combinatorics, and measure theory" and to SERGEI P. NovrKov of the University of Mary land, College Park, and the L. D. Landau In stitute for Theoretical Physics "for his fun damental and pioneering contributions to algebraic and differential topology on one Gregory A. Margulis Sergei P. Novikov hand, and to mathematical physics on the other hand." Margulis and Novikovwill share with both algebraic and analytic methods. The work the $100,000 prize, which will be presented by the has subsequently reshaped the ergodic theory of president of the State of Israel, Moshe Katsav, at a spe general group actions on manifolds. cial ceremony at the Kn esset (parliament) in In a second tour de force, Margulis solved the Jerusalem on May 22, 2005. 1929 Oppenheim Conjecture, stating that the set irrational Gregory A. Margulis of values at integer points of an indefinite nondegenerate quadratic form in more than three At the center of the work of Gregory Margulis lies variables is dense in Rn. This had been reduced (by his proof of the Selberg-Piatetskii-Shapiro Conjec Rhagunathan) to a c onjecture about unipotent ture, affirming that lattices in higher rank Lie groups flows on homogeneous spaces, proved b y Mar are arithmetic, a question whose origins date back to gulis. This method transformed to this ergodic set Poincare. This was achieved by a remarkable tour de ting a family of questions till then investigated force, in which probabilistic ideas revolving around only in analytic number theory. a noncommutative version of the ergodic theorem A third dramatic breakthrough came when were combined with p-adic analysis and with Margulis showed that Kazhdan's "Property T" algebraic geometric ideas showing that "rigidity" (known to hold for rigid lattices) could be used in phenomena, earlier established by Margulis and a single arithmetic lattice construction to solve others, could be formulated in such a way ("super two apparently unrelated problems. One was the rigidity") as to imply arithmeticity. This work dis solution to a problem posed by Rusiewicz, about plays stunning technical virtuosity and originality, MAY 2005 NOTICES OF THE AMS 539 finitely additive measures on spheres and Euclid Novikov made a fundamental and striking con ean spaces. The other was the first explicit con tribution to two separate fields in mathematics, struction of infinite families of expander graphs of while he is one of those rare mathematicians who bounded degree, a problem of practical application brings deep, key mathematical ideas to bear on dif in the design of efficient communication networks. ficult pivotal problems of physics, in ways that are Margulis's work is characterized by extraordi stunning and compelling for both mathematicians nary depth, technical power, creative synthesis of and physicists. ideas and methods from different areas of math Born in Russia in 1936, Sergei P. Novikov gradu ematics, and a grand architectural unity of its final ated from Moscow State University in 1960. In 196 5, form. Though his work addresses deep unsolved he received his Ph.D. in physics and mathematics problems, his solutions are housed in new con from the Steklov Institute of Mathematics in Moscow. ceptual and methodological frameworks of broad Since 1971, Novikov has been head of the Mathe and enduring application. He is one of the mathe matical Division at the L. D. Landau Institute for matical giants of the last half century. Theoretical Physics in Moscow. Since 1992, he has Born in 1946 in Russia, Margulis received his been a professor in the Department of Mathematics Ph.D. in 1970 from Moscow State University. Starting and at the Institute for Physical Science and Tech in 1970, he was associated with the Institute for nology at the University of Maryland, College Park. Novikovreceived the Lenin Prize of the USSR in 1967 Problems in Information Transmission at that uni and the Fields Medal in 1970.ln 1981, he was elected versity, first as junior scientific worker, later as as a full member of the Academy of Sciences of the senior staff member, and from 1986 until he left USSR. He is a foreign associate of the U.S. National in 1991, as leading scientist. Since 1991, Margulis Academy of Sciences. has been a professor of mathematics at Yale University. He received the Fields Medal in 1978. He About the Wolf Prize is a foreign honorary member of the American The Israel-based Wolf Foundation was established Academy of Arts and Sciences and a member of the by the late German-born inventor, diplomat, and U.S. National Academy of Sciences. philanthropist Ricardo Wolf. A resident of Cuba for many years, Wolf became Fidel Castro's ambas Sergei P. Novikov sador to Israel, where he lived until his death in Sergei P. Novikov is awarded the Wolf Prize for his 1981. Five annual Wolf Prizes have been awarded fundamental and pioneering contributions to topol since 1978 to outstanding scientists and artists ogy and to mathematical physics. His early work "for achievements in the interest of mankind and in algebraic and differential topology includes such friendly relations among peoples, irrespective of milestones as the calculation of cobordism rings nationality, race, color, religion, sex, or political and stable homotopy groups, proof of the topo view." The prizes of $100,000 in each area are logical invariance of rational Pontrjagin classes, given every year in four out of five scientific fields formulation of the "Novikov Conjecture" on higher in rotation: agriculture, chemistry, mathematics, signature invariants, and proof of the existence of medicine, and physics. In the arts, the prize rotates closed leaves in two-dimensional foliations of the among architecture, music, painting, and sculp 3-sphere. ture. To date, a total of 224 scientists and artists In the early 1970s Novikov turned his attention from twenty-one countries have been honored. to mathematical physics, initially contributing to -From Wolf Foundation announcements general relativity and conductivity of metals. He constructed a global version of Morse theory on For a list of previous Wolf Prize winners in math manifolds and loop spaces that had novel appli ematics, see the Web page http: I /www. aquanet. cations to quantum field theory (multivalued action co.il/wolf/wolfS.html. functionals). His most significant achievements in mathematical physics flow from his introduction of algebraic-geometric methods to the study of completely integrable systems. These include a systematic study of finite-gap solutions of two dimensional integrable systems, formulation of the equivalence of the classification of algebraic geometric solutions of the KP equation with the conformal classification of Riemann surfaces, and work (with Krichever) on "almost commuting" op erators that appear in string theory and matrix models ("Krichever-Novikov algebras", now widely used in physics). 540 NOTICES OF THE AMS VOLUME 52, NUMBER 5 2005]PBM Communications Award The 2005 Communications Award of the Joint Pol While some of his audience icy Board for Mathematics (JPBM) was presented at undoubtedly consists of math the 111 th Annual Meeting of the AMS in Atlanta in ematicians themselves, he January 2005. writes for scientists and schol The JPBM Communications Award is presented ars who are mathematically lit annually to reward and encourage journalists and erate. In this way, he has other communicators who, on a sustained basis, reached many thousands of bring accurate mathematical information to non scientists. Dr. Cipra's work has mathematical audiences. The award carries a cash educated mathematicians and prize of $1,000. nonmathematicians alike by Previous recipients of the JPBM Communica exposing them to current and tions Award are James Gleick (1988), Hugh White deep mathematical ideas more (1990), Ivars Peterson (1991), Joel Schneider about the beauty and power (1993), Martin Gardner (1994), Gina Kolata (1996), of mathematics. Barry Cipra Philip]. Davis (1997), Constance Reid (1998), Ian has given his readers a greater Barry Cipra Stewart (1999), John Lynch and Simon Singh (spe understanding of the ideas of mathematics, but most importantly he has changed cial award, 1999), Sylvia Nasar (2000), Keith]. De their perception of the nature of mathematics. vlin (2001), Claire and Helaman Ferguson (2002), and Robert Osserman (2003). The 2005 JPBM Communications Award was Biographical Sketch presented to BARRY CrrRA. The text that follows pre Barry Cipra received his doctoral degree in math sents the award citation, a brief biographical sketch, ematics from the University of Maryland in 1980. and the recipient's response upon receiving the After a brief career as an academic, he turned to award. freelance writing, and he has continued with that work for the past 15 years. He has written many Citation articles for Science magazine, one of the premier The Joint Policy Board for Mathematics presents its journals of scientific exposition. Examples of the 2004 Communications Award to Dr. Barry Cipra intriguing titles of his articles are "Simple recipe who, for nearly twenty years, has written about creates acid test for primes" and "How to play pla mathematics of every kind-from the most ab tonic billiards". He is a regular contributor to SIAM stract to the most applied. His lucid explanations News, writing many dozens of articles that are ac of complicated ideas at the frontiers of research cessible and illuminating. He has authored five have appeared in dozens of articles in newspa volumes of What's Happening in the Mathematical pers, magazines, and books. Sciences for the AMS, each including a compilation MAY 2005 NOTICES OF THE AMS 541 of expository articles on recent mathematical de "Let me cut to the chase: every mathematics velopments aimed at the mathematically literate library requires a copy of this book. ... Every public. Those volumes have been widely distributed supervisor of higher degree students requires a (and admired) in the scientific community and in Washington. copy on theirshel£ Welcome to the rich world Cipra received the 1991 Merten M. Hasse Prize of computer-supported mathematics!" from the Mathematical Association of America for -Mathematical Reviews an expository article on the Ising model, published in the December 198 7 issue of the American Math ·· ······ ··········· ··· ·· ·· ·· · .. .· . ····· ematical Monthly. He is the author of Misteaks .. .and how to find them before the teacher does ... (a cal culus supplement), published by A K Peters, Ltd. mathematics Cipra completed his Ph.D. degree under the direction of Michael Razar, with much help from by EHperiment Steve Kudla. He was a Moore Instructor at the Plausible Reasoning in the ~1 51 Century Massachusetts Institute of Technology, a research instructor at the Ohio State University, and an assistant professor at St. Olaf College in North Jonat~an Dorwein. Oavi~ Dailey field, Minnesota, before turning to freelance S4S.00; H98 pp. writing. Response "Whether you take sides in the debate about It is a· great honor to receive the JPBM Communi the validity of experimental mathematics or cations Award. To be able to write about mathe are not sure what all the fuss is about, the matics for a living-to meet so many first-rate book is a rewarding read." mathematicians and learn about their exciting - The Berkeley Science Review work-is a pleasure beyond description. This is an amazing age in which to be reporting on mathe ····· ·· ·············· ······ ········ ······ ···· · matics and its applications. I never would have guessed, in 1987, that I would wind up reporting on the proofs of Fermat's Last Theorem and the EHperimentation Kepler Conjecture (and, very possibly, the Poin in mathematics care Conjecture). I have witnessed an incredible growth in the applications of mathematics, espe Computational Paths to Discovery cially in biology, which fifteen years ago was barely a whisper at math meetings and now is a promi Jonat~an Dorwein . Oavi~ Dailey, nent theme at many. Perhaps most surprisingly, I've Rolan~ Girgenso~n seen mathematics go from a virtual nonentity in popular culture to become the basis (or McGuffin) $49.00; 3G8 pp. of award-winning plays and movies. I've been helped by many people over the years. Chief among them are Klaus Peters, Lynn Steen, Ed "The authors ... explain experimental mathe Block, Paul Sally, and Sam Rankin. I would like to matics in a lively, surprisingly accessible thank my editors, especially Gail Corbett, Tim Ap fashion." penzeller, and Paul Zorn, who have made the final, -L'Enseignement Mathematique published versions of my articles so much better than their first drafts. Indeed, the key to writing, "These are such fun books to read! ... But do I've found, is expressible in a familiar mathemati not be fooled by the lighthearted, immensely cal term: iteration. The hard part, as mathematicians entertaining style. You are going to learn well know, is making sure the iterative process more math (experimental or otherwise) than converges to the desired result you ever did from any two single volumes. Not only that, you will learn by osmosis how to become an experimental mathematician." -American Scientist ; www.a~peters.com 542 NOTICES OF THE AMS VOLUME 52, NUMBER 5 MAAPrizes Presented in Atlanta At the Joint Mathematics Meetings in Atlanta in has been shown to have had influence beyond their January 2005, the Mathematical Association of own institutions. America presented the following honors. The 2005 Haimo awards were presented to GERALD L. ALEXANDERSON of Santa Clara University, Gung and Hu Award for Distinguished APARNA HIGGINS of the University of Dayton, and Service DEBORAH HuGHES HALLETT of the University of Arizona The Yueh-Gin Gung and Dr. Charles Y. Hu Award and Harvard University. for Distinguished Service to Mathematics is the "Jerry Alexanderson is a master teacher, an most prestigious award made by the Association. inspiration to both students and colleagues," The 2005 Gung and Hu Award was presented to the citation states. "In his 4 7 years of teaching at GERALD L. ALEXANDERSON of Santa Clara University. Santa Clara University (3 5 years of which he was The award citation states, "Jerry has a long record department chair), he has consistently had the of able service to mathematics as a practitioner, reputation for being not only the best, but also one teacher, administrator, professional organization of the most demanding teachers. His classes are leader, publicist, advocate, and enthusiast whose amusing, entertaining, and highly informative, an love for mathematics and its people comes through impressive mix of challenging mathematics and clearly in his public talks and widely-read books." historical anecdotes, delivered clearly and con cisely. Many mathematicians (and former students Alexanderson is known for his work on the Math in other careers) discovered the excitement of ematical People volumes of interviews with math mathematics in the first course they took with ematicians. Since 1975 he has been the associate Jerry, and his personal advice and encouragement director of the William Lowell Putnam competi continues to guide many of those careers today." tion. He served as chair of his department for "Aparna Higgins is one of the dynamos of the thirty-five years and received a President's Special U.S. mathematical community," the award citation Recognition Award in 1996 for this service to his states. "Her ease with and genuine connection to institution. Since 1994 he has been chair of the students is remarkable; her dedication to teaching Board of Trustees of the American Institute of and mentoring is recognized by colleagues near and Mathematics. During his fifty years of MAA mem far .... [Her]love of all things mathematical and the bership he has served in many different capacities, desire to encourage others fuels her charisma, en including serving as MAA president and secretary. ergy, and enthusiasm. Her joy is contagious in the classroom, at MAA student chapter meetings, in her Haimo Awards for Distinguished College REU [Research Experiences for Undergraduates] or University Teaching summer programs, and with Project NExT Fellows." The Deborah and Franklin Tepper Haimo Awards Higgins has received two other teaching awards, for Distinguished College or University Teaching of from her own university and from the Ohio section Mathematics honor college or university teachers of the MAA. who have been widely recognized as extraordi "Deborah Hughes-Hallett is known for her su narily successful and whose teaching effectiveness perb skills in the classroom," the prize citation MAY 2005 NOTICES OF THE AMS 543 states. "She excels at all scales, from the class Princeton=Math room to the international educational scene. Her pioneering programs at the University of Arizona and at Harvard will continue to support and inspire COMPLEXITIES the worldwide teaching of mathematics for Women in Mathematics decades." The best known of these programs is Bettye Anne Case and Anne M. Leggett, the Harvard-based Calculus Consortium. Her role Editors in designing mathematics courses for students at Harvard's Kennedy School of Government has This eye-opening book presents the stories of dozens of won high praise. "[These courses] have involved a women who have pursued careers in mathematics. fundamental rethinking of either curriculum or The contributors offer their own narratives, recount the method, and are driven by her uncompromising experiences of women who came before them, and offer devotion to her students and her rigorous under guidance for those who will follow in their career paths. standing of how they think," The one thing they share in common is a genuine the prize citation passion for mathematics. says. Cloth $35.95 ISBN 0-691-11462-5 BeckenbachBookPrize The Beckenbach Book Prize is awarded to an author SCIENTIFIC PARALLEL of a distinguished, innovative book published by the MAA. The award is not given on a regularly COMPUTING scheduled basis but only when a book appears that L. Ridgway Scott, Terry Clark, is judged to be truly outstanding. and Babak Bagheri The 2005 Beckenbach Book Prize was presented to ]AMEs TANTON, a high school mathematics teacher Scientific Parallel Computing is the first textbook to inte at St. Mark's School in Southborough, Massachu grate all the fundamentals of parallel computing in a setts. He received the prize for his book Solve This: single volume. Designed for graduate and advanced Math Activities for Students and Clubs, published undergraduate courses in the sciences and in engineer by the MAA. The book, the prize citation states, "is ing, computer science, and mathematics, it focuses on much more than just a collection of ideas to use the three key areas of algorithms, architecture, languages, with math clubs. It causes, coerces, and induces and their crucial synthesis in performance. The new paradigm of cluster computing is fully addressed. the reader to think about mathematics in non conventional ways, exploring diverse topics from A supporting website provides access to oil the codes number theory, geometry, combinatorics, proba and software mentioned in the book, and offers topical bility, knot theory, topology, tiling, and many more. information on popular parallel computing systems. The sections on 'Take it Further' and 'Solutions and Cloth $55.00 ISBN 0-69 1-11935-X Discussions' expand upon many of the problems and activities to suggest additional directions of REAL ANALYSIS exploration, provide notes and proofs on some Measure Theory, Integration, and Hilbert Spaces important areas and theorems of mathematics, Elias M. Stein and Rami Shakarchi and present problems whose solutions are still unknown. Even professional mathematicians are Real Analysis is the third volume in the Princeton Lectures likely to encounter questions and problems that will in Analysis, a series of four textbooks that aim to pres capture their interest in this creative, innovative, ent, in an integrated manner, the core areas of analysis. and delightful exposure to mathematics and math Here the focus is on the development of measure and ematical thinking." integration theory, differentiation and integration, Hilbert spaces, and Hausdorff measure and fractals. Certificates of Meritorious Service 1 Cloth $59.95 ISBN0-691-11386-6 DueMay The Certificate of Meritorious Service is presented A/so avoilable, the first two volumes in the Princeton Lectures in Analysis: for service at the national level or for service to an •Fourier Analysis: An Introduction MAA section. Those honored in 2005 were CHARLES Cloth $49.95 ISBN 0-691-11384-X CABLE of Allegheny College, Allegheny Mountain •Complex Analysis Section; ]ON Cloth $49.95 ISBN 0-691-11385-8 Scorr of Montgomery College, Maryland District of Columbia-Virginia Section; BARBARA OSOFSKY of Rutgers University, New Jersey Section; Celebrating 100 Years of Excellence RoY DEAL ]R., of Oklahoma State University, PRI;NC~TON soo-m-4726 Oklahoma-Arkansas Section; and ERNIE SOLHEID of Untverstty :Press math.rurress.rrinceton.edu California State University at Fullerton, Southern California-Nevada Section. 544 NOTICES OF THE AMS VOLUME 52, NUMBER 5 AWMPrizes Presented in Atlanta At the Joint Mathematics Meetings in Atlanta in Schafer Prize January 2005, the Association for Women in Math The Alice T. Schafer Prize for Excellence in Math ematics (AWM) presented the following honors. ematics by an Undergraduate Woman honors outstanding achievement in mathematics by a Hay Award female undergraduate. The Louise Hay Award for Contributions to The 2005 Schafer Prize was awarded to MELODY Mathematics Education recognizes outstanding CHAN, a senior at Yale University. She has excelled in achievements in any area of mathematics educa a wide variety of mathematics courses and was tion, to be interpreted in the broadest possible awarded the prestigious Hart Lyman Prize. She has sense. made presentations at the Yale Math Club, earned The 2005 Hay Award was presented to SusANNA an honorable mention in the Putnam Competition, S. EPP of DePaul University. "For the past twenty and is vice president of theYale chapter of Phi Beta five years, she has committed herself to helping stu Kappa. She participated in the Budapest Semesters in Mathematics in Hungary and in Research Experi dents come to understand the unspoken logic and ences for Undergraduates (REU) programs. At the language that underlie mathematical thought," the University of Minnesota at Duluth REU, she wrote award citation states. She has written a very pop three professional-level papers on the concept of ular and well-regarded textbook, Discrete Mathe the distinguishing number. matics with Applications, and has coauthored a Runners-up for the Schafer Prize are MARGARET I. volume in the University of Chicago School Math DOIG, University of Notre Dame, and ELENA FucHS, ematics Project (UCSMP) Secondary Series. Epp has University of California, Berkeley. ANNALIES VuoNG, written many articles on the teaching and learning University of California, Santa Barbara, received an of mathematics and has been active in national honorable mention. efforts to improve mathematics education. At DePaul University, she developed more than a dozen successful courses. The prize citation says, "For her selfless contributions to mathematics education, her role as a mentor, her scholarship, her administrative skills, her human qualities of kindness, absolute honesty and trustworthiness, and her willingness to listen, the Association for Women in Mathematics is pleased to designate Susanna S. Epp as the Fifteenth Annual Louise Hay Awardee." MAY 2005 NOTICES OF THE AMS 545 Happy 1OOth, Baley Price! G. Baley Price, During the 1950s Price worked to get the math Distinguished ematical community involved in undergraduate Professor Emeri and school education. While he was president of tus of the Uni the Mathematical Association of America (MAA) versity of Kansas, in 1957-58, Price collaborated withAMS president turned 100 years Richard Brauer and National Council of Teachers old on March 14, of Mathematics president Harold Fawcett to appoint 2005. Known for a committee to establish the School Mathematics his dedication to Study Group (SMSG). This group launched the "new teaching and to math" program that transformed mathematics public service, teaching in schools across the country. While some Price had a hand times dismissed as a failure, SMSG was often suc G. Baley Price in class, 1960s. in many key de- velopments in the cessful when teachers were properly trained. Price American mathematical community during the participated in writing sessions for SMSG and also twentieth century. He has been a member of the taught in a summer institute designed to train AMS since January 1, 1929. teachers to use the "new math" curriculum. Price served as an AMS associate secretary from Griffith Baley Price was born in 1905 and re 1946 until1949 and was editor of the Bulletin from ceived his bachelor's degree from Mississippi Col 1950 until1957. He was also very active in the MAA, lege in Clinton, Mississippi, in 1925. He received his doctorate in 1932 from Harvard University, serving not only as president but also as a vice pres where, like his fellow students C. B. Morrey and ident and as a longtime member of the MAA Board Hassler Whitney, he was a student of G. D. Birkhoff. of Governors. During the 1950s he was on the The title of Price's dissertation was "Double pen MAA Committee on the Undergraduate Program in dulum and similar dynamical systems". After a Mathematics. He received the MAA's Distinguished short stint teaching at Brown University, in 193 7 Service Award in 1970. He was the first chairman he joined the faculty of the University of Kansas, of the Conference Board of the Mathematical Sci where he remained for the rest of his career. In 1970 ences (CBMS) (1959-60) and also served as CBMS he was named the first E. B. Stouffer Distinguished executive secretary (1960-62). Professor of Mathematics. He retired in 1975. A University of Kansas alumnus who became a Price devoted a great deal of time and effort to publishing sales representative was so impressed service on behalf of the mathematics profession and with Price when they first met in the 1960s that in the general public. While at Brown University he had 2004 he committed nearly a million dollars toward become acquainted with R. G. D. Richardson, who the establishment of an endowed chair in Price's was then AMS secretary and who was instrumen honor. Not long thereafter, Price himself made a tal in launching Mathematical Reviews, which began gift of $500,000 to the university for an endowed publication in 1940. Price served on the publicity professorship in honor of his wife, Cora Lee Beers committee that helped drum up subscriptions for Price, who held a Ph.D. in English and was on the the fledgling journal. Around this time he served faculty of the University of Kansas. She passed on the AMS Council and was also named an asso away in late 2004. ciate secretary of the Society, but his World War II Still active in his retirement, Price wrote a history service prevented his serving in that capacity: in of the mathematics department at the University of 1943 he began a three-year tour of service as a Kansas as well as historical pieces for the AMS cen civilian member of the Operational Section of the tennial celebration in 1988 and the MAA's seventy Eighth Air Force. His job was to use mathematics fifth anniversary in 1990. His long career of service to improve the accuracy of bombing attacks. In to mathematics has made him a beloved member of 2003 the United States Congress paid him a spe the community. Happy Birthday, Professor Price! cial tribute for this service. -Allyn jackson 546 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Scientific WorkPlace® Scientific Word· Version 5 Sharing Your Work Is Easier • Typeset PDF in the only software . three ., \: lwidth . that allows you to transform IbTEX .; 1uations . --Stanton )anc . · , l Monte Carlo ;:;Ull LSCV banclwlclth. the . ltl The fust choKe 1::; files to PDF fully hyperlinked and n . t' 1y The ) banclwlc 1. with embedded graphics in over 50 parall\eter combma 1\.all v distributed (UD f LSCVfun.C). The UD formats ill(lel)enclent allCl iclel~ 1Cv~';'clation probletn (l:e/5 : 1 ·e & is the salnple ·e·· cross (UJ. d /'y-l w 1e1 • Export documents as RTF with the least SCluar ::;1 t . , clefinecl as h" = 6 . banclwiclth is the one for UD clata, allC r 1S _, . IJle size. 