ISSN 0002-9920 (print) ISSN 1088-9477 (online)

of the American Mathematical Society

April 2013 Volume 60, Number 4

Remembering (1923—2005) page 398 Errors and Corrections in Literature page 418 Mathematical Methods in the Study of Historical Chronology page 441 Newton’s Laws and Coin Tossing page 450

Visualizing the impact of humans on the planet (see page 425) Open Access Journals Abstract and in Mathematics Applied Analysis

Your research wants to be free!

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013

Advances in Decision Sciences

Advances in Mathematical Physics

Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 Hindawi Publishing Corporation Volume 2013 http://www.hindawi.com 7 6 Advances in 9 15 Operations Algebra 768 Research 7 Advances in Submit your manuscripts at Numerical Analysis Hindawi Publishing Corporation3 Volume 2013 Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com 2 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 Discrete Dynamics in Nature and Society

Computational and Mathematical Methods International Journal of in Medicine Game Theory Analysis Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013

International Journal of Mathematics and Mathematical Sciences

International Journal of International Journal of International Journal of Differential International Journal of Partial Differential Combinatorics Equations Engineering Mathematics Equations

Hindawi Publishing Corporation Volume 2013 Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 Mathematics Awareness Month - April 2013

Balancing needs and seeking solutions for a complex changing world

To learn more about the connections between mathematics and sustainability, visit www.mathaware.org Solve the differential equation. Solve the differential equation. t ln t dr + r = 7tet dt t ln t dr + r = 7tet dt 7et + C r = 7et + C ln t ✓r = ln t ✓

Who has the #1 homework system for calculus? The answer is in the questions. When it comes to online calculus, you need a solution that can grade the toughest open-ended questions. And for that there is one answer: WebAssign. WebAssign’s patent pending grading engine can recognize multiple correct answers to the same complex question. Competitive systems, on the other hand, are forced to use multiple choice answers because, well they have no choice. And speaking of choice, only WebAssign supports every major textbook from every major publisher. With new interactive tutorials and videos offered to every student, it’s not hard to see why WebAssign is the perfect answer to your online homework needs. It’s all part of the WebAssign commitment to excellence in education. Learn all about it now at webassign.net/math.

800.955.8275 webassign.net/math

WA Calculus Question ad Notices.indd 1 11/29/12 1:06 PM Notices of the American Mathematical Society April 2013 Communications 390 A Photographic Look at the Joint Mathematics Meetings, San Diego, 2013 461 426 2012–2013 Faculty Salaries Report Richard Cleary, James W. Maxwell, and Colleen Rose 434 What Is New in ? VI . on an iPad . Empire G. Grätzer 463 Quantitative Approaches to 398 434 Sustainability Seminars Rachel Levy The Notices offerings for April are diverse and fascinating . 466 Remembering Steve Rallis We have an article about the development of ergodic theory . Jim Cogdell and Dihua There is an article about the level of errors in the mathemati- Jiang, Coordinating Editors cal literature and how we deal with them . Another piece treats 470 Doceamus: An Update on the issue of historical chronology, or how we can be sure of Jobs for Doctorates in the dates of important events . And there is a memory of dis- Mathematics Education tinguished Raoul Bott . April is Mathematics Robert Reys, Barbara Reys, Awareness Month, and the theme this year is Mathematics and Anne Estapa and Sustainability . The Opinion Column by Simon Levin and 474 Aschbacher and Caffarelli the Communication by Rachel Levy treat this topic . Awarded 2012 Wolf Prize —Steven G. Krantz, Editor Elaine Kehoe 475 Platonism Is the Law of the Land Features David A. Edwards 480 2013 Steele Prizes 398 Remembering Raoul Bott 484 2013 Conant Prize (1923–2005) 486 2013 Morgan Prize Loring W. Tu, Coordinating Editor 488 2013 E . H . Moore Prize 489 2013 Robbins Prize 418 Errors and Corrections in Mathematics 490 2013 Satter Prize Literature 492 2013 Wiener Prize Joseph F. Grcar 494 2013 Veblen Prize Commentary 441 Mathematical Methods in the Study of 392 Opinion: The Mathematics Historical Chronology of Sustainability Simon Levin Florin Diacu 395 Letters to the Editor (and Correction) 450 Newton’s Laws and Coin Tossing 461 A Mathematician’s Lament Don Ornstein —A Book Review Reviewed by William Schmidt NotNoticiceses Departments ofof the the American American Mathematical Mathematical Society Society About the Cover...... 425

EDITOR: Steven G . Krantz Mathematics People ...... 497 ASSOCIATE EDITORS: Jerison and Lee Awarded 2012 Bergman Prize, 2012 Poincaré Prizes Krishnaswami Alladi, David Bailey, Eric Bedford, Awarded, Simon Awarded Sacks Prize, Mawhin Awarded Schauder Jonathan Borwein, Susanne C . Brenner, , (Graphics Editor), Jennifer Prize, Prizes of the Accademia Nazionale dei Lincei, (1913– Chayes, Gerald Folland, Susan Friedlander, Robion 2012). Kirby, Rafe Mazzeo, Harold Parks, Mark Saul, Carla D . Savage, Steven Strogatz, James Walker Mathematics Opportunities...... 500 SENIOR WRITER and DEPUTY EDITOR: Math in Moscow Scholarship Program, NSF-CBMS Regional Conferences Allyn Jackson 2013; DMS Workforce Program in the Mathematical Sciences; NSF MANAGING EDITOR: Sandra Frost Scholarships in Science, Technology, Engineering, and Mathematics; CONTRIBUTING WRITER: Elaine Kehoe Project NExT 2013–2014; AWM Gweneth Humphreys Award; News from CONTRIBUTING EDITOR: Randi D . Ruden the IMA. EDITORIAL ASSISTANT: David M . Collins PRODUCTION: Kyle Antonevich, Anna Hattoy, Inside the AMS...... 503 Teresa Levy, Mary Medeiros, Stephen Moye, Erin Murphy, Lori Nero, Arlene O’Sean, Karen Ouellette, AMS Hosts Congressional Briefing, National Who Wants to Be a Donna Salter, Deborah Smith, Peter Sykes, Patricia Mathematician Competition, From the AMS Public Awareness Office. Zinni ADVERTISING SALES: Anne Newcomb Reference and Book List ...... 504 SUBSCRIPTION INFORMATION: Subscription prices for Volume 60 (2013) are US$547 list; US$437 60. Mathematics Calendar ...... 511 institutional member; US$328 20. individual mem- ber; US$492 30. corporate member . (The subscription New Publications Offered by the AMS...... 516 price for members is included in the annual dues ). A late charge of 10% of the subscription price will be Classified Advertisements...... 521 imposed upon orders received from nonmembers after January 1 of the subscription year . Add for postage: Surface delivery outside the Meetings and Conferences of the AMS...... 524 and India—US$27; in India—US$40; expedited deliv- ery to destinations in North America—US$35; else- Meetings and Conferences Table of Contents...... 535 where—US$120 . Subscriptions and orders for AMS publications should be addressed to the American Mathematical Society, P O. . Box 845904, Boston, MA 02284-5904 USA . All orders must be prepaid . ADVERTISING: Notices publishes situations wanted and classified advertising, and display advertising for publishers and academic or scientific organizations . Advertising material or questions may be sent to [email protected] (classified ads) or notices-ads@ From the ams.org (display ads) . SUBMISSIONS: Articles and letters may be sent to AMS Secretary the editor by email at [email protected], by fax at 314-935-6839, or by postal mail at Department of Mathematics, Washington University in St . Louis, Call for Nominations for the 2014 Bôcher Memorial Prize, Campus Box 1146, One Brookings Drive, St . Louis, MO 63130 . Email is preferred . Correspondence with the Frank Nelson Cole Prize in Number Theory, and Levi L. Conant managing editor may be sent to [email protected] . Prize ...... 508 For more information, see the section “Reference and Book List” . Call for Nominations for the 2014 Joseph L. Doob Prize, NOTICES ON THE AMS WEBSITE: Supported by the AMS membership, most of this publication is freely Leonard Eisenbud Prize for Mathematics and Physics, and the available electronically through the AMS website, the Distinguished Public Service Award ...... 509 Society’s resource for delivering electronic prod- ucts and services . Use the URL http://www.ams. org/notices/ to access the Notices on the website . Call for Nominations for the 2014 Frank and Brennie Morgan [Notices of the American Mathematical Society (ISSN 0002- AMS-MAA-SIAM Prize ...... 510 9920) is published monthly except bimonthly in June/July by the American Mathematical­ Society at 201 Charles Street, Providence, RI 02904-2294 USA, GST No . 12189 2046 RT**** . Applications Invited for Position of / Periodicals postage paid at Providence, RI, and additional MathSciNet Associate Editor ...... 523 mailing offices . POSTMASTER: Send address change notices to Notices of the American Mathematical Society, P .O . Box 6248, Providence, RI 02940-6248 USA . ] Publication here of the Society’s street address and the other information in brackets above is a technical requirement of the U . S . Postal Service . Tel: 401-455-4000, email: [email protected] . © Copyright 2013 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 es- tablished 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 . AMERICAN MATHEMATICS SOCIETY

˜ e AMS is pleased to spotlight several new and classic titles in our growing collection of Mathematics Education books. ˜ ese publications are valuable resources for current and future mathematics teachers, and collectively span the educational spectrum from the preschool through collegiate levels.

CBMS Integers, Conference Board of the Mathematical Sciences The Understanding Issues in Mathematics Education Volume 17 Fractions and The Mathematical Mathematical Numbers in Education Arithmetic of Teachers II Education of Elementary

THEMAT A IC M A L N ΤΡΗΤΟΣ ΜΗ ΕΙΣΙΤΩ American Mathematical Society A S

O C I

C

R

I

E E ΑΓΕΩΜΕ

T in cooperation with

M

Y A Teachers II School

F O 8 Mathematical Association of America A Guide for Teachers U 88 NDED 1 Judith D. Sally, This report is an important Mathematics Northwestern University, Evanston, IL, resource for those who teach mathematics Hung-Hsi Wu, University of , and Paul J. Sally, Jr., University of and statistics to current and future PreK–12 Berkeley, CA Chicago, IL mathematics teachers. It makes recommenda- tions for the mathematics that teachers should [This book] delivers the mathematical knowl- This book, which consists of twelve interactive know and how they should come to know that edge that elementary-grades teachers need. seminars, is a comprehensive and careful study mathematics. It also urges greater involvement of the fundamental topics of K-8 arithmetic. of and statisticians in teacher —American Educator The guide aims to help teachers understand education so that the nation’s mathematics the mathematical foundations of number teachers have the knowledge, skills, and dis- This is a textbook for pre-service elementary theory in order to strengthen and enrich positions needed to provide students with school teachers and for current teachers to their mathematics classes. Five seminars are a mathematics education that ensures high refer to for explanations of well-known and dedicated to fractions and decimals, and the school graduates are college- and career-ready until now unexplained facts. Wu provides a remaining seminars cover standard topics in as envisioned by the Common Core State comprehensive treatment of all the standard detail, albeit in a slightly unconventional order. Standards. topics about numbers in the school mathe- The book is intended for the professional matics curriculum: whole numbers, fractions, development of teachers and is appropriate This series is published in cooperation with the and rational numbers. Assuming no previous for teacher education programs, as well as for Mathematical Association of America. knowledge of mathematics, the presentation enrichment programs such as Mathematical CBMS Issuesin MathematicsEducation , Volume develops the basic facts about numbers from Circles for Teachers. 17; 2012; 86 pages; Softcover; ISBN: 978-0-8218-6926- the beginning and thoroughly covers the sub- 0; List US$33; AMS members US$26.40; Order code ject matter for grades K through 7. Titles in this series are co-published with the CBMATH/17 Mathematical Sciences Research Institute (MSRI). 2011; 551 pages; Hardcover; ISBN: 978-0-8218-5260- MSRI MathematicalCircles Library, Volume 10; 6; List US$79; AMS members US$63.20; Order code 2012; 208 pages; Softcover; ISBN: 978-0-8218-8798-1; Axiomatic MBK/79 List US$39; AMS members US$31.20; Order code Geometry MCL/10 John M. Lee, University of Math from Washington, Seattle, WA Three to Seven Geometry The story of geometry is the The Story of a A Guide for Teachers story of mathematics itself. This Mathematical Circle book tells the story of how the axiomatic Judith D. Sally, for Preschoolers Northwestern University, method has progressed from Euclid’s time to ours, as a way of understanding what Evanston, IL, and Paul Alexander Zvonkin, Université Bordeaux mathematics is, how we read and evaluate , University of I, Talence, France J. Sally, Jr. mathematical arguments, and why mathe- Chicago, IL matics has achieved the level of certainty it As anyone who has taught or raised young children knows, mathematical education for Concepts in plane and solid geometry are has. It is designed primarily for advanced little kids is a real mystery. This book is a cap- carefully explained, and activities that teachers undergraduates who plan to teach secondary tivating account of a professional mathema- can use in their classrooms are emphasized. school geometry, but it should also provide tician’s experiences conducting a math circle The book should give teachers a firm founda- something of interest to anyone who wishes for preschoolers in his apartment in Moscow tion on which to base their instruction in the to understand geometry and the axiomatic in the 1980s. elementary and middle grades. In addition, it method better. should help teachers give their students a solid Pure and AppliedUndergraduate Texts , Volume Titles in this series are co-published with the basis for the geometry that they will study in 21; 2013; approximately 473 pages; Hardcover; ISBN: Mathematical Sciences Research Institute (MSRI). high school. 978-0-8218-8478-2; List US$75; AMS members US$60; Order code AMSTEXT/21 MSRI MathematicalCircles Library, Volume 5; Titles in this series are co-published with the 2011; 300 pages; Softcover; ISBN: 978-0-8218-6873-7; Mathematical Sciences Research Institute (MSRI). List US$39; AMS members US$31.20; Order code MCL/5 MSRI MathematicalCircles Library, Volume 3; 2011; 202 pages; Softcover; ISBN: 978-0-8218-5362-7; List US$39; AMS members US$31.20; Order code MCL/3

To order visit ams.org/bookstore, facebook.com/amermathsoc or call (800) 321-4267 (U.S. & ), (401) 455-4000 (Worldwide) twitter: @amermathsoc 390 Photographs by Sandy Huffaker, photographer, and Annette Emerson (AMS).

N otices of the AMS V olume 60, N umber 4 See photo key on page 493

April 2013 Notices of the AMS 391 Opinion

• Develop schemes for robust governance in these The Mathematics of multiscale systems. None of these is unexplored, and all are foci of much Sustainability current activity. For example, there has been a great deal Assuring a sustainable future for our children and of work on collective motion in animal populations and on grandchildren is, arguably, the greatest challenge facing moving from Lagrangian models that describe the actions humanity and raises a plethora of scientific and mathemat- of individuals to Eulerian models that capture population ical challenges. In the language of the Brundtland Report statistics [Flierl, Grünbaum, Levin, and Olson, 1999]. The World Commission on Environment and Development, methods require writing equations of the form 1987, it means leaving for future generations the same n(x, v, t + δt) options we have for how we want to live our lives. However, Z 0 0 0 0 0 0 operationalizing that concept is easier said than done = dx dv PδX(x − x − v δt; x , v , t) [Solow, 1991]. 0 0 0 0 0 For energy sustainability alone, mathematics has much ∗ PδV(v − v − aδt; x , v , t)n(x , v , t) to contribute in finding better and less polluting ways for the evolution of the spatial position/velocity densi- to explore for new energy, in increasing combustion ties and then using an appropriate closure scheme to efficiency, in the development of alternative energy, in derive a continuum description. An alternative approach the management of energy grids and networks, and in [Kevrekidis, 2002], using microscopic simulators in the minimizing the climate consequences of energy use. For absence of knowledge of specific closure schemes, is an sustainability of financial markets and economic systems, example of exciting new mathematical technique develop- the role of mathematics is equally ubiquitous and essential. ment. Collective motion also has much to teach us about For the remainder of this article, however, I will focus on collective action more generally, especially regarding a less appreciated aspect: the services, direct and indirect, group decision making, and new mathematical tools are that ecosystems provide humanity [Daily, 1997], [Ehrlich proving useful there [Brams, 1978], [Couzin et al., 2011], and Mooney, 1983]. Sustaining these services—including [Saari, 2008a, 2008b]. food, fiber, fuel, pharmaceuticals, climate mediation, the Ecosystems, the biosphere, and the coupled human- sequestration of toxics, and simply aesthetics like the environmental systems in which they are embedded are protection of charismatic species—may represent the complex adaptive systems [Holland, 1995], [Levin, 2003], most urgent challenge of all. in which individual agents interact locally, producing The protection of many ecosystem services is tied to emergent patterns that feed back to affect individual protecting broad patterns of species abundance and nutri- behaviors. In such systems, scaling from the microscopic ent cycling. These macroscopic features, while insensitive to the macroscopic is essential for understanding the to much detail about particular organisms or species, drivers of pattern, the potential for collapse or regime obviously emerge from the collective properties of large shift, and the conflicts that arise between the interests numbers of interactions among individual organisms; and of individual agents (whether organisms or institutions understanding how this occurs is at the core of protecting or nations) and the collective good. Pattern formation services. Similar issues relate to the governance of the in such systems has long been a topic of mathematical biosphere, where the microscopic interactions are among interest [Turing, 1952], but new challenges remain. So people or governments pursuing their own selfish agendas, too has there been historical mathematical interest in with macroscopic consequences that affect all. Evolution critical transitions and regime shifts [Thom, 1969], albeit repeatedly has confronted this challenge at multiple levels, not without controversy; but attention to early warning for example, in the emergence of multicellularity [Levin, indicators has received new attention recently [Lenton, 2010], and it is crucial for us to learn how to do so Livina, Dakos, van Nes, and Scheffer, 2012], [Scheffer, at the global level. Hence, initially, I identify a set of 2009], [Scheffer et al., 2009], [Scheffer et al., 2012]. mathematical challenges towards achieving sustainability: Although understanding the science of the emergence • Develop a statistical mechanics of ecological commu- and robustness of ecological pattern is essential to nities, socio-economic systems, and the biosphere. developing sustainability, it is far from sufficient. The • Model the emergence of an ecological pattern. greatest challenge facing us is to achieve cooperation in • Determine indicators of impending critical transitions dealing with problems of the Global Commons [Hardin, between states. 1968], [Levin, 1999], [Skyrms, 1996], especially as regards public goods and common pool resources. This brings to the fore a different set of mathematical tools—control DOI: http://dx.doi.org/10.1090/noti982 theory, game theory, voting theory, and mechanism

392 Notices of the AMS Volume 60, Number 4 Opinion design [Barrett, 2007], [Dixit, Levin, and Rubenstein, E. S. Maskin (2008), Mechanism design: How to implement 2012], [Akcay, Meirowitz, Ramsay, and Levin, 2012], social goals, American Economic Review 98(3), 567–576. doi: [Maskin, 2008], [Myerson, 2008], [Saari, 2008a, 2008b]— 10.1257/Aer.98.1.567. for identifying under what conditions cooperation is R. B. Myerson (2008), Perspectives on mechanism design in economic theory, American Economic Review 98(3), 586–603. possible and how best to achieve it. doi:10.1257/Aer.98.3.586. The problems of achieving sustainability are urgent and J. Rehmeyer (2011), Mathematical and Statistical Challenges for huge and will require complementary inputs of diverse Sustainability, Providence, RI: American Mathematical Society. disciplines [Levin and Clark, 2010], [Rehmeyer, 2011]. D. G. Saari (2008a), Complexity and the geometry of voting. Obviously mathematics has a great deal to contribute in Mathematical and Computer Modelling 48(9-10), 1335–1356. addressing these problems; but it is equally certain, as doi:10.1016/J.Mcm.2008.05.033. has always been the case, that new mathematics will be (2008b), Disposing dictators, demystifying voting para- doxes, Mathematical and Computer Modelling 48(9-10), 1671–1673. stimulated by the energy and freshness that comes from doi:10.1016/J.Mcm.2008.05.016. new applications and new challenges. The discipline of M. Scheffer (2009), Critical Transitions in Nature and Society, mathematics has much to contribute, and much to gain, Princeton, NJ: Press. from engagement. M. Scheffer, J. Bascompte, W. A. Brock, V. Brovkin, S. R. Carpenter, V. Dakos, and G. Sugihara (2009), Early- References warning signals for critical transitions, Nature 461(7260), 53–59. E. Akcay, A. Meirowitz, K. Ramsay, and S. A. Levin (2012), doi:10.1038/Nature08227. Evolution of cooperation and skew under imperfect information, M. Scheffer, S. R. Carpenter, T. M. Lenton, J. Bascompte, Proc. Natl. Acad. Sci. USA. W. Brock, V. Dakos, and J. Vandermeer (2012), Antici- S. Barrett (2007), Why Cooperate: The Incentive to Supply Global pating critical transitions, Science 338(6105), 344–348. doi: Public Goods, Oxford, UK: . 10.1126/Science.1225244. S. J. Brams (1978), The Presidential Election Game, New Haven: Yale B. Skyrms (1996), Evolution of the Social Contract, New York: University Press. Cambridge University Press. I. D. Couzin, C. C. Ioannou, G. Demirel, T. Gross, C. J. Torney, R. M. Solow (1991), Sustainability: An Economist’s Perspective, A. Hartnett , and N. E. Leonard (2011), Uninformed individ- Eighteenth J. Seward Johnson Lecture in Marine Policy, June 14, uals promote democratic consensus in animal groups. Science 1991, Woods Hole, Massachusetts: Woods Hole Oceanographic 334(6062), 1578–1580. doi: 10.1126/science.1210280. Institution, Marine Policy Center. G. C. Daily, ed. (1997), Nature’s Services: Societal Dependence on R. Thom (1969), Topological models in biology, Topology 8, 313– Natural Ecosystems, Washington, DC: Island Press. 335. A. Dixit, S. A. Levin, and D. I. Rubenstein (2012), Reciprocal A. M. Turing (1952), The chemical basis of morphogenesis, insurance among Kenyan pastoralists, Theoretical Ecology. Philosophical Transactions of the Royal Society, Series B, 237, 37–72. P. R. Ehrlich and H. A. Mooney (1983), Extinction, substitution World Commission on Environment and Development (1987), and ecosystem services, Bioscience,33, 248–254. Our Common Future, New York: Oxford University Press. G. Flierl, D. Grünbaum, S. Levin, and D. Olson (1999), From individuals to aggregations: The interplay between behavior and —Simon Levin physics, Journal of Theoretical Biology 196, 397–454. Princeton University G. Hardin (1968), The tragedy of the commons, Science 162, 1243– [email protected]. 1248. J. Holland, (1995), Hidden Order. How Adaptation Builds Complexity, Reading, MA: Addison Wesley. I. G. Kevrekidis, C. W. Gear, J. M. Hyman, P. G. Kevrekidis, O. Runborg, and C. Theodoropoulos (2002), Equation-free mul- tiscale computation: enabling microscopic simulators to perform system-level tasks, Commun. Math. Sci. T. M. Lenton, V. N. Livina, V. Dakos, E. H. van Nes, and M. Scheffer (2012), Early warning of climate tipping points from critical slowing down: comparing methods to improve robustness, Philosophical Transactions of the Royal Society A—Mathematical Physical and Engineering Sciences 370(1962), 1185–1204. doi: 10.1098/Rsta.2011.0304. S. Levin (2010), Crossing scales, crossing disciplines: collective motion and collective action in the Global Commons, Philosophical Transactions of the Royal Society B—Biological Sciences 365(1537), 13–18. doi:10.1098/Rstb.2009.0197. S. A. Levin (1999), Fragile Dominion: Complexity and the Commons, Reading, MA: Perseus Books. (2003), Complex adaptive systems: Exploring the known, the unknown and the unknowable, Bulletin of the American Mathematical Society 40, 3–19. S. A. Levin and W. C. Clark, eds. (2010), Toward a Science of Sustainability: Report from the NSF toward a Science of Sustainabil- ity Conference, Warrenton, VA, November 29–December 2, 2009, Princeton, NJ: Princeton University Printing and Mailing Services.