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' \tappeare m cKichan SOFTWARE , I NC. Tools for Scientific Creativity since 1981 Mathematics People Fedkiw Receives NAS Award Lander Receives Public for Initiatives in Research Understanding Award RoNALD FEDKIW of Stanford University has been awarded the The American Association for the Advancement of Science National Academy of Sciences (NAS) Award for Initiatives (AAAS) has named ERic S. LANDER to receive its Public Un in Research. He has been honored for "his many innova derstanding of Science and Technology Award for 2004. tions in the modeling and numerical simulation of flows Lander is the founding director of the newly created and his pioneering contributions to physically based com Broad Institute of the Massachusetts Institute of Tech puter graphics." nology and of Harvard University. He was cited by AAAS The award carries a cash prize of $15,000 and is awarded "for his excellence in communicating complex scientific annually to recognize innovative scientists under thirty ideas, and their implications for society, to the general five years old and to encourage research likely to lead public and policy-makers, while actively engaged in a toward new capabilities for human benefit. The prize demanding and aggressive research program." alternates among the fields of physical sciences, engi A former Rhodes scholar, Lander earned his under neering, and mathematics. graduate degree in mathematics with highest honors from Princeton University in 1978 and then received his Ph.D. -From an NAS announcement in mathematics from Oxford University in 1981. His hon ors and awards include a MacArthur Fellowship in 1987. He was elected a member of the U.S. National Academy of Hughes Receives AAAS lifetime Sciences in 199 7 and the U.S. Institute of Medicine in 1999. The AAAS Award for Public Understanding of Science Mentor Award and Technology, established in 1987, recognizes scientists and engineers who make outstanding contributions to the RHONDA ]. HuGHES of Bryn Mawr College has been named popularization of science. It carries a monetary prize of the recipient of the 2004 AAAS Lifetime Mentor Award of $5,000. the American Association for the Advancement of Science (AAAS). She has helped fifty-seven women and minority - From an AAAS news release students earn graduate degrees in mathematics, including seventeen at the doctoral level. Hughes and Sylvia Bozeman of Spelman College devel oped the Spelman-Bryn Mawr Summer Mathematics Pro Boyd Awarded CRM-Fields gram and the Enhancing Diversity in Graduate Education Prize (EDGE) Program, both designed to help young women through college and graduate school. Hughes is a former DAVID BoYD of the University of British Columbia has been president of the Association for Women in Mathematics. awarded the 2005 CRM-Fields Prize. The prize, awarded The AAAS Mentor Award for Lifetime Achievement hon annually by the Centre de Recherches Mathematiques ors AAAS members who have mentored and guided sig (CRM) in Montreal and the Fields Institute in Toronto, rec nificant numbers of underrepresented students toward a ognizes exceptional contributions by a mathematician Ph.D. degree in the sciences. The recipient receives $5,000 working in Canada. The prize carries a cash award of and a commemorative plaque. 5,000 Canadian dollars (approximately US$3,850), and the recipient is expected to present a lecture at the CRM and -From an AAAS announcement at the Fields Institute. 548 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Mathematics People Boyd was recognized for seminal contributions to an receive an unrestricted research grant of $625,000 over five alytic number theory, particularly his explorations of the years. deep connections between the Mahler measure of poly The fellowships are awarded to researchers in mathe nomials and special values of their associated I-functions. matics, natural sciences, computer science, and engineer Boyd received his B.Sc. from Carleton University in 1963 ing who are in the first three years of a faculty appoint and his M.A. in 1964 and Ph.D. in 1966 from the Univer ment. sity of Toronto. He has taught at the University of Alberta and the California Institute of Technology. He has been -From a Packard Foundation announcement teaching at the University of British Columbia since 1971 and is currently a full professor. Boyd is a fellow of the Royal Society of Canada. His awards include the E. W. R. Mazya Awarded Celsius Gold Steacie Prize and both the Coxeter-James and Jeffery Williams prize lectureships of the Canadian Mathematical Medal Society (CMS). His service to the Canadian mathematical community includes terms as vice president of the CMS, VLADIMIR MAz'yA of the Ohio State and Liverpool Universi as chair of the NSERC mathematics grant selection com ties received the Celsius Gold Medal of the Royal Society mittee, and as acting director of the Pacific Institute for of Sciences of Uppsala on August 31, 2004. The citation states that Maz'ya received the medal "for his outstand the Mathematical Sciences. ing research in partial differential equations and hydro The CRM and the Fields Institute established the CRM dynamics". Fields prize in 1994 to recognize exceptional research in Maz'ya authored more than four hundred papers and the mathematical sciences. Previous recipients of the prize fifteen books in various branches of pure, applied, and nu are H. S. M. (Donald) Coxeter, George A. Elliot, James merical analysis. In particular, as early as 1960 he dis Arthur, Robert V. Moody, Stephen A. Cook, Israel Michael covered the equivalence of Sobolev embeddings and isoperi Sigal, William T. Tutte, John B. Friedlander, John McKay, metric/ isocapacitary inequalities, which had great impact Edwin Perkins, and Donald A. Dawson. on further development on the theory of function spaces, spectral theory, and differential geometry. In 1968 he con -From a Fields Institute announcement structed counterexamples related to the nineteenth and twentieth Hilbert problems. He solved F. John's problem on time-harmonic waves above a submerged body in 1978. AIM Five-Year Fellow Recently he found regularity criteria of Wiener's type for higher order elliptic equations. Announced The Royal Swedish Society was founded in 1710 and is JoEL KAMNITZER of the University of California, Berkeley, has the oldest scholarly society in Sweden. The Gold Medal is been named the recipient of the 2005 American Institute the Society's highest award and was initiated in 1960. It of Mathematics (AIM) Five-Year Fellowship. is awarded every two years in different areas of science, Kamnitzer will receive his Ph.D. in 2005 from Berkeley. going to a mathematician every sixth year. Among previ His research interests include algebraic geometry, repre ous recipients in the mathematical sciences are L. Carle son, sentation theory, and combinatorics. His thesis, titled L. Hbrmander and ]. Peetre. "Mirkovic-Vilonen cycles and polytopes", gives a combi -from a Royal Society of Sciences of Uppsala natorial description of a family of varieties arising in geo announcement metric representation theory. The fellowship will provide sixty months of full-time research, as well as funds for travel and equipment. The runners-up for the AIM Fellowship are Spyros Alex akis (Princeton University), Beth Samuels (Yale University), Lior Silberman (Princeton University), and Lauren Williams (Massachusetts Institute of Technology). -From an AIM announcement Packard Fellowships Awarded The David and Lucile Packard Foundation has awarded six teen Fellowships for Science and Engineering for the year 2004. Two mathematical scientists were among the awardees. MANJUL BHARGA VA of Princeton University and ALEXANDRU D. IoNEscu of the University of Wisconsin, Madison, will each MAY 2005 NoTICES OF THE AMS 549 Mathematics Opportunities maximal impact on their future scientific development. Enhancing the Mathematical Awards are made in the form of either Research Fellowships Sdences Workforce or Research Instructorships. The Research Fellowship option provides full-time support for any eighteen in the Twenty-First Century academic-year months in a three-year period, in intervals In an effort to increase the number of U.S. citizens, na not shorter than three consecutive months. The Research tionals, and permanent residents who are well prepared Instructorship option provides a combination of full-time in the mathematical sciences and who pursue careers in and half-time support over a period of three academic the mathematical sciences and other scientific disciplines, years, usually one academic year full time and two acade the Division of Mathematical Sciences (DMS) of the National mic years half time. Under both options, the award in Science Foundation (NSF) has instituted a program called cludes six summer months; however, no more than two Enhancing the Mathematical Sciences Workforce in the summer months of support may be received in any calendar Twenty-First Century. This program builds on the Verti year. Under both options, the stipend support for twenty cal Integration of Research and Education (VIGRE) program four months (eighteen academic-year months plus six sum and includes a broadened VIGRE activity as well as addi mer months) will be provided within a forty-eight-month tional components for Research Training Groups in the period. Mathematical Sciences (RTG) and for Mentoring through The deadline for applications is October 19, 2005. For Critical Transition Points (MCTP) in the Mathematical more information and application instructions, see the Sciences. NSF website at http://www.nsf.gov/pub~ys/ods/ VIGRE grants are designed to allow departments in the getpub.cfm?nsf05510. mathematical sciences to carry out innovative educational programs in which research and education are integrated and in which undergraduates, graduate students, post -From an NSF announcement doctoral fellows, and faculty are mutually supportive. Integrating research and education for graduate students and postdoctoral associates, involving undergraduates in NSF Graduate Teaching substantial learning by discovery, and developing a team approach are keys to successful VIGRE projects. VIGRE Fellowships in K-12 Education student and postdoctoral associates and their mentors The National Science Foundation (NSF) has instituted the may participate in international research and education col Graduate Teaching Fellowships in K-12 Education (GK-12) laborative activities, including activities in other countries to support fellowships and associated training that enable that are integrated into and benefit the overall VIGRE program at the institution. graduate students in NSF-supported science, technology, The DMS expects to make eighteen or nineteen awards engineering, and mathematics (STEM) disciplines to acquire under this program in 2005. The deadline for proposals additional skills that will broadly prepare them for pro is September 16, 2005. For more information about the fessional and scientific careers in the twenty-first cen program and all of its components, see the website tury. Through interactions with teachers in kindergartens http://www.nsf.gov/pubs/2003/nsf03575/nsf03575. through high schools, graduate students can improve htm. communication and teaching skills while enriching STEM instruction in these schools. Expected outcomes include -From an NSF announcement improved communication, teaching, and team-building skills for the fellows; professional development opportu nities for K-12 teachers; enriched learning for K-12 NSF Postdoctoral Research students; and strengthened partnerships between insti tutions of higher education and local school districts. Fellowships The deadline for letters of intent is May 4, 2005, and The National Science Foundation (NSF) awards Mathematical full proposals are due June 2, 2005. For more details, Sciences Postdoctoral Research Fellowships (MSPRF) for ap see http://www.nsf.gov/pubsys/ods/getpub. propriate research in areas of the mathematical sciences, cfm?nsf05553. including applications to other disciplines. Awardees are permitted to choose research environments that will have -From an NSF announcement 550 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Mathematics Opportunities September 20-24, 2005: Workshop on percolation, SASTRA Ramanujan Prize Schramm Loewner evolution, and related topics. The ShamnughaArts, Science, Technology, Research Acad October 18-22, 2005: Workshop on renormalization emy (SASTRA), based in the state of Tamil Nadu in South and universality in mathematical physics. India, has instituted a SASTRA Ramanujan Prize of $10,000 November 29-December 3, 2005: Workshop on renor to be awarded annually to a mathematician not exceeding malization in dynamical systems. the age of thirty-two for outstanding contributions in an A second Coxeter Lecture Series will be given by Lai-Sang area of mathematics influenced by the late Indian mathe Young. matical genius Srinivasa Ramanujan. Young mathemati The winter semester program, on holomorphic dynamics, cians all over the world are eligible for this award. The age laminations, and hyperbolic geometry, is organized by limit has been set at thirty-two because Ramanujan Bruce Kleiner, Mikhail Lyubich, Yair Minsky, Mike Shub, and Michael Yampolsky. The program will focus on the interac achieved so much in his brief life of thirty-two years, and tion between three-dimensional hyperbolic geometry and also to encourage doctoral and post-doctoral research. holomorphic dynamics. Activities and dates follow. SASTRA, based in Tanjore in South India, started in january 5-9, 2006: Workshop on partially hyperbolic 1984 as a college of engineering. It grew considerably in dynamics, laminations, and Teichmiiller flow. size in a short span of time and began also offering de March 7-11,2006: Workshop onholomorphic dynamics. grees in areas outside of engineering, attaining the status May 23-27, 2006: Workshop on hyperbolic geometry. of a university in 2001. In 2003 SASTRA opened the Srini The current year's thematic program on the geometry vasa Ramanujan Center and a branch campus in Kum of string theory will close with a series of Coxeter lectures bakonam, the hometown of Ramanujan, and also pur by Renata Kallosh of Stanford University, May 9-11, 2005. chased Ramanujan's home to maintain it as a museum. In The Strings 2005 conference in string theory will be held this connection SASTRA conducted an international con at the University of Toronto, July 11-16, 2005. A summer ference in December 2003 in Kumbakonam. The Presi school program in strings, gravity, and cosmology will be dent of India, Abdul Kalam, inaugurated the conference held June 20-July 8, 2005, at the Perimeter Institute, and declared Ramanujan's home a musuem and national Waterloo, Ontario, Canada. treasure. SASTRA will conduct an international confer The thematic program for the 2006-2007 academic ence each year in Kumbakonam during December 20-22 year will be on cryptography and on geometric applications to coincide with Ramanujan's birthday which is on De of homotopy theory. The fall program will be organized cember 22. by Hugh Williams, Ian F. Blake, Alfred Menezes, Michele Starting in December 2005, the SASTRA Ramanujan Mosca, Kumar Murty, Renate Scheidler, Douglas Stinson, Prize will be awarded at each of these annual conferences. and Ramarathnam Venkatesan. The winter program will The winner will be invited to give a talk at the conference. be organized by J. F. Jardine, G. Carlsson, and J. D. Nominations for the first SASTRA Ramanujan prize must Christensen. be made by July 31, 2005. A panel of experts will select The Seventh IMACS International Symposium on Itera the winner from the nominations. The nomination must tive Methods in Scientific Computing will be held at the include the vita of the nominee, some selected papers, three Fields Institute and the University of Toronto May 5-8, letters supporting the nomination, and be sent to: SASTRA 2005. Organizers are Christina Christara, Peter Forsyth, Ramanujan Prize, Department of Mathematics, University Tamas Terlaky, and Justin W. L. Wang. Invited speakers in of Florida, Gainesville, FL 32611, USA; or to sastrapri ze@ clude Tony Chan, Tom Coleman, Andy Conn, Paul Fischer, math. ufl . edu. Information is available on the webpage Ilse Ipsen, Van Henson, Tim Kelley, Kees Oosterlee, and http:llwww.math.ufl .edulsastra-prizel. Andy Wathen. The Fields Institute Summer School in Operator Alge -Krishnaswami Alladi, University of Florida bras will be held at the University of Ottawa, June 7-17, 2005. For details on all Fields Institute activities, see the News from The Fields Institute website http: I lwww. fields. utoronto. ca. -From a Fields Institute announcement The Fields Institute for Research in the Mathematical Sci ences has announced its thematic program for the 2005-2006 academic year, on renormalization and uni News from Oberwolfach versality in mathematics and mathematical physics and on The Mathematisches Forschungsinstut Oberwolfach (MFO) holomorphic dynamics, laminations, and hyperbolic geom has announced its scientific program for 2006. The new etry. The fall program, organized by Pavel Bleher, Mikhail program is available on the website http: I lwww. mfo. de Lyubich, and Michael Yampolsky, will aim to give a broad or http: I lwww. oberwo l fach. or g. There one can also perspective of applications of renormalization ideas. Ac find information on the scheduled weeks for miniwork tivities and dates for the fall program on renormalization shops and guidelines for proposals. The deadline for and universality follow. proposals for workshops in 2007 is the end of July 2005. September 13-15, 2005: Coxeter Lecture Series. Lec turer: Oded Schramm. -MFO announcement MAY 2005 NOTICES OF THE AMS 551 For Your Information the real news is that despite cultural barriers, women are CorrrrnentsonFenllieMaili entering these fields in greater and greater numbers. About a third of all United States citizens who have received Ability Spark Reaction Ph.D.'s in mathematics recently are women. About half of In remarks made during an economics conference in Cam all undergraduate mathematics degrees in the United bridge, Massachusetts, in January 2005, Harvard Univer States go to women. Yes, there is still a shortage of women sity President Lawrence Summers suggested that innate on the mathematics and sciences faculties of many Amer differences might account for the disparity in men's and ican universities, including Harvard. So universities should women's achievement in mathematics and science. After hire more of these excellent women and then treat them his remarks elicited a torrent of protest, Summers posted as if they value them. We call on Lawrence Summers, as an apology on the Harvard website. well as the leaders of all educational institutions, to take Among those registering protest were leaders of several positive action to encourage the influx of women and mi mathematics organizations, including the AMS. Below are norities into mathematics, science, and engineering. [This excerpts from some of the statements by these leaders. statement was endorsed by the AWM Executive Commit tee and appeared as a letter to the editor in the New York David Eisenbud, AMS president, and James Times on January 28, 2005 .] Arthur, AMS president elect: The speculations made by Lawrence Summers, President International Council for Industrial and Applied of Harvard University, at a conference on January 14, 2005, Mathematics (ICIAM): about the causes of the current shortage of women in sci Recent remarks of the President of Harvard University ence were inappropriate. His high position at Harvard have led to media speculation that innate differences in places on him a high burden of responsibility. His remarks the mathematical abilities of men and women make it less may be damaging and counterproductive to a cause he likely that women will succeed in science and mathemat and all educators should support. We who strive to make ics. IClAM does not accept this notion. IClAM members are our subject areas attractive and accessible to all express well aware that there are many barriers (whether financial, our dismay at such remarks. cultural, or practical) that face women who want to pur january 21, 2005 sue mathematical or scientific careers at the highest lev els. The unbroken career paths that are typical of successful Martin Golubitsky, president, Society for male careers in mathematics take no account of the spe Industrial and Applied Mathematics (SIAM): cific responsibities of women related to childbearing and The widely reported speculations made by the president family. As an international organization representing the of Harvard University, Lawrence Summers, about possible world's applied mathematicians, IClAM is committed tore causes of the current shortage of women in science have moving the educational inequalities in mathematics that fueled controversy and had some potentially unfortunate exist in many parts of the world and to improving the ac effects. One of the most serious is the possible discour cess to careers in the mathematical sciences for all men agement of talented women with strong potential for ex and women. cellence in science, engineering, and mathematics. SIAM's position is that it is essential to encourage women (as well February 1, 2005 as men) to pursue studies in science and mathematics. SIAM wishes to emphasize strongly the many outstanding - Allyn jackson accomplishments of women in mathematics and its ap plications. February 2, 2005 Carolyn Gordon, president, Association for Women in Mathematics: Regarding Lawrence H. Summers's remarks on the un derrepresentation of women in mathematics and science, 552 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Inside the AMS facing departments today, including personnel issues (staff Login Now Required for Online and faculty), long range planning, hiring, promotion and Notices tenure, budget management, assessments, outreach, stew ardship, junior faculty development, communication, and Following what has become a standard procedure for many departmental leadership. online publications, the AMS has begun requiring users to If you are interested in attending a future workshop, log into an AMS user account to access the online version please look for registration information sent out in advance of the Notices. The Notices remains freely available to all, of the Joint Meetings or contact the AMS Washington office and setting up an online account does not require mem [email protected]. bership in the AMS. This change was made primarily as a way to emphasize to online Notices readers that the world -AMS Washington office wide availability of the online Notices is supported by dues paid by the AMS membership. - Allyn jackson Deaths of AMS Members HAYK BADALYAN, professor, Yerevan State University, Arme nia, died on October 27, 2004. Born on June 22, 1915, he AMS Holds Workshop was a recent member of the Society. MALCOLM K. BRACHMAN, president, Northwest Oil, Dallas, for Department Chairs died on January 11, 2005. Born on December 9, 1926, he The AMS hosted a one-day workshop for mathematical sci was a member of the Society for 4 7 years. ences department chairs atthe 2005 Joint Mathematics Meet VINCENT McBRIEN, retired, Harvard University, died on Feb ings in Atlanta, Georgia. This year's workshop focused on a ruary 2, 2005. Born onApril21, 1916, he was a member of number of areas of importance to department chairs in the Society for 60 years. cluding faculty evaluations, strategic planning, effective use MARILYN MOLLOY, professor, Our Lady of the Lake Univer of resources, and accountability. Over thirty department sity, San Antonio, died on March 16, 2003.BornonApril24, chairs and leaders came together to share ideas and expe 1931, she was a member of the Society for 36 years. riences in a form of" department chair therapy", thus creating DONAlD R. MORRISON, professor emeritus, UniversityofNew an environment that enabled attendees to address depart Mexico, Albuquerque, died on September 5, 2004. Born on mental matters from new perspectives. May 3, 1922, he was a member ofthe Society for 57 years. Workshop leaders included Krishnaswami Alladi, GEORGE W. PEGLAR, professor emeritus, from San Francisco, department chair of mathematics, University of Florida; died on January 7, 2005. Born on September 2, 1922, he Deanna Caveny, department chair of mathematics, College was a member of the Society for 51 years. of Charleston; Peter March, department chair of mathe WERNER ScHOMBURG, of Bochum, Germany, died on january matics, Ohio State University; and Robert Olin, dean of arts 3, 2005. Born on June 9, 1926, he was a member of the and sciences, University of Alabama-Tuscaloosa. Society for 7 years. The Department Chairs Workshop is an annual event ] ozEF JoACHIM TELEGA, professor, Polish Academy of hosted by the AMS prior to the start of the Joint Meetings. Sciences, Warsaw, died on January 28, 2005. Born on Past workshop sessions have focused on a range of issues March 24, 1943, he was a member of the Society for 13 years. MAY 2005 NOTICES OF THE AMS 553 Reference and Book List The Reference section of the Notices Upcoming Deadlines telephone: 301-405-7892; email: is intended to provide the reader with April15, 2005: Nominations for Maria [email protected]. frequently sought information in Mitchell Women in Science Award. See May 4, 2005: Letters of intent for an easily accessible manner. New http:ll209.68.19.123lmuseumsl NSF Graduate Teaching Fellowships in information is printed as it becomes wmninsc. php, or contact the Maria K-12 Education (GK-12). See "Mathe matics Opportunities" in this issue. available and is referenced after the Mitchell Women in Science Award May 31, 2005: Registration for In first printing. As soon as information Committee at the Maria Mitchell As ternational Mathematics Competition is updated or otherwise changed, it sociation, 4 Vestal Street, Nantucket, for University Students. See http: 11 will be noted in this section. MA 02554; telephone 508-228-9198. www. i me-math. org or contact John E. May 1, 2005: Applications for AWM Contacting the Notices Jayne, Department of Mathematics, Travel Grants. See http: I lwww. University College London, Gower The preferred method for contacting awm-math.orgltravelgrants.html; Street, London WClE 6BT, United the Notices is electronic mail. The editor is the person to whom to send Where to Find It articles and letters for consideration. A brief index to information Articles include feature articles, that appears in this and previous issues of the Notices. memorial articles, communications, AMS Bylaws-November 2003, p. 1283 opinion pieces, and book reviews. The editor is also the person to whom to AMS E-mail Addresses-December 2004, p. 1365 send news of unusual interest about AMS Ethical Guidelines-]une/ ]uly 2004, p. 675 other people's mathematics research. AMS Officers 2004 and 2005 (Council, Executive Committee Publications The managing editor is the person Committees, Board of Trustees)-May 2005, p. 564 ' to whom to send items for "Mathe AMS Officers and Committee Members-October 2004, p. 1082 matics People", "Mathematics Op Conference Board of the Mathematical Sciences-September 2004, p. 921 portunities", "For Your Information", "Reference and Book List", and "Math Information for Notices Authors-june/July 2004, p. 670 ematics Calendar". Requests for Mathematics Research Institutes Contact Information-August 2004, p. 810 permissions, as well as all other inquiries, go to the managing editor. National Science Board-January 2005, p. 76 The electronic-mail addresses are New Journals for 2003-]une/]uly 2004, p. 672 noti ces@math. ou. edu in the case of NRC Board on Mathematical Sciences and Their Applications__; March 2005 p. 361 the editor and noti ces@ams. org in ' the case of the managing editor. The NRC Mathematical Sciences Education Board-April 2005, p. 465 fax numbers are 405-325-7484 for NSF Mathematical and Physical Sciences Advisory Committee-February 2005, p. 261 the editor and 401-331-3842 for the managing editor. Postal addresses Program Officers for Federal Funding Agencies-October 2004, p. 1078 (DoD, DoE); December 2004, p. 1368 (NSF) may be found in the masthead. 