April 2013 Notices of the AMS 393 Do the Math

The Golden Ticket Invisible in the Storm P, NP, and the Search for the The Role of Mathematics in Impossible Understanding Weather Lance Fortnow Ian Roulstone & John Norbury “You will love this book. It’s “With illuminating descriptions and completely accessible and captures minimal technicality, Invisible in the thrill, potential, and heartbreak the Storm provides a vivid historical of an edgy mathematical problem in perspective on how the development terms that nonmathematicians will of mathematical ideas, together with appreciate. After reading The Golden modern computer technology, has Ticket, I sort of hope P isn’t NP completely transformed our ability to after all.” understand and predict the weather. —, Google Vice President This is a gripping and highly and Chief Internet Evangelist and informative book.” one of the “Fathers of the Internet” —Roger Penrose, author of Cycles Cloth $26.95 978-0-691-15649-1 of Time Cloth $35.00 978-0-691-15272-1 Heavenly Mathematics The Forgotten Art of Spherical Trigonometry Glen Van Brummelen “Written by the leading expert on the subject, this engaging Digital Dice book provides an in-depth Computational Solutions to Practical historical introduction to spherical Probability Problems trigonometry. Heavenly Mathematics breathes new and interesting life into Paul J. Nahin a topic that has been slumbering for With a new preface by the author far too long.” “An enjoyable read, as [Nahin] writes —June Barrow-Green, associate clearly, with humour and is not afraid editor of The Princeton Companion to to include equations where necessary. Mathematics Nahin spices the book throughout Cloth $35.00 978-0-691-14892-2 with factual and anecdotal snippets. Digital Dice will appeal to all who like Arithmetic recreational mathematics.” Compactifications of —Alan Stevens, Mathematics Today Paper $18.95 978-0-691-15821-1 PEL-Type Shimura Varieties Towing Icebergs, Falling Kai-Wen Lan Dominoes, and Other By studying the degeneration of abelian varieties with PEL Adventures in Applied structures, this book explains the Mathematics compactifications of smooth integral Robert B. Banks models of all PEL-type Shimura varieties, providing the logical “There is something here for every foundation for several exciting recent mathematically inclined reader. The developments. The book is designed aerodynamics of balls in sport, the to be accessible to graduate students spread of diseases, traffic flow, the who have an understanding of effect of meteor impacts—[Banks] schemes and abelian varieties. deals with these and much more in London Mathematical Society Monographs, 36 engaging, well-judged detail.” Martin Bridson, Ben Green, and , —Robert Matthews, New Scientist Series Editors Cloth $150.00 978-0-691-15654-5 Paper $16.95 978-0-691-15818-1

See our E-Books at press.princeton.edu Letters to the Editor

Egalitarianism of Mathematics think that the letter writer is closer did I ask for (or receive) a reduction In the February 2013 issue of the to the truth than the editor. in my university teaching load. My Notices, it is claimed in an Opinion The credentialed mathematics pro- sixth grade students benefited from piece by the editor and in a Letter fession has built a wall around Ameri- my knowledge and I was allowed to to the Editor that mathematics as a can mathematics as a subject. The retain my knowledge of the public subject is egalitarian. It may appear profession is the gate-keeper of the school classroom dynamics. It was journals, the grants, the conferences, a positive experience for everyone so to those inside the wall as they and even online repositories such as except for some of the mathemat- greet each other within the cloister’s arXiv. Mathematics as a subject is not ics educators in our department. walls but it certainly does not appear egalitarian because how it is defined They complained that I was being to be so to those of us outside. and what it contains is determined counterproductive since they could The editor assumes that mathe- solely by a nonegalitarian profession. not hold meetings when I was at the matics-as-subject is independent of intermediate school. I believe the mathematics-as-profession. He then —Scott Guthery reason for their negativity was they argues that while the profession is Independent scholar were afraid to be asked to duplicate not egalitarian, because of this in- Boston, Massachusetts what I was doing. dependence, the subject can be and, [email protected] If you are a university mathemati- in his view, is. The author of the cian teaching future teachers, what Letter to the Editor says mathemat- (Received January 20, 2013) you should do is replicate Professor ics is egalitarian because it is “open Yong’s experience. Ideally universi- to all with the requisite talent and ties should recognize that faculty training.” For this writer, the work An “Adventure” That All Should teaching in the local schools is more itself—the talent—does not suffice. Experience valuable than writing an article on It seems to me that the requirement Darryl Yong’s article, “Adventures teaching factoring quadratic trinomi- of training asserted by the second in Teaching”, in the November 2012 als to a group of pre-service teachers. author denies the independence of Notices should be required reading subject and profession asserted by for any professor, administrator, —Murray H. Siegel the first. Which one is closer to the politician, or business executive Arizona State University truth? who decides to wade in to solve the Polytechnic campus There is no empirical foundation problems of teaching mathematics. [email protected] to mathematics against which con- Professor Yong’s description of the tributions can be measured in an high school environment is deadly (Received November 30, 2012) objective manner. This is not true of accurate. My wife and I are currently other sciences. If one discovers a new college faculty but the majority comet or a new bird species one is of our combined seventy-nine Some Fine Points about Henri years of teaching mathematics was acknowledged as having contributed Matisse accomplished in the elementary, mid- In reference to the January 2013 issue to astronomy or ornithology and is dle, and high school grades. of the Notices, page 27: The articles considered thereby to be a member In the 1980s there was the Holmes about I. M. Gelfand are undoubtedly of the community. It would be absurd Group, which attempted to create very solid in the large, but not always to assert that the discovery of a new relationships between universities in fine detail. comet is void because the person that prepared teachers and school The editors have done an injustice making the discovery is unable to districts. The sharing that would take to I. M. Singer by printing the follow- present credentials of their training place would benefit both organiza- ing statement about the painter Henri in astronomy. The empirical sciences tions. Somehow, the Holmes Group Matisse: “Particularly outstanding is can successfully argue that subject faded away but Professor Yong’s ad- his late work: Jazz and the remark- and profession are independent and venture should be experienced by all able ‘papier-decouples’, efforts done that it is therefore possible for the university mathematicians who teach in the early 1880s.” subject to be egalitarian even if the courses to future teachers. Matisse would have been in his profession is not. Years ago, when I was a math- early teens in the early 1880s, hav- Without a subject-based means to ematics professor at a large state ing been born on the last day of measure contribution, mathematics university, I spent an hour every 1869. Thus in “1880s”, two of the takes training as a proxy. It is the day teaching at a school in the local four digits must be incorrect. In fact requirement of a credential in order school district. For example, one year “Jazz” is dated 1947, and that is the to participate that binds subject to I taught sixth grade mathematics period of the “papiers découpés”— profession. Because of this binding, every morning. I was my students’ note that there are also a number of the nonegalitarianism of the latter math teacher. I did this as a vol- typographical errors in that phrase: comes to cloud the former. Thus I unteer. I was not paid a salary nor a missing plural, an unnecessary

April 2013 Notices of the AMS 395 Letters to the Editor hyphen, two missing accents, and a Missing the Point it is important to critically extract superfluous “L”. Possibly Professor This is in reply to a letter by Alex- as much useful information as we Singer wrote the second accent and ander Eremenko (Notices, Vol. 59, can from them, while pointing out it was mistaken for an L? We could No. 5, May 2012), titled “Uncritical their weaknesses and deficiencies. also hope that Professor Singer had use of citation database”. This letter Doing so is much more likely to written “in his early ‘80s”, but that referred to our paper in the Notices communicate the special aspects of would be barely correct for some of (J. Panaretos and C. Malesios “Influ- our subject to decision-makers and the cut-paper works, and incorrect ential Mathematicians: Birth, Educa- the general public and to advocate for for Jazz. tion and Affiliation”, February 2012, better quantitative measures.

I am surprised that the editors did vol. 59 (2), pp. 274–286). —J. Panaretos and C. Malesios not have any idea of Matisse’s dates A. Eremenko criticizes the inad- Department of Statistics (1869–1954), which might have pre- equacies of the Thomson Database Athens University of Economics vented some of this. And they should HCR list. He also questions whether and Business the number of citations is a reason- know to check their French. Greece able measure of the scientific influ- [email protected] —Martin C. Tangora (emeritus) ence of a mathematician. Unfortu- nately, he is completely missing the University of Illinois at Chicago (Received January 10, 2013) [email protected] point of the paper. As is quite clearly stated in its introduction and conclusions, the (Received December 21, 2012) purpose of our paper was not to rank mathematicians (e.g., by selecting Review of Manin Documentary the “most influential”) or to argue Correction The January 2013 Notices had a re- in favor of citations as a measure of The photograph of John G. Hock- view by Gunther Cornelissen of the assessing the quality of mathemati- ing (Notices, March 2013, page DVD Late Style—Yuri­ I. Manin Looking cians—on the contrary we made ex- 335), should have been cred- Back on a Life in Mathematics. tensive mention of the shortcomings ited “Photo © 2010 Susan Marie In his review, Gunther Cornelissen of the use of citation statistics. Davis”. The Notices apologizes for criticizes the fact that the film does Its purpose was to point out that, this omission. not achieve the goal of documenting when citations are used as an insti- —Sandy Frost Yuri Manin’s outstanding work in tutional/national indicator (whether mathematics. we like it or not), they reflect only the We, the creators of the film, would current affiliation of the scientists; like to say that from the beginning we wanted to investigate whether we were aware that it is not possible a different picture emerges when the for us, as nonmathematicians, to mobility patterns of these influential document Yuri Manin’s life work. mathematicians are taken into con- But does this put a ban on a bio- sideration. graphical approach? For three years For better or worse, a database we accompanied Yuri Manin and his that is often used for citation in- wife Xenia Semenova from Bonn to dicators is the Thomson database. Moscow, Paris, and Simferopol, where Notwithstanding the weaknesses that this or any other similar database he was born and grew up. Yuri Manin may have, it would be hard to ques- gave us impressive insights speaking tion the fact that most of the math- about his life. These documents give ematicians included in the list have information which alone makes the had noteworthy mathematical and/ video worth watching. or scientific influence. Are they the “most” influential? This we did not —Agnes Handwerk presume to judge anywhere in our Freelance journalist paper. Are they even the most cited? Hamburg We did not claim this either, and [email protected] referred to them as “highly cited”. Would the inclusion or exclusion of —Harrie Willems a few names drastically change the Freelance journalist emerging mobility patterns? Cer- Amsterdam tainly not. [email protected] Indicators and databases are here to stay, and instead of simply re- (Received January 16, 2013) jecting them as “almost useless”,

396 Notices of the AMS Volume 60, Number 4 The creators of MathJobs.Org welcome you to: Announcing...MathPrograms.Org

Receive, read, rate, and respond to electronic applications for your mathematical sciences programs, such as undergraduate summer research programs and travel grant competitions.

Customize your settings and control the application form; also set secure access for the admissions committee.

Enter program announcements for public display.

Download data to personal computers for use in word processing and spreadsheets or as a full permanent storage file.

Service is FREE to applicants. Institutions pay annually for one program or for multiple programs. Remembering Raoul Bott (1923–2005)

Loring W. Tu, Coordinating Editor With contributions from Rodolfo Gurdian, , , Arthur Jaffe, Shing-Tung Yau, and Loring W. Tu

Raoul Bott passed away The contributions are listed in the order in on December 20, 2005. which the contributors first met Raoul Bott. As Over a five-decade career the coordinating editor, I have added a short he made many profound introductory paragraph (in italics) to the beginning and fundamental contri- of each contribution. —Loring Tu butions to geometry and topology. This is the sec- Rodolfo Gurdian ond part of a two-part article in the Notices to Rodolfo Gurdian was one of Raoul Bott’s room- commemorate his life and mates when they were undergraduates at McGill. work. The first part was The imaginary chicken-stealing incident in this arti- an authorized biography, cle is a reference to a real chicken leg incident they “The life and works of experienced together at Mont Tremblant, recounted Raoul Bott” [4], which in [4]. he read and approved What follows is an account of some of the mischief a few years before his that Raoul Bott and I carried out during our days Photo by Bachrach. at McGill. Figure 1. Raoul Bott in 2002. death. Since then there have been at least three I met Raoul in 1941, when we were in our volumes containing remembrances of Raoul Bott first year at McGill University. Both of us lived in by his erstwhile collaborators, colleagues, students, Douglas Hall, a student dormitory of the university, and friends [1], [2], [7]. I have also written elsewhere but we were in different apartments. He paid about my experiences working with him [5]. This attention to me because I got a higher grade in second part presents some personal recollections trigonometry. He also noticed that I played the that do not overlap with what has already appeared guitar, and I appreciated his piano playing. in print. More reminiscences and appreciations The following year we shared an apartment, to- of his work may be found in the upcoming final gether with Frazer Farlinger, a student in medicine. volume of the Collected Papers of Raoul Bott [6]. Raoul majored in and I in Bott had a passion for mathematics, which he chemical engineering. The difference among us kept to the very end, even after his retirement from was that Frazer and I had to study very hard, while Harvard. At the same time, he was firmly planted Raoul didn’t. He used to say that attending the in the real world. As many of his acquaintances lectures was sufficient for him, since electrical would agree, he exemplified the French phrase joie engineering was a very logical subject. His marks de vivre. His mathematical work speaks for itself, were satisfactory but could have been much better but it is hoped that the following reminiscences if he had only worked harder. can give some idea of his personality, his zest for I met Oskar and Celia Pfeffer, his charming life, and his humanity. stepparents. Realizing that they were not very well off, I called Raoul’s attention to the fact that, being Loring W. Tu is professor of mathematics at Tufts University. His email address is [email protected]. Rodolfo Gurdian, who has since passed away, wrote this DOI: http://dx.doi.org/10.1090/noti977 piece around the year 2000 and Raoul Bott read it.

398 Notices of the AMS Volume 60, Number 4 so talented in mathematics, he could easily obtain shout back, “Of course, I a scholarship by studying just a little more. I think recognize you,” and would I influenced him, because he improved his grades shoot him dead anyway. and became one of the best students in the class. One reason we became In his last years at McGill, I believe he did obtain a good friends might have scholarship. been that, during our Rooming together, we became close friends. We childhood, we both en- loved to make mischief. On Saturdays we used to gaged in similar mischief. go to the movies, often to a theater called The Moreover, Latin Ameri- System. Buying just one ticket, we could watch cans may have more in three movies in a row. Although the ticket price was common with Europeans low, both of us being broke, we found a “system” than with North Amer- to sneak into The System without paying by taking icans. So we enjoyed advantage of the fact that only one person was in making pranks together. charge of the theater’s two entrances. One of the entrances gave access to the upper floor through a Stephen Smale wooden staircase, and the other went to the first At a conference in 1967 floor. The trick we devised was for one of us to talk Stephen Smale orga- to the ticket clerk, while the other would distract nized a beach hike in him by running upstairs and making a lot of noise. which Raoul Bott nearly As the ticket clerk followed the noisemaker, the drowned. Afterwards, Bott Figure 2. Raoul Bott (left) in the other one took advantage of the situation to sneak sometimes joked that 1930s. into the theater’s first floor. Of course, once we Smale tried to kill him. were in, it was difficult for the ticket clerk to find us, When Bott said Smale was because we sat in the first available seats, feigning his “worst” student, it was to be regular customers. not in the mathematical During the summer months every engineering sense but in the moral student was required to work to get some practical sense. Here is Smale’s ver- experience. Since Douglas Hall was closed for sion of the event. the summer, we rented a room together near the university. Raoul was six foot two, and I just Raoul often introduced me under five foot six. So you can imagine what a with the words “Steve was strange-looking pair we made! One summer Sunday my first student” and then we decided on a prank. Raoul put on his gray and added with great empha- red bathrobe and a turban and armed himself with sis “and my worst!” He a small dagger. I put on my red short pants and described our relationship (often tumultuous) in his a green T-shirt and carried a tambourine. When 1 Raoul went out to the street, I followed, playing talk at the conference for the tambourine and dancing in circles around him. my sixtieth birthday. Raoul says there, “Steve tried to People in the street were shocked. Suddenly, Raoul Figure 3. Rodolfo Gurdian and drown me,” as he describes approached an old lady and threatened her with Raoul Bott as undergraduates at one of our excursions. the dagger. When she started screaming, we ran McGill, c. 1942. like hell, realizing that the joke and the fun were It is true that I mis- getting out of hand. calculated the timing and We used to talk about our future careers. I dangers of an incoming tide at Taylor’s Point on told him that, due to my facility with money, I the Olympic Peninsula. It was at a 1967 Battelle would dedicate myself to business, which, in fact, conference in Seattle on general relativity that I I successfully achieved in life. He joked that, since put together a group of about a dozen people I would become a wealthy man while he, as a for a three-day hike, camping along the ocean professor, would be very poor, eventually he would beach. Toward the end of the trip we came to be forced to come to Costa Rica to seek my help. (the notorious) Taylor’s Point and had to make Our first meeting would be in the backyard of my a decision. My wife, Clara; daughter, Laura; and house, where he, out of hunger, would be stealing a few others decided on a detour. I convinced my chickens. Finding a thief in my backyard, I the remainder, including Raoul; his wife, Phil; his would come out with a gun. Upon seeing me, Raoul Stephen Smale is professor emeritus of mathematics at would shout, “Please, Rodolfo, don’t shoot. It’s me, the University of California, Berkeley. His email address is Raoul, your old friend.” By that time, I would have [email protected]. become an insensitive wealthy man, so I would 1The Collected Papers of Stephen Smale, Vol. 1, p. 8.

April 2013 Notices of the AMS 399 Figure 6. At the Bombay Airport on the occasion Figure 4. International Symposium on Algebraic of the Colloquium on Differential Analysis, Tata Topology, Mexico City, 1956. Front row, from left Institute, Bombay, 1964. Left to right: Mr. Jalihal to right: 1 = William Massey (?), 3 = Friedrich (Public Relations Officer, Tata), Deane Hirzebruch (?), 4 = Hans Samelson, 5 = Raoul Bott, Montgomery, Donald C. Spencer, Georges de 6 = J. H. C. Whitehead. Second row: 5 = Witold Rham, Mrs. Gårding, Lars Gårding, Komaravolu S. Hurewicz (who was killed a few days later falling Chandrasekharan, Raoul Bott, , Mr. off a pyramid), 7 = . Third row: Puthran (Registrar, Tata). 3 = Morris Hirsch, 7 = Leopoldo Nachbin. According to Michael Atiyah, since he was not in the photo, he might have been the lecturer. was thinking, “This is how one drowns.” In fact, we all survived (my backpack was lost to the seas). Raoul also writes that, after an excursion with me, he often got on his knees to give thanks: “Back home again and still alive!” Raoul Bott and I were close friends for over five decades. Already in 1953 we would meet for weekly lunches, as I was taking his class in advanced . I have often spoken of how it was Bott who started me on the road to serious mathematics. He was a great teacher and a great inspiration. There were sometimes divergences in our ap- proaches to mathematics as well as divergences in which fields we worked in. Very early on, Raoul seemed a bit upset with my working in ordinary Figure 5. Symposium on Differential Topology in differential equations. That subject was for those Honor of , Institute for Advanced who found partial differential equations too diffi- Study, Princeton, April 2–5, 1963. Back row (left cult. Moreover, at that time Raoul was not too crazy to right): Raoul Bott, , G. A. Hedlund, about even p.d.e. In he preferred T. T. Frankel, Stephen Smale, N. H. Kuiper, J. F. using finite-dimensional approximations, in con- Adams, William Browder, J. W. Milnor, M. A. trast to my using infinite-dimensional Kervaire. Front row: S.-S. Chern, R. G. Pohrer, Atle (Palais–Smale). Selberg, Marston Morse, Walter Leighton, Morris The last extended time we spent together was Hirsch, S. S. Cairns, . on the occasion of his one- to two-month visit to the City University of at my invitation. He and Phil were still drinking martinis with Clara daughter, Renee; Mike Shub, his wife (at that time), and me after all those years. He was redoing in his Beth; and my son, Nat (ten years old), that my own framework my joint result with Mike Shub on tide calculations would justify an attempt to take the number of real zeros of a system of polynomial the sea route. Success was at hand when I looked equations. Later I saw his writeup with Cliff Taubes. ahead to see Raoul being battered against the cliffs Living in the world of Raoul Bott was a wonderful by the heavy sea. He writes in [3, vol. 2] that he part of my life.

400 Notices of the AMS Volume 60, Number 4 proceeded to mesmerize the assembled crew of senior professors, first explaining Morse theory, a brand new tool for the conservative audience, and then his periodicity theorem. Of course I didn’t know it, but it was a job talk—he was coming to Harvard! To put this in context, it’s important to know what sort of a place Harvard was at the time. Dave Widder and Joe Walsh were the most senior and, together with Garrett Birkhoff, represented the tradition from the 1930s. At a dinner party with the Walshes, although a tuxedo was no longer expected, the women still left the dining room after Figure 7. Raoul Bott and Joseph H. Sampson, dessert so the men could light their cigars and Conference on Manifolds, Tokyo, mid-1960s. discuss masculine topics. , , and were the European stars who had put the department on the map. The younger generation was represented by the three American functional analysts: Andy Gleason, , and Lynn Loomis. But it was still a very conservative place run by conservative gentlemen. Faculty wives put on white gloves to serve tea before the colloquium. had left for Chicago, so his fancy ideas in topology, cohomology, sheaves, and so on were unknown areas at Harvard. Semi-simple Lie groups were a distant concept. Enter Raoul. He was not a breath of fresh air; he was a gale—not merely the new and wonderful mathematical topics that he brought in his tool kit, but his spark, his energy, his fearlessness. You know how universal it is to refrain from asking Figure 8. Explaining topology on the TV show what you fear may be a stupid question when a Science and Engineering Television Journal, seminar speaker begins to lose you? Not Raoul! He c. 1965. regularly raised his hand and asked that “stupid question”, half knowing the answer but wanting to slow down the speaker, hear it again, and bring the David Mumford rest of us into the circle of mutual appreciation present in the best seminars. Being ashamed not David Mumford describes Bott’s “job talk” at Har- to know something vard and the transformation in the culture of the basic was com- Harvard mathematics department upon Bott’s ar- pletely alien to his rival in 1959. temperament. My first encounter with Raoul was most memorable. And then there I was still a lowly undergraduate at Harvard, though were the “colloquium taking some advanced courses. One day in 1958, parties”. I think these Raoul came to give the colloquium talk, a weekly became a regular affair Thursdays at 4:30 in 2 Divinity Avenue fixture in the 1960s: (rotating with MIT but not, if I recall correctly, every Thursday the with Brandeis as yet). The stage at 2 Divinity was speaker’s host in- on a low platform, maybe three feet high with vited a large part of low stairs on each side. After being introduced, the Boston research Raoul did what no one else had ever done: he community and their marched directly onto the stage with a small jump. spouses to a party. His enthusiasm was both physical and mental. He Here were these clus- ters, typically the Figure 9. Lecturing at the David Mumford is professor emeritus of mathematics guys talking math at Brown University. His email address is david_mumford@ and their wives talking Universität Bonn, 1969. brown.edu.