554 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Reference and Book List Kingdom; telephone +44-20-7679 January 1, 2006: Submissions for Alpha and Omega: The Search for 7322; fax +44-20-7419 2812; email: Competition 2006 of the European the Beginning and End of the Universe, [email protected]. Mathematical Society. See http: I I by Charles Seife. Viking, July 2003. June 1, 2005: Applications for fall www.mat.dtu.dk/people/V.L.Hansen/ ISBN 0-6 70-03179-8. program of the Christine Mirzayan rpa/secondartcomp.html. Automated Reasoning and the Dis Science and Technology Policy Gradu January 1, 2006: Applications for covery of Missing and Elegant Proofs, ate Fellowship Program of the National ICM 2006 Travel Grants. See http: I I by Larry Wos and Gail Pieper. Rinton Academies. See http: I /www7. www. i cm2006. org or email: grants@ Press, December 2003. ISBN 1-58949- nationalacademies.org/ i cm2006. o rg. 023-1. pol i cyfe ll ows or contact The Na Beyond Coincidence, by Martin tional Academies Christine Mirzayan Book List Plimmer and Brian King. Icon Books, Science and Technology Policy Gradu The Book List highlights books that March 2004. ISBN 1-840-46534-4. ate Fellowship Program, 500 5th Street, have mathematical themes and are Beyond Reason: Eight Great Prob NW, Room 508, Washington, DC 20001; aimed at a broad audience potentially lems That Reveal the Limits of Science, telephone: 202-334-2455; fax: 202-334- including mathematicians, students, by A. K. Dewdney. Wiley, April 2004. 1667. and the general public. When a book ISBN 0-471-01398-6. June 2, 2005: Full proposals for has been reviewed in the Notices, a Cogwheels of the Mind: The Story of NSF Graduate Teaching Fellowships in reference is given to the review. Gen Venn Diagrams, by A. W. F. Edwards. K-12 Education (GK-12). See "Mathe erally the list will contain only books matics Opportunities" in this issue. Johns Hopkins University Press, April published within the last two years, 2004. ISBN 0-801-87434-3. June 2, 2005: Applications for NSF though exceptions may be made in University-Industry Cooperative Re Constantin Caratheodory: Mathe cases where current events (e.g., the matics and Politics in Turbulent Times, search Programs in the Mathematical death of a prominent mathematician, Sciences (UICRP). See http: I /www. by M. Georgiadou. Springer, Septem coverage of a certain piece of mathe ber 2004. ISBN 3-540-44258-8. nsf.gov/pubsys/ods/getpub. matics in the news) warrant drawing cfm?nsfOS 504. The Constants of Nature: From readers' attention to older books. Sug Alpha to Omega-The Numbers That June 30, 2005: Nominations for gestions for books to include on the list Encode the Deepest Secrets of the Uni the 2005 Fermat Prize. See http: I I may be sent to noti ces-bookl i st@ www.ups-tlse.fr/ACTUALITES/ verse, by John D. Barrow. Jonathan ams .org. Sciences/Prix_Fermat_2004/ Cape, September 2002. Pantheon ''Added to "Book List" since the Aregl ement. html. Books, January 2003. ISBN 0-375- list's last appearance. July 31, 2005: Nominations and 42221-8. (Reviewed November 2004.) applications for the Monroe H. Martin Count Down: Six Kids Vie for Glory 13: The Story of the World's Most Prize. Contact R. Roy, Director, Insti at the World's Toughest Math Compe Popular Superstition, by Nathaniel tute for Physical Science and Technol tition, by Steve Olson. Houghton Mif ogy, University of Maryland, College Lachenmeyer. Thunder's Mouth Press, October 2004. ISBN 1-568-58306-0. flin, April2004. ISBN 0-618-25141-3. Park, MD 20742-2431. (Reviewed August 2004.) August 1, 2005: Submissions for 1089 and All That. A journey into Mathematics, by David Acheson. Ox ,, The Curious Incident of the Dog Competition 2005 of the European in the Nighttime, by Mark Haddon. Mathematical Society. See http: I I ford University Press, July 2002. ISBN 0-19-851623-1. (Reviewed February Vintage, May 2004. ISBN 1-400-03271-7. www.mat.dtu.dk/people/V.L.Hansen/ The Curious Life of Robert Hooke, rpa/secondartcomp.html. 2005.) Across the Board: The Mathematics the Man Who Measured London, by September 16, 2005: Proposals for Lisa Jardine. HarperCollins, February NSF program on Enhancing the Math of Chessboard Problems, by John ]. 2004. ISBN 0-060-53897-X. ematical Sciences Workforce in the Watkins. Princeton University Press, *Dark Hero of the Information Age: Twenty-First Century. See "Mathe April2004. ISBN 0-691-11503-6. In Search of Norbert Wiener, by Flo matics Opportunities" in this issue. ''Action This Day, edited by Michael Conway and Jim Siegelman. Basic October 1, 2005: Nominations for Smith and Ralph Erskine. Random Lucien GodeauxPrize. Contact]. Aghion, House of Canada, February 2003. Books, December 2004. ISBN 0-738- c/o Secretariat of the Royal Society of ISBN 0-593-04910-1. 20368-8. Sciences of Liege, Institute of Mathe Adam Spencer's Book of Numbers, '' The Essential Turing, edited by matics of the University of Liege, 12 by Adam Spencer. Four Walls Eight B. Jack Copeland. Oxford University Grande Traverse, Sart Tilman Bat. B 3 7, Windows, January 2004. ISBN 1-568- Press, September 2004. ISBN 0-198- B-4000 Liege 1, Belgium; email: 58289-7. 25080-0. j aghi on@ul g. ac. be. Alfred Tarski: Life and Logic, by Everything and More: A Compact October 19, 2005: Applications for Anita Burdman Feferman and History of Infinity, by David Foster NSF Postdoctoral Research Fellow Solomon Feferman. Cambridge Uni Wallace. W. W. Norton, October 2003. ships (MSPRF). See "Mathematics Op versity Press, October 2004. ISBN ISBN 0-393-00338-8. (Reviewed portunities" in this issue. 0-521-80240-7. June/July 2004.) MAY 2005 NoTICES OF THE AMS 555 Reference and Book List The Fabric of the Cosmos, by Brian The Knot Book: An Elementary In Mathematics in Nature: Modeling Greene. Knopf, February 2004. ISBN troduction to the Mathematical Theory Patterns in the Natural World, by John 0-3 75-41288-3. of Knots, Colin C. Adams. AMS, Sep Adam. Princeton University Press, No Fields Medalists' Lectures, edited tember 2004. ISBN 0-8218-3678-1. vember 2003. ISBN 0-691-11429-3. by Sir Michael Atiyah and Daniel The Liar Paradox and the Towers Meta Math! The Quest for Omega, Iagolnitzer. World Scientific, 2nd of Hanoi: The Ten Greatest Math Puz by Gregory ]. Chaitin. April 2004. edition, December 2003. ISBN 9-812- zles of All Time, by Marcel Danesi. Available at http: I jwww. cs. 38259-3. Wiley, August 2004. ISBN 0-471- umaine.edu/~chaitin/omega.html. From Eudoxus to Einstein: A His 64816-7. The (Mis)Behavior of Markets: A tory of Mathematical Astronomy, by Masters of Theory: Cambridge and Fractal View ofRisk, Ruin and Reward, C. M. Linton. Cambridge University the Rise of Mathematical Physics, by by Benoit Mandelbrot and Richard Press, August 2004. ISBN 0-521- Andrew Warwick. University of Hudson. Basic Books, August 2004. 82750-7. Chicago Press, July 2003. ISBN 0-226- ISBN 0-465-043 5 5-0. From Newton to Hawking: A History 87375-7. More Damned Lies and Statistics: of Cambridge University's Lucasian *Math and the Mona Lisa: The Art How Numbers Confuse Public Issues, Professors of Mathematics, edited by and Science of Leonardo da Vinci, by by Joel Best. University of California Kevin C. Knox and Richard Noakes. Bulent Atalay. Smithsonian Books, April Press, August 2004. ISBN 0-520- Cambridge University Press, Novem 2004. ISBN 1-588-34171-2. 23830-3. ber 2003. ISBN 0-521-66310-5. Math Magic: How to Master Every More Mathematical Astronomy Gamma: Exploring Euler's Constant, day Math Problems, by Scott Flansburg. Morsels, by Jean Meeus. Willmann-Bell by Julian Havil. Princeton University Perennial Currents, revised edition, Inc., 2002. ISBN 0-943396-743. Press, May 2003. ISBN 0-691-09983-9. August 2004. ISBN 0-060-72635-0. The Music of the Primes: Searching (Reviewed August 2004.) Math through the Ages: A Gentle to Solve the Greatest Mystery in The Golden Ratio: The Story of Phi, History for Teachers and Others, by Mathematics, by Marcus du Sautoy. the World's Most Astonishing Number, WilliamP. Berlinghoff and Fernando Q. HarperCollins, April 2003. ISBN by Mario Livio. Broadway Books, Gouvea. Oxton House, 2002. ISBN 0-066-21070-4. October 2002. ISBN 0-767-90815-5. 1-881929-21-3. (Reviewed October Number Theory from an Analytic 2004.) (Reviewed March 2005.) Point of View, by Badih Ghusayni. The Mathematical Century: The 30 A Handbook of Mathematical Dis Komati, December 2003. ISBN 9953- Greatest Problems of the Last 1 00 0-0282-7. course, by Charles Wells. Infinity Years, by Piergiorgio Odifreddi and '' The Oxford Murders, Publishing Company, 2003. ISBN by Guillermo Arturo Sangalli. Princeton University Martinez. Abacus, January 2005. ISBN 0-7414-1685-9. (Reviewed September Press, May 2004. ISBN 0-691-09294-X. 0-349-11721-7. 2004.) Mathematical journeys, by Peter D. Phase Change: The Computer Rev The Heart of Mathematics: An Schumer. Wiley lnterScience, February olution in Science and Mathematics, Invitation to Effective Thinking, by 2004. ISBN 0-471-22066-3. by Douglas S. Robertson. Oxford Edward B. Burger and Michael A Mathematician at the Ballpark: University Press, March 2003. ISBN Starbird. Key College Publishing Odds and Probabilities for Baseball 0-195-15748-6. (Springer-Verlag), April 2000. ISBN Fans, by Ken Ross. Pi Press, July 2004. Prime Obsession: Bernhard Rie 0-555953-407-9. (Reviewed February ISBN 0-131-47990-3. mann and the Greatest Unsolved 2005.) Mathematicians as Enquirers: Learn Problem, by John Derbyshire. Joseph * Incompleteness: The Proof and ing about Learning Mathematics, edited Henry Press, March 2003. ISBN Paradox of Kurt Godel, by Rebecca by Leone Burton. Kluwer, April2004. 0-309-08549-7. Goldstein. W. W. Norton & Company, Hardbound, ISBN 1-4020-7853-6; Probability Theory: The Logic of February 2005. ISBN 0-393-05169-2. paperback, ISBN 1-4020-7859-5; eBook, Science, by E. T. Jaynes, edited by '' The Infinite Book: Where Things ISBN 1-4020-7908-7. G. Larry Bretthorst. Cambridge Uni Happen That Don't, by John D. Barrow. Mathematicians under the Nazis, by versity Press, April2003. ISBN 0-521- Jonathan Cape, February 2005. ISBN 0- Sanford L. Segal. Princeton University 59271-2. 224-06917-9. Press, July 2003. ISBN 0-691-00451-X. The Reader of Gentlemen's Mail: Karl Pearson: The Scientific Life in (Reviewed April2005.) Herbert 0. Yardley and the Birth of a Statistical Age, by Theodore M. Mathematics: A Very Short Intro American Codebreaking, by David Porter. Princeton University Press, duction, by Timothy Gowers. Oxford Kahn. Yale University Press, March February 2004. ISBN 0-691-11445-5. University Press, October 2002. ISBN 2004. ISBN 0-300-09846-4. Kepler's Conjecture: How Some of 0-192-85361-9. (Reviewed February The Riemann Hypothesis: The Great the Greatest Minds in History Helped 2005.) est Unsolved Problem in Mathematics, Solve One of the Oldest Math Prob Mathematics and War, edited by byKarlSabbagh.FarrarStraus&Giroux, lems in the World, by George G. Szpiro. Bernheim Booss-Bavnbek and ]ens April2003. ISBN 0-374-25007-3. Wiley, January 2003. ISBN 0-471- H0yrup. Birkhauser, December 2003. * The Road to Reality: A Complete 08601-0. (Reviewed January 2005.) ISBN 3-764-31634-9. Guide to the Laws of the Universe, by 556 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Reference and Book List About the Cover Roger Penrose. Knopf, February 2005. Extreme 3D visualization ISBN 0-679-45443-8. The background image of this month's cover is a photograph included Signs of the Inka Khipu: Binary Cod ing in the Andean Knotted-String ?Y Jo~athan Borwein and David Bailey, perhaps somewhat whimsically, m their article on experimental mathematics. The photograph was taken Records, by Gary Urton. University of for a publicity brochure for the now defunct New Media Innovation Texas Press, August 2003. ISBN 0-292- 78540-2. ~entre in downtown Vancouver, British Columbia, an organization par tially sponsored by Simon Fraser University, to which Borwein is affili '' Stalking the Riemann Hypothe ated. The two young men, who are graduate students in the the de sis: The Quest to Find the Hidden Law of Prime Numbers, by Dan Rockmore. pa~t.ment of Electrical and Computer Engineering at the University of Bntish Columbia, are in a kind of box with what might be called Pantheon, April 2005. ISBN 0-375- 42136-X. surround-~rojection. The approximate spheres are displayed in dupli cate at rapidly alternating times in synchronization with the goggles they Strange Curves, Counting Rabbits, and Other Mathematical Explorations, are wearmg.' so that what they see is a simulated 3D image, not just the flat proJections on the walls on their box. The projections are interac by Keith Ball. Princeton University tive, controlled by input through a key pad held by Timothy Chen, the Press, November 2003. ISBN 0-691- student on the right. The project the students are involved in is part of 113 21-1. (Reviewed in December 2004.) Towards a Philosophy ofReal Math Mr. Chen's student work at U. B. C. What is being projected is a flow field ematics, by David Corfield. Oxford ?f spheres in a cylin~er with various obstacles interactively superimposed mto the flow. The mset photographs are screen displays produced by University Press, April 2003. ISBN 0-521-81722-6. Mr. Chen from the same project. It's hard to imagine exactly what role such high end visualization tech The Transformation of Mathemat ics in the Early Mediterranean World: ~ology will play in mathematical research, but not impossible. One likely ai?plication for sinlilar, but not quite so sophisticated, display sys From Problems to Equations, by Reviel Netz. Cambridge University Press, May tems nught very well be in public presentations. The effects can be spec tacular. 2004. ISBN 0-521-82996-8. Brian Corrie of Simon Fraser University provided us with the digital The Universal Book ofMathematics: From Abracadabra to Zeno's Para version of the background photograph. doxes, by David Darling. Wiley, July -Bill Casselman, Graphics Editor 2004. ISBN 0-471-27047-4. (noti ces-cover@ams. org) '' The Works of Archimedes: Trans lation and Commentary. Volume I: The Two Books on the Sphere and the Cylinder. Translated by Reviel Netz. Cambridge University Press, April 2004. ISBN 0-521-66160-9. (Reviewed in this issue.) A World without Time: The Forgot ten Legacy of Gddel and Einstein, by Palle Yourgrau. Basic Books, January 2005. ISBN 0-465-09293-4. You Can Do the Math: Overcome Your Math Phobia and Make Better Financial Decisions, by Ron Lipsman. Praeger Publishers, November 2004. ISBN 0-275-98341-2. MAY 2005 NOTICES OF THE AMS 557 American Mathematical Society Contributions Dear Friends and Colleagues, During 2004 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 fifth 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 indefinitely, and we hope to reach our goal of two million dollars over the next few years. Young scholars programs work and supporting them is important for mathematics. The Centennial Fellowships play a key role for outstanding young mathematicians at the forma tive stages of their careers, from three to twelve years beyond the degree. The fellowships are funded directly 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 devel opment@ams. org. Roy L. Adler Isidore Fleischer Ralph Mansfield William and Theda Salkilld Kathleen Baxter Ramesh Gangolli Trevor McMinn Henry M. Schaerf Shirley and Gerald Bergum Rosalind Guaraldo Cathleen Synge Morawetz Margaret W. Taft Shirley Cashwell Yanguang Li Franklin P. Peterson B. A. Taylor Carl Faith JosephS. Mamelak Moshe Rosenfeld Steven H. Weintraub KyFan Bequests Received Barbara ]. Beechler 558 NOTICES OF THE AMS VOLUME 52, 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: Gifts were made in honor of the following individuals: Maurice Auslander by Bernice L. Auslander Mike Breen by Rhode Island College Subinoy Chakravarty by Office of Quality Assurance, Michigan Jane Kister by Louis L. Lerner Family of Independence Agency The Mathematical Profession - Anonymous William G. Chinn on behalf of the William & Grace Chinn Trust Franklin Tepper Raimo by Deborah Tepper Haimo Irving Reiner by Irma M. Reiner Contributors to the AMS During 2004 ,., Donors who have given for three years consecutively. E 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 £ Deborah Tepper Haimo E*Bengt G. Carlson e*David Harbater £*Jeanne LaDuke £*Robert v. Kohn £ Theodore S. Chihara ,, Carl E. Harrell £ Jean Pierre Lafon ASSOCIATES £*George F. Leger £*Stuart Citrin £*Adam O'Neill Hausknecht £*Joseph A. Langsam (Gifts of $5,000 and above) £ Ralph Mansfield £*Alfred Clark Jr. Carl A. Haver! £*Richard Snyder Laugesen Estate of Barbara J Beechler £ David B. Massey £*William A. Clee ·• Henry Helson E*H. Blaine Lawson Jr. * Roger E. Howe £'' M. Susan Montgomery £*Edward T. Cline Jr. £*Leon A. Henkin £*Walter R. Lawson £*Harry Lucas Jr. Vaughan R. Pratt £ John Coffey £*James B. Herreshoff £ Peter D. Lax £*Thomas R. Savage £''Paul ]. Sally Jr. e*Daniell. A. Cohen £*Gerald A. Heuer e* Alan C. Lazer *• Robert and Maria W. £*Richard M. Schoen £*Thomas Carney Corrigan * Gloria C. Hewitt £''Joan R. Leitzel Steinberg £*Joel H. Spencer £''Louis J. Cote £ Stephen R. Hilbert £*H. W. Lenstra Jr. Alan and Katherine Stroock £"Jean E. de Valpine £*Stephen H. Crandall £*Peter]. Hilton William]. LeVeque Fund £ Wolfgang L. Walter e* Albert W. Currier £ Melvin Hochster £ Bingxi Li * Sally Whiteman- Albert Leon £ Buck Ware £''Everett C. Dade £*Samuel S. Holland Jr. £ Zvie Liberman Whiteman Prize £*Susan Schwartz Wildstrom £ David B. Damiano £ Charles S. Holmes £"Elliott H. Lieb Anonymous (l) Anonymous (5) Ingrid Daubechies £ Raymond T. Hoobler e*Walter L. Lok E*Robert ]. Daverman £ James E. Householder £"'Russell D. Lyons £*Paul L. Davis £*James G. Huard £ Mark Mahowald ASSOCIATES PATRONS £*Guy M. De Primo * Joseph A. Hughes £"Joseph S. Mamelak (Gifts of $1,000 and above) (Gifts of $100 and above) £ Michael E. Detlefsen £''George W. Hukle £*Stefano Marchiafava £*Thomas]. Marlowe Jr. £*Clarence M. A blow £*Robert L. Devaney £*Craig L. Huneke * Richard D. Anderson £ John Norman Mather Ax £ Robert D. M. Accola £*Charles E. Dickerson £ Thomas W. Hungerford £ James * Jacob R. Matijevic £*Roy L. Adler £*Gerald P. Dinneen £ James F. Hurley £ Felix E. Browder R. Daniel Mauldin £*Alfred Aeppli £*Martin]. Dowd £*Franklin T. Iha £*Nathaniel Chafee £''Stephen B. Maurer £*Ali R. Amir-Moez £*Robert L. Druet E'~Hiroshi Inoue £ Richard A. Cleveland £*James W. Maxwell ··· Neal R. Amundson £ Loyal Durand E*Eugene Isaacson £*William Craig £ James W. Maynard £*Peter L. Duren £*Lloyd K. Jackson £*John H. Ewing £*George E. Andrews £ Patricia Tulley McAuley £*Earl F. Ecklund Jr. £ Bradford W. Johnson £''George F. Haddix e*·Kenneth I. Appel £ William G. McCallum £*Sigurd Elkjaer £ David W. Jonah *• Phyllis and Donald Kahn £ Abraham Arad Hochman £ Michael M. McCrea ,., Richard A. Askey £ Paul Feit £*James H. Justice Philanthropic Fund £ Leon R. McCulloh £*Gerald B. Folland £*Henry Price Kagey * Joseph E. Kist * Walter 0. Augenstein E'' Dusa McDuff Paul Fong £*Herbert M. Kamowitz £*Greg Kuperberg and Rena ]. £"Donald G. Babbitt '' Lionel W. McKenzie £"'William Fulton Yoshikazu Karamatsu Zieve Claude W. Bardos £ Bruce E. Meserve £* Joseph M. Gani £ Julian R. Karelitz Louis L. Lerner £ Domingo Barrera-Rosillo £ John ]. Michels e··'William James Lewis £*Hyman Bass £*Richard L. Gantos £''Jerome Karle * David Middleton E* Jacques Neveu e*Frances B. Bauer £*John B. Garnett £'' John F. Kellaher £ Russell G. Miller E Morris Newman £ Jack Bazer * Irving E. Gaskill £ Herbert B. Keller £*Guido Mislin £*Vladimir V. Piterbarg £ Robert Beig £ Hillel H. Gershenson £ Wayne G. Kellner * Richard A. Moore £*Samuel Murray Rankin III £ Gary D. Berg £ Murray Gerstenhaber E'~Harry Kesten £*Cathleen Synge Morawetz * Norton Starr £"Gerald E. Bergum £*Stephen P. Gill £*Barbara Lee Keyfitz and £ Frank Morgan £*Steven H. Weintraub £ Leonard D. Berkovitz £*Anthony A. Gioia Martin Golubitsky * Yasuhiro Morita Anonymous (5) e*• David S. Berry £*Samuel Gitler E Davar Khoshnevisan E Robert A. Morris £ Joan S. Birman £''Richard P. Goblirsch Andrei Khruzin Lee D. Mosher £''Seth I. Goldberg £ Allan M. Kirch SPONSORS £*Richard L. Bishop £ Andrew P. Mullhaupt Jerry L. Bona £*Carolyn S. Gordon E'' Jane Kister £''Albert A. Mullin (Gifts of $500 and above) * Malcolm K. Brachman £*'Curtis Greene E*Maria Margaret Klawe '' Paul ]. Nikolai '' Arielle and Jerald Brodkey £*David M. Bressoud £ Wilfred Martin Greenlee £''Roland R. Kneece Jr. E'~Nobuo Nobusawa Philanthropic Fund £ John Bromback £''Edward H. Grossman £ Michael Hugh Knowles £ Mitsuyuki Ochiai William & Grace Chinn Trust £ John E. Brothers £ Elizabeth Grossman and £''Eric]. Kostelich £*Donald S. Ornstein £ Paul Dana Cole £ W. Dale Brownawell Joshua Boorstein £ Jerald]. Kovacic £*Robert Osserman * William L. Duren Jr. £*Robert L. Bryant £*Victor W. Guillemin £ Thomas R. Kowalski £*Scott C. Otermat £*Timothy and Laurie Francis- £*Adam Buraczewski £ Ruth M. Hailperin £ Henry A. Krieger £* Istvan Ozsvath Wright £ Almut Burchard £ Heini Halberstam £ Masatake Kuranishi Hiram Paley * Andrew M. Gleason £* James L. Calvert £*Mary-Elizabeth Hamstrom £"Gary ]. Kurowski Alexander Perlin MAY 2005 NOTICES OF THE AMS 559 AMS Contributions <*William L. Perry * Alun Wyn-Jones e*Erik Jan Balder e*Rupert D. Boswell Jr. e*Ruth M. Charney e I. Piatetski-Shapiro e Tatsuh.iko Yagasaki e* John T. Baldwin • Truman A. Botts * Ronald J. Chase <*Gilles Pisier e*Masayuki Yamasaki E Karen Teresa Ball • Djamel Bourbia • John E. Chavez <*Thane Earl Plambeck e Ann Yasuhara • John W. Ballard e* Aldridge K. Bousfield e''Pak Soong Chee e*Vera S. Pless e* J. Michael Yohe e*William R. Ballard e*Ward D. Bouwsma •*Jeff Cheeger John C. Polking e Genji Yoshino <*Donald H. Ballou Paul J. Bowron e* Concordia C. Chen * Aleksey Popelyukhin E Charles T. Zahn E Arienne S. Balser e* JohnS. Bradley * Kwan-Wei Chen • Sidney C. Port e* Allen D. Ziebur <*William David Banks e*Richard C. Bradley E Sh.iing S. Chern e*Paul H. Rabinowitz E Paul F. Zweifel * S. F. Barber • Steven B. Bradlow * Robert Chew James V. Ralston Anonymous (54) • Carlo Bardaro * Louis R. Bragg E Peter H. S. Chiang e*M. M. Rao e* Julio Edgardo Barety e*Kenneth A. Brakke e* Jayanth.i Ch.idambaraswamy e* John Elton Rawson • Nikola Baricevic E Alberto Branciari e*Poon Chit E Coke Stevenson Reed FRIENDS * Bruce H. Barnes • Larry J. Brant E Sergei V. Chmutov E Irving S. Reed (Gifts of less than $100) e* Jose Barros-Neto e*Fred Brauer * Choong Yun Cho e*Robert J. Reynolds Johan F. Aarnes e David J. Barsky e*George U. Brauer e Jal R. Choksi E Tong-Sh.ieng Rhai • Ian M. Aberbach * Karl F. Barth John C. Breckenridge e*Charles C. Chouteau E Rhode Island College William Abikoff e* Alexander Barvinok E Joseph P. Brennan E Hubert E. Chrestenson e Stephen J. Ricci <*William P. Abrams e*Felice D. Bateman E Richard P. Brent * William E. Christilles • Richard S. Rich e* Colin C. Adams e*Paul T. Bateman e* James G. Bridgeman e*Demetrios Christodoulou * John F. Richards e*William W. Adams £''Patricia E. Bauman * Joseph Edward Brierly * Philip T. Church e*Marc A. Rieffel e*Iain T. A. C. Adamson e* J. Thomas Beale * Judith E. Broadwin <*Richard C. Churchill • Vijay K. Rohatgi <*Irving Adler e*R. Michael Beals e* Jerald S. Brodkey Philippe G. Ciarlet e Guillermo Romero Melendez • Jeffrey D. Adler E Homer F. Bechtell James 0 . Brooks e*Paul Civin e*Gerald Rosenfeld e*Max K. Agoston <*William A. Beck e William Browder • Kevin F. Clancey t Emilio 0. Roxin e*T. Aikou e* Edward Beckenstein e*Edgar H. Brown Jr. E Chester Dodge Clark Alexander A. Rylov <*Michael!. Aissen e*David S. Becker E Farris Wayne Brown e* Jeffrey W. Clark • Jeffrey R. Sachs <*Ethan J. Akin * William H. Beckmann e George B. Brown e*Robert A. Clark • Hector N. Salas E Yousef Alavi e* John A. Beekman e*Kenneth S. Brown E Sally Ann Clark e*Habib Salehi e*Michael 0 . Albertson E Wolfgang Bell e*Lawrence G. Brown e* Jack D. Clayton e Anita J. Salem e* Stephanie B. Alexander • Katalin A. Bencsilth <*Robert F. Brown • C. Herbert Clemens e*Hans Samelson <*Gerald L. Alexanderson e* Julius S. Bendat e* Sharan Inez Brown E Philip A. Cobb e* Paul T. Schaefer e*M. Kursheed Ali E Robert L. Benedetto • Gary Grant Brownell e* Alan Cobham E Norman L. Schryer JohnS. Alin • James W. Benham E Paulo Brumatti E James A. Cochran Cedric F. Schubert e* Anne H. Allen E Carlos Benitez E Rogier Brussee e* John C. Cock e Werner Schuett e J. Thomas Allen III e* Georgia Benkart <*Billy F. Bryant e* James Wesley Cogdell Jerry D. Schuur e Henrique Mariano C. Amaral • George Benke • Clifford M. Bryant Jr. e*Amy Cohen E Lester J. Senechal * Alain Amiouni E Lynn S. Bennethum e* Joseph T. Buckley e Frederick R. Cohen e*Norman E. Sexauer e Dinu D. Amzar e*Dave Benson <*Richard S. Bucy • Paula Cohen-Tretkoff E Abdulalim A. Shabazz e Fredric Davis Ancel • Carlos A. Berenstein • Steven Allen Buechler e* Donald L. Cohn e*Yuji Sh.imizuike Allan G. Anderson e*George M. Bergman e*Royce E. Buehler E Harvey Cohn E Frederic W. Shultz e*Douglas R. Anderson • Dennis D. Berkey • Sebastian loan Buhai <*Charles A. Cole e* Allan J. Silberger e* Joel H. Anderson E Elwyn R. Berlekamp Pierre Victor Bulens e*George Cole e* John R. Smart E John D. Anderson * Salvatore D. Bernardi e* Stephen S. Bullock E Vincent E. Coli Jr. E Roy C. T. Smith Marlow E. Anderson e* Chris Bernhardt £*Robert Bumcrot * Susan Jane Colley <*Louis Solomon e*Michael T. Anderson e Joseph Bernstein * Daniel Willis Bump * Daniel Comenetz Steven Robert Spencer E Susan Andima e* Swanhild Bernstein • R. B. Burckel e*Frank F. Connor <*Olaf P. Stackelberg * Peter P. Andre • John A. Berton • James E. Burke * Peter S. Constantin e*lvar