April 2013 Notices of the AMS 401 about their fami- lies (there were no women on the faculty then). Raoul often came a bit late and would raise his voice and say something like, “What’s going on here? Is this a party?” or ironically, “Is this Harvard?” He wanted music and often got Tom Lehrer to play and even sing by the end of the evening. Figure 11. Freeman Dyson, Raoul Bott, and Raoul taught me Valentine Bargmann on the occasion of the many things about Oppenheimer Memorial, Institute for Advanced both mathematics Study, Princeton, 1971. and life. I know no one who didn’t feel a lift, a rush when he came into the room. the time Raoul was a visitor at the Institute for He was not merely Advanced Study. While they discussed many things, Figure 10. Raoul Bott and David large in stature but, connections were not on the agenda. Only later Mumford in the early 1970s. more than that, large did one realize their central role in Yang-Mills in spirit. We miss him immensely. theory. In the late 1970s we had a long discussion while driving together from Cambridge to an Arthur Jaffe AMS summer meeting in Providence about how important it was to have a good dictionary to Arthur Jaffe, president of the American Mathemati- translate between as physics and cal Society from 1997 to 1998, reminisces about his differential geometry as mathematics in order for forty-year relationship with Raoul Bott as colleague, people in the two subjects to communicate. friend, and confidant. While I had first met Raoul in a mathematics I first encountered Raoul Bott in 1964. While a conference before I came to Harvard in 1967, student in Princeton I came across the beautiful I really got to know Raoul well during the Les Bott-Mayberry paper on matrices and graphs and Houches summer school in 1970. Cecile DeWitt had used their representation of a determinant in established a famous summer school of theoretical analyzing a problem in quantum theory. But that physics after World War II; it was located in a experience did little to prepare me for our first small French village in the Alps, not far from Mont face-to-face meeting. Raoul’s personality and spirit Blanc. The only problem about trying to work in struck me with awe; it left an indelible mark in my Les Houches was the distraction of a striking view memory. of the Aiguille du Midi. In 1970 the focus of the Later we became colleagues, and it was then school was mathematical quantum field theory. that Raoul evolved into a very special and dear Raoul was officially an “observer” at the school, friend. We shared many mathematical discussions sent by the Battelle Institute, which sponsored the together. We also spent hours talking about the event. world, laughing over an amusing story, listening Cecile had an interesting philosophy about Les to music, or sharing a meal with a good wine. Houches: in order to maximize interaction, the Raoul had always had an interest in physics, but participants at Les Houches should come at the that had some unusual twists. For example, when beginning of the meeting and remain there until Raoul learned of normal ordering (a simple form of the end. And this school lasted two full months! renormalization), he used to ask me if it could have Both in the lectures and at the meals in the large something to do with the resolution of singularities dining room, the participants interacted like a in algebraic geometry, still an intriguing question. large family for sixty days. Raoul brought his wife, Raoul described discussions he had with his neighbor Chen Ning Yang during 1955–57 at Phyllis, and their three young girls, and their son, Tony, also visited on occasion. So, over the course Arthur Jaffe is professor of mathematics at Harvard Uni- of that summer, I really got to know the Botts. In versity. His email address is [email protected]. fact, George and Alice Mackey and their daughter,

402 Notices of the AMS Volume 60, Number 4 Ann, were there too, so there was a big contingent friends at Dunster: with from Harvard. students and with sci- After returning to Cambridge that fall, I began entific colleagues in a mathematical physics seminar at which Raoul the dining hall, with became a regular attendee. Several of the students members of the “Se- from Les Houches also came to Harvard. Raoul nior Common Room” enjoyed them all but became especially fond of at their regular meet- Konrad Osterwalder, who eventually spent six ings, at the Dunster years in Cambridge and became one of Raoul’s concerts, at the Red regular confidants before he left for the E.T.H. Tie dinners, and during Züuntil sometimerich. many other occasions in In 1976 I helped a friend in Paris organize a the master’s residence summer school in Cargèse, Corsica, where I had with Raoul and Phyl- been during the summer of 1964. The experience lis. Raoul sometimes left me with a lasting impression of the beauty had his mathematical and of this Greek-French village by the friends stay in Dunster, Mediterranean Sea. The success of the resulting such as Fritz Hirze- gathering led to our having five more schools bruch or Michael Atiyah, and he would enjoy in Cargèse. The second school in 1979 brought Figure 12. Michael Atiyah and letting us know when together an interesting group of mathematicians Raoul Bott at Stonehenge in the some mathematician and physicists, including Raoul, Michael Atiyah, early 1970s. Jürg Fröhlich, Jim Glimm, Gerard ’t Hooft, Harry friend of his might Lehmann, , Kurt Symanzik, Ken be making an unan- Wilson, , and Jean Zinn-Justin. The nounced visit to accompanying photographs (Figures 16 and 17) Cambridge. show that Raoul was in good form in the school: not With our frequent in- tersection at Dunster only did he give beautiful lectures but he animated House, we often made less formal moments. The combination of the plans to do things productive and interactive scientific atmosphere, together. We both en- along with an inviting beach, brought Raoul and joyed music and often Phyllis back to Cargèse in 1987 and 1991. had met at undergrad- Originally I had been appointed professor of uate concerts. I recall physics at Harvard, although some of my courses our discussing the con- were cross-listed in mathematics. But in the spring cert at Sanders Theater of 1975 the mathematics department invited me to when Yo-Yo Ma’s under- become a member. Raoul was chairman at the time, graduate quartet played and I recall the pleasure with which he described Brahms. I went with to me that vote. Raoul also enlivened the ensuing Raoul to the first per- faculty meetings following Thursday lunch at the formance that the Tallis Faculty Club. Until sometime in the 1980s, the Scholars sang in Boston department decided on teaching assignments in an in a concert at the old-fashioned way: discuss this at a faculty meeting, Church of the Advent. which always seemed to have full attendance! The Much after that, I sat Figure 13. Raoul Bott with Henri chair wrote on the board a list of necessary courses; with Raoul in his music Cartan—Bott modeling Cartan’s the persons present filled in their names in order room in his apartment birthday gift T-shirt—at the of seniority in the department. This gave the more on Richdale Avenue seventieth birthday fête for senior members of the department an elevated while he played Bach Cartan, I.H.E.S., Bures-sur-Yvette, status, in which Raoul amusedly reveled. on his Steinway. Later France, 1974. In 1978 Raoul and Phyllis became Masters in that room we dis- of Dunster House. I had been a happy mem- cussed the differences between recordings of the ber of Lowell House ever since some students Goldberg Variations made by Glenn Gould and brought me there during my first year at Har- András Schiff. (András is another genius from vard. But Raoul asked me to switch and be Hungary whom I admire. I have also come to know with him at Dunster, which I eventually did. I him as a friend, and I wish that I could have brought along a few of my own collaborators, introduced Raoul and András to each other, for and I have many fond memories of evenings with they certainly would have hit it off well.)

April 2013 Notices of the AMS 403 Mac computer at Youville Hospital in Cambridge, where Phyllis was recuperating. For a while George Mackey was nearby in the same hospital. During those days Raoul often came to my home for dinner. We sat around the kitchen table over swordfish or bluefish from the broiler. The conversation sometimes turned to music. Raoul admired the virtuosity of my harpist friend, Ursula. Many people regarded Raoul as a father figure, but Ursula was drawn to him for the empathy he expressed for others in need of assistance and for his understanding of the problems of the world. Raoul explained that life would be much easier for him and Phyllis in California rather than in their Figure 14. Israel M. Gelfand, Robert MacPherson, multistory townhouse. In Cambridge many friends Raoul Bott, and David Kazhdan at Bott’s home in were sad to see them go. Newton, Massachusetts, 1976. The mathematics department held a dinner in the Faculty Club the night before Raoul and Phyllis left Cambridge. After most other persons went home that evening, I gave Raoul a bottle of excellent Bordeaux. Occasionally I spoke with Raoul by telephone, a nice link between Cambridge and Carlsbad, California. It was wonderful to hear his voice and to get some news. During one of those conversations late in 2005, Raoul let me know that he had left that bottle of Bordeaux with his daughter Candace in Cambridge. She was bringing it to California the next day so he could share it during a small family reunion. At the time I did not realize how Raoul was telling me that his end was upon him. Shortly afterward I cried at the news. Shing-Tung Yau

Figure 15. Raoul Bott as a pirate king, Phyllis Shing-Tung Yau reflects on Raoul Bott’s influence Bott as a maiden, and two students on him—mathematical, personal, and professional. impersonating Phyllis and Raoul at a Dunster I first met Raoul Bott about forty years ago House Halloween party, late 1970s. when he briefly visited Shiing-Shen Chern and the Berkeley mathematics department. Bott was a great and famous mathematician then, while I Raoul did not always remember the seminar was merely a graduate student. My teacher, Chern, schedule, so he enjoyed having his office across was interested in his paper on the localization of from the main seminar room in the department. Chern numbers for Kähler manifolds and went One felt in the middle of things with people through it several times during a seminar. I was, passing by, and with the glass window along the of course, very impressed by Bott’s elegant theory. hall corridor, one could always see at a glance what Little did I know that much later this theory seminars were taking place. would be developed into an extremely valuable Raoul could turn up at the most unusual time or tool for computations in geometry. I used this place. When I married in 1992, Raoul was an usher theory myself, along with my coauthors, Bong Lian in the wedding. I recall how proud he was to escort and Kefeng Liu, in solving the mirror symmetry my daughter, Margaret. And I was not surprised conjecture (independently solved by Givental), in the summer of 2002 to arrive at the airport which was part of the broader theory of Calabi–Yau in Vienna and find Raoul and his granddaughter, manifolds. Vanessa Scott, there too. They were on their way to In 1971 there was a special program on make a film about Raoul’s life, a wonderful story I led by Bott at the Institute for Advanced Study in saw in 2006. Raoul’s life turned upside down when Phyllis Shing-Tung Yau is professor of mathematics at Harvard had a stroke. He began to spend every day with his University. His email address is [email protected].

404 Notices of the AMS Volume 60, Number 4 Princeton. Since I was graduating from Berkeley, the IAS was an attractive place for me to go. When I applied to several universities, I got a few good offers. Although I could have gotten a higher salary elsewhere, Chern urged me to spend some time at the IAS, partly because of Bott’s program. So I went and enjoyed my year tremendously. At the IAS, I became interested in ways of constructing metrics with special curvature properties and applying them to solve questions in topology. For example, I thought about constructing metrics with positive scalar curvature to create obstructions for a nonabelian group action on a . (I later wrote a paper with Lawson based on this idea.) By studying the wedge product of differential forms under a circle action, I found obstructions Figure 16. Michael Atiyah and Raoul Bott looking to the existence of a topological circle action at a colony of ants in Cargèse, Corsica, July 1979. on a manifold; these obstructions exist in the cohomology ring of the manifold. I showed my work on group actions to Raoul. He was pleased, Of course, I learned much more from Raoul and his encouragement was really important to a during my years at Harvard, no less in statesman- young man like me. ship than in mathematics—he was extraordinarily Later on, Raoul had many more interactions with skilled in handling departmental affairs. I felt truly me. After I proved the Calabi conjecture and the sad when he passed away. I gave a talk on his positive mass conjecture, the latter with Richard life’s work at a Journal of Differential Geometry Schoen, he tried hard to convince me to come conference. In preparation for the talk, I researched to Harvard, which offer I did not accept at first. his contributions to mathematics. I was amazed During that process he invited my wife and me to to learn how much he had accomplished and how his home for dinner several times. At the time, he much he had done that I did not know about. was the master of Dunster House at Harvard. It Raoul certainly ranks among the most influential was inspiring to see how much time and energy mathematicians of the last century. His work was he invested in college undergraduates. I was truly deep, his vision far reaching, and his impact grateful for his hospitality during my visits to durable. May his spirit always be with us! Harvard. In return, I tried to entertain him well when he visited Beijing at the invitation of Chern Loring W. Tu in 1980. During that visit I proposed the Chinese name Bo Le to him. Bo Le was a famous personage Loring W. Tu coauthored Differential Forms in Al- in Chinese history reputed to have the ability to gebraic Topology with Raoul Bott. A second volume, recognize excellent horses, those that can run a Elements of , in the works thousand miles. Apart from the aptness of its long before Bott’s passing, is due to appear in 2014. meaning, the name was appropriate phonetically also: “Bo” is the Chinese surname closest to “Bott,” Making a Problem Your Own and “Le” is about as close to “Raoul” as a Chinese The first time I met Raoul was at an orientation lunch character can sound. Raoul told me he liked this for incoming graduate students in mathematics Chinese name. at the Harvard Faculty Club. Raoul gave us some The pivotal moment of my life was the time advice on how to write a Ph.D. . He said it when I was having some trouble in the mathematics was like doing a homework problem, but a harder department at the University of California in San problem. He ended by saying, “Make the problem Diego. I needed help with a decision. Raoul was your own.” It puzzled me what it meant to “make visiting Berkeley, and I flew to Oakland to have a problem my own,” but I was too intimidated dinner with him. After dinner we had a long to ask. I thought it was one of those things, like discussion about my future. A true statesman, he the taste of a certain fruit, that is impossible to laid out the pros and cons of what I should do. explain except to those who have experienced it I felt greatly relieved after talking with him and themselves. made the most important decision of my career, A few years later, when I was an assistant which was to come to Harvard, a decision that I professor at the , my Ph.D. have never regretted. thesis advisor, Phil Griffiths, came to visit. I

April 2013 Notices of the AMS 405 by “making a problem your own.” I think it meant to be so absorbed by the problem that you forget everything else—to be possessed, so to speak. It has happened to me a few more times, missing a subway stop on my way to the airport or jumping out of bed at night with a solution. Each time I feel that I have finally made a mathematical problem my own.

Bott as a Lecturer Bott’s lectures were legendary. He had a knack for explaining ideas in simple, easily understood terms, no matter how abstruse, complicated, or abstract Figure 17. Left to right: Raoul Bott, John Imbrie the topic. His lectures were always clear and exciting. (in white shirt, face partially hidden), Michael They were magical in that they gave you the feeling Atiyah, and Konrad Osterwalder at a summer you had understood something, sometimes even school on mathematical physics in Cargèse, when you had not. Not surprisingly, his lectures Corsica, July 1979. were popular and his courses heavily enrolled. His courses had impact beyond mathematics students at Harvard, for they were attended also by students and faculty from other departments and other universities. The physicist Cumrun Vafa cited Bott’s courses for changing his perception of modern mathematics and profoundly influencing his later studies [2, p. 277]. Likewise, Edward Witten credited Bott’s lectures with teaching him techniques of geometry and topology, such as Morse theory and equivariant cohomology, which have proven pivotal in his work on supersymmetry. Bott always seemed glad to be in the classroom. His courses were a lot of fun. In every lecture there were spontaneous moments of laughter. This came about not through preparation and Figure 18. Foreground, left to right: Susan canned jokes but because of his innate sense of Bombieri, , Phyllis and Raoul Bott, humor, unique perspective, colorful phrases, and Shing-Tung Yau, Michael Atiyah. Background: superb delivery. In his hands, the construction of Lars Gårding in the back on far right. Forbidden a spectral sequence could become entertaining. He City, Beijing, 1980. always focused on the central idea and simple but illuminating examples.

picked him up from the airport and drove him Authority to a restaurant. While in the car, we started One year Bott taught the second semester of talking about a mathematical problem. I became so complex analysis, and the textbook he chose was engrossed that I lost all sense of time, place, and Lars Ahlfors’s Complex Analysis. At some point orientation. The next thing I knew, a policeman he departed from the book and gave a different was handing me a ticket for driving the wrong way definition. Now students often revere the textbook on a one-way street. Griffiths advised me helpfully, “Go tell the judge as the ultimate authority, so a hand shot up and that you were thinking about mathematics.” So I a student blurted out, “But Ahlfors says this, not showed up in court to dispute the charge, and I that!” Bott replied calmly, “Yes, but Bott says that.” did as Griffiths told me. The judge took a look at As usual, Bott understood things his own way and my driver’s license and said, “You live only one was not about to faithfully follow any book. In fact, block away from this street. You have no excuse!” in topology courses he did not even follow his own He upheld the fine of seventy-five dollars. At that books, because usually his understanding of the moment, it dawned on me what Raoul had meant subject had evolved since the book appeared.

406 Notices of the AMS Volume 60, Number 4 A Conscripted Lecture One day in the early 1980s a poster appeared on the bulletin board of the Harvard mathematics department on the third floor of the Science Center. It looked just like any other announcement, but with a twist. On the top it said, “By popular demand, Professor Raoul Bott will give a lecture on ‘The Atiyah–Singer Index Theorem: What It Really Means’ ”. The date, time, and place of the lecture were all clearly spelled out. What was unusual about this poster was the presence of an asterisk next to Raoul Bott’s name and a footnote at the bottom: “*Please inform the speaker.” A few minutes before the scheduled time on Figure 19. Left to right: Tsai-Han Kiang, the appointed day, the room was packed. No one Shiing-Shen Chern, Hsio-Fu Tuan, Shan-Tao Liao, had the temerity to inform the speaker about the Raoul Bott, Shing-Tung Yau, Guang-Lei Wu (in lecture, so we were all wondering if Raoul Bott white shirt), Mrs. Yau (in red dress) in Beijing, was going to show up. At the appointed time, he 1980. showed up, made a few jokes, and then proceeded to deliver a wonderful lecture on the Atiyah–Bott fixed point theorem and the Atiyah–Singer index theorem, all in the allotted hour.

Finder’s Fee Nowhere were Bott’s powers of persuasion more evident than at the seventy-fifth anniversary of the Institute for Advanced Study at Princeton in March 2005. On that occasion he gave a talk reminiscing about how the institute in the fifties changed his life and launched his career (Figure 37). A few days after the conference, Bob MacPherson, a professor at the institute, called him to say that a couple in the audience that day were so moved by Bott’s talk that they donated two million dollars to the institute. Bott recounted the story to me and added, “I should have asked for a finder’s fee.” Figure 20. Raoul Bott, , and Jean-Pierre Serre at a reception in honor of O. Zariski and Liquors L. Ahlfors, Library of Harvard mathematics Coming from a family of teetotalers, I knew nothing department, winter 1981–82. about alcohol as a graduate student. At one point I thought it would be good to repair my ignorance in this domain. Raoul had the look of a bon vivant student I would know first-hand the difficulties a who might be knowledgeable about such things. student would encounter in learning the subject. I Just as some students might ask him for good think Raoul did not anticipate that it would end references in topology, when I ran into him in the up taking up so much of my time. In the end I was elevator one day I asked him, “Professor Bott, can glad to have written the book with him. For me you recommend some liquors to me?” He gave me it was a form of apprenticeship, and I felt that I a sly look sideways, and said, “Candy is dandy, but had learned a tremendous amount of mathematics liquor is quicker!” before mentioning a few brands. from a master. To this day I remember the aphorism but not the Raoul was pleased with the resulting book. Once brands of liquor he recommended. in a lecture I attended, he mentioned some facts—I forget about what, maybe Joint Books or spectral sequences—and told the audience that When I first started working on the book Differential they could find them all in the “Bible”. There Forms in Algebraic Topology with Raoul, I was a was a momentary perplexity among the audience, graduate student. He thought that we made a and then it transpired that Bott was referring to great pair working together, because as a graduate our joint book. For a devout Catholic like Bott to

April 2013 Notices of the AMS 407 good friend of mine and a student of his of whom he thought highly. I remember at a conference, Raoul once put his arm around her shoulder and exclaimed to the public, “My finest student!” On the day that Nancy got married, Raoul said to me, “Loring, you missed your chance.”

Dust Bunnies In my first year as an assistant professor at Michigan, I worked long distance with Raoul on the book Differential Forms in Algebraic Topology. That summer I returned to Harvard to facilitate our collaboration. At the time Raoul and his wife, Phyllis, were comasters of Dunster House, a Figure 21. Left to right: Joan Glashow, Dorothy Harvard undergraduate house with three hundred Haag, Rudolf Haag, Sheldon Glashow, Arthur undergraduates. Too cheap to rent a place of my Jaffe, Barbara Dauschke, Raoul Bott, Phyllis Bott, own, I asked Raoul if he had a guestroom for me in Klaus Hepp, Konrad Osterwalder, Walter the Dunster House master’s residence. Bott readily Kaufmann-Bühler at a dinner for Rudolf Haag, agreed. founding editor of Communications in The guestroom was a room attached to the Mathematical Physics, at Harvest Restaurant, master’s residence but with a separate entrance. Cambridge, MA, September 1982. This way I had my privacy, but I could go into the master’s residence to use the kitchen and dining room. To afford Raoul and Phyllis their privacy, I compare our book to the Bible must have been the normally did not do that except when they were highest form of compliment. away. The Botts by then had a house on Martha’s Although we had projected a second volume, Raoul did not mention it after the completion of the first, possibly because he did not want to put me through the experience again. It was many years later that I brought it up. The book would be called Elements of Equivariant Cohomology. We worked on it for many years. My chief regret is that we did not finish it while he was alive, but I have hope that it will soon see the light of day. While working on the books, Raoul often told me to be “generous with credit to others.” Human nature being what it is, we probably all have the tendency to overestimate our own contribution and, conversely, to underestimate that of others. These days, whenever my baser nature threatens to come to the fore, I remember this lesson from Raoul. One reason we got along so well I think is that with my strict Confucian upbringing, in which every edict is serious, I found Raoul’s wit and irreverence refreshing. As for Raoul, he said that as he got older, he liked more and more the Confucian reverence for the aged.

Personal Happiness Raoul had a playful streak that persisted through- out his life. He liked to tease everyone: his wife, children, friends, colleagues, and even students. Figure 22. The poster for a conference in honor His interaction with me was no exception. of Raoul Bott in 1984, with an ink painting by His concern for me extended to my personal the artist and topologist Anatoly T. Fomenko happiness. My time as a graduate student at depicting the . Harvard overlapped with that of Nancy Hingston, a

408 Notices of the AMS Volume 60, Number 4 Vineyard and would often spend a large part of the summer there. I worked with Raoul on occasional trips to the Vineyard or when he returned to Cambridge from time to time. As comasters of Dunster House, Raoul and Phyllis often had to entertain on a large scale, holding receptions for students and parents, for example, and so Harvard provided them with live-in help, who were usually graduate students in fields other than mathematics. The live-in help lived upstairs from the Botts, so that summer I found myself living in the Dunster House master’s residence with three young women, the live-in help of the year. The first time Raoul came back in the summer, he got very mad at the four of us; apparently we had been living in squalor (though not in sin). Figure 23. Receiving the National Medal of Science from President Reagan in 1987. Pointing to dust bunnies everywhere, he said, “Look at this!” The three young women were not used to cleaning the house, because during the school year there was a cleaning staff from Harvard. As for me, at that point of my life I was oblivious to dust bunnies; they were simply invisible to me. It was strange that as in mathematics, where, after Raoul showed me his fixed-point theorems, I began to see fixed-point phenomena everywhere, in the same way, after Raoul pointed out those dust bunnies, I began to notice dust bunnies everywhere. After that, each time just before Raoul was to return to Cambridge, my three housemates and I would clean the master’s residence from top to bottom.

Book Contract The dust-ball incident was one of only two times Figure 24. At the Harvard Science Center, with Lars Ahlfors in the background, 1988. that I saw Raoul get mad. The other time had to do with the contract for our book. While working on the book, we circulated the manuscript to some colleagues and students for feedback. Possibly For Raoul, I think the royalty was not an issue because of Raoul’s fame, the book was heavily at all, but for me, a low-paid assistant professor at courted by publishers. Both Walter Kaufmann- the time, it was much more significant. With the Bühler, the mathematics editor at Springer, and letter in hand, I walked into Raoul’s office, looking Klaus Peters, the editor at Birkhäuser, at the time frantic. When Raoul saw me and read the letter, he an independent publisher,2 came to Harvard to got quite mad. He said, “They signed a contract. lobby us for their book series. We chose Springer, Tough luck.” He then picked up the phone and not only because of its long history and excellent called the editor. In his usual authoritative voice, reputation for quality but in part because of the he told the editor firmly that we had no intention of better royalty Springer offered. renegotiating the contract. That was the end of it. After the book was published, Kaufmann-Bühler Springer backed off and seems to have flourished. was quite happy, because as he told me, “The book was selling like hotcakes.” He passed away a few Style years later and was replaced by a succession of At a conference in in 2008, Michael Atiyah editors at Springer. At one point, one of the new said that someday historians of mathematics may editors sent me a letter, pleading difficult financial want to decipher joint papers to figure out who circumstances at Springer and asking Raoul and wrote what. In some cases this may be quite easy. me to sign a new contract with a lower royalty rate. Raoul was a consummate stylist. His writings were pithy. He had a colorful, inimitable way of 2Birkhäuser has since become part of Springer. expressing himself. People have often come up

April 2013 Notices of the AMS 409 the night I had spent in the erstwhile bed of Queen Elizabeth had something to do with it!” According to a recent message from Atiyah, the queen was Victoria, not Elizabeth. Raoul had stayed with the Atiyahs in the Master’s Lodge at Trinity College, Cambridge, where Atiyah was then the master. In her time, Queen Victoria and her consort, Prince Albert, did in fact stay as guests at Trinity College, and the four-poster bed that they used became a guest bed.