Stakgold E Irina Anno * Vittorio Berzi e*Donald L. Burkholder Roberto Conti e Richard P. Stanley E Philip M. Anselone * James S. Bethel Daniel M. Burns Jr. Kenneth L. Cooke * Robert L. Stanley <*Michael Anshel e* Gerhard Betsch e*Warren T. Burns Jr. * Thomas A. Cootz Benjamin Austin Steele e*Peter H. Anspach e*Edward James Bevan e''Ralph Stevens Butcher e* Arthur H. Copeland Jr. e John R. Stock e* Stuart S. Antman e*William A. Beyer e* James S. Byrnes * Heinz 0. Cordes <*Wilhelm F. Stoll E Benedict G. Archer e Nicholas J. Bezak Luciano Caccianotti Constantin Corduneanu e*Richard W. Sullivan e*Myla M. Archer E Gau tam Bharali E Robert Calderbank e*H. Cornet <*Margaret W. Taft e Richard F. Arenstorf e*Meempat Bhaskaran M. Carme Calderer Chris Cosner <*Nicholas F. Taussig e Arturo Fernandez Arias <*Marilyn S. Bickel e Thomas E. Caldwell Ovidiu Costin e*B. A. Taylor e* Susumu Ariki * Klaus D. Bierstedt e* James J. Callahan <*Malcolm A. Coulter <*Laurence R. Taylor e* Dmitri Avenirovich Louis J. Billera <*Massimo Campanino e Lenore J. Cowen • John A. Thorpe Arkhanguelski e''Martin Billik * L. Lorne Campbell E Walter Craig e* James G. Tlrnourian e*Martin Arkowitz e Russell G. Bilyeu e Eric Cane! E Michael G. Crandall E Selden Y. Trimble V e* Judith M. Arms • John J. Bircher E Sylvain E. Cappell e*Thomas M. Creese * Walmer V. Vasconcelos <*Thomas E. Armstrong e Terrence Paul Bisson e Corrado Cardarelli Jacqueline P. Criscenti e*David A. Vogan Jr. Patrick Artaud '' Peter B. Bjorkiund * Bille C. Carlson * Florence G. Critchlow * Karen Vogtmann Esperanza Blancaflor Arugay e* Jerome Blackman E James A. Carlson e*Ernest S. Croat III e* John Thomas Walsh * Winifred A. Asprey e*Denis L. Blackmore E Jon F. Carlson E D. J. Crowley <''Frank W. Warner ill PanAtela <*William D. Blair E Amos Joel Carpenter * Helen F. Cullen e*William E. Warren E Joseph N. Atwater e* J. A. Rod Blais James B. Carrell <*Frederic Cunningham Jr. e*Mutsuo Watabe e*Bernice L. Auslander E Albert A. Blank Ron M. Carroll '' Donald L. Curlovic e* Anthony A. Weidner E Jeremy Avigad • John D. Blanton e*David W. Carter e Bradley N. Currey E Alexander H. Weintraub Luchezar L. A vramov e* Steven E. Bias berg Thomas J. Carter e Charles W. Curtis <*Ellen Westhelrner e*Sheldon Axler E Kbenesh W. Blayneh e Paolo Casati <*Herbert]. Curtis e*Thomas A. Weston <*Christine W. Ayoub Michael N. Bleicher E Charles Case e*Philip C. Curtis Jr. e*Kath.ieen B. Whitehead * Raymond G. Ayoub * David S. Bloom e*Burtis G. Casler • Robert Turner Curtis e*Roger A. Wiegand e*Kiyosh.i Baba • Antonia W. Bluher Jennie Virginia Cassell Raul E. Curto e* John F. Wilkinson e* Albert E. Babbitt Jr. E Dorothee Jane Blum Phyllis J. Cassidy <*David Scott Cyphers e* Alfred B. Willcox e William W. Babcock e''John M. Boardman e*Erio A. Castagnoli • Robert C. Dalang e G. Kenneth Williams e* George Bachman <''Istvan Bodnar E Alfred S. Cavaretta e John P. Dalbec Margaret A. Williams E William G. Bade e M. V. Bodnarescu e* Gulbank D. Chakerian e John W. Dalida E lzaak Wirszup <*Richard J. Bagby e* Gebhard Biieckle e* Jag dish Chandra <*James N. Damon e*Scott A. Wolpert t*Paul M. Bailyn e* S. Elwood Bohn e*Chao-Ping Chang <''Martin P. Dana Jay A. Wood e*Charles R. Baker GuyBoillat E 1-Lok Chang John P. D'Angelo e Alan C. Woods * Kirby A. Baker <*Theodore S. Balis e* Sun-Yung Alice Chang George B. Dantzig • Bostwick F. Wyman Matthew Howard Baker e*Francis Bonahon Paul Jackson Channell * Donald A. Darling 560 N OTICES OF THE AMS VOLUME 52 N UMBER 5 AMS Contributions e George Dassios e Leonard Evens £ Sarah Glaz e* james H. Hedlund e*Henry G. Jacob £ Boris A. Datskovsky E W. Norrie Everitt £*Raoul F. Gloden £ Jean-Francais L. Heitz * William Burkley Jacob •* Chandler Davis E* Carl Faith £"]. D. Goddard e Sigurdur Helgason £''Robert E. Jaffa * Donald M. Davis E Edson de Faria Kazimierz A. Goebel £*Leo Hellerman E Martin Jakob £ Martin D. Davis £ Bruno Farina E Abraham Goetz e Simon Hellerstein Thomas F. Jambois £ Jane M. Day Frank A. Farris £ Robert Gold e LaRae C. Helliwell £''Melvin F. Janowitz £*Anthony T. Dean £*Ruth G. Favro E"Samuel Goldberg * John P. Hempel e Heera La! Janwa Stephen M. DeBacker * George F. Feeman * Dorian Goldfeld E* judson Hendelman E Herbert Jarszick £ Lokenath Debnath £*Solomon Feferman £ ]. K. Goldhaber E*Francis McVey Henderson E*Trevor M. Jarvis Michael D. Delahunty E*Mark E. Feighn Lawrence Goldman E* James V. Herod E Leslie Stephen Jennings £ Dominick Del Casale £ Mikhail Feldman E*William Mark Goldman £ Andre Hertkorn E Charles H. Jepsen £* Aristide Deleanu £ Norman Feldman E Daniel A. Goldston E john 0. Herzog Vikram Jha Franklin D. Demana E john C. Fenley * Michael Golomb £*Georg Hetzer E*Paul J, Johanson £*Ralph E. DeMarr E*Rosanne M. Ferdico E*Martin Golubitsky £*Michael]. Heumos E David J. john Sergei S. Demidov E*lan M. Ferris E Jose Luis Gomez Pardo E Herbert K. Heyer £ Eugene C. johnsen £ Darinka Dentcheva £*Victor G. Feser E* Kenneth R. Good earl e Donald G. Higman * Trygve Johnsen £ Etienne Desquith * Maurice C. Figueres £*Roe W. Goodman £*Hugh M. Hilden £*Charles N. johnson * Dennis DeTurck E Steven R. Finch £ Suzanne Dinga Goodrich e* Shirley A. Hill £ Chris Lee Johnson £*Fred I. Diamond E*Benjarnin Fine £*Rudolf Gorenflo e Alexandrou A. Himonas E D. Randolph Johnson e Charles R. Diminnie E Leib Finkelstein £"Richard P. Gosselin e*Neil B. Hindman * Dale Martin Johnson £''Glen Allen Dobbs E*Benji N. Fisher E John A. Goth Charles W. Hines £ David R. Johnson £*Theresa K. Dodds e*Uri Fixman E Yasuhiro Goto £*Nancy Hingston E*Donald G. Johnson £ Jozef Dodziuk E*Harley Flanders * Rainer Gi:ittfert £*Peter David Hislop £''Guy johnson Jr. e* Heinz Deitrich Doebner £ Leopold Flatto £*Claude Goutier e Chungwu Ho e Harold H. Johnson £ Ogun Dogru Bernard A. Fleishman £ Sidney W. Graham e Arthur M. Hobbs £*jon L. Johnson £ Pierre E. Dolbeault £''Richard J. Fleming £ Bernhard Gramsch e Stephen L. Hobbs £ Kenneth Walter Johnson Igor Dolgachev £ Wendell H. Fleming £*Kevin A. Grasse * Harry Hochstadt e*Norman W. Johnson * joanne M. Dombrowski £ Elias G. Flytzanis E Jack E. Graver £ Scott H. Hochwald e Theodore D. Johnson £ Simon K. Donaldson £*Micah E. Fogel E*Larry K. Graves e*Richard E. Hodel e* Charles H. Jones £*James P. Donaly '' Susana F. L. de Foglio E Richard ]. Greechie E* Jonathan P. E. Hodgson £ Kathryn A. jones £ Donald R. Doolittle £*Julie A. Fondurulia E Richard Mutegeki Green E*Helmut H. W. Hofer £*Vaughan F. R. Jones £ F. M. W. Doove * Marc F. Fontaine E*William L. Green e··'Michael E. Hoffman £*William B. Jones Robert S. Doran E Benjamin J. Ford E Peter H. Greene TomH0holdt £ Jacob C. Joosten £ Alberto M. Dou E Hans-Bj0rn Foxby £*Frederick P. Greenleaf Harald R. A. Holmann E*Troels jorgensen •* Jim Douglas Jr. * Walter L. Foxworth E* John Gregoriou E*Philip John Holmes e*Virginia V. ]ory £ Ronald G. Douglas * Michael Lee Frame Phillip A. Griffith E*Richard B. Holmes Joseph Etienne juarez £ Karl Heinz Dovermann Arne Fransen E*Leonard Gross £ Jean MacGregor Horn * Richard K. Juberg £*Robert E. Dowd £ Timothy]. Frawley E Robert Andrew Grossman E* john M. Hosack E Joaquim]. A. Judice Diane M. Dowling Scott Gordon Fraze E*Edward A. Grove Jim E. Haste E Winfried Just £* ]. Scott Downing e*Michael W. Frazier E Larry C. Grove E Philip K. Hotchkiss E John Kahila •* Alex J. Dragt E*Daniel S. Freed Gerd Grubb E*Fredric T. Howard E* Jeffry N. Kahn Alexander N. Dranishnikov £ Thomas Hall Freeman E* A. F. Gualtierotti * Henry C. Howard E*Peter]. Kahn £*Gerald C. Drew E Christopher L. Frenzen E*Rosalind ]. Guaraldo e*W. L. Hoyt E* ]oji Kajiwara e* Arthur A. Drisko £ Marianne Smith Freundlich E* Michel Guillerault * ]. S. Hsia E Yilichir6 Kakihara £ Bruce K. Driver * Peter]. Freyd * Samuel L. Gulden £*Pao-sheng Hsu E Agnes M. Kalemaris * Thomas L. Drucker E Stephen H. Friedberg * Robert D. Gulliver II e*Tiao-Tiao Hsu E N.]. Kalton Thomas E. Duchamp E*Eugene M. Friedman E*Richard K. Guy E*Verena Huber-Dyson E*Yoshinobu Karnishima £ James S. Dukelow Jr. E* joyce B. Friedman £*Wynne Alexander Guy E* Archibald Perrin Hudgins Edgar D. Kann £*SteveN. Dulaney E*Merwyn M. Friedman E*Kwang Chul Ha e*Denise Huet E*Richard A. Kanner £*William Dart Dunbar Jr. £*Daniel E. Frohardt £*Seymour Haber E*Ruth L. Hughes E* Stanley Kaplan T. E. Duncan £* Jurg M. Frohlich A. Glen Haddock E*Mark E. Huibregtse E*Wilfred Kaplan £ Alan H. Durfee e*Hisanori Fujita £ Lee 0. Hagglund £*Birge K. Huisgen- e*Ioannis Karatzas Kanat Durgun e*Hidenori Fujiwara E*Peter Hagis Jr. Zimmermann E*Martin Lewis Karel £*Clifford J. Earle £*Koji Fukuda E james N. Hagler * james E. Humphreys £* Johan Karlsson £''Timothy R. Eaton E William R. Fuller * Susan G. Hahn Bruce Hunt * Svetlana R. Katok £''Patrick Barry Eberlein £*Jeffry B. Fuqua £*William D. Hahn e*Karen C. Hunt E*Victor ]. Katz £ Ernest ]. Eckert * Lis! Novak Gaal £ Gerhard E. Hahne £ Paul M. Hunt E* Arthur Kaufman £ Allan L. Edmonds E*Richard F. Gabriel * John A. Haight Walker E. Hunt Eric Roger Kaufmann £''H. P. Edmundson E*Steven Allen Gabriel Dawit Haile • Beryl E. Hunte E* John P. Kavanagh Harold M. Edwards £*Andrei Gabrielov £ Mark Haiman £*Marcel Hupperich E jerry L. Kazdan Paul ]. Eenigenburg E Michael E. Gage E* Richard M. Hain £*Michael G. Hurley £ Anthony ]. Keeping £ William I. Eggers E*Luis Gallardo ·• Alfred W. Hales Taqdir Husain £ James E. Keisler £ Kurt Merrell Ehlers E joseph A. Gallian E*R. Stanton Hales Jr. * Carol Hutchins £*Edward L. Keller £ David L. 0. Ehren E* Jean H. Gallier £*Robert Joseph Halk e Edward D. Huthnance Jr. E David C. Kelly Edwin G. Eigel Jr. E Carl C. Ganser E Brian C. Hall e Edward L. Hutton £ Jerry S. Kelly * Stanley Mamour Einstein * Howard Garland £ james E. Hall £* Jang C. Hwang £*John B. Kelly Matthews E Charles Ray Garner Jr. E''David Handel £ John Jew-Chen Hwang £ Daniel C. Kemp Sylvan H. Eisman £*Daryl Neil Geller £ Kit Hanes Francesco Iachello E"]. H. B. Kemperman e Elmer Eisner £ Stuart Alan Geman £*John L. Hank e*Tom Ilmanen E* Edward S. Kennedy £*Alexander S. Elder £*Jean Raymond Genet * ]. Ray Hanna Stanko Indihar '' George A. Kent e*David Eliezer £ Constantinos Georghlou E John Michael Harby E Nancy Jane Ingram e R. Keown E* Joanne Elliott * John C. Georgiou E*Beverly Bailey Hargraves * Ron Irving £ Eugene Thomas Kernan £ Steven P. Ellis * Stephen R. Gerig Bruno Harris Richard E. Isaac £ Panayotis Kevrekidis * Richard S. Elman £''Margaret P. Gessaman •* Steven Guy Harris E Godfrey L. Isaacs £'' Efim Khalimsky * Paul W. Eloe E David Gieseker £*Fred F. Harrop E* l. Martin Isaacs £* Kazuo Kid a '' Gerard G. Emch E*David S. Gillman £ Juris Hartmanis E Noburo Ishii * Rudger W. Kieboom E*Thomas J. Emerson E Jane P. Gilman £ Ki-ichiro Hashimoto * Mourad E. H. Ismail E Peter C. Kiessler £''Hans P. Engler E*Maurice Eugene Gilmore Kazuyuki Hatada * Masanori Itai Steven]. Kifowit '' Philip G. Engstrom Jack E. Girolo e'~ Akio Hattori e*Noboru Ito e'' jun Kigami E''Benjamin Epstein £ Colin M. Glanville e'~ Furnio Hazama E*Eric Robert Jablow E John Killeen £ John M. Erdman E George Glauberman £ Melvin L. Heard e Allyn Jackson £ Djun Maximilian Kim E'~ Kumar Eswaran £ John Daniel Glaus e William S. Heck E William H. jaco Tatsuo Kimura MAY 2005 NOTICES OF THE AMS 561 AMS Contributions * Wilfred M. Kincaid £ John M. Lee £ Edward Manougian E*Tetsuro Miyakawa £*Peter]. Olver Donald R. King £ Robert N. Leggett Jr. £*Margaret 0 . Marchand £ Shigeaki Miyoshi £*Philip]. O'Neil £ L. Richardson King £*Shawn Pierre Legrand * Scott A. Markel Victor ]. Mizel Barrett O'Neill £*Walter W. K.irchherr R. Sherman Lehman * David E. Marker E*Hisao Mizumoto £Ken Ono £*Alexandre A. K.irillov £ Gustav I. Lehrer Marat Markin £ Michael ]. Molnar £ Francis 0. Onukwuli £ Ellen E. Kirkman Gerald M. Leibowitz £*Charles Michel Marie £*]. Donald Monk * Yoshitsugu Oono * Jan Kisyitski E*L. Carl Leinbach £ Edwin L. Marsden Jr. £ Hugh L. Montgomery £ Seth F. Oppenheimer E*Kouichi Kiyokawa £*Cecil E. Leith £ David Imler Marshall £*Peter L. Montgomery £*Edward T. Ordman £*Peter H. Kleban E*Manoel Jose M. S. Lemos £*Murray Angus Marshall £*Barbara B. Moore Kent Orr £ Moses Gabriel Klein £ George F. Lenz £ John M. Marstrand £*Hal G. Moore £*MasonS. Osborne * Bruce A. Kleiner E*Hernan Leon Mario Umberto Martelli £ Robert Laurens Moore £*Steve G. Oslon £ Bradford Kline £*]ames I. Lepowsky £*Daniel Martin £ W. Keith Moore £*James M. Osterburg £*Julia Knight £*Raymond Leroide £*Donald A. Martin £ Marjorie A. Moretz £ T. G. Ostrom £ Ronald]. Knill £*Steven C. Leth £*George E. Martin £ Christopher L. Morgan £ Javier Otal £ Alan P. Knoerr '' Robert ]. Levit £*Wallace S. Martindale Ill £ John W. Morgan £*James C. Owings Jr. £*Marvin I. Knopp £ Joel Levy Bernard Maskit Larry]. Morley Takashi Oyabu £ Charlotte A. Knotts-Zides £ Andrew D. Lewis * Robert M. Mason £ Peter C. Morris E*Michio Ozeki £*John R. Knudsen £*Andrew M. Lewis £*Frank H. Mathis * John A. Morrison Raymond E. Ozimkoski £*Donald E. Knuth E*D. ]. Lewis E*Yasuo Matsushita E james Morrow £ Cardenas Agustin Pacheco E*Yukio Kobayashi £*George M. Lewis £*Farley Mawyer £*Joseph G. Moser £*Joseph W. Paciorek Richard M. Koch E*H. L. Lewis £*Charles N. Maxwell Pierre Marie Moussa £*Judith A. Packer £*John G. Koelzer E*L Gaunce Lewis Jr. , • John C. Mayer £*Benjamin N. Moyls Lowell ]. Paige Alfred Koestner Roger T. Lewis £*Raymond A. Mayer ]r. Motohico Mulase £* Jingyal Pak E*Yoshiharu Kohayakawa £ Frederick W. Leysieffer £ Jose Bernardo Mayorga £*Marvin G. Mundt £ Bruce P. Palka £*Kurt Siegfried Kolbig * Yanyan U E*Rafe Mazzeo £*James R. Munkres E*Diethard Ernst Pallaschke E*Yasuo Komori Pierre Liardet £*Stephen ]. McAdam £ David C. Munton John H. Palmieri Sabine Koppelberg Richard ]. Libera £ Byron Leon McAllister E*Nobuyuki Murai Jagdish N. Pandey Gabor Korchmaros * Stephen Lichtenbaum * Michael ]. McAsey £ Brian H. Murdoch E*Taxiarchis Papakostas £ Antoni A. Kosinski E*F. W. Light Jr. Vincent 0 . McBrien £ Grattan P. Murphy £ Diego Pareja-Heredia £ P. Robert Kotiuga Eduardo Lima de Sa £ Dennis ]. McCaughan £ M. K. Venkatesha Murthy £ Kyoo-Hong Park Manfred Kotter E*Shen Lin £ Mark W. McConnell £ ]an Mycielski £*George D. Parker £* Zdislav v. Kovarik £ Wen-Hsiung Lin £ Thomas McConnell * Takasi Nagahara £ Thomas H. Parker E*Yoshihiro Koya £*John E. Lindgren £*Robert A. McCoy £*Alexander Nagel £*Alberto Parmeggiani £ Andrzej Kozek £*Peter A. Linnell £ Thomas L. McCoy £ Oleg N. Naikine * Walter R. Parry • Irwin Kra £ Miriam A. Lipschutz-Yevick £ Bernard R. McDonald E*Hirotada Naito £ William Parry £*David P. Kraines £*Sally Irene Lipsey £*John G. McDonald E*Kuniaki Nakamitsu e* Antonio Pasini £*Allan M. Krall £*William G. Lister £*William D. Mcintosh £ Masataka Nakamura * Nicholas Passel! £* Jurg Kramer £ Chiu-Chu Melissa Liu £*Thomas G. McKay E*Kazumi Nakano £*Henry]. Passerini £*Raymond F. Kramer Jr. Ming Chit Liu £ Henry P. McKean Jr. £ John F. Nash Jr. £*Donald S. Passman £ Herbert C. Kranzer E*Tsai-Sheng Liu £*Robert W. McKelvey £*Joseph Neggers * John]. Pastor £ Bernd Krauskopf E*Xiaoyan Liu £ James P. McKeon £ Edward 0. Nelson £ Nicholas ]. Patterson £ Donald L. Kreider £ Albert E. Livingston E*T. G. McLaughlin Roger B. Nelson Walter M. Patterson lli £*Gary R. Krumpholz £*Stuart P. lloyd £ Elizabeth W. McMahon E*Csaba Nemethi £ Charles M. Patton £ Peter Kuchment £*George W. Lofquist £ George Joseph McNinch * Umberto Neri £*Sandra 0 . Paur E*Keisaku Kumahara £*William C. Lordan £*Robert C. McOwen £*Walter D. Neumann £*Lawrence E. Payne Gulsen Kurem £ George G. Lorentz Janet M. McShane £ Lee P. Neuwirth £*Robert G. Payton £*Robert P. Kurshan £ Martin Lorenz * Ralph D. McWilliams £ Paul Neva! £ Rachel H. Pearlman £*Herbert Kurss * Michael P. Loss £*John C. Meakin £*Charles W. Neville £*Edward A. Pedersen £*John F. Kurtzke Jr. £ Benjamin A. Lotto £ David Meier £ Charles M. Newman ]ames M. Peek £ Nosup Kwak llppo Simo Louhivaara e*Paul Meier £*Peter E. Ney £ Lambertus A. Peletier Leong-Chuan Kwek £ L 562 N OTICES OF THE AMS VOLUME 52 N UMBER 5 AMS Contributions £''David S. Protas £*RobertS. Rumely e*Dev Sinha e*Kazuaki Taira £*Mark E. Watkins £* J6zef H. Przytycki £*Leon W. Rutland Jr. £ Hardiv H. Situmeang e*Lajos F. Tak<\cs Antoni Wawrzyli.czyk £*Eric Todd Quinto £*Dorothy S. Rutledge E WalterS. Sizer E Richard B. Talmadge Edward C. Waymire £ Serban Raianu Patrick J, Ryan e*Thomas Skill E*Dov Tamari £ Cary H. Webb £*Louis B. Rail £ Robert D. Ryan £ David L. Skoug Peter C. Tan £ Fu-Chan Wei e*Melapalayam S. Ramanujan £ Charles John Rykken £ Eric V. Sind £ Sao Tang Tan £ Clifford E. Well £*George N. Raney £ Cihan K. Saclioglu e Bryan A. Smith e*Hisao Tanaka £*Lawrence R. Weill •• R. Michael Range £*Anthony Sacremento E Karen E. Smith Yoshihiro Tanaka £''Hans F. Weinberger £ Salvatore Rao E Albert W. Saenz £*Richard A. Smith £ Peter Tannenbaum £*Michael!. Weinstein E Louise Arakelian Raphael E Bruce E. Sagan * Wayne Stewart Smith E James J, Tattersall £ Walter Weinstein £*Maxwell 0. Reade £ Gershon Sageev £*Wilbur L. Smith £ Silviu Teleman £ Guido L. Weiss Douglas C. Reber £''Salem A. Sahab William K. Smith £*Paul M. Terwilliger £*Michael S. Weiss Raymond M. Redheffer e*Takashi Sakai * William H. Smoke £*Andrew]. Terzuoli £*Richard M. Weiss £* Don Redmond £ Toshio Sakata * Joel A. Smaller £*Lawrence E. Thomas John Weissman £ Timothy Redmond £*Salim W. Salem £*William F. Smyth E Abigail A. Thompson £*David M. Wells * Christopher L. Reedy £*William Salkind * Ernst Snapper E Gino Tironi E Raymond 0. Wells Jr. £''David E. Reese £*Laurent Saloff-Coste £ William M. Snyder Jr. E Nobushige Toda E David V. V. Wend '~ Ernestine Reeves-Hicks £ Mary F. Salter £''Emilio del Solar-Petit Eugene R. Tomer £*James G. Wendel £ Eugenio Regazzini £ Hatem Abdul-Mohsin £ Bruce Solomon * Daniel B. ]. Tomiuk £ John C. Wenger E Ryan C. Reich Samman E David Reed Solomon E Jun Tomiyama e*Henry C. Wente £''Michael Reid E*Pierre Samuel £ Boris Solomyak E*Hing and Mary Powderly Tong £* Aric]. Werly £*William H. Reid E Oscar Adolfo Sanchez- E Baorui Song E Nicolo Goodrich Torre £"Elisabeth Werner £*Irma M. Reiner Valenzuela e*Linda R. Sons e*Charles R. Traina E Robert]. Wernick E* John H. Reinoehl £*Robert W. Sanders E John]. Sopka E Francois Treves E Eric M. West £*Robert B. Reise! * Angel San Miguel E* John R. Sorenson £*Timothy Guy Trucano E* john E. Wetzel Peter A. Rejto e*Ulderico Santarelli E Michael J. Sormani E*Tamotsu Tsuchikura Arthur T. White * Richard S. Rempel * Jose Cloves Verde Saraiva E Raymond A. Sosnowski E Kaz6 Tsuji £ Brian Cabell White Estate of james G. Renno Jr. £*Donald E. Sarason E Ian N. Spatz £*Ralph P. Tucci E Kenneth G. Whyburn E Pedro Manuel Resende e*Chelluri C. A. Sastri E Edward ]. Specht E Howard G. Tucker £ Bill Wickless E* Guillermo Restrepo £*Hiraki Sato £*Dennis Spellman E Steven Mark Turadek £*Roger A. and Sylvia M. Wie- E Michael Bela Revesz William G. Saunders E* Stephen E. Spielberg E* Joann Stephanie Turisco gaud E Edgar N. Reyes * Stanley A. Sawyer E* John]. Spitzer £*Edward C. Turner E Arthur S. Wightman '' Fazlollah Reza £*Richard C. Scalzo Aravamuthan Srinivasan E Helene R. Tyler * Robert A. Wijsman e*Bruce Reznick E* juan jorge Schaffer Ram P. Srivastav E* Johan Tysk * Clinton Curtis Williams e*Martin G. Ribe E Doris W. Schattschneider E Hellmuth Stachel e*Karen Uhlenbeck E* Susan Gayle Williams E Barbara Slyder Rice £*Michelle Schatzman E Ross E. Staffeld! E William Ullery E Charles K. Williamson e*Norman ]. Richert E* Samuel Schechter * ]. T. Stafford e*Tomio Umeda e*Paul A. Willis E Lawrence B. Richmond * Gideon Schechtman £*William L. Stamey E Harald Upmeier E Alan Stephen Wilmshurst E* John H. Rickert josef Schmid E Paul H. Stanford E* john A. W. Upton E Robert L. Wilson E* Eleanor G. Rieffel E Markus Schmidmeier £*Lee james Stanley E Angehrn Urban * Robert Lee Wilson £ Georg ]. Rieger E Harvey]. Schmidt Jr. * Charles S. Stanton E Colleen A. Vachuska E Beth A. Wingate E Ronald Edgar Rietz £*Wolfgang M. Schmidt £*Christopher W. Stark E Sreeram Valluri * Eric]. Wingler * Robert D. Rigdon E Franz]. Schnitzer £*James Stasheff Johannes A. Van Casteren £*Heinz-Wolfgang Wissner Pete E. Riley e*Roberto H. Schonmann E'' John Q. St Clair E*Diederik T. Van Daalen £ Hans S. Witsenhausen E Timothy Rupert Riley Richard M. Schori £ Alfred T. Steele e*R. Lee Van de Wetering t:*Louis Witten e*]ose Rio £*Bertram M. Schreiber Jan D. Stegeman e*Bob Van Rootselaar James]. Woeppel E Thomas W. Rishel E* John Schue E Sherman K. Stein E* A. H. VanTuyl e*Peter john Wolfenden E Craig William Roberts * George W. Schueller * Leon Steinberg £ Fred S. Van Vleck £*Elliot S. Walk E Joseph B. Roberts e*Paul E. Schupp E* Charles I. Steinborn £''Dale E. Varberg £ Stephen D. Wolthusen e*Lois]. Roberts E jacob T. Schwartz Ellen M. Stenson E Joseph c. Varilly E James S. W. Wong E* James B. Robertson * Gerald W. Schwarz e*Ronald]. Stern Nick H. Vaughan £*John W. Wood E Charles D. Robinson E Eric Schweitzer £ Peter]. Sternberg £ Juan L. Vazquez N. M.]. Woodhouse * JohnRoe £''Michael]. Schwietzer e*Glenn H. Stevens E Sjoerd M. Verduyn Lunel e*George V. Woodrow Ill Jonathan Rogawski £ Stanley L. Sclove E T. Christine Stevens £ Divakar Viswanath E Christopher T. Woodward £*David E. Rohrlich £*Dana S. Scott E Katherine F. Stevenson £*Michael Voichick E Anatoly Yagola E* Judith Roitman E* Leonard L. Scott Jr. E George Stoica e*Dan Voiculescu * Hiroyoshi Yamaki E josephine Jardin Romero E Ridgway Scott e*Murray Marvin Stokely III £*Paul A. Vojta £*Michael Yanowitch £*William L. Root * Anthony Karel Seda E H. Christine B. Stokes e*Hans W. Volkmer Fawzi M. Yaqub E Colin Raymond Rose e*George Seifert E Russell A. Stokes e*Hidekazu Wada e*Mitsuru Yasuhara £*Nicholas]. Rose E* George B. Seligman E H. A. Stone e*Philip D. Wagreich £ Suresh Yegnashankaran * Robert A. Rosenbaum * Stuart A. Seligson £*Lawrence D. Stone £ Jonathan M. Wahl £ Peter Yff E David Rosenberg £ Mohamed W. I. Sesay Emil J, Straube £ Sebastian Walcher * Ken-Jchi Yoshihara E* Jonathan M. Rosenberg £ Robert E. Seydel Jr. £*Walter A. Strauss e*David B. Wales £*Paul M. Young £*John Rosenknop Freydoon Shahid.i Kurt Strebel £*Homer F. Walker * Noriko Yui £*Kenneth A. Ross E Peter B. Shalen £ Ralph Strebel E justin Clement Walker e*lon Zaballa Michael A. Roth £ Gerald Peter Shannon * Robert S. Strichartz £ William Wallace Ron H. Zagami £*Richard L. Roth Michael V. Shapiro £*G. Ralph Strohl Jr. E Nolan R. Wallach e*Nobuo Zama £*Mitchell]. Rothstein £*Henry Sharp Jr. E Gerhard 0. Strohmer £ Lawrence]. Wallen E Jean-Claude Zambrini E* Christel Rotthaus £*Desmond Sheeran £ Ulrich Stuhler E Hans Ulrich Walther e*Franc;ois Zara £ Cecil C. Rousseau Eugene P. Shelly E Kelly John Suman * PaulK. C. Wang * Michel M. Zarka E* James Rovnyak £*John C. Shepherdson E Patrick Suppes £*Evelyn K. Wantland £*Karl Zeller £*Virginia G. Rovnyak E*Kenichi Shiraiwa £*Andrei A. Suslin ·• Lesley A. Ward £ Li Zhou £ Ranjan Roy £ William Ivan Shorter £*Myron M. Sussman £ Sven Ole Warnaar E Jennifer joy Ziebarth E James Samuel Royer E Steven E. Shreve £ Ueli Suter £*Seth L. Warner e*Neal Zierler £''Melvin Glenn Royer e*Stuart ]. Sidney £*Vladimir Sverak £* Bette L. Warren E Philip W. Zipse Wimberly C. Royster £*Martha]. Siegel £*Richard G. Swan Lawrence C. Washington Paul Zorn £*Daniel Ruberman £*DanielS. Silver Ellen E. Swanson Arthur G. Wasse_rman E* John A. Zweibel ··· Robert ]. Rubin £ Evelyn Marie Silvia E Charles W. Swartz Robert H. Wasserman Paul]. Zwier £ Alexander M. Rubinov E Anne-Marie Simon e*William]. Sweeney * Michiaki Watanabe Anonymous (264) E* Zalman Rubinstein e*Iakov G. Sinai E Glen Dewane Swiggart £ Toshihiro Watanabe £*Bernard D. Rudin £ Surjeet Singh E Mabel Szeto Herbert Harold Wathan £*Wolfgang M. Ruess £ David B. Singmaster £*Earl]. Taft £ DavidS. Watkins MAY 2005 NOTICES OF THE AMS 563 From the AMS Secretary Officers of the Sodety 2004 and 2005 Updates Except for the members at large of the Sylvia T. Bozeman Paul J. Sally Jr. 2/ 05-1/ 08 Council, the month and year of the Irene Gamba Journal ofthe AMS Editorial first term and the end of the present Henri A. Gillet Committee term are given. For members at large David R. Morrison Ingrid Daubechies 2/ 04-1/07 of the Council, the last year of the 2005 Mathematical Reviews Editorial present term is listed. Susan Hermiller Committee Brian Marcus B. A. Taylor 2/ 02-1/ 06 Council John E. McCarthy Mathematical Surveys and President Paul J. Sally Jr. Monographs Editorial Committee James G. Arthur 2/ 05-1/ 07 Paul Zorn Peter S. Landweber 2/ 03-1/ 05 David Eisenbud 2/ 03-1/ 05 2006 J. Tobias Stafford 2/ 05-1/ 08 President elect James W. Cannon Mathematics of Computation James G. Arthur 2/04-1/ 05 Sylvain E. Cappell Editorial Committee Immediate Past President Beverly E. J. Diamond Chi-Wang Shu 2/ 02-1/ 08 Mark Goresky David Eisenbud 2/ 05-1/ 06 Proceedings Editorial Committee Alejandro Uribe Vice Presidents Eric D. Bedford 2/ 01-1/ 06 2007 Hai:m Brezis 2/ 05-1/ 08 Transactions and Memoirs Editorial Sara Billey Vaughn F. R. Jones 2/ 04-1/07 Committee Carolyn S. Gordon Hugo Rossi 2/ 02-1/ 05 William Beckner 2/ 00-1/ 05 Sheldon H. Katz Karen Vogtmann 2/ 03-1/ 06 Robert Guralnick 2/ 05-1/09 Michael F. Singer Secretary Catherine H. Yan Robert J. Daverman 2/ 99-1/ 07 Board of Trustees Associate Secretaries Members of Executive James G. Arthur (ex officio) 2/ 05-1/ 07 John L. Bryant 2/99-1/ 05 Committee John B. Conway 2/ 01-1/06 David Eisenbud (ex officio) 2/ 03-1/ 05 Susan J. Friedlander 2/ 96-1/ 08 Members of the Council, as provided MichelL. Lapidus 2/ 02-1/ 08 John M. Franks (ex officio) 2/ 99-1/ 07 for in Article 7, Section 4 (last sen Eric M. Friedlander 2/ 00-1/ 10 Matthew Miller 2/ 05-1/ 08 tence), of the Bylaws of the Society. Linda Keen 2/ 99-1/ 09 Lesley M. Sibner 2/ 93-1/ 07 Walter L. Craig 2/ 03-1/ 07 Donald E. McClure (ex officio) Treasurer David R. Morrison 2/ 02-1/05 2/ 03-1/ 07 John M. Franks 2/ 99-1/ 07 Hugo Rossi 2/ 02-1/ 06 Jean E. Taylor 2/ 03-1/ 08 Associate Treasurer Paul J. Sally Jr. 2/ 04-1/ 08 Carol S. Wood 2/ 02-1/ 07 Donald E. McClure 2/ 03-1/ 07 Members at Large Publications Committees All terms are for three years and ex Bulletin Editorial Committee pire on January 31 following the year Donald G. Saari 2/ 98-6/ 05 given. Susan J. Friedlander 7/ 05-1/ 09 2004 Colloquium Editorial Committee Colin C. Adams Susan J. Friedlander 2/ 96-1/ 05 564 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Mathematics Calendar The most comprehensive and up-to-date Mathematics Calendar information is available one-MATH at http://www. ams.org/mathcal/. May 2005 and ideal theory that interact with commutative algebra. '' 2-4 2nd Symposium on Networked Systems Design and Imple Support: We have some support for travel and local expenses from mentation (NSDI'OS), Boston Park Plaza Hotel, Boston, Massachu the UNL Department of Mathematics; additional support from the setts. National Science Foundation is anticipated. In the allocation of NSF Description: NSDI '05 is a symposium focused on the design funds, particular consideration will be given to advanced graduate principles of large-scale distributed and networked systems. We students, recent Ph.D.s, and members of underrepresented groups. believe systems as diverse as scalable Web services, peer-to-peer In order to be considered for support you need to register. file sharing, sensor nets, and distributed network measurement Deadline: Deadline of registration is extended to April15. share a set of common challenges. Progress in any of these areas Information: For more information, contact the organizers: Luchezar requires a deep understanding of how researchers are addressing L. Avramov: email: avramov@math. unl. edu. David A. Jorgensen: the challenges of large-scale systems in other contexts. Our goal is email: dj orgens@uta. edu; tel: 817-272-5507. Aihua Li: email: lia@ to bring together researchers from across the systems community mail. montclair. edu; tel: 973-65 5-7271. Conference web site: including operating systems, distributed systems, and computer http://dreadnought .uta. edu/-dave/WiegandFest.html. networking-to foster a cross-disciplinary approach to addressing our common research challenges. NSDI will provide a high-quality, * l 2-l 4 Groups in Galway 2005, National University of Ireland, single-track forum for presenting new results and discussing ideas Galway, Ireland. that overlap these disciplines. Information: The annual conference 'Groups in Galway' will be held lnformation:http: //www. usenix. org/nsdi05/;email: conference at National University of Ireland, Galway, this year on 12-14 May. @usenix . org; tel: 510-528-8649. The current conference is in honour of Professor Martin Newell. The scope of the conference covers all areas of group theory, '' 7-9 Nebraska Commutative Algebra Conference: WiegandFest, applications, and related fields. The University of Nebraska-Lincoln, Lincoln, Nebraska. Speakers: Rex Dark (NUl, Galway), Warwick de Launey (Center for Description: The conference is open to anyone working in com Communications Research, USA), Colin Campbell (St Andrews), Alla mutative algebra. The purpose is for recent Ph.D.s and graduate Detinko (NUl, Galway/Belarus), Francisco de Giovanni (Universita students in the field to meet experienced researchers, to learn about di Napoli, Hermann Heineken (Universitaet Wuerzburg), Luise recent developments in commutative algebra, to exchange ideas Charlotte Kappe (SUNY), Tom Laffey (UC Dublin), John McDermott with one another, and to develop collaborations for future research. (NUl, Galway), Des MacHale (UC Cork), Mike Newman (Australian Plus, it gives us in the commutative algebra community a wonder National University), Rachel Quinlan (UC Dublin/Galway), Stewart ful opportunity to celebrate the many important contributions of Stonehewer (Warwick). Roger and Sylvia Wiegand. The primary topics will be aspects of Registration and Contact Information: All who are interested representation theory, homological algebra, computational algebra, are invited to attended. There is no formal registration (and in respect to participation in the meeting, this fact should be noted. This section contains announcements of meetings and conferences of interest to some segment of the mathematical public, including ad All communications on meetings and conferences in the mathematical sciences should be sent to the Editor of the Notices in care of the American hoc, local, or regional meetings, and meetings and symposia devoted Mathematical Society in Providence or electronically to notices@ams . org to specialized topics, as well as announcements of regularly scheduled or mathcal@ams. org. meetings of national or international mathematical organizations. A In order to allow participants to arrange their travel plans, organizers of complete list of meetings of the Society can be found on the last page of meetings are urged to submit information for these listings early enough each issue. 