Lecture Preparation Figure 25. Phyllis and Raoul on Martha’s One year when I was at the University of Michigan, Vineyard in the 1980s. Raoul was invited to give a lecture in a prestigious series. During his visit to Ann Arbor, Raoul stayed with me in my one-bedroom apartment. The morning of the lecture, he was writing his lecture notes. After writing seven pages, he said, “That’s enough. I will not be able to cover more than five pages in an hour.” I have found this to be a useful rule of thumb: five to seven pages of handwritten notes are about right for an hour lecture on the blackboard. I learned more from Raoul’s leisurely but well-timed pace of five handwritten pages in an hour than from other people’s fifty slides, each densely packed with information.

Figure 26. “Raoul, Raoul, Raoul your Bott.” Another Narrow Escape Squibnocket Pond, Martha’s Vineyard, 1989. Raoul’s life seemed to be blessed. He left his native Hungary/Slovakia before the Nazi invasion, survived near-drowning in an expedition organized to me to tell me how much they like our book. Sometimes, as if to prove that they have read it, by Stephen Smale, and visited India without a visa they cite specific passages that they like best. Much at a time when visas were required. In Ann Arbor to my chagrin, these are usually not the ones I he also had a narrow escape. wrote. At the end of his visit to Ann Arbor, I drove him to the Detroit International Airport, twenty Sleeping in Another Woman’s Bed miles away, in my Ford Maverick. It was a used car Jane Kister was a young logician at the University that I had bought from a departing postdoc at the of Oxford in the seventies. In the fall of 1978, University of Michigan. Soon after I purchased the just after marrying the topologist Jim Kister, Jane car, I noticed that it was leaking transmission fluid, spent a sabbatical semester at MIT. At a reception but the rate of the leak was so slow—just one or at Harvard, Raoul put his arm around her and two drops a day—that it did not seem worthwhile announced, “I’ve slept in this woman’s bed.” Jane’s to replace the entire transmission. On the highway face turned beet red. What happened was that Jane as we were heading towards the airport, the car was also on sabbatical in the spring of 1977 and started smoking under the hood. We were alarmed, had rented her house in Oxford to the Botts. It but Raoul had a plane to catch and the airport was indeed true that Raoul had slept in Jane’s bed, was not so far away, so I continued driving at full though not simultaneously with her. speed. While visiting in the early eighties, Raoul thought that he had also slept in Queen Elizabeth’s Just as we arrived at the airport, dense white bed, but of course without the queen in it. In his smoke billowed from under the hood and the car Collected Papers he credited this experience with went dead. It looked like it could explode. Raoul his sudden joint insight with Michael Atiyah into hurriedly ran to his flight, and I jumped out of the relation between equivariant cohomology and the car. After his return to Boston he called me to the moment map [3, Volume 4, p. xiii]: “Possibly make sure that I was still alive.

410 Notices of the AMS Volume 60, Number 4 The Toaster Incident at Dunster House Lin was always Raoul navigated the perils of academic politics with so quiet when consummate skill. He and Phyllis were comasters I was the mas- of Dunster House for six years. After they stepped ter. He must have down, another professor was appointed as the thought that I was master. To distinguish him from Raoul, I will call doing a good job.” him the new master. The new master was a very It so happened nice man, but his term was marked by controversy. that the new mas- I will give one example. It stemmed from a toaster ter was an ethnic oven. Chinese from In- Some Jewish students did not want to eat the donesia, a resident food in the dining hall for reasons of keeping tutor whom he kosher. They asked the new master for a toaster particularly liked oven so that they could heat up their own kosher and was accused food. The new master bought a toaster oven for of being partial them. One of the tutors (academic advisors) at to was a Chinese- Dunster House, an activist with strong principles, American, and the wrote a letter to the student paper, the Harvard professor calling Crimson, criticizing the use of house funds to buy for the master’s ouster was a Chi- the toaster oven, because in his view this was an nese from . act of favoritism towards one particular religion, Raoul turned to akin to a violation of the separation of church and me and asked, “Is Figure 27. Raoul steering a boat (not state, a founding principle of our republic. this one of those his own). The new master fired this tutor. More letters Chinese battles so followed in the Crimson. It was no longer about the inscrutable to us toaster oven, but about the new master’s leadership. Westerners?” Other tutors wrote letters, accusing the master of I do not know autocracy and partiality, of favoring some tutors what he said to over others. There were calls for the master’s President Ruden- ouster. Students organized demonstrations in stein. Rudenstein Harvard Yard supporting the fired tutor. Professor renewed the con- Edmund Lin, a former chair of the Department of tract of the new Molecular Genetics at Harvard Medical School and master. The con- a member of the Senior Common Room of Dunster troversy died down House, wrote a letter to President Rudenstein of after the stu- Harvard, calling for the master’s resignation. Only dents graduated. at Harvard could there be a raging debate about Edmund Lin told constitutional principles arising from a toaster me afterwards, “I Figure 28. Raoul Bott in 1991. oven. This was when the new master’s five-year am sure it was term was up for renewal. President Rudenstein Raoul who saved the new master’s skin.” asked to meet with Raoul, evidently because he valued Raoul’s judgment. Knowing that I was a Foreign Languages close friend of Edmund Lin, Raoul asked me if I Raoul had a wonderful self-deprecating sense knew what was going on. I did, not only because of of humor. He was a talented linguist. He spoke my friendship with Edmund Lin but also because German, Hungarian, and Slovak fluently, not to I read the Crimson every day. Raoul did not read mention English, of which he was a master. But the Crimson. there is a limit to the number of languages one can When I explained the incident to Raoul, his learn or need to learn. I like his experience with immediate reaction was “An activist troublemaker? Italian. Before a conference in Italy, he bought a You should never fire someone like that. If you do, cassette course on Italian. Repeating the sentences there is no end to the trouble. You should give him on the cassette tape, he studied Italian for two tenure!” Raoul had a very good nose for staying weeks. When he got to Italy, he found that he had out of trouble. Of course, this did not mean that forgotten all the sentences except for one. He told he would give every activist tenure. It just meant me that the one sentence he could say in Italian that in this case the stakes were not high enough was “Ascolti e ripeta,” which means “Listen and to fire the tutor. Raoul then said pensively, “Ed repeat.”

April 2013 Notices of the AMS 411 sport, he took part in an undergraduate theater production, playing a Hungarian linguist in My Fair Lady. At one Halloween party, Raoul and Phyllis dressed up as a pirate king and a young maiden, but two students upstaged them by dressing up as Raoul and Phyllis Bott! The male student sported a big beard and was chock-full of gray hair, and to top it off, he was carrying Raoul’s signature briefcase (Figure 15). An avid swimmer and a regular on the clothing- optional beach of Martha’s Vineyard, Raoul earned himself the sobriquet “The Mayor of Lucy Vincent Beach”. He played tennis and bicycled to work. Once when I visited his home, he showed me with great pride some kitchen renovation, saying that he did it all with a router.

Material Enjoyment Figure 29. Phyllis and Raoul on Martha’s From Raoul, I learned that a lifelong dedication Vineyard in the 1990s. to intellectual pursuits is not incompatible with enjoyment of material things. Raoul bought a beautiful house on Martha’s Vineyard. Although the house was not right on the water, it was surrounded by an expanse of wild vegetation and had an unobstructed view of the ocean. There was even a brook on the property. Since most of the houses there were hidden in dense foliage, Raoul’s house had a view of nature with no other sign of human habitation. One day another house rose up, towering above the canopy of trees in full view from Raoul’s window, the only house visible in otherwise pristine nature. Raoul said it stuck out like a sore thumb, but he was philosophical about it. After all, his own house might be a sore thumb to the other owner. While we were working on the book Differential Forms in Algebraic Topology, he teased me about the enormous amount of time I was spending on it, asking me if I thought that with the expected royalty it would come out to minimum wage. Then he said, “I want to buy a boat with it.” I thought he was joking, but years later he did buy a boat. Raoul had a fascination with cars, and on one visit he proudly showed me his collection, Figure 30. Raoul Bott, George Mackey, and a single 2-inch exact replica of a Jaguar that he Arthur Jaffe at Arthur Jaffe’s wedding, Lime Rock, said a student of his gave him. Finally, at the age Connecticut, September 12, 1992. of seventy-four, he bought a BMW, exemplifying another piece of advice he gave me: “Live it up!”

Nonmathematical Activities Mineral Collection In spite of his prodigious output in mathematics, One of the pleasures of talking to Raoul was Raoul found time to do other things. As comasters the unexpected insight that he often offered. of Dunster House, Raoul and Phyllis actively Sometime in the early nineties, Raoul received in participated in the life of the undergraduates, the mail a calendar of Steve and Clara Smale’s sharing meals with them, meeting with their priceless collection of natural crystals, lovingly and parents, and organizing and attending cultural beautifully photographed by Steve Smale himself. activities in the house. Raoul played the piano well Raoul showed me the calendar in his office, and enough to give public performances. Ever the good while admiring the breathtaking beauty of the

412 Notices of the AMS Volume 60, Number 4 minerals, he said, “What a way to avoid inheritance tax! You just have to slip a few of these to your children.” Of course, he did not mean it as an estate-planning tip; besides, I had neither a fortune nor children to benefit from this advice, but it was so characteristic of Raoul to have a unique perspective on everything.

Practical Advice Fresh out of graduate school, I once visited Raoul on Martha’s Vineyard to work on our joint book. Sitting on a bench surveying his beautiful estate, he said to me, “Loring, buy land.” At the time I was too poor to buy anything, but time has borne out the wisdom of his advice, especially when the land is in a well-chosen location like Martha’s Vineyard. Figure 31. Lecturing at Harvard in the 1990s. One of Raoul’s observations on life has played a crucial role in my mental equilibrium. When he was at the Institute for Advanced Study at Princeton in 1949–51, he once had a conversation with , a fellow Hungarian who was at the time a professor at the institute. Von Neumann told Raoul that he had known only one great mathematician, David Hilbert, and that having been a prodigy in his youth, he never felt that he had lived up to his promise. Raoul wrote in [3, Volume 4, p. 270], “So you see, it is not difficult to be found wanting—one just needs an appropriate measuring rod.” If even von Neumann felt inadequate in his achievement Figure 32. Raoul Bott, Isadore M. Singer, in comparison with Hilbert’s, what chance for , and Michael Atiyah at a professional satisfaction do we ordinary mortals Journal of Differential Geometry reunion dinner, have? After Raoul recounted this incident to me, Cambridge, MA, 1999. The four founders of I resolved never to compare myself with anyone index theory are holding paintings by Milen else, especially not with my friends and classmates Poenaru depicting their work. who have achieved greatness. I was fortunate to be in the job market during a brief window of opportunity when there were have been one of Raoul’s top five favorite theorems. many jobs available, and so I actually had a few I think Raoul would have agreed. The list of five was choices. Tufts had a fine reputation and excellent colleagues, but what clinched the deal was what a rather artificial framework and should probably Raoul said to me, “It will be nice to have you not be taken too literally. It was what came to in the backyard.” The physical proximity made Raoul’s mind on the spur of the moment, but he collaboration easier, and after moving to Tufts, I simply could not fit all of his favorite theorems worked on a few more joint projects with him and in there. In the end, my article included another had the pleasure of attending more of his courses. thirteen in addition to the top-five list.