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 the meeting in question. To achieve this, listings should be received in for papers and specifies the place, date, subject (when applicable), and Providence eight months prior to the scheduled date of the meeting. the speakers; a second announcement will be published only if there The complete listing of the Mathematics Calendar will be published are changes or necessary additional information. Once an announcement only in the September issue of the Notices. The March, June/July, and has appeared, the event will be briefly noted in every third issue until December issues will include, along with new announcements, references it has been held and a reference will be given in parentheses to the to any previously announced meetings and conferences occurring within month, year, and page of the issue in which the complete information the twelve-month period following the month of those issues. New appeared. Asterisks (*) mark those announcements containing new or information about meetings and conferences that will occur later than revised information. In general, announcements of meetings and conferences held in North the twelve-month period will be announced once in full and will not be repeated until the date of the conference or meeting falls within the America carry only the date, title of meeting, place of meeting, names of twelve-month period. speakers (or sometimes a general statement on the program), deadlines The Mathematics Calendar, as well as Meetings and Conferences of for abstracts or contributed papers, and source of further information. the AMS, is now available electronically through the AMS website on the Meetings held outside the North American area may carry more detailed World Wide Web. To access the AMS website, use the URL: http://www. information. In any case, if there is any application deadline with ams. org/. MAY 2005 NOTICES OF THE AMS 565 Mathematics Calendar particular no registration fee!), although it would be appreciated challenges in their modeling, simulation, and integration in the if intending attendees send an email to one of the organizers medical and surgical processes. It will be structured around the (addresses below). Details of the talks and their scheduling will be following themes: medical imaging and geometrical modeling, posted at http ://www. maths. nuigalway. ie/gig05. html closer fluid-structure interactions in health problems, static/dynamical to the event. For further information, please contact one of the design and control of (implantable) medical devices, finite element conference organizers, Jim Cruickshank ( james . cruickshank@ based computer aided design/manufacturing. nuigalway. ie) or Dane Flannery (dane. flanneryl!lnuigalway. ie). Information: http://www. crm. umontreal. ca/Med05/. * 1 3-1 5 1st KAIST International Symposium on Enhancing Univer '' 23-june 3 11th European Intensive Course on Complex Analysis sity Mathematics Teaching, Korea Advanced Institute of Science and its Generalizations (with applications to partial differential and Technology, Daejeon, Korea. equations), University of Coimbra, Coimbra, Portugal. Program: The Department ofMathematics of the Korea Advanced In Program: Week 23-27: Matrix Differential Riccati Equations- Ger stitute of Science and Technology announces its initial international hard Jank (Aachen University, Germany); Orthogonal Polynomials symposium on the enhancement of the teaching of mathematics in on the unit circle. From Gabor Szego to Barry Simon - Francisco universities and colleges. This symposium provides an opportunity Marcellan Espafi.ol, (Universidad Carlos III de Madrid, Spain); Week for university and college faculty to give presentations of their 30-3: Monogenic, hypermonogenic and holomorphic Cliffordian successful approaches to teaching undergraduate mathematics. functions - Eric Lehman (Universite de Caen, France); Transform Presentations may include teaching styles, uses of technology, Analysis: The Hilbert Transform- Fred Brackx (University of Ghent, successful classroom demonstrations, and much more. Hands-on Belgium); Transform Analysis: The Continuous Wavelet Transform workshops will highlight specific methods. There are specially - Nele De Schepper (University of Ghent, Belgium). designed sessions for graduate students, with presentations both Deadlines: There is no deadline for the registration. Living expenses by and for teaching assistants. can be partially covered for some Ph.D. students if they do not have Coordinators: Young Han Choe, Deane Arganbright. support from their own institution and if there is enough money Abstract Deadline: April1, 2005. available. Information: The following symposium Web page provides details Organizer: H. Malonek, A. Branquinho, ]. Carvalho e Silva, P. for submissions and registration, together with the program outline, Cerej eir as. fees, featured speakers, contacts, and other useful information: Information: http: I /www .mat . uc .pt;-ajplb/11 .htm. http://math.kaist.ac.kr/2005. * 29-June 4 (REVISED) Spring School in Analysis: Function Spaces * 16-20 Mini-invasive Procedures in Medicine and Surgery: Math and Applications, Paseky nad Jizerou, Czech Republic. ematical and Computational Challenges, Centre de recherches Topics: Hardy Operators, Function Spaces, Embeddings, Whitney mathematique, Montreal, Quebec, Canada. Extension Problem, Helly's intersection theorem. Information: Workshop and Spring School. The objective of this tan Main speakers: W. Desmond Evans (University of Wales, Cardiff, dem School-Workshop is to explore several facets of mini-invasive UK); Pavel Shvartsman (Technion, Haifa). procedures in medicine and surgery, identify issues, problems, Organizers: Jaroslav Lukes, Lubas Pick (Charles University, Prague, trends, and mathematical and computational challenges in their Czech Republic). modeling, simulation, and integration in the medical and surgi lnformation:http: I /www .karlin.mff. cuni. cz/katedry/kma/ss/ cal processes. It will be structured around the following themes: jun05/; email: [email protected]. medical imaging and geometrical modeling, fluid-structure inter actions in health problems, static/dynamical design and control June 2005 of (implantable) medical devices, finite element based computer aided design/manufacturing. More information can be found on the * 1-3 Workshop on POE and Harmonic Analysis, Norwegian Uni conference website at http://www. crm. umontreal . ca/Med05/. versity of Science and Technology, Trondheim, Norway. Aim: The aim is to bring together, in an informal, workshop setting, '' 16-27 Mini-course on Nonconvex Variational Problems and a small group of researchers working on problems related to PDE Applications, University of Utah, Salt Lake City, Utah. and Harmonic Analysis. Ph.D. students are also very welcome. Speakers: Andrea Braides (Principal Lecturer), Georg Dolzmann Organizing committee: S. Selberg, E. Malinnikova, P. Hasto. (Principal Lecturer), Andrej Cherkaev, Elena Cherkaev, David Dob Information: http://www .math. ntnu. nor sselberg/Workshop/ son, Graeme Milton, Klaus Schmitt, Andrejs Treibergs, Marian Info3 .htm. Bocea. Mini-course Topics: From Discrete Systems to Continuous Vari * 1-4 Classics in POE: A meeting in Honor of Nina Nikolaevna ational Problems, Macroscopic Models for Phase Transforming Uraltseva 70th Birthday, KTH, Stockholm, Sweden. Materials, Minimal Extensions of Nonconvex Lagrangians, Mini Organizing Committee: Ari Laptev, Gregory Seregin, Henrik max Problems, Regularization, Bifurcations, Nonconvex Variational Shahgholian. Problems, Young measures; Applications to Geometry, Compos Tentative list of speakers: D. Apushkinskaya, Saarbrii.cken; A. ites, Thermodynamics, Band-gap materials, Inverse problems, and Arkhipova, St. Petersburg; H. Brezis, Paris; L. A Caffarelli, Austin, Optimal design. Texas; A. Chang, Princeton; M.]. Esteban, Paris; A. Friedman, Ohio; I. Financial Support: Some financial support is available through Gamba, Austin, Texas; M. Giaquinta, Pisa; A. Grigoryan, London; C. a NSF-VIGRE grant, for graduate students who are U.S . citizens, Kenig, Chicago; I. Kondrat'ev, Moscow; A. Nazarov, St. Petersburg; nationals, or permanent residents. Application deadline: May 1, L. Nirenberg, NY; A. Petrosyan, West Lafayette; G. S. Weiss, Tokyo; 2005. V. Zhikov, Russia. Information: http://www.math. utah. edu/vigre/minicourses/ Information: Henrik Shahgholian, chairman, email: henrikshl!l nvpa/speak .html. math. kth. se; Marie Lundin, Administrative Secretary, email: mal undl!l math. kth. se. '' 16-27 Mini-Invasive Procedures in Medicine and Surgery: Math ematical and Computational Challenges, Centre de recherches * 4 8th Annual Workshop on Applications and Generalizations of mathematiques, Universite de Montreal, Montreal, Quebec, Canada. Complex Analysis, University of Aveiro, Aveiro, Portugal. Objective: Of this tandem School-Workshop is to explore several Registration: Besides a formal registration there will be no fee. facets of mini-invasive procedures in medicine and surgery, identify In case of interest please register until May 16. Please send the issues, problems, trends, and mathematical and computational registration to pceresl!lmat. ua. pt. 566 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Mathematics Calendar Tentative Program: A preliminary version of the detailed Program Seoull51-747, Korea. will be posted in the web page until May 23. VVorkshop Topics: Recent development in inverse problems, multi General Information: Indications such as "Technical support", scale analysis, and homogenization reveals that these fields share "How to arrive to ... " and others will be posted on the web page several fundamental concepts in common. The main purpose of until May 16. this workshop is to bring together researchers coming from these VVeb page:http:llwww.mat.ua.ptlpcereslws8_2005.html. fields to share their new ideas and to earn benefit from others Organizers: H. Malonek, A. Branquinho, ]. Carvalho e Silva, P. different viewpoints. Cerejeiras. Invited Speakers: Elena Beretta, Eric Bonnetier, Yves Capdeboscq, Andrej Cherkaev, Soon Young Chung, Mathias Fink, Leslie Green '' 4-9 Second International Conference on Abstract and Applied gard, Tomas Hou, Masaru Ikehata, Jun Yub Lee, Mikyoung Lim, Analysis: ICAAA 2005, Quy Nhon, Vietnam. Graeme Milton, Gen Nakamura, Jin Keun Seo, Gunther Uhlmann, Organizer: Institute of Mathematics, Vietnamese Academy of Michael Vogelius. Science and Technology, and Quy Nhon University. Information: http : I lwww .math. snu. ac .krlbk21lworkshopl. Sponsors: Institute ofMathematics, Vietnamese Academy of Science and Technology, National Basic Research Program inNatural Science, '' 22-2 5 International Seminar "Control Theory and Theory of Quy Nhon University. Generalized Solutions of Hamilton-Jacobi Equations" (CGS- Topics: ODE and Dynamical Systems, Wavelet Analysis, p-adic 2005), Institute of Mathematics and Mechanics of the Ural Branch Analysis, PDE and Microlocal Analysis, Numerical Analysis, Com of the Russian Academy of Sciences, Yekaterinburg (Ekaterinburg), plex Analysis, Harmonic Analysis, Fractional Analysis, Stochastic Russia. Analysis, Nonlinear Analysis and Differential Geometry, Ill-posed Dedication: To the 60th Anniversary of Academician Andrei and Inverse Problems. Izmailovich Subbotin (February 16, 1945-0ctober 14, 1997). Information: Please send the Registration Form to Dr. Nguyen Huu Description: It is planned to discuss the present state of control Dien, Institute of Mathematics, VAST 18 Hoang Quoc Viet Road, theory and theory of generalized solutions of Hamilton-Jacobi CauGiay District, 10307, Hanoi Vietnam; email: nhdien©math . ac . equations. vn;http:llwww.math.ac .vnlconferencelicaaa2005l. Topics: Scopes of the seminar include: Generalized solutions of Hamilton-Jacobi equations. Control of dynamic systems under *5-9 Representation Theory, Geometry and Automorphic Form. conditions of conflict and uncertainty. Problems of estimation and International Conference in honor of j. Bernstein's 60th birthday, identification in dynamic systems. Inverse problems and distributed Tel-Aviv University, Tel-Aviv, Israel. control systems. Numerical algorithms for constructing solutions Organizers: Alexander Beilinson, Alexander Braverman, Dennis of optimal control problems and boundary value problems for Gaits gory, Alexander Goncharov, David Kazhdan, Wilfried Schmid, Hamilton-Jacobi equations. Marie-France Vigneras. Organizers: The Ural Branch of the Russian Academy of Sciences, Speakers (tentative): A. Beilinson, R. Bezrukavnikov, A. Braverman, the Institute of Mathematics and Mechanics of the Ural Branch of V. Drinfeld, B. Feigin, D. Gaitsgory, A. Goncharov, D. Kazhdan, L. the Russian Academy of Sciences, and the Ural State University. Lafforgue, G. Lusztig, P. Sarnak, W. Schmid, P. Schneider, W. Soergel, Travel Information: Visit http : I I cgs . imm . uran. rul . . M. -F. Vigneras, A. Zelevinsky. Languages: Russian and English. Information: For more information as well as for support appli Information: http : I legs. imm. uran.rul. cations please contact Alexander Braverman (braval©math. huj i. ac. il) or Dennis Gaits gory (gai tsgde©math. uchicago . edu). *27-30 14th International Scientific Congress CNIC 2005: 40 Years at the Service of Science and Technology, Havana, Cuba. '' 5-july 21 joint Summer Research Conference, Snowbird Resort, Information: WEB pages with more information: Our site: http : I I Snowbird, Utah. www.cnic.edu . cui14CongresoiBienvenida.htm and Information: http : I lwww. ams. orglmeetingslsrc .html. http:llwww.loseventos . culseminariocnic2005. Venue: http: I lwww. cpalco. com. Contact: email: seminario©cnic . edu. * 1 0-11 CAM 2005, University of Central Oklahoma, Edmond, cu;feria©cnic . edu.cu. Oklahoma. VVorkshop Topic: A cryptology workshop designed for the non july 2005 specialist. Organizers: Dr. Jesse Byrne, 100 N University Drive, Edmond, '' 4-8 MODELLING 2005-Third IMACSConference on Mathematical OK 73034; email: jbyrne©ucok.edu; tel: 405-974-5575 and Dr. Modelling in Applied Sciences and Engineering, University of Charlotte Simmons, 100 N University Drive, Edmond, OK 73034; West Bohemia, Pilsen, Czech Republic. email: cksimmons©ucok. edu; tel: 405-974-5316. Information: The event is organized by Department ofMathematics, Speakers: Confirmed speakers: D.]. Bernstein (University of Illinois Faculty of Applied Sciences, University of West Bohemia in Pilsen, at Chicago), Robert Lewand (Goucher College), and Alice Silverberg Czech Republic, in cooperation with other Czech universities and (University of California at Irvine). research institutions in the week preceding the 17th IMACS World Travel Funds: This workshop is supported by the NSA. Travel Congress IMACS 2005 in Paris. funds are available to deter the expenses of participants. Graduate ·Purpose: The purpose of the Conference is to stimulate research students, junior faculty, women, minorities, and persons with and, in an informal setting, to foster the interaction of researchers disabilities are especially encouraged to participate and to apply in different fields of science. for support. Early application for support is encouraged. Topics: Computational modelling in general, Computer models in Information: http: I lwww .math. ucok. edu or contact an organizer. fluid dynamics and biomechanics, Computer models in geomatics engineering, Optimal design problems and structural optimization, '' 1 0-1 5 Summer School on Harmonic, VVavelet, P-adic analysis, Multiscale modelling, Qualitative properties of nonlinear boundary Quy Nhon, Vietnam. value problems, Advanced numerical methods related to the above Description: Continuation of the Second International Conference fields. on Abstract and Applied Analysis ICAAA 2005; see June 4-9, 2005. Invited Speakers: A. Cepek, Czech Technical University, Czech Information: http: I lwww . math. ac . vnl conferencelicaaa2005l. Republic, Z. Dostal, Technical University Ostrava, Czech Republic, C. Farhat, Stanford University, USA, R. Glowinski, University of '' 22-24 Inverse Problems, Multi-Scale Analysis and Homogeniza Houston, USA, ]. Jarusek, Mathematical Institute, Academy of tion, Seoul National University, San56-l Shinrim-dong, Kwanak-gu, Sciences of the Czech Republic, S. Korotov, University of Jyvaskyla, MAY 2005 NOTICES OF THE AMS 567 Mathematics Calendar Finland, Y. Kuznetsov, University of Houston, USA,]. Lopez-Gomez, Geometry and Applications in Finance, Thermo Aitoloakarnanias, Universidad Complutense de Madrid, Spain, ]. Malik, Institute of Greece. Geonics, Academy of Sciences of the Czech Republic, J. Mandel, Program: Designed for graduate students of mathematics with a University of Colorado at Denver, USA, B. Miara, ESIEE, France, U. background in stochastic analysis. Previous knowledge of differ Sultangazin, Space Research Institute, Republic of Kazakhstan, M. ential geometry (manifold theory) is desirable but not necessary. Tabata, Kyushu University, Japan, M. Tvrdy, Mathematical Institute Topics covered include Brownian motions and stochastic equations AS CR, Czech Republic, R. Van Keer, University of Ghent, Belgium. on manifolds, interest rate curves and stochastic movements on Website: The MODELLING website has now be opened up at http : manifolds, Riemannian Brownian motion? connections? stochastic I /www . modelling. zcu. cz/ and more details may be found there. flows, martingale fields. Sponsors: Under the auspices of the University of Patras, Greece, '' 6-23 35th Saint-Flour Probability Summer School, Saint-Flour, and exclusive sponsoring of the National Bank of Greece. France. Lecturers: Freddy Delbaen, David Elworthy, Michel Emery, Darnir Program: Fluctuation theory for Levy processes (R. Doney). Prob Filipovic, Xue-Mei Li, Marc Yor. ability and real trees (S.N. Evans). Optimal transport, old and new Funding: Available for all participants. (C. Villani). Deadline: May 20, 2005. Information: http : I /math. uni v-bpclermont. fr /stflourI. Information: A. Arvanitoyeorgos ( arvani to@math. upatras . gr), '' 4-8 Workshop on Coagulation-Fragmentation Processes: Theory V. Papakonstandinou ( papakonb@math. upatras. gr), http: I /www. and Applications, International Centre for Mathematical Sciences, math.upatras.gr/dimitsana. Edinburgh, UK. Aim: The aim of this workshop is to bring together the many dis '' 1 8-24 Methods of Logic in Mathematics 2005, Euler International parate groups of researchers who work on coagulation-fragmentation Mathematical Institute, St. Petersburg, Russia. (CF) equations and related processes. A key feature will be the Topics: Its main themes include computability theory, model theory, exchange of ideas between theoreticians and practitioners involved set theory and their connections with other areas of mathematics, with CF equations. This should promote a greater awareness of most notably algebra, arithmetic, computation and geometry. The existing results and outstanding problems in this area and provide meeting will consist of Invited Talks, Contributed Paper sessions mathematicians with new mathematical challenges and experi and tutorials. Two of the tutorials will be given by Simon Thomas mental scientists, engineers and industrialists with new analytical, (Rutgers) and Zlil Sela (Hebrew University). modelling, and numerical techniques. Invited Speakers: IncludeM.Arslanov(Kazan), Su Gao (North Texas), Organizing Committee: Fernando da Costa (CAMGSD, Institute E. Gordon (Eastern Illinois), E. Griffor (Michigan), W. Hodges (QMC Superior Tecnico, Lisboa). Michael Grinfeld (Mathematics, Uni London), T. Hyttinen (Helsinki),]. Iovino (San Antonio), I. Kalimullin versity of Strathclyde). Wilson Lamb (Mathematics, University of (Kazan), I. Kazatchkov (Omsk), Z. Sela (Hebrew University), R. Soare Strathclyde). Jonathan Wattis (Mathematical Sciences, University of (Chicago), S. Thomas (Rutgers), M. Viljanen (Helsinki), A. Weiermann Nottingham). The meeting is organized in collaboration with The (Munster, Utrecht), B. Zilber (Oxford), Yi Zhang (GuangZhou). Centro de Amilise Matematica, Geometria e Sistemas Dinamicos Information: See http : I /www . pdmi . ras. ru/EIMI/2005/2lag/. (CAMGSD) of Institute Superior Tecnico, Lis boa, Portugal. For more information contact Edward Griffor by email at egriff or@ . Deadlines: Those wishing to contribute a talk and/or apply for umich. edu or Elena Novikova at novikova@pdmi. ras. ru. financial support should complete their application by May 2, 2005. Applications will not be accepted after June 6th which is also '' 25-August 12 Summer Research Institute on Algebraic Geome the date by which payment of the registration fee must be made try, University of Washington, Seattle, Washington. in order to secure a place. Online registration is available at the Information: email: rha@ams. org. meetings' website. Information: http: I /www. icms . org. uk/meetings/2005/ August 2005 coagfrag. '' 2005 Mathematical Modeling of Infectious Diseases: Dynam * 7-9 OTFUSA2005: Conference on Operator Theory, Function ics and Control, Institute for Mathematical Sciences, National Spaces and Applications-Dedicated to the 60th birthday of University of Singapore, Singapore 118402. Frank-Oime Speck, Department of Mathematics, University of Information: KP Chua, Administrative Officer, Institute for Mathe Aveiro, Aveiro, Portugal. matical Sciences, National University of Singapore, 3 Prince George's Description: To celebrate the 60thbirthday ofProfessor Frank Speck Park, Singapore 118402; tel: (+65) 6874 1893; Facsimile: (+65) 6873 and to bring together those enrolled in the research activities related 8292; http: I /www. ims. nus. edu. sg . . with operator theory, function spaces and related applications. Topics: Convolution type operators and related classes of singular '' 1-5 14th USENIX Security Symposium, Baltimore, Maryland. operators, Bessel potential and pseudo-differential operators, fac Description: The USENIX Security Symposium brings together torisation theory, operator relations and normalisation problems; researchers, practitioners, system administrators, system pro The theory of function spaces and distributions around Lebesgue, grammers, and others interested in the latest advances in security Lorentz, Sobolev, Besov and Triebel-Lizorkin spaces, embeddings, of computer systems. interpolation, traces and extensions, representation formulas, os Information: Contact: Conference Department, email: conference@ cillation; Applications to mathematical physics, wave diffraction usenix . org; tel: 510-528-8649; http: I /www . usenix. org/sec05/. and scattering theory, elliptic boundary value problems, mixed problems in canonical domains, localisation, interface problems, * 1-9 XVI Coloquio Latinoamericano de Algebra, Colonia, Uruguay. boundary integral methods, boundary-domain methods, explicit Description: This biannual event is the premier conference linking solutions, regularity, singularities, fractal analysis and asymptotic algebraists and algebraic geometers from all of Latin America. behaviour. Topics: Besides the plenary talks and general courses this meeting Invited Speakers: ]. Cerda, M. Costabel, R. Duduchava, S. Jaffard, will have seven thematic parallel sessions on the following topics: Yu. Karlovich, V. Maz'ya, D. Natroshvilli, S. Samko, B. Silbermann, Commutative Algebra and Algebraic Geometry, Non-associative H. Triebel, H. Upmeier, W. L. Wendland. Algebras and Ring theory, Group Theory, Hopf Algebras and Information: http: I /www. mat. ua. pt/ otfusa2005/. Algebraic Combinatorics, Homological Methods and Representation Theory, Number Theory, Operator Algebras. A special session on '' 1 0-1 7 Dimitsana Summer School on Stochastic Differential Applications of Algebra will also be held. 568 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Mathematics Calendar Speakers: A list of a few of the confirmed speakers is the following: system problems. Furthermore, this session will provide partici Nicolas Andruskiewitsch, Raymundo Bautista, Michel Brion, Ken pants opportunities to interact and establish research connections Brown, Antonio Campillo, Max Karoubi, Jean-Louis Loday, Susan with each other. The deadline to submit your paper to this special Montgomery, Adrian Ocneanu, Jose Antonio de la Pefla, Vladimir session is AprillS, 2005. Please send your paper directly to Aihua Popov, Hans-Jurgen Schneider, Aron Simis, Frank Sottile, Richard Li, the organizer. Email submission is preferred. The address is: Stanley, Boris Tsygan, Mariusz Wodzicki. lial!lmail. montclair. edu. Please see the conference web site for Organizing and Scientific Committee: Walter Ferrer Santos (Co instruction. ord.), Gerardo Gonzalez-Sprinberg, Alfredo Jones, Alvaro Rittatore, Organizer: Aihua Li, Department of Mathematical Sciences, Mont Andrea Solotar. clair State University, 1 Normal Avenue, Montclair, NJ 07043, USA. Deadline: May 1st, 2005. Office phone: 973-655-7271. lnformation:http: I /www. cmat. edu. uy I cmat/ eventos/16cla/ en; lnformation:http : I /www. iasted. org/ conferences/2005/ aruba/ Walter Ferrer: email: wrferrerl!lcmat. edu. uy c471.htm '' 1 5-19 Eisenstein Series and Applications, AlM Research Confer '' 29-September 2 Numerical Methods for Optimal Control in High ence Center, Palo Alto, CA. Dimensions, AlM Research Conference Center, Palo Alto, CA. Organizers: Wee Teck Gan, Steven Kudla, and Yuri Tschinkel. Organizers: Doron Levy, Ian Mitchell, and Adam Oberman. Workshop Topics: This workshop, sponsored by AlM and the Workshop Topics: This workshop, sponsored by AlMand the NSF , NSF, will consider some recent applications of Eisenstein series to will be devoted to problems of optimal control, broadly interpreted problems in arithmetic geometry and number theory. A central goal to include stochastic control problems and differential games. It of the workshop will be to try to understand the common structural is a standard practice to formulate these problems in terms of properties of the Eisenstein series occuring in applications. a multi-dimensional Hamilton-jacobi-Bellman (HJB) equation. The Application Deadline: May 15, 2005. workshop will focus on computational methods for tackling high Information: http: I I aimath. org/ ARCC/workshops/ eisenstein. dimensional HJB and related equations. html. Application Deadline: May 29, 2005. Information: Visit http: I /aimath. org/ ARCC/workshops/ * 1 5-1 9 Workshop on POE and and Mathematical Finance, Institut optimalcontrol .html. Mittag-Leffler, Stockholm, Sweden. List of Speakers: K. H. Karlsen (CMA/Univ. of Oslo), P. Laurence September 2005 (Univ. of Rome), M. Avellaneda (New York Univ.), B. Oksendal (Univ. * 5-9 IVth International Workshop on Functional Analysis, Liege, of Oslo), J-P. Fouque (North Carolina State Univ.), K. Solna (Univ. of Belgium. California, Irvine), N. Touzi (ENSEA), C. Schwab (Swiss Fed. inst. of Description: This Functional Analysis Conference is a joint venture Tech., Zurich), M. Jonsson (Royal Inst. of Tech.), T. Zariphopoulou of the University of TRlER (Germany) and of the University of LIEGE (Univ. of Texas at Austin), A. Ilhan (Princeton Univ.), D. Talay (Belgium); it is organized on the occasion of the 65th birthday of (INRIA), ]. Tysk (Uppsala Univ.), W. Schachermayer (Vienna Univ. Professor Jean Schmets (University of LIEGE). of Tech.), I. Karatzas (Columbia Univ.), H. M. Soner (Koch Univ.), H. Organizers: Francoise Bastin (University of LIEGE, Belgium), Su Pham (Jussieu), ]. Teichmann (Vienna Univ. of Tech.), S. Howison sanne Dierolf (University of TRIER, Germany), Jochen Wengenroth (Univ. of Oxford). (University of TRIER, Germany). Information: Contact: Teitur Arnarson, Dept. of Math. KTH, 100 Information: http: I /www. ulg. ac. be/sectmath/Sept05 .html; 44 Stockholm, Sweden; email: tei turl!lmath. kth. se. http://http://www.afo.ulg.ac .be/. '' 28-September 25th Conference on Differential Geometry, Man '' 12-1 7 International Conference on Mathematical Analysis of galia, Romania. Random Phenomena, Hotel Abou Nawas Hammamet, Hammamet, Invited Participants: Geometers; Members of BSG-AMS, EMIS. Tunisia. Institutional Organizers: Balkan Society of Geometers, University Scientific Programe: Stochastic Analysis, Mathematical Finance, Politehnica of Bucharest-Department of Mathematics, University Poisson Analysis, Intersection local times, Hydrodynamics equa of Bucharest-Faculty of Mathematics and Informatics, Society of tions, Statistical mechanics, Probability in Quantum Physics. Mathematical Sciences from Romania, Callatis High School of Invited Speakers: Luigi Accardi, Helene Airault, Sergio Albeverio, Mangalia. . Philippe Blanchard, Eric Carlen, Fernanda Cipriano, Ana Bela Topics: Differential geometry, dynamical systems, optimization Cruzeiro , Custodia Drumond, Diogo Gomes, Hanno Gottschalk, on Riemannian manifolds, Theory of Relativity, celebration of Martin Grothaus, Takeyuki Hida, Yuri Kondratiev, Tobias Kuna, Paul Dan Barbillian (1895-1961), mathematics in High Schools and Lescot, Paul Malliavin, Rui Vilela Mendes, Nobuaki Obata, Bernt Universities, BSG elections. Oksendal, Maria ]. Oliveira, Giulia di Nunno, Habib Ouerdiane, Deadlines: Submission of applications: June 20, 2005. Selected Nicolas Privault, Michael Rockner, Francesco Russo, Jose L. Silva, scientific papers will be published in BSG journals. Isabel Simao, Ludwig Streit, Jean Claude Zambrini. Coordinators: Constantin Udriste, Gabriel Pripoae. Inscription deadline: June 15, 2005. Information: email: vbalanl!lmathem. pub . ro. udristel!lmathem. Information: Ana Bela Cruzeiro, Grupo de Fisica-Matematica U.L. pub. ro (scientific); callatisl!lseanet. ro (educational). and Dep. de Matematica I.S.T. Av. Rovisco Pais 1049-001 Lis boa, Portugal; tel: +351 218417063; fax: +351 218417048; email: '' 29-31 Algebraic Methods and Applications in Dynamical Systems abcruzl!lmath. ist . utl. pt or Habib Ouerdiane, Departement de Special session in the 5th lASTED International Conference on Mathematiques. Faculte des Sciences de Tunis. Campus universitaire- Modeling, Simulation, and Optimization (MSO 2005), Oranjestad, 1060 Tunis. Tunisie; tel.: + 216-71-872-020, fax: + 216 1 885 Aruba. · 350, email: habib . ouerdianel!lfst. rnu. tn; http: I /www. uma. pt/ Description: Investigacao/Ccm/hammamet .html. This special session will focus on the algebraic methods and applications in continuous or discrete dynamical systems. In '' 20-22 International Conference: Problems of Modern Mathemat recent years, algebraic techniques are increasingly applied to many ics and Mechanics, Institute of Mathematics, Almaty, Kazakhstan. real world problems producing dynamic systems. Through the Topics: Theory of functions and functional analysis. Differential presentations we propose to identify common themes and recent equations and mathematical physics. Algebra and mathematical developments regarding algebraic approach to solving dynamical logic. Computational mathematics. Mechanics of solids &structures. MAY 2005 NOTICES OF THE AMS 569 Mathematics Calendar Hydrodynamics, mechanics of gas and plasma. Information tech Conference Webpage: http: I /www . siam. org/meetings/gd05/. nologies. Program of Conference consists of plenary (40 minutes) For additional information, contact SlAM Conference Department and sectional (20 minutes) talks. Working languages of Conference at meetingsl!lsiam . org. are Kazakh, Russian and English. Important Dates: July 10, 2005-Submission of abstracts. July 20, November 2005 2005-The Second announcement. '' 3-4 (NEW DATE) DIMACS Workshop on The Epidemiology and Organizing Committee: Postal address: Institute of Mathematics, Evolution of Influenza, DIMACS Center, CoRE Bldg, Rutgers Uni Pushkinstr., 125,Almaty, 050010, Kazakhstan; tel: +7(3272)913 764; versity, Piscataway, New Jersey. (Oct. 2004, p. 1096) fax:+7(3272)913740; email: [email protected]; http : I /www .math.kz/ confen.htm. December 2005 '' 21-26 International Conference of Computational Methods in *5-9 30th Australasian Conference in Combinatorial Mathematics Sciences and Engineering 2005 (ICCMSE 2005), Hotel Poseidon, and Combinatorial Computing (30ACCMCC), The University of Loutraki, Korinthos, Greece. Queensland, Brisbane, Australia. Information: Contactinformation: Secretary ICCMSE 2005 (Mrs Eleni Invited Speakers: Simon Blackburn, Royal Holloway, University of Ralli-Simou); email: iccmsel!luop. gr, Postal Address: 26 Menelaou London, U.K.; Matthew Brown, The University of Adelaide, Australia; Street, Amfithea Paleon Faliron, GR-175 64, Athens, Greece, Fax: Mike Grannell, The Open University, U.K.; LilyKhadjavi, Loyola Mary +30210 94 20 091 or+ 30 2710 237397; http://www.uop.gr; mount University, U.S.A.; Curt Lindner, Auburn University, U.S.A.; iccmse/. Brendan McKay, The Australian National University, Canberra; Wal Wallis, Southern Illinois University, Carbondale, U.S.A. October 2005 Contributed talks are welcome in all areas of combinatorics, graph theory, combinatorial computing and applications. * 7-8 Twenty-Fifth Annual Southeastern-Atlantic Regional Con Deadline: A closing date for abstracts and registration will be ference on Differential Equations, University of Dayton, Dayton, announced later; this will be around late October 2005. Ohio. lnformation:http: I /www .maths. uq. edu. au/cdmc/30accmcc .html. Organizer: Muhammad N. Islam. Email the Directorate jbl!lmaths. uq. edu. aufor further information, Principal Speakers: T. A. Burton (Northwest Research Institute, or check the conference web page. Washington, Retired from Southern illinois University), Functional Differential Equations; Srdjan Stojanovic (University of Cincinnati, Ohio), Nonlinear Partial Differential Equations-methods in Financial January 2006 Mathematics; Avner Friedman (Ohio State University, Ohio), Partial * 1 2-1 5 joint Mathematics Meetings, San Antonio, Texas. Differential Equations. (tentative); Konstantina Trivisa (University Information: http: I /www. ams. org/amsmtgs/national.html. Maryland), Nonlinear Partial Differential Equations and Applied Mathematics. (tentative) April 2006 In addition to the principal speakers, there will also be sessions of twenty minute contributed talks. Pending funding from the '' 1-2 AMS Southeastern Section Meeting, Florida International National Science Foundation, travel support funds will be available University, Miami, Florida. for advanced graduate students and recent Ph.D. recipients. Women Information: http: I /www. ams. org/amsmtgs/sectional.html. and minority participants are especially encouraged to participate '' 8-9 AMS Central Section Meeting, University of Notre Dame, Notre in this conference and to apply for support. Dame, Indiana. Information: http: I /academic. udayton. edu/ searcde-25/. To get Information: http: I /www . ams. org/amsmtgs/sectional.html. instructions on registration, lodging, submission of abstracts, and application for support. Please pass this announcement along to '' 22-23 AMS Eastern Section Meeting, UniversityofNewHampshire, all who might be interested in participating in the conference. Durham, New Hampshire. If you have questions about the conference, please send e-mail Information: http: I /www. ams. org/amsmtgs/sectional. html. to searcde251!lnotes. udayton. edu or call Muhammad Islam (93 7 229-2109) or Mark Oxley (937 255-3636 Ex 4515). '' 29-30 AMS Western Section Meeting, San Francisco State Univer sity, San Francisco, California. '' 20-22 3rd Symposium on Stochastic Algorithms: Foundations Information: http: I /www. ams. org/amsmtgs/sectional.html. and Applications (SAGA'OS), Moscow State University, Moscow, Russia. Scope: The 3rd Symposium on Stochastic Algorithms, Foundations and Applications (SAGA'05) will be held in Moscow (Russia) from The following new announcements will not be repeated until 20th to 22nd October 2005. The symposium offers the opportu the criteria in the next to the last paragraph at the bottom of nity to present original research on the analysis, implementation, the first page of this section are met. experimental evaluation and real-world application of stochastic algorithms. In particular, the focus of SAGA' OS is on new algorithmic June 2006 ideas involving stochastic decisions and the design and evaluation '' 12-1 5 2006 International Conference on Applied Mathematics of stochastic algorithms within realistic scenarios. Thus, the sym and lnterdisci pli nary Research-Nankai, Nankai University, Tianjin, posium wants to foster the co-operation between practitioners and P.R. China. theoreticians from this research area. Information: The website of the meeting is at http: I /www . is am. Information: Further information can be found at http://mech. nankai . edu. en. Please check out the website for further informa math.msu.su/department/dm/SAGA2005orhttp://www.dcs.kel . tion. ac.uk/events/saga05/. * 21-231CNPAA-2006: Mathematical Problems in Engineering and * 30-November 3 SIAM, Hilton Phoenix East, Phoenix, Arizona. Aerospace Sciences, Budapest, Hungary. Information: The Call for Presentations deadlines for GD05 are Contact and Information: ICNPAA-2006, S. Sivasundaram, 104 fast approaching. Snow Goose Ct., Daytona Beach, FL 32119; email: infol!licnpaa. Dead Ii ne Dates: Minisymposium proposals: April1, 200 5; Abstracts com, [email protected]; http://www.icnpaa. com. for all contributed and minisymposium presentations: May 2, 2005. 570 NOTICES OF THE AMS VOLUME 52, NUMBER 5 New Publications Offered by the AMS group as an automata group; F. Dahmani, An example of non Algebra and Algebraic contracting weakly branch automaton group; A. Akhmedov, Geometry Travelling salesman problem in groups. Contemporary Mathematics, Volume 372 May 2005, 230 pages, Softcover, ISBN 0-8218-3362-6, LC Geometric Methods 2004065011, 2000 Mathematics Subject Classification: 20-06, 20F65, 20F36, 20F10, 20E05, 20E06, 20E08, 20E26, 20E22, in Group Theory 57M60, All AMS members $55, List $69, Order code CONM/372 Geometric Methods Jose Burillo, Universitat in Group Theory Politecnica de Catalunya, Jose Burillo Sean Cleary Barcelona, Spain, Sean Cleary, Murray Elder Analysis Jennifer Toback The City College of New York Enrlc Ventura Editors (CUNY), Murray Elder, University of St. Andrews, Fife, Analyzable Scotland, Jennifer Taback, Bowdoin College, Brunswick, Functions and ME, and Enric Ventura, Universitat Politecnica de Analyzable Applications Catalunya, Barcelona, Spain, Editors Functions and Applications 0. Costin and M. D. Kruskal, This volume presents articles by speakers and participants in 0. Costin Rutgers University, Piscataway, M.D.Kruskol two AMS special sessions, Geometric Group Theory and A. Macintyre N], and A. Macintyre, Editors Geometric Methods in Group Theory, held respectively at University of London, UK, Northeastern University (Boston, MA) and at Universidad de Editors Sevilla (Spain). The expository and survey articles in the book cover a wide range of topics, making it suitable for The theory of analyzable functions is researchers and graduate students interested in group theory. a technique used to study a wide class Contents: M. Cardenas and F. F. Lasheras, Properly 3- of asymptotic expansion methods and their applications in realizable groups: a survey; A. Martino and S. 0 Rourke, Free analysis, difference and differential equations, partial actions on 2n-trees: a survey; G. Levitt, Characterizing rigid differential equations and other areas of mathematics. simplicial actions on trees; J. Gonzalez-Meneses, Improving an Key ideas in the theory of analyzable functions were laid out algorithm to solve multiple simultaneous conjugacy problems by Euler, Cauchy, Stokes, Hardy, E. Borel, and others. Then in in braid groups; E. Godelle and L. Paris, On singular Artin the early 1980s, this theory took a great leap forward with the monoids; 0. Bogopolski, A surface groups analogue of a work of J. Ecalle. Similar techniques and concepts in analysis, theorem of Magnus; V. Addepalli and E. C. Turner, Shift logic, applied mathematics and surreal number theory automorphisms of finite order; V. Shpilrain, Counting emerged at essentially the same time and developed rapidly primitive elements of a free group; R. Weidmann, A rank through the 1990s. The links among various approaches soon formula for amalgamated products with finite amalgam; became apparent and this body of ideas is now recognized as D. Kahrobaei, A simple proof of a theorem of Karrass and a field of its own with numerous applications. Solitar; S. W. Margolis, J. Meakin, and Z. Sunik, Distortion This volume stemmed from the International Workshop on functions and the membership problem for submonoids of Analyzable Functions and Applications held in Edinburgh groups and monoids; J. Belk and K.-U. Bux, Thompson's group (Scotland). The contributed articles, written by many leading F is maximally nonconvex; S. Cleary and J. Taback, Seesaw experts, are suitable for graduate students and researchers words in Thompson's group F; X. Martin, Piecewise-projective interested in asymptotic methods. representation of Thompson's group T; T. Dymarz, Bijective quasi-isometries of amenable groups; I. Bumagin, On Contents: S. Ait-Mokhtar, A singularly perturbed Riccati definitions of relatively hyperbolic groups; G. Baumslag, equation; T. Aoki, T. Kawai, T. Koike, and Y. Takei, On global Embedding wreath-like products in finitely presented groups. aspects of exact WKB analysis of operators admitting infinitely I; S. Cleary and J. Taback, Metric properties of the lamplighter many phases; M. Aschenbrenner and L. van den Dries, MAY 2005 NOTICES OF THE AMS 571 New Publications Offered by the AMS Asymptotic differential algebra; W. Balser and V. Kostov, May 2005, 202 pages, Hardcover, ISBN 0-8218-3810-5, LC Formally well-posed Cauchy problems for linear partial 2005042811, 2000 Mathematics Subject Classification: 46L08; differential equations with constant coefficients; F. Blais, 46Lxx, All AMS members $68, List $85, Order code R. Moussu, and J.·P. Rolin, Non-oscillating integral curves and MMON0/ 226 a-minimal structures; B. Braaksma and R. Kuik, Asymptotics and singularities for a class of difference equations; 0. Costin, Topological construction of transseries and introduction to generalized Borel summability; E. Delabaere, Addendum to Differential Equations the hyperasymptotics for multidimensional Laplace integrals; F. Diener and M. Diener, Higher-order terms for the de Moivre-Laplace theorem; J. Ecalle, Twisted resurgence Nonlinear Partial monomials and canonical-spherical synthesis of local objects; Differential A. Fruchard and E. Matzinger, Matching and singularities of canard values; B. Mudavanhu and R. E. O'Malley, Jr., On the Nonlinear Partial Equations and renormalization method of Chen, Goldenfeld, and Oono; Differential Equations S. P. Norton, Generalized surreal numbers; C. Olive, D. Sauzin, and Related Analysis Related Analysis and T. M. Seara, Two examples of resurgence. GUK:liOI"\QChen George Gasper Gui-Qiang Chen, George JosephJel'ome Contemporary Mathematics, Volume 373 Editors Gasper, and joseph jerome, May 2005, 371 pages, Softcover, ISBN 0-8218-3419-3, LC Northwestern University, 2004062790, 2000 Mathematics Subject Classification: 03C64, Evanston, IL, Editors 34M3?, 34M30, 58]37, 34D15, 40G10, 39A05, All AMS members $71, List $89, Order code CONM/ 373 The Emphasis Year on Nonlinear Partial Differential Equations and Related Analysis at Northwestern University produced this fine collection of original research and survey articles. Many well Translations of Hilbert C;~Modules MATHEMATICAL known mathematicians attended the events and submitted MONOGRAPHS V. M. Manuilov and E. V. their contributions for this volume. Troitsky, Moscow State Eighteen papers comprise this work, representing the most Hilbert c•-Modules University, Russia significant advances and current trends in nonlinear PDEs and V. M. MariuJlov their applications. Topics covered include elliptic and E. V. Troitsky Based on lectures delivered by the parabolic equations, Navier Stokes equations, and hyperbolic authors at Moscow State University, conservation laws. Important applications are presented from this volume presents a detailed incompressible and compressible fluid mechanics, introduction to the theory of Hilbert combustion, and electromagnetism. Also included are articles · ® ...... -- C* -modules. on recent advances in statistical reliability in modeling, Hilbert C* -modules provide a natural simulation, level set methods for image processing, shock generalization of Hilbert spaces waves, free boundaries, boundary layers, errors in numerical arising when the field of scalars C is replaced by an arbitrary solutions, stability, instability, and singular limits. C* -algebra. The general theory of Hilbert C* -modules The volume is suitable for researchers and graduate students appeared more than 30 years ago in the pioneering papers of interested in partial differential equations. W. Paschke and M. Rieffel and has proved to be a powerful tool in operator algebras theory, index theory of elliptic Contents: M. Bendahmane and K. H. Karlsen, Uniqueness of operators, K- and KK-theory, and in noncommutative entropy solutions for doubly nonlinear anisotropic degenerate geometry as a whole. Alongside these applications, the theory parabolic equations; A. Bertozzi, J. Greer, S. Osher, and of Hilbert C* -modules is interesting on its own. K. Vixie, Nonlinear regularizations of TV based PDEs for image processing; J. L. Bona and V. V. Varlamov, Wave In this book, the authors explain in detail the basic notions generation by a moving boundary; G.-Q. Chen and K. Trivisa, and results of the theory, and provide a number of important Analysis on models for exothermically reacting, compressible examples. Some results related to the authors' research flows with large discontinuous initial data; P. Constantin, interests are also included. A large part of the book is devoted Eulerian-Lagrangian hydrodynamic equations; P. Degond, to structural results (self-duality, reflexivity) and to F. Mehats, and C. Ringhofer, Quantum hydrodynamic models nonadjointable operators. derived from the entropy principle; E. Feireisl, Mathematics of Most of the book can be read with only a basic knowledge of viscous, compressible, and heat conducting fluids; functional analysis; however, some experience in the theory of A. Friedman, Symmetry-breaking bifurcations for free operator algebras makes reading easier. boundary problems; J. Glimm, J. W. Grove, Y. Kang, T. Lee, X. Li, D. H. Sharp, K. Q. Ye, Y. Yu, and M. Zhao, Errors in Contents: Basic definitions; Operators on Hilbert modules; numerical solutions of spherically symmetric shock physics Hilbert modules over W * -algebras; Reflexive Hilbert C* problems; P.-E. Jabin and C. Klingenberg, Existence to modules; Multipliers of A -compact operators. Structure solutions of a kinetic aerosol model; J. W. Jerome, Functional results; Diagonalization of operators over C* -algebras; Homotopy triviality of groups of invertible operators; Hilbert analytic methods for evolution systems; Y. Li, Stability of Riemann solutions with large oscillation for the Euler modules and KK-theory; Bibliography; Notation index; Index. equations; Z. Lin, Some recent results on instability of ideal Translations of Mathematical Monographs, Volume 226 plane flows; A. Rousseau, R. Temam, and J. Tribbia, Boundary conditions for an ocean related system with a small parameter; D. Serre, A remark on Y. Brenier's approach to 572 NOTICES OF THE AMS VOLUME 52, NUMBER 5 New Publications Offered by the AMS Born-Infeld electro-magnetic fields; R. Shvydkoy and june 2005, 191 pages, Softcover, ISBN 0-8218-3459-2, LC S. Friedlander, On recent developments in the spectral 2005040968, 2000 Mathematics Subject Classification: 52B20; problem for the linearized Euler equation; K. Song, Transonic OSA15, 11H06, 11P21, 13P10, 14M25, 52C07, 62H17, 68W30, flow arising from 2-D Riemann problems; X. Wang, A note on 90C10, All AMS members $47, List $59, Order code long-time behavior of solutions to the Boussinesq system at CONM/374 large Prandtl number. Contemporary Mathematics, Volume 371 Probability March 2005, 321 pages, Softcover, ISBN 0-8218-3533-5, LC For 2004062338, 2000 Mathematics Subject Classification: 35-06, Clau,.ss room 35-02, 35AOS, 35BOS, 65M12, 65N12, 76N10, 76POS, 76D03, 76B03, All AMS members $71, List $89, Order code Heads or Tails: CONM/371 An Introduction Discrete Mathematics and to Limit Theorems Combinatorics in Probability Emmanuel Lesigne, Universite Integer Points in Fram;ois Rabelais, Tours, Polyhedra France Integer Points in Polyhedra-Geometry, Geometry, Number Everyone knows some of the basics of Number Theory, probability, perhaps enough to play Algebra, Optimization Theory, Algebra, cards. Beyond the introductory ideas, there are many Alexander Borvinok Matthias Beck wonderful results that are unfamiliar to the layman, but which Christian Ho ose Bruce Reznick Optimization Volkmar Welker are well within our grasp to understand and appreciate. Some Editors Alexander Barvinok, of the most remarkable results in probability are those that University of Michigan, Ann are related to limit theorems- statements about what happens Arbor, Matthias Beck, San when the trial is repeated many times. The most famous of these is the Law of Large Numbers, which mathematicians, Francisco State University, engineers, economists, and many others use every day. Christian Haase, Duke In this book, Lesigne has made these limit theorems accessible University, Durham, NC, Bruce Reznick, by stating everything in terms of a game of tossing of a coin: University of Illinois at Urbana-Champaign, and heads or tails. In this way, the analysis becomes much clearer, Volkmar Welker, Philipps-Universitiit Marburg, helping establish the reader's intuition about probability. Germany, Editors Moreover, very little generality is lost, as many situations can be modelled from combinations of coin tosses. The AMS-IMS-SIAM Summer Research Conference on Integer This book is suitable for anyone who would like to learn more Points in Polyhedra took place in Snowbird (UT). This about mathematical probability and has had a one-year proceedings volume contains original research and survey undergraduate course in analysis. articles stemming from that event. Topics covered include commutative algebra, optimization, discrete geometry, Contents: Prerequisites and overview; Modeling a probabilistic statistics, representation theory, and symplectic geometry. The experiment; Random variables; Independence; The binomial book is suitable for researchers and graduate students distribution; The weak law of large numbers; The large interested in combinatorial aspects of the above fields. deviations estimate; The central limit theorem; The moderate deviations estimate; The local limit theorem; The arcsine law; Contents: J. Agapito, A weighted version of quantization The strong law of large numbers; The law of the iterated commutes with reduction for a toric manifold; M. Beck, logarithm; Recurrence of random walks; Epilogue; Biographies; J. A. De Loera, M. Develin, J. Pfeifle, and R. P. Stanley, Bibliography; Index. Coefficients and roots of Ehrhart polynomials; B. Chen, Ehrhart polynomials of lattice polyhedral functions; Y. Chen, Student Mathematical Library, Volume 28 I. Dinwoodie, A. Dobra, and M. Huber, Lattice points, June 2005, approximately 160 pages, Softcover, ISBN 0-8218- contingency tables, and sampling; C. Cochet, Kostka numbers 3714-1, LC 2005041058, 2000 Mathematics Subject and Littlewood-Richardson coefficients; C. Haase, Polar Classification: 60-01, 60Fxx, All AMS members $23, List $29, decomposition and Brion's theorem; P. Hersh and V. Welker, Order code STML/ 28 Grabner basis degree bounds on Tor~[AJ (k, k). and discrete Morse theory for posets; J. B. Lasserre, Integer programming duality and superadditive functions; F. Santos, The Cayley trick and triangulations of products of simplices; M. Beck, B. Chen, L. Fukshansky, C. Haase, A. Knutson, B. Reznick, S. Robins, and A. Schlirmann, Problems from the Cottonwood Room. Contemporary Mathematics, Volume 374 MAY 2005 NOTICES OF THE AMS 573 New AMS-Distributed Publications Analysis New AMS-Distributed Publications COURS S PECIALI S E S Introduction a Introduction l'etude des espaces aI' etude des espaces de Banach Analyse et Probabilitis de Banach Algebra and Algebraic Daniel LI Herve QUEFFELEC Analyse et probabilites Geometry Daniel Li, and Herve Queffelec, Universite de Lille 1, Une introduction aux France 12 Devoted to the study of Banach UNE INTRODUCTION motifs AUXMOTIFS spaces, this book emphasizes (MOTIFS PURS, MOTIFS MIXTES, !'tRIODES) Motifs purs, motifs connections with classical analysis, Yves Andre harmonic analysis, and probability theory. It is suitable for mixtes, periodes beginning graduate students. The study is taken from the Yves Andre, Ecole Normale beginning and then worked out thoroughly, presenting several Superieure, Paris, France fundamental results which were obtained during the period Panoramas et Syntheses 1950-2000: Grothendieck's theorem, Dvoretzky's theorem, 2004 "Motives" were introduced 40 years ago Rosenthal's dichotomy theorem, and Gowers's dichotomy by A. Grothendieck as "a systematic theorem, etc., with applications. theory of arithmetic properties of A publication of the Societe Mathematique de France, Marseilles (SMF), algebraic varieties as embodied in their distributed by the AMS in the U.S., Canada, and Mexico. Orders from groups of classes of cycles". This text provides an exposition of other countries should be sent to the SMF. Members of the SMF receive the geometric foundations of the theory (pure and mixed) and a a 30% discount from list. panorama of major developments that have occurred in the last Contents: Preliminaire. Topologies faible et prefaible. Filtres, 15 years. The last part is devoted to a study of periods of ultrafiltres. Ordinaux; Notions fondamentales de probabilites; motives, with emphasis on examples (polyzeta numbers, Bases dans les espaces de Banach; Convergence notably). It is suitable for graduate students and research inconditionnelle; Variables aleatoires banachiques; Type et mathematicians interested in number theory. cotype des espaces de Banach. Factorisation par un espace de This item will also be of interest to those working in number Hilbert; Operateurs p-sommants. Applications; Quelques theory. proprietes des espaces LP; L'espace l1; Sections euclidiennes; Espaces de Banach separables sans la propriete A publication of the Societe Mathematique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from d'approximation; Processus gaussiens; Sous-espaces reflexifs other countries should be sent to the SMF. Members of the SMF receive de I 1; Quelques exemples d'utilisation de la methode des a 30% discount from list. selecteurs; Espace de Pisier des fonctions presque sfuement continues. Applications; Annexe. Algebres de Banach. Groupes Contents: Partie I. Motifs purs: Sources: geometrie enumerative, abeliens compacts; Bibliographie; Index des notations; Index cohomologie, theorie de Galois; ®-Categories rigides, categories des noms cites; Index terminologique. tannakiennes; Cycles algebriques et cohomologies (cas des varietes projectives lisses); Motifs purs de Grothendieck; Les Cours Specialises-Collection SMF, Number 12 conjectures standard; Groupes de Galois motiviques; Les January 2005, 627 pages, Softcover, ISBN 2-85629-155-4, 2000 conjectures de plenitude et de serni-simplicite des realisations Mathematics Subject Classification: 42A55, 42A61, 43A46, enrichies; Effectivite; Comment contourner les conject:uTes 46B03,46B07,46B09,46B15,46B20,46B25,46B28, 52A21, standard; Applications de la theorie des cycles motives; 52A38, 60D05, 60G42, 60G46, 60G50, Individual member Filtrations sur les anneaux de Chow et nilpotence; Structure de $94, List $104, Order code COSP/12 la categorie des motifs purs pour une equivalence adequate quelconque; Motifs purs virtuels attaches aux k -varietes (transition vers la mixite); Partie II. Motifs mixtes: Pourquoi des motifs mixtes?; Le formalisme elementaire des morphismes multivalues; Motifs mixtes de Voevodsky; Twists et cohomologie motivique; Proprietes fondamentales de DMgm (k); Complexes de faisceaux motiviques; Exemples: 1- motifs et motifs de Tate mixtes; Vers le coeur de DMgm (k); Realisations mixtes et regulateurs; Partie III. Periodes: Relations de periodes; Motifs et valeurs speciales de la fonction r; Motifs et nombres polyzeta; Bibliographie; Index terminologique. Panoramas et Syntheses, Number 17 January 2005, 261 pages, Softcover, ISBN 2-85629-164-3, 2000 Mathematics Subject Classification: 14F42, 19£15, 32G20, 11]91, Individual member $33, List $37, Order code PASY/17 574 NOTICES OF THE AMS VOLUME 52, NUMBER 5 New AMS-Distributed Publications graduate students and research mathematicians interested in Memoires Geometry and number theory, algebra, and algebraic geometry. Topology This item will also be of interest to those working in algebra and algebraic geometry. COBORDISME COMPLEXE DES ESPACES PRO FINIS A publication of the Societe Mathematique de France, Marseilles (SMF), ET FONCTEUR T DE LANNES distributed by the AMS in the U.S., Canada, and Mexico. Orders from Cobordisme other countries should be sent to the SMF. Members of the SMF receive Fran<;o is-Xavier DEHON complexe des a 30% discount from list. Contents: J.-M. Fontaine, Arithmetique des representations 2004 espaces profinis et galoisiennes p-adiques; K. Kato, p-adic Hodge theory and ~CH II II II \1111 \)\ 11!)1 I Ill Ill\'\{ I foncteur T de Lannes values of zeta functions of modular forms; P. Schneider and J. Teitelbaum, Correction to "p-adic boundary values". Franc;ois-Xavier Dehon, Asterisque, Number 295 Universite de Nice, France November 2004, 300 pages, Softcover, ISBN 2-85629-158-9, The author shows that the continuous MU-cohomology of the 2000 Mathematics Subject Classification: 11F11, 11F67, 11F80, mapping spaces from the classifying space Err of some 11F85, 11G05, 11Gl6, 11G40, 11R33, 11R39, 11R56, 11Sl5, commutative compact Lie group to the pro-p-completion of a 11S20, 11S25, 11S80, 11S99, 14F30, 14F40, 14F42, 14Gl0, space whose p-adic cohomology is torsion free is the image of 14G35, 14G40, 22E50, Individual member $74, List $82, the p-completed MU-cohomology of the target space by a Order code AST/295 functor TBrr analogous to the functor T associated to the classifying space of the cyclic group of order p. The book is suitable for graduate students and research mathematicians interested in geometry and topology. A publication of the Societe Mathematique de France, Marseilles (SMF), distributed by the AMS in the U.S., Canada, and Mexico. Orders from other countries should be sent to the SMF. Members of the SMF receive a 30% discount from list. Contents: Introduction; Cobordisme complexe des espaces profinis; Cohomologie des espaces fonctionnels de source le classifiant d'un groupe de Lie compact commutatif; Appendice; Bibliographie; Index. Memoires de la Societe Mathematique de France, Number 98 December 2004, 138 pages, Softcover, ISBN 2-85629-162-7, 2000 Mathematics Subject Classification: 55Q05, 18C15, 55N22, 55R37, 55S25, 55Uxx, Individual member $33, List $37, Order code SMFMEM/98 Number Theory 295 Cohomologies 2004 p-adiques et COHOMOLOGIES applications p-ADIQUES ET APPLICATIONS arithmetiques (III) ARITHMETIQUES (III) •-••••HELOT.J.-M . mNTA>NE . L'LL""~ Pierre Berthelot, Universite de K. KATO, l'>L RAPOPORT, editeu.,.. Rennes I, France, Jean-Marc Fontaine and Luc Illusie, Universite de Paris-Sud, Orsay, France, Kazuya Kato, Kyoto University, japan, and Michael Rapoport, University of Koln, Germany, Editors This volume contains survey papers on p-adic methods in arithmetic geometry. Topics covered include Galois representations, p-adic L -functions of modular forms, and Iwasawa theory of modular forms. The book is suitable for MAY 2005 NOTICES OF THE AMS 575 journals at de Gr r Advances in Geometry ~ · · ~··· • Advances in Geometry • Forum Mathematicum Managing Editors: T. Grundhofer, K. Stram bach Editors: M. Brin, F. R. Cohen, R. Fintushel, M. Fliess, M. Fukushima, G. Gallavotti, J. Garnier, R. Gobel, ISSN 1615-715X J. Lindenstrauss, K.-H. Neeb, j. Noguchi, A. Ranicki, P. Sarnak, 2005. Volume 5 (4 issues). D. S. Scott, D. Segal, C. D. Sogge, K. Strambach, H. J. Sussmann, Approx. 530 pages G. Talenti, G. Wustholz Annual subscription rate: Print only or online only: € 268.00 ISSN 0933-7741 Print + online: € 290.00 2005. Volume 17 (six issues). Price per issue: € 70.00 Approx. 960 pages. Annual subscription rate: Print only or online only: € 498.00 Print + online: € 538.00 Single issues: € 88.00 Journal of Group Theory --~ 7 · Sunobu4 · 100.1 • Journal fiir die reine und angewandte Mathematik (Crelle's journal) • Journal of Group Theory Managing Editor: R. Weissauer Managing Editor: j . S. Wi lson Editors: j. Cuntz, G. Huisken, Y. I. Manin, E. Viehweg, P. Vojta Editors: A.V. Borovik, M. R. Bridson, M. Broue, F. Buekenhout F. De Giovanni, R. Gobel, C. K. Gupta, R. Gural ni ck, ' ISSN 0075-4102 A.A. Ivanov, W.M . Kantor, E.!. Khukhro, G. Malle, S.A. Morris, 2005. Volumes 578-589. A.YU. Olshanskii, C.W. Parker, D.j.S. Robinson, Y. Segev, Approx. 2900 pages. A. Shalev, S. Sidki Annual subscription rate: Print only or online only: € 2298.00 ISSN 1433-5883 2005. Volume 8 (six issues). Print + online: € 2482.00 Single issues: € 200.00 Approx. 480 pages. Annual subscription rate: Print only or online only: € 348.00 Print + online: € 376.00 Single issues: € 62.00 E Test a free online sample copy at de Gruyter www.degruyter.com/journals/extenza Berlin· New York Prices are subject to change. Classified Advertisements Positions available, items for sale, services available, and more University, 1 University Parkway, Univer terdisciplinary research and graduate ed ILLINOIS sity Park, IL 60466. AA/EOE ucation involving ecosystem processes is If you would like more information about desirable. The appointee will be expected GOVERNORS STATE UNIVERSITY the university, please visit our website at: to maintain a vigorous research program Mathematics Faculty www. govst. edu/h r . in the field of numerical analysis and sci 00004 5 entific computing and participate in teach ing, advising and mentoring at the gradu Governors State University invites appli ate and undergraduate levels. Applicants cations for a full-time, tenure-track faculty should send a letter of interest and a de position in applied mathematics at the as MICHIGAN tailed curriculum vitae including a de sistant/ associate professor level begin scription of current and future research in ning August 2005. Our desired candidate MICHIGAN STATE UNIVERSITY terests and a list of publications to: will have the ability and desire to work East lansing, Ml48824 Search Committee: Numerical Analy across disciplines. It is expected that the proMSc Program in sis will faculty member collaborate with the Industrial Mathematics Department of Mathematics Division of Science (which also includes Oregon State University computer science, chemistry, and biology) Direct your students toward one of the Corvallis, OR 97331-4605 and education faculty. The successful can professional M.Sc. programs. Industry Additionally three letters of recommen didate will teach courses at the upper needs business-savvy mathematicians. See division level in a new BA in Mathematics dation, one of which addresses teaching, http://www.sciencemasters.com/. are required. They should be sent directly (which includes a Secondary Teacher Ed 000019 to the above address. For full considera ucation sequence); serve as coordinator tion, complete application materials must for both programs; provide leadership in arrive by April15, 2005. Further informa developing and implementing the curric OREGON tion is available at http: I /www. math. ula; advise undergraduate students and oregonstate.edu/hiring. supervise their research projects; and pur OSU is an Affirmative Action/Equal Op sue scholarly activity in mathematics or Oregon State University Department of Mathematics portunity Employer. mathematics education. The current fac 000044 ulty in mathematics consists of a team of Corvallis, OR 97331-460S experienced university lecturers who de liver the general education mathematics The Department of Mathematics invites curriculum. Qualifications: a Ph.D. in math applications for a full-time, nine-month, TEXAS ematics and a strong commitment to both tenure-track Assistant Professor position specializing in numerical analysis or sci teaching and research. To apply, send a let TEXAS TECH UNIVERSITY entific computing. Applicants should have ter of application, curriculum vitae, a brief a Ph.D in mathematics or a closely related Department of Mathematics and statement of research results and interests, field, significant active research engage Statistics transcripts, and three professional letters ment in numerical analysis or scientific of reference to: Dr. Gary Lyon, Chair, Math The Department of Mathematics and Sta computing, and excellence in teaching. In tistics at Texas Tech University invites ap ematics Faculty Search Committee, Col terest in participating in team-based in- lege of Arts and Sciences, Governors State plications and nominations for the newly Suggested uses for classified advertising are positions available, books or 2005 issue-June 27, 2005; October 2005 issue-July 25, 2005; November lecture notes for sale, books being sought, exchange or rental of houses, 2005 issue-August 26, 2005; December 2005 issue-September 28, 2005. and typing services. 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Advertisers will be Upcoming deadlines for classified advertising are as follows: June/ July billed upon publication. 2005 issue-April 27, 2005; August 2005 issue-May 26, 2005; September MAY 2005 NOTICES OF THE AMS 577 Classified Advertising endowed Dick and Martha Brooks Regents Endowed Professorship in Mathematics and Statistics. Applicants for this position should have outstanding research cre dentials and a proven commitment to teaching and graduate student direction. The Department of Mathematics and Statistics includes 4 7 full-time faculty members, more than 200 undergraduate majors, and 80 masters and Ph.D. stu dents. Faculty have established a suc cessful record of generating funded re search and engaging in cooperative research efforts, within the department and with other departments and univer sities. The successful candidate will be ex pected to provide leadership in advancing research and academic programs and to enhance the national and international reputation of the department. Exceptional candidates from all areas of mathematics and statistics will be considered. Institutional funds will provide the aca demic salary for the chair holder, enabling funds generated by the endowment to be used for the purposes of research pro gram enhancement activities. Applications, with curriculum vitae and addresses of five references, and nomi nations should be sent to Lawrence Schovanec, Chair, Department of Mathe matics and Statistics, Texas Tech Univer sity, Lubbock, TX 79409. For additional information see http: I /www. math. ttu. edu/hi ring. htm 1. Texas Tech is an Af firmative Action/Equal Opportunity em ployer. 000043 578 NOTICES OF THE AMS VOLUME 52, NUMBER 5 ~Springer springeronline.com the language of science Recognized for their Outstanding Contributions to Mathematics Springer Congratulates its Award-Winning Authors of 2005! Peter D. Lax New York University Winner of the Abel Prize Branko Griinbaum University of Washington Winner of the Leroy P. Steele Prize for Mathematical Exposition Gregory Margulis and Sergei Novikov Yale University University of Maryland Winners of the Wolf Prize in Mathematics Harold M. Edwards New York University Winner of the Albert Leon Whiteman Memorial Prize Robert P. Lang lands Institute for Advanced Study, Princeton Winner of the Leroy P. Steele Prize for a Seminal Contribution to Research Paul Glasserman Columbia University Winner of the Wilmott Award for Cutting-Edge Research in Quantitative Finance Steven E. Shreve Carnegie Mellon University Winner of the Wilmott Award for New Book of the Year in Quantitative Finance Find books by these exceptional authors on springeronline.com Easy Ways to Order ~ Call: Toll-Free 1-800-SPRINGER ·Web: springeron line.com · E-mail: [email protected] · Write: Springer, Dept. S8507, PO Box 2485, Secaucus, NJ 07096-2485 • Visit: your local scientific bookstore or urge your librarian to order. S8507 £Q Whether they become Please give generously. t.!!!f!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) school students. email: [email protected] www.ams.org 09/04 AMERICAN MATHEMATICAL SOCIETY A Rich Introduction to I(nots The Knot Book An Elementary Introduction to the For Mathematical Theory of Knots Classroom Use Colin C. Adams, Williams College, Williamstown, MA From reviews of the first edition: This book belongs in every mathematical library. -Charles Ashbacher, Book Reviews Editor, journal of Recreational Mathematics Throughout the book there are lots of exercises of various degrees of difficulty. Many "unsolved questions" provide opportunity for further research. -Zentralblatt MATH We use knots to moor our boats, to wrap our packages, to tie our shoes. Yet the mathematical theory of knots quickly leads to deep results in topology and geometry. The Knot Book is an introduction to this rich theory, starting with our familiar understanding of knots and a bit of college algebra and finishing with exciting topics of current research. 2004; 307 pages; Softcover; ISBN 0-8218-3678-1 ; List $29; All AMS members $23; Order code KNOTN049 For Classroom Supersymmetry for Mathematicians: An Introduction Use V. S. Varadarajan, University of California, Los Angeles In this book,V: S.Varadarajan presents a cogent and self-contained exposition of the foundations of supersym metry for the mathematically-minded reader. Titles in this series are copublished with the Courant Institute of Mathematical Sciences at New York University. Courant Lecture Notes, Volume II; 2004; 300 pages; Softcover; ISBN 0-8218-3574-2; List $39;AII AMS members $31; Order code CLN/ IIN049 For (!] Classroom The Stationary Tower Use Notes on a Course by W. Hugh Woodin Paul B. Larson, Miami University, Oxford, OH Notes on a Course · by W. Hugh Woodin Written by a leading figure in modern set theory, this book is the first detailed treatment of his method of the Paul B. Larson stationary tower that is accessible to graduate students. (~! University Lecture Series. Volume 32; 2004; 132 pages; Softcover; ISBN 0-8218-3604-8; List $29;AII AMS members $23; Order code ULECT/32N049 Transformation Groups for Beginners S. V. Duzhin, Steklov Institute of Mathematics, St. Petersburg, Russia, and B. D. Chebotarevsky, Minsk, Belarus The modern mathematical way of treating symmetry is through transformation groups. This book offers an easy introduction to these ideas for the relative novice. Student Mathematical Library,Volume 25; 2004; approximately 256 pages; Softcover; ISBN 0-8218-3643-9; List $39;AII AMS members $31 ; Order code STMU25N049 I-800-321-4AMS (4267), in the U.S. and Canada, or 1-401-455-4000 (worldwide); fax: 1-40 1-455-4046; email: [email protected]. American Mathematical Society, 20 I Charles Street, Providence, Rl 02904-2294 USA For many more publications of interest, visit the AMS Bookstore www.ams.org/bookstore AMERICAN MATH EMATICAL SOCIETY 8/04 AMERICAN MATHEMATICAL SOCIETY Mathematical Reviews on the Web Your premier source for searching the world's mathematical literature! www.arns.org/rnathscinet ®AMS AMERICAN MATHEMATICAL SOCIETY J www.ams.org J. 1/05 Meetings & Conferences oftheAMS IMPORTANTINFORMATIONREGARDING 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 withlinkstotheabstractforeachtalkcanbefoundo:n theAMSwebsite.Seehttp: I /wv-NJ. ams. org/meeti ngs/.Programs and abstracts will continue to be displayed on the AMS website in the Meetings and Conferences section until about three weeks after the meeting is over. Final programs for Sectional Meetings will be archived on the AMS website in an electronic issue of the Notices as noted below for each meeting. Michael Kapovich, University of California Davis, Gener Santa Barbara, alized triangle inequalities and their applications. Mihai Putinar, University of California Santa Barbara, California Positive polynomials, a hilbertian perspective. University of California Santa Barbara James Sethian, University of California Berkeley, Advances in advancing interfaces: New techniques for propagating Aprill6-l 7, 2005 fronts in wave propagation and materials sciences. Saturday - Sunday Special Sessions Meeting #1 007 Algebraic Geometry and Combinatorics, Alexander Yong Western Section and Allen Knutson, University of California Berkeley. Associate secretary: Michel L Lapidus Arithmetic Geometry, Adebisi Agboola, University of Announcement issue of Notices: February 2005 California Santa Barbara, and Cristian Dumitru Popescu, Program first available on AMS website: March 3, 2005 University of California San Diego. Program issue of electronic Notices: April 2005 Issue of Abstracts: Volume 26, Issue 3 Automorphisms of Surfaces, Anthony Weaver, Bronx Community College of the City University of New York, and Deadlines Peter Turbek, Purdue University Calumet. For organizers: Expired Complexity of Computation and Algorithms, Mark Burgin, For consideration of contributed papers in Special Sessions: University of California Los Angeles. Expired Curvature in Group Theory and Combinatorics, Laura M. For abstracts: Expired Anderson, State University of New York at Binghamton, Noel Patrick Brady, University of Oklahoma, Robin The scientific information listed below may be dated. For Forman, Rice University, and Jonathan P. McCammond, the latest information, see www. ams. o rg/amsmtgs/ sectional.html. University of California Santa Barbara. Dynamical Systems in Neuroscience, Eugene M. Izhikevich, Invited Addresses The Neurosciences Institute. Mei-Chu Chang, University of California Riverside, Set Function Theory, Mihai Putinar and Stephan R. Garcia, addition and set multiplication. University of California Santa Barbara. MAY 2005 NOTICES OF THE AMS 583 Meetings & Conferences Geometric Methods in Three Dimensions, Daryl Cooper, Michael J. Hopkins, Massachusetts Institute of Technology, David Darren Long, and Martin G. Scharlemann, Univer Title to be announced. sity of California Santa Barbara. Christian Krattenthaler, University of Lyon-I, Exact and Geometry and Physics, Xianzhe Dai, University of Califor asymptotic enumeration of vicious walkers with a wall nia Santa Barbara, and Zhiqin Lu, University of California interaction. Irvine. Frank Natterer, University of Muenster, Imaging and History ofMathematics, Shawnee L. McMurran, California inverse problems for partial differential equations. State University San Bernardino, and James J. Tattersall, Horng-Tzer Yau, New York University and Stanford Uni Providence College. versity, Dynamics of Bose-Einstein condensate. Noncommutative Geometry and Algebra, Kenneth R. Good earl, University of California Santa Barbara, J. T. Stafford, Special Sessions University of Michigan, and].]. Zhang, University of Wash Affine Algebraic Geometry, Shreeram Abhyankar, Pur ington. due University, Hubert Flenner, Ruhr University Bochum, Recent Advances in Combinatorial Number Theory, Mei and Makar Limanov, Wayne State University. Chu Chang, University of California Riverside, and Van Ha Algebraic Combinatorics, Patricia Hersh, Indiana Univer Vu, University of California San Diego. sity-Bloomington, Christian Krattenthaler, University of Representation Theory of Algebras (in Honor of Claus Lyon-I, and Volkmar Welker, Philipps University Marburg. Michael Ringel), Alex Martsinkovsky, Northeastern Uni Algebraic Cryptography, Dorian Goldfeld, Columbia versity, Dan Zacharia, Syracuse University, Birge K. University, Martin Kreuzer and Gerhard Rosenberger, Huisgen-Zimmermann, University of California Santa Universitiit Dortmund, and Vladimir Shpilrain, The City Barbara, and Edward L. Green, Virginia Polytech Institute College of New York. & State University. Algebraic Cycles, Eric Friedlander and Marc Levine, North Ricci Flow/Riemannian Geometry, Guofang Wei and western University, and Fabien Morel, Universite Paris. Rugang Ye, University of California Santa Barbara. Algebraic Geometry, Yuri Tschinkel, Georg-August Universitat Gottingen, and Brendan E. Hassett, Rice University. Mainz, Germany Dirac Operators, Clifford Analysis and Applications, Klaus Giirlebeck, University of Weimar, Mircea Martin, Baker Uni June 16-19,2005 versity, John Ryan, University of Arkansas, and Michael Thursday - Sunday Shapiro, IPN Mexico. Discrete Geometry, Jacob Eli Goodman, The City College Meeting #1 008 of New York (CUNY), Emo Welzl, Eidgen Technische ]oint International Meeting with the Deutsche Mathematiker Hochschule, and Gunter M. Ziegler, Technical University Vereinigung (DMV) and the Oesterreichische Mathematis of Berlin. che Gesellschaft (OMG) Function Spaces and Their Operators, Ernst Albrecht, Associate secretary: Susan]. Friedlander Universitat des Saarlandes, Raymond Mortini, Universite Announcement issue of Notices: February 2005 de Metz, and William Ross, University of Richmond. Program first available on AMS website: Not applicable Program issue of electronic Notices: Not applicable Functional Analytic and Complex Analytic Methods in Issue of Abstracts: Not applicable Linear Partial Differential Equations, R. Meise, University of Dusseldorf, B. A. Taylor, University of Michigan, and Deadlines Dietmar Vogt, University of Wuppertal. For organizers: Expired Geometric Analysis, Victor Nistor, Pennsylvania State For consideration of contributed papers in Special Sessions: University, and Elamr Schrohe, Universitat Hannover. Expired Geometric Topology and Group Theory, Cameron MeA For abstracts: Expired Gordon, The University of Texas at Austin, Cynthia Hog-Angeloni, Johann Wolfgang Goethe-Universitat, and The scientific information listed below may be dated. For Wolfgang Metzler, University of Frankfurt. the latest information, see www.ams.org/amsmtgs/ Group Theory, Luise-Charlotte Kappe, SUNY at Bingham i nternmtgs. html. ton, Robert Fitzgerald Morse, University of Evansville, · and Gerhard Rosenberger, Universitat Dortmund. Invited Addresses Hilbert Functions and Syzygies, Uwe Nagel, University of Helene Esnault, University of Essen, Deligne's integrality Kentucky, Irena Peeva, Cornell University, and Tim Romer, theorem in unequal characteristic and rational points over Universitat Osnabruck. finite fields. History of Mathematics (including a special workshop on Richard Hamilton, Columbia University, The Ricci flow. Mathematics and War), Thomas W. Archibald, Acadia 584 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Meetings & Conferences University, John H. McCleary, Vassar College, Moritz Topics in Applied Mathematics: Multiscale Problems, Oscilla Epple, University of Stuttgart, and Norbert Schappacher, tions in Partial Differential Equations, and Homogenization, Technische Universitat Darmstadt. Alexander Mielke, University of Hannover. Homotopy Theory, Paul G. Goerss, Northwestern University, Topics in Applied Mathematics: Numerical Partial Differen Hans-Werner Henn, Institut de Recherche Mathematique tial Equations/Equations with Inherent Conditions, Rolf Avancee, Strasbourg, and Stefan Schwede, Universitat Jeltsch, Eidgen Technische Hochschule, Maria Lukacova Bonn. Medvidova, Technical University of Hamburg, and J. Mac HopfAlgebras and Quantum Groups, M. Susan Montgomery, Hyman, Los Alamos National Laboratory. University of Southern California, and Hans-Jurgen Topology of Manifolds, Matthias Kreck, University of Schneider, University of Munich. Heidelberg, and Andrew Ranicki, University of Edinburgh. Mathematical Physics, Laszlo Erdos, Mathematisches Institut der Albert Ludwigs UniversWit, and Michael P. Loss, Georgia Institute of Technology. Annandale-on Mathematics Education, Gunter Torner, Universitiit Duisburg-Essen, and Alan Schoenfeld, School of Education, Hudson, New York Berkeley. Bard College Modules and Comodules, Sergio L6pez-Permouth, Ohio University, and Robert Wisbauer, University of Dusseldorf. October 8-9, 2005 Multiplicative Arithmetic ofIntegral Domains and Mono ids, Saturday - Sunday Scott Chapman, Trinity University, San Antonio, Franz Halter-Koch, University of Graz, and Ulrich Krause, Uni Meeting #l 009 versitat Bremen. Eastern Section Nonlinear Elliptic Boundary Value Problems, Thomas Associate secretary: Lesley M. Sibner Bartsch, Universitaet Giessen, and Zhi-Qiang Wang, Utah Announcement issue of Notices: August 2005 State University. Program first available on AMS website: August 25, 2005 Nonlinear Waves, Herbert Koch, Universitat Dortmund, and Program issue of electronic Notices: October 2005 Daniel I. Tataru, University of California Berkeley. Issue of Abstracts: Volume 26, Issue 4 Ordinary Differential, Difference, and Dynamic Equations, Deadlines Werner Balser, Universitat Ulm, Martin Bohner, University For organizers: Expired of Missouri-Rolla, and Donald Lutz, San Diego State For consideration of contributed papers in Special Sessions: University. June 21, 2005 Quantum Knot Invariants, Anna Beliakova, Universitat For abstracts: August 16, 2005 Zurich, and Uwe Kaiser, Boise State University. Representations and Cohomology of Groups and Algebras, The scientific information listed below may be dated. For Dave Benson, University of Georgia, and Henning Krause, the latest information, see www. ams. o rg/ amsmtgs/ Universitat Paderborn. sectional . html. Set Theory, Joel Hamkins, City University New York, Peter Invited Addresses Koepke, Universitat Bonn, and Benedikt Lowe, Univer siteit van Amsterdam. Persi Diaconis, Stanford University, Title to be announced (Erdos Memorial Lecture). Spectral Analysis of Differential and Difference Operators, Evgeni Korotyaev, Humboldt-University Berlin, Boris Harold Rosenberg, University of Paris VII, Title to be Mityagin, The Ohio State University, and Gerald Teschl, announced. University of Vienna. Alice Silverberg, University of California Irvine, Title to be Stochastic Analysis on Metric Spaces, Laurent Saloff-Coste, announced. Cornell University, Karl-Theodor Sturm, University of Christopher Sogge, Johns Hopkins University, Title to be Bonn, and Wolfgang Woess, Graz Technical University. announced. Topics in Applied Mathematics: Algebraic Approaches to Benjamin Sudakov, Princeton University, Title to be Preconditioning, Heike Fassbender, Technical University announced. of Braunschweig, and Andreas Frommer, University of Wuppertal. Special Sessions Topics in Applied Mathematics: Control Theory, Peter Algebraic and Geometric Combinatorics (Code: SS 12A), Benner, Technical University of Chemnitz. Cristian P. Lenart, State University of New York at Albany, Topics in Applied Mathematics: Mathematical Problems of and Lauren L. Rose and Sheila Sundaram, Bard College. Mechanics, Friedrich Pfeiffer and Jurgen K. Scheurle, Extremal and Probablistic Combinatorics (Code: SS 11A), Technical University of Munich. Benjamin Sudakov, Princeton University. MAY 2005 NOTICES OF THE AMS 585 Meetings & Conferences Geometric Group Theory (Code: SS lA), Sean Cleary, The Invited Addresses City College of New York, and Melanie I. Stein, Trinity Alberto Bressan, Pennsylvania State University, Title to be College. announced. Geometric Transversal Theory (Code: SS 3A), Richard Pollack, Courant Institute, New York University, and Jacob Assaf Naor, Microsoft Research, Title to be announced. Eli Goodman, The City College of New York. Prasad V. Tetali, Georgia Institute of Technology, Title to Global Theory of Minimal Surfaces (Code: SS 6A), David A. be announced. Hoffman, Mathematical Sciences Research Institute, and Rekha R. Thomas, University of Washington, Title to be Harold Rosenberg, University of Paris VII. announced. History of Mathematics (Code: SS 2A), Patricia R. Allaire, Special Sessions Queensborough Community College, CUNY, Robert E. Bradley, Adelphi University, and Jeff Suzuki, Bard College. Commutative Ring Theory (Code: SS lA), David F. Anderson Homological Aspects of Commutative Algebra (Code: SS 4A), and David E. Dobbs, University of Tennessee at Knoxville. Alexandre Tchernev, University of Albany, SUNY, and Mathematical Aspects of Wave Propagation Phenomena Janet Vassilev, University of Arkansas. (Code: SS 2A), Boris P. Belinskiy, University of Tennessee Infinite Groups (Code: SS lOA), Anthony M. Gaglione, at Chattanooga, and Anjan Biswas, Tennessee State United States Naval Academy, Benjamin Fine, Fairfield University. University, and Dennis Spellman, Philadelphia University. Invariants of Graphs and Matroids (Code: SS 8A), Gary Gordon and Lorenzo Traldi, Lafayette College. Lincoln, Nebraska Measurable, Symbolic, and Tiling Dynamical Systems (Code: University of Nebraska in Lincoln SS 9A), Natalie Preibe Frank, Vassar College, and Samuel J. Lightwood, Western Connecticut State University. October 21-23, 2005 Special Functions and Orthogonal Polynomials: Theory and Friday - Sunday Applications (Code: SS 7A), Diego Dominici, State Univer sity of New York at New Paltz. Meeting #1 011 Theory of Infinite-Dimensional Lie Algebras, Vertex Oper Central Section ator Algebras, and Related Topics (Code: SS SA), Antun Associate secretary: Susan]. Friedlander Milas, SUNY at Albany, Alex J. Feingold, Binghamton Announcement issue of Notices: August 2005 University, and Yi-Zhi Huang, Rutgers University. Program first available on AMS website: September 8, 2005 Program issue of electronic Notices: October 2005 Issue of Abstracts: Volume 26, Issue 4 Johnson City, Deadlines For organizers: Expired Tennessee For consideration of contributed papers in Special Sessions: East Tennessee State University July 5, 2005 For abstracts: August 30, 2005 October 1 5-16, 2005 Saturday - Sunday The scientific information listed below may be dated. For the latest information, see www. ams . o rg/ amsmtg s I Meeting #1 010 sectional . html. Southeastern Section Invited Addresses Associate secretary: Matthew Miller Announcement issue of Notices: August 2005 Howard A. Masur, University of Illinois at Chicago, Billiards Program first available on AMS website: September 1, 2005 in Polygons: Connections of Geometry and Complex Analy Program issue of electronic Notices: October 2005 sis to Dynamical Systems. Issue of Abstracts: Volume 26, Issue 4 Alejandro Uribe, University of Michigan, Title to be announced. Deadlines Judy Walker, University of Nebraska, Title to be announced. For organizers: Expired For consideration of contributed papers in Special Sessions: Jack Xin, University of Texas, Title to be announced. June 28, 2005 For abstracts: August 23, 2005 Special Sessions The scientific information listed below may be dated. For Algebraic Geometry (Code: SS lA), Brian Harbourne, Uni the latest information, see www. ams. org/amsmtgs/ versity of Nebraska-Lincoln, and Bangere P. Purnaprajna, sectional . htm l. University of Kansas. 586 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Meetings & Conferences Combinatorial Matrix Theory (Code: SS lOA), Leslie The scientific information listed below may be dated. For Hogben, Iowa State University, and Bryan L. Shader, Uni the latest information, see www. ams. o rg/amsmtgs/ versity of Wyoming. sectional.html. Dynamic Equations on Time Scales (Code: SS SA), Lynn H. Invited Addresses Erbe and Allan C. Peterson, University of Nebraska Lincoln. Matthew Foreman, University of California Irvine, Title to Geometric Methods in Group Theory and Semigroup Theory be announced. (Code: SS 6A), Susan M. Hermiller and John C. Meakin, Mark Haiman, University of California Berkeley, Title to be University of Nebraska-Lincoln, and Zoran Sunik, Texas announced. A&M University. Wilhelm Schlag, California Institute of Technology, Title Geometry of Differential Equations (Code: SS llA), jeanne to be announced. Nielsen Clelland, University of Colorado, Irina A. Kogan, Hart H. Smith, University of Washington, Title to be North Carolina State University, and Zhijun Qiao, University announced. of Texas-Pan American. Graph Theory(Code: SS 8A), Andrew]. Radcliffe, University Special Sessions of Nebraska-Lincoln, Zsuzsanna Szaniszlo, Valparaiso Algebraic Combinatorics and Geometry (Code: SS 7A), Sara University, and Jonathan Cutler, University of Nebraska C. Billey, University of Washington, and Mark Haiman, Lincoln. University of California Berkeley. Large Cardinals in Set Theory (Code: SS 4A), Paul B. Larson, K- Theory in M-Theory (Code: SS 6A), Gregory D. Landweber, Miami University, Justin Tatch Moore, Boise State University, University of Oregon, and Charles F. Doran, University of and Ernest Schimmerling, Carnegie Mellon University. Washington. Mathematical Ecology (Code: SS 9A), David Logan, University Noncommutative Algebra and Noncommutative Birational of Nebraska-Lincoln, and William Robert Wolesensky, Geometry (Code: SS 3A), Arkady Dmitrievich Berenstein, College of St. Mary. University of Oregon, and Vladimir Retakh, Rutgers Mathematical and Engineering Aspects of Coding Theory University. (Code: SS 3A), Lance Perez and judy Walker, University Partial Differential Equations with Applications (Code: SS of Nebraska-Lincoln. 4A), Alexander Panchenko, Washington State University, Randomness in Computation (Code: SS 7A), John M. Hitch R. E. Showalter, Oregon State University, and Hong-Ming cock, University of Wyoming, Aduri Pavan, Iowa State Yin, Washington State University. University, and Vinodchandran Variyam, University of Regular Algebras and Noncommutative Projective Geom Nebraska-Lincoln. etry (Code: SS 2A), Brad Shelton, University of Oregon, Michaela Vancliff, University of Texas at Arlington, and Recent Progress in Operator Algebras (Code: SS 2A), Allan P. James J. Zhang, University of Washington. Donsig and David R. Pitts, University of Nebraska-Lincoln. Representations of Groups and Algebras (Code: SS SA), Jonathan W. Brundan, Alexander S. Kleshchev, and Eugene, Oregon Viktor Ostrik, University of Oregon. Resolutions (Code: SS lA), Christopher Alan Francisco, Uni University of Oregon versity of Missouri, and Irena Peeva, Cornell University. November 12-13, 2005 Saturday - Sunday Taiwan Meeting #1 012 December 14-18, 2005 Western Section Wednesday - Sunday Associate secretary: Michel L. Lapidus Announcement issue of Notices: September 200S Meeting #1 013 Program first available on AMS website: September 29, First ]oint International Meeting between the AMS and the 200S Taiwanese Mathematical Society. Program issue of electronic Notices: November 200S Associate secretary: John L. Bryant Issue of Abstracts: Volume 26, Issue 4 Announcement issue of Notices: June 200S Program first available on AMS website: Not applicable Deadlines Program issue of electronic Notices: Not applicable For organizers: Aprill2, 200S Issue of Abstracts: Not applicable For consideration of contributed papers in Special Sessions: July 26, 200S Deadlines For abstracts: September 20, 200S For organizers: To be announced MAY 2005 NOTICES OF THE AMS 587 Meetings & Conferences For consideration of contributed papers in Special Sessions: To be announced Notre Dame, Indiana For abstracts: To be announced University of Notre Dame April8-9, 2006 San Antonio, Texas Saturday - Sunday Henry B. Gonzalez Convention Center Meeting #1 016 January 12-1 5, 2006 Central Section Thursday - Sunday Associate secretary: Susan]. Friedlander Announcement issue of Notices: February 2006 Meeting #1 014 Program first available on AMS website: To be announced ]oint Mathematics Meetings, including the 112th Annual Program issue of electronic Notices: To be announced Meeting of the AMS, 89th Annual Meeting of the Mathe Issue of Abstracts: To be announced matical Association ofAmerica, annual meetings of the As sociation for Women in Mathematics (AWM) and the National Deadlines Association of Mathematicians (NAM), the winter meeting For organizers: September 9, 2005 of the Association for Symbolic Logic (ASL), with sessions For consideration of contributed papers in Special Sessions: contributed by the Society for Industrial and Applied Math To be announced ematics (SIAM). For abstracts: To be announced Associate secretary: Matthew Miller Announcement issue of Notices: October 2005 Program first available on AMS website: November 1, 2005 Durham, Program issue of electronic Notices: January 2006 Issue of Abstracts: Volume 27, Issue 1 New Hampshire Deadlines University of New Hampshire For organizers: Expired For consideration of contributed papers in Special Sessions: April22-23, 2006 August 3, 2005 Saturday - Sunday For abstracts: September 28, 2005 Meeting #1 017 AMS Invited Addresses Eastern Section Charles L. Fefferman, Princeton University, Title to be an Associate secretary: Lesley M. Sibner nounced. Announcement issue of Notices: To be announced Dusa McDuff, SUNY at Stony Brook, Title to be announced. Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced Miami, Florida Deadlines Florida International University For organizers: September 22, 2005 For consideration of contributed papers in Special Sessions: April1-2, 2006 To be announced Saturday - Sunday For abstracts: To be announced Meeting #1 01 5 Southeastern Section San Francisco, Associate secretary: Matthew Miller Announcement issue of Notices: To be announced California Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced San Francisco State University Issue of Abstracts: To be announced April29-30, 2006 Deadlines Saturday - Sunday For organizers: September 1, 2005 For consideration of contributed papers in Special Sessions: Meeting #1 018 To be announced Western Section For abstracts: To be announced Associate secretary: Michel L. Lapidus 588 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Meetings & Conferences Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Storrs, Connecticut Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced University of Connecticut October 28-29, 2006 Deadlines Saturday - Sunday For organizers: September 30, 2005 Eastern Section For consideration of contributed papers in Special Sessions: Associate secretary: Lesley M. Sibner 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 Special Sessions Issue of Abstracts: To be announced History ofMathematics (Code: SS lA), Shawnee L. McMurran, Deadlines California State University, San Bernardino, and James J. Tattersall, Providence College. For organizers: March 28, 2006 For consideration of contributed papers in Special Sessions: To be announced Salt Lake City, Utah For abstracts: To be announced University of Utah Fayetteville, October 7-8,2006 Saturday - Sunday Arkansas Western Section University of Arkansas Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced November 3-4, 2006 Program first available on AMS website: To be announced Friday - Saturday Program issue of electronic Notices: To be announced Southeastern Section Issue of Abstracts: To be announced Associate secretary: Matthew Miller Announcement issue of Notices: To be announced Deadlines Program first available on AMS website: To be announced For organizers: March 7, 2006 Program issue of electronic Notices: To be announced For consideration of contributed papers in Special Sessions: Issue of Abstracts: To be announced To be announced Deadlines For abstracts: To be announced For organizers: April 3, 2006 For consideration of contributed papers in Special Sessions: To be announced Cincinnati, Ohio For abstracts: To be announced University of Cincinnati October 21-22, 2006 New Orleans, Saturday - Sunday Central Section Louisiana Associate secretary: Susan]. Friedlander New Orleans Marriott and Sheraton New Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Orleans Hotel Program issue of electronic Notices: To be announced January 4-7,2007 Issue of Abstracts: To be announced Thursday - Sunday joint Mathematics Meetings, including the 113th Annual Deadlines Meeting of the AMS, 90th Annual Meeting of the Mathe For organizers: March 21, 2006 matical Association of America (MAA), annual meetings of For consideration of contributed papers in Special Sessions: the Association for Women in Mathematics (A tVM) and the To be announced National Association of Mathematicians (NAM), and the For abstracts: To be announced winter meeting of the Association for Symbolic Logic (ASL), MAY 2005 NOTICES OF THE AMS 589 Meetings & Conferences with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Washington, District Associate secretary: Susan]. Friedlander Announcement issue of Notices: October 2006 of Columbia Program first available on AMS website: November 1, 2006 Program issue of electronic Notices: January 2007 Marriott Wardman Park Hotel Issue of Abstracts: Volume 28, Issue 1 and Omni Shoreham Hotel Deadlines january 7-10,2009 For organizers: April1, 2006 Wednesday - Saturday For consideration of contributed papers in Special Sessions: ]oint Mathematics Meetings, including the !15th Annual To be announced For abstracts: To be announced Meeting of the AMS, 92nd Annual Meeting of the Mathe matical Association ofAmerica (MAA), annual meetings of the Association for Women in Mathematics (A WM) and the Oxford, Ohio National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL). Miami University Associate secretary: Lesley M. Sibner Announcement issue of Notices: October 2008 March 16-1 7, 2007 Program first available on AMS website: November 1, 2008 Friday - Saturday Program issue of electronic Notices: January 2009 Central Section Issue of Abstracts: Volume 30, Issue 1 Associate secretary: Susan]. Friedlander Announcement issue of Notices: To be announced Deadlines 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 Deadlines For abstracts: To be announced For organizers: To be announced For consideration of contributed papers in Special Sessions: To be announced San Francisco, For abstracts: To be announced California San Diego, California Moscone Center West and the San Francisco Marriott San Diego Convention Center January 6-9,2010 january 6-9, 2008 Wednesday - Saturday Sunday - Wednesday ]oint Mathematics Meetings, including the !16th Annual ]oint Mathematics Meetings, including the 114th Annual Meeting of the AMS, 93rd Annual Meeting of the Mathe Meeting of the AMS, 91st Annual Meeting of the Mathe matical Association of America (MAA), annual meetings of matical Association ofAmerica (MAA), annual meetings of the Association for Women in Mathematics (A WM) and the the Association for Women in Mathematics (A WM) and the National Association of Mathematicians (NAM), and the National Association of Mathematicians (NAM), and the winter meeting of the Association for Symbolic Logic (ASL). winter meeting of the Association for Symbolic Logic (ASL). L. Associate secretary: Michel L. Lapidus Associate secretary: John Bryant Announcement issue of Notices: October 2007 Announcement issue of Notices: October 2009 Program first available on AMS website: November 1, 2007 Program first available on AMS website: November 1, 2009 Program issue of electronic Notices: January 2008 Program issue of electronic Notices: January 2010 Issue of Abstracts: Volume 29, Issue 1 Issue of Abstracts: Volume 31, Issue 1 Deadlines Deadlines For organizers: April1, 2007 For organizers: April1, 2009 For consideration of contributed papers in Special Sessions: For consideration of contributed papers in Special Sessions: To be announced To be announced For abstracts: To be announced For abstracts: To be announced 590 NOTICES OF THE AMS VOLUME 52, NUMBER 5 Meetings & Conferences New Orleans, Louisiana New Orleans Marriott and Sheraton New Orleans Hotel January 5-8,2011 Wednesday - Saturday ]oint Mathematics Meetings, including the 117th Annual Meeting of the AMS, 94th Annual Meeting of the Mathe matical Association ofAmerica, annual meetings of the As sociation for Women in Mathematics (A WM) and the National Association ofMathematicians (NAM), and the winter meet ing of the Association for Symbolic Logic (ASL). Associate secretary: Susan]. Friedlander Announcement issue of Notices: October 2010 Program first available on AMS website: November 1, 2010 Program issue of electronic Notices: January 20ll Issue of Abstracts: Volume 32, Issue 1 o:x;q;w 4£. + *"'*"'""a:;o*""' Deadlines For organizers: April 2, 20ll For consideration of contributed papers in Special Sessions: "qw epo 1 ' t•c+ To be announced For abstracts: To be announced )Ill J YQ iff{ $4 it MAY 2005 NOTICES OF THE AMS 591 Meetings and Conferences of the AMS Associate Secretaries of the AMS Western Section: MichelL. Lapidus, Department of Math Eastern Section: Lesley M. Sibner, Department of Mathe ematics, University of California, Sproul Hall, Riverside, CA matics, Polytechnic University, Brooklyn, NY 11201-2990; 92521-0135; e-mail: l api dus@math. ucr. edu; telephone: 951- e-mail: lsi bne r@duke. poly. edu; telephone: 718-260-3 505. 827-5910. Southeastern Section: Matthew Miller, Department of Math Central Section: Susan J. Friedlander, Department of Math ematics, University of South Carolina, Columbia, SC 29208- . ematics, University of illinois at Chicago, 851 S. Morgan (M/C 0001, e-mail: mill er@math. sc. edu; telephone: 803-777-3690. 249), Chicago, lL 60607-7045; e-mail: susan@math. nwu. edu; tele phone: 312-996-3041. The Meetings and Conferences section of the Notices March 16-17 Oxford, Ohio p. 590 gives information on all AMS meetings and conferences approved by press time for this issue. Please refer to the page 2008 numbers cited in the table of contents on this page for more January 6-9 San Diego, California p. 590 detailed information on each event. Invited Speakers and Annual Meeting Special Sessions are listed as soon as they are approved by 2009 the cognizant program committee; the codes listed are needed January 7-10 Washington, DC p. 590 for electronic abstract submission. For some meetings the list Annual Meeting may be incomplete. Information in this issue may be dated. 2010 Up-to-date meeting and conference information can be January 6-9 San Franciso, California p. 590 found at www. ams. orglmeeti ngsl. Annual Meeting Meetings: 2011 January 5-8 New Orleans, Louisiana p. 591 Annual Meeting 2005 April16-17 Santa Barbara, California p. 583 Important Information regarding AMS Meetings June 16-19 Mainz, Germany p. 584 Potential organizers, speakers, and hosts should refer to October 8-9 Annandale-on-Hudson, page 100 in the January 2005 issue of the Notices for general New York p. 585 information regarding participation in AMS meetings and October 15-16 Johnson City, Tennessee p. 586 conferences. October 21-23 Lincoln, Nebraska p. 586 Abstracts November 12-13 Eugene, Oregon p. 587 Speakers should submit abstracts on the easy-to-use interactive December 14-18 Taiwan p. 587 Web form. No knowledge of ID"£X is necessary to submit an electronic form, although those who use ID"£X may submit 2006 abstracts with such coding, and all math displays and sirni January 12-15 San Antonio, Texas p. 588 larily coded material (such as accent marks in text) must Annual Meeting be typeset in ID"£X. Visit http:llwww.ams.orglcgi-binl April1-2 Miami, Florida p. 588 abstractslabstracts.pl. April8-9 Notre Dame, Indiana p. 588 Questions about abstracts and requests for paper forms April 22-23 Durham, New Hampshire p. 588 may be sent to abs-i nfo@ams. org. April29-30 San Francisco, California p. 588 Paper abstract forms must be sent to Meetings & Confer ences Department, AMS, October 7-8 Salt Lake City, Utah p. 589 P.O. Box 6887, Providence, RI 02940. There is a $20 processing fee for each paper abstract. There October 21-22 Cincinnati, Ohio p. 589 is no charge for electronic abstracts. Note that all abstract dead October 28-29 Storrs, Connecticut p. 589 lines are strictly enforced. November 3-4 Fayetteville, Arkansas p. 589 Close attention should be paid to specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated. 2007 January 4-7 New Orleans, Louisiana p. 589 Annual Meeting Conferences: (see http: I lwww. ams. orglmeeti ngsl for the most up-to-date information on these conferences.) June 5-July 21, 2005: Joint Summer Research Conferences in the Mathematical Sciences, Snowbird, Utah (see November 2004 Notices, page 1294). July 25-August 12, 2005: Summer Research Institute on Algebraic Geometry, Seattle, Washington (see November 2004 Notices, page 1293). Co-sponsored conference: June 2006: Fifth Conference on Poisson Geometry, Tokyo, Japan (watch http: I ltmugs. math. metro-u. ac. j pigeneral. html for future information). 592 NOTICES OF THE AMS VOLUME 52, NUMBER 5 1 ((i~ AMS ADVERTISERS' fORUM AMERICAN MATHEMATICAL SOCIETY c"' 0 ·;::; 1!1 .!::!"' -.::;"' "'G.l ::0 ·;::; ns ;j ..0 ASA "' a. G.l"'- ~"' G.l G.l AMERICAN STATISTICAL ""l ~:I-- rn u <;, ·;;; ~ ~Ill Ill ASSOCIATION E CIJ ns I ~ ..,0 u The nation 's leading professional association for statistics I c: ... and statisticians since 1839 " 1429 Duke Street, Alexandria, VA 22314-3415 .. • • Toll-free: 1 (888) 231-3473 Phone: (703) 684-1221 • • • Fax: (703) 684-8037 [email protected] • www.amstat.org www.ams.org/bookstore Discrete ovnam_ics- in Nature and societY ,_...t.,..r~of matfjemat\CS ..,610G Index of Advertisers AMS New and Recently Published ...... inside front cover AIM Publications ...... 497 Birkhaiiser ...... ••...... 498 AMS Book and journal Donation Program ...... 515 Cambridge University Press ...... 531 lOS Press ...... ••...... 538 For more information, email [email protected] or cal/800-225-6911 Bathsheba Grossman (sculpting geometry) ...... 538 A K Peters ...... 542 Princeton University Press ...... ••...... 544 POWELL'S MacKichan Software, Inc ...... 547 TECHNICAL BOOKS de Gruyter ...... 576 33 NW Pork Avenue, Portland, OR 97209 powells. com Springer ...... •...... ••...... 579 AMS Epsilon Fund ...... 580 AMS Books ...... ••...... ••...... 581 MathSciNet ...... ••...... 582 Springer ...... ••...... ••...... back cover To be featured in or to find out more about Advertisers' Forum, contact Anne Newcomb by phone at 1.800.321.4267 (1.401.455.4084, worldwide) or send email to [email protected]. ------4f1 Springer springeronline.com the language of science Springer for Mathematics Linearity, Symmetry and Prediction in NOW IN SOFTCOVER Mathematics Is Not a THE HYDROGEN ATOM Stochastic Calculus for Spectator Sport Finance I George Phillips, University of St. Stephanie F. 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