Favorite Theorems The Wolf Prize When I was writing “The life and works of Raoul Raoul used to say that there were two kinds Bott” in 2001, I interviewed Raoul and asked him of mathematicians, smart ones and dumb ones. to list three of his own theorems that he liked the The smart ones were people like Michael Atiyah best. He had trouble doing it, saying that it was and Jean-Pierre Serre, who understood new ideas like asking him which of his children he liked best. quickly. He classified himself as a dumb math- Eventually he came up with a list of the top five. ematician, because understanding came to him The Atiyah–Bott fixed point theorem for elliptic slowly. This may be so, but his understanding was complexes was not one of them. profound, as his corpus of many beautiful and After the memorial service for Raoul in January deep theorems attests. If he did not understand 2006, Michael Atiyah gave a compelling lecture on something, he had no hesitation in saying so. When why the Atiyah–Bott fixed point theorem should he was awarded the Wolf Prize, he told me that he

April 2013 Notices of the AMS 413 be delighted—as well as humbled—to join a list that, so to speak, starts with Marc Chagall! My first words of thanks here are in tribute to Ricardo and Francisca Wolf for setting up a founda- tion so much in tune with the most essential need of our ever-shrinking planet. The universality of their purpose speaks for itself: “To promote science and art for the benefit of mankind.” And how inspired of them to see the commonality of art and science, and to include mathematics, where these two spheres of endeavor are well nigh indistinguishable, in their generous bequest. Figure 33. Jean-Pierre Serre and Raoul Bott, But we feel doubly honored that a small and c. 2000. relatively new country, with so many pressing and highly nontrivial—as we say in our mathematical jargon—problems on its agenda, nevertheless finds time to bestow this award at its highest level. This act alone is a moving tribute to the life of the spirit in a world mostly concerned with more mundane things. Unfortunately, the very term “Mathematics” strikes terror in most mortal hearts, and so it is possibly appropriate here to put our subject into some sort of perspective. And I can think of no better way of doing this than to divulge to you just how my junior, but much wiser, colleague Jean-Pierre Serre cajoled me into being the one to deliver this acceptance speech. “For if I were to give the speech,” he argued, “then all I would Figure 34. Receiving the Wolf Prize from say is that while the other sciences search for the President Ezer Weizman of Israel in 2000. rules that God has chosen for this Universe, we mathematicians search for the rules that even God has to obey.” And I certainly couldn’t let him get was in very good company, because he was sharing away with that! the prize with Serre. But, after this little tongue in cheek, my time is One of them had to give a speech in the Knesset, definitely up! the Israeli parliament. According to Raoul, Serre Still, please permit me two more words of thanks. wanted him to give the speech, because Serre The first is to the committee that had a long enough thought that Raoul “had a better stage presence” memory to settle on us from amongst so large an and that Raoul “looked more like a mathematician.” array of worthy and younger candidates. And our But how to explain to the Israeli lawmakers the final thank you is to our families and especially research for which they were being awarded the our wives, who for a lifetime have put up with our prize? This is the usual conundrum of pure absent-minded ways and have been our anchors in mathematicians called upon to explain their work. the real world. Serre came up with a gem that Bott incorporated into his speech: Final Years Mr. President of the State, Mr. Speaker of the After Phyllis became partially disabled following Knesset, Mr. Minister of Education, Members of the an operation, the Botts moved to California in the Diplomatic Corps, Dear Colleagues and Guests: fall of 2004, where the year-round good weather It is a great honor for me to rise in this beauti- permitted Phyllis more opportunities for outdoor ful chamber and in so distinguished a company to mobility in a wheelchair. In [5] I mentioned some accept the Wolf Prize in Mathematics on behalf of of the coincidences in Raoul’s life and my own in Jean-Pierre Serre and myself. terms of the places where we ended up—McGill, Thank you. Princeton, Harvard, Michigan—wherever he went, I In our field alone the previous winners of this followed a few decades later, if only in the vicinity Prize include both heroes of our youth and cherished sometimes. The final coincidence was that the friends. And if we look beyond, well, who would not town the Botts moved to, Carlsbad, California, was

414 Notices of the AMS Volume 60, Number 4 only twenty-five miles from my parents’ house! So it was easy for me to continue to visit the Botts. Soon after their move, Raoul was diagnosed with lung cancer. In spite of the poor prognosis, he was his usual cheerful self. He explained the principle of chemotherapy to me this way: “It tries to kill the cancer faster than it kills you.” He faced the prospect of death with equanimity. When I asked him if he would be returning to Massachusetts at some point, he pointed to the ground and said, “I am going in here.”3 It has often been said that mathematics is a young person’s game. Raoul’s life is a particularly inspiring counterexample. I saw him three weeks before he passed away. I had been working on a problem with him on the volume of a symplectic quotient. He was in top form mentally. He explained to me a new way of looking at the problem that greatly simplified it. I cried, “This is so simple!” He said, “That’s the way I like it.” At the age of eighty-two, battling cancer, he was still trying to understand integration on a Figure 35. Visiting his childhood home in symplectic quotient. There was a paper of Victor Dioszeg, Slovakia, in 2002. Guillemin and Jaap Kalkman on the subject, but he wanted to understand it in his own way. Clearly, his motivation was not any external reward, like an NSF grant or more honors. He simply wanted to understand. He was a true mathematician. His life showed us what is humanly possible. He continued to make beautiful discoveries and publish important papers to the very end.

Royal Society In the final year of his life, Bott was inducted into the Royal Society. The Royal Society dates back to 1660 and is a roster of luminaries in the history of science. Each new fellow signs in a book that has the signatures of all former and current fellows. For health reasons, Bott was not able to travel to London for the signing, but Michael Atiyah, a former president of the Royal Society, brought to California the actual page from the book Bott was to sign. For good measure, Atiyah also brought Figure 36. Loring Tu with Phyllis and Raoul Bott, Raoul a scanned and bound copy of the preceding Boston, 2004. pages. An induction ceremony was held at the Kavli Institute for Theoretical Physics, University of California, Santa Barbara, in October 2005. References When I visited Raoul in California a month later, [1] M. F. Atiyah, Raoul Harry Bott, Biographical Memoirs he excitedly showed me pages from his copy of of Fellows— Royal Society 53 (2007), 63–76. Reprinted the Royal Society book, exclaiming, “Look at this! in Bull. Lond. Math. Soc. 42 (2010), no. 1, 170–180. Christopher Wren! Isaac Newton! George Stokes! [2] P. R. Kotiuga, editor, A Celebration of the Mathemat- Lord Kelvin!” For a man of science, this may be the ical Legacy of Raoul Bott, CRM Proceedings & Lecture Notes, vol. 50, American Mathematical Society, 2010. ultimate good company. [3] R. D. MacPherson, editor, Raoul Bott Collected Papers, Volumes 1–4, Birkhäuser, 1994–95. 3Raoul was buried in the Chilmark cemetery on his beloved [4] L. W. Tu, The life and works of Raoul Bott, in The Martha’s Vineyard, so he did return to Massachusetts after Founders of Index Theory: Reminiscences of Atiyah, all. Bott, Hirzebruch, and Singer, edited by S.-T. Yau, In-

April 2013 Notices of the AMS 415 32, 33, 34, 35: from the Bott Family Collection. Figure 5 (Photographer: Alan Richards), Figure 11 (Photog- rapher unknown): from The Shelby White and Leon Levy Archives Center, Institute for Advanced Study, Princeton, NJ. Figure 10: from David Mumford. Figures 13 (Photographer: Carol Tate), 14 (Photographer: Carol Tate), 20 (Photographer: Carol Tate): from Carol Tate. Figures 16, 17, 21, 24, 30: from Arthur Jaffe. Figures 18 (Photographer: Shiu-Yuen Cheng), 19 (Photogra- pher: Shiu-Yuen Cheng): from Shiu-Yuen Cheng. Figure 22 (Painting by Anatoly T. Fomenko). Figure 36 (Photographer: Mary Moise): from Loring W. Tu. Figure 37. Speaking at the 75th Anniversary of Figure 37 (Photographer: Cliff Moore). the Institute at Princeton, March 2005. Figure 38 (Photographer: KITP Staff): from the Kavli Insti- tute for Theoretical Physics, UC Santa Barbara. Photos are used by permission of photographers or owners for this article only, and the images may not be transferred to other settings.

Figure 38. Induction into the Royal Society, ceremony at UC Santa Barbara, October 2005.

ternational Press, Somerville, MA, 2003, pp. 85–112. An updated version appeared in the Notices of the American Mathematical Society 53 (2006), 554–570. [5] , Reminiscences of working with Raoul Bott, in The Founders of Index Theory: Reminiscences of Atiyah, Bott, Hirzebruch, and Singer, 2nd ed., edited by S.-T. Yau, International Press, Somerville, MA, 2009, 157–160. [6] , editor, Raoul Bott: Collected Papers, Volume 5, Springer Basel AG, to appear in 2014. [7] S.-T. Yau, editor, The Founders of Index Theory, 2nd ed., International Press, Somerville, MA, 2009. (A fine book, except that Loring Tu’s name is misprinted throughout the two articles he authored.)

Acknowledgment, Photo Sources and Credits: Loring Tu is grateful to Phyllis Bott, Jocelyn Bott Scott, and Candace Bott for verifying the essential accuracy of his memories of Raoul Bott; to Michael Atiyah, Jeffrey D. Carl- son, George Leger, and Stephen J. Schnably for helpful comments; to Christine di Bella, Shiu-Yuen Cheng, M. S. Narasimhan, Bernard Shiffman, and Carol Tate for the iden- tification of personages and dating of photos; and to the following sources for providing the photos: Figure 1 (Photo by Bachrach): from Robert Bachrach. Figures 2, 3, 4, 6, 7, 8, 9, 12, 15, 23, 25, 26, 27, 28, 29, 31,

416 Notices of the AMS Volume 60, Number 4 www.bristol.ac.uk

School of Mathematics

Salary on Professorial scale to be agreed on appointment (or between £47,314 - £52,233 per annum if appointed at Reader)

The University of Bristol seeks an outstanding candidate for a post as Heilbronn Professor/Reader in Combinatorics. The successful candidate will become a member of the Pure Mathematics Group within the School of Mathematics. We are seeking applicants who have demonstrated, or shown potential for, the highest achievement in any area of combinatorics, broadly interpreted.

This position forms part of the University’s strategic expansion in research areas that complement existing strengths in Pure Mathematics at Bristol. The successful candidates will be expected to build up and lead a research group of the highest international level, and to interact with Heilbronn postdoctoral fellows.

The Pure Mathematics Group at the University of Bristol is internationally recognised with interests spanning pure mathematics, including number theory, arithmetic geometry, arithmetic combinatorics, ergodic theory, random matrix theory, representation theory, complex analysis, partial differential equations, set theory and logic. The Group has strong links with the Heilbronn Institute for Mathematical Research, which focuses on discrete mathematics, funds about 20 postdoctoral fellows in the School of Mathematics, and runs a co-ordinated series of research programmes, conferences and workshops.

The School of Mathematics is one of the leading centres for research and teaching in mathematics in the UK. There are groups working in Applied Mathematics, Pure Mathematics and Statistics. In the 2008 Research Assessment Exercise (RAE), Applied Mathematics ranked third, Pure Mathematics ranked joint fi fth and Statistics ranked joint fourth in terms of grade-point average.

Women are currently under-represented in the School of Mathematics, therefore we particularly welcome applications from women for this post. The University has numerous family friendly policies which can be found on the Positive Working Environment website.

In order to receive full attention, applications should be received by 12:00 pm (i.e. midnight) on Monday 1 April 2013.

To discuss any aspect of the posts, please contact Professor Trevor Wooley, FRS, +44 (0)117 331 5240, [email protected]

Alternatively, please feel free to contact any of the people listed below for informal discussions with respect to the post:

Lynne Walling, Head of Pure Mathematics, +44 (0)117 331 5245, [email protected] Professor Noah Linden, Head of the School of Mathematics, +44 (0)117 928 8633, [email protected]

To apply please visit our web site at www.bris.ac.uk/jobs enter the vacancy number ACAD100199 into the job search and follow the link to the on line application process.

The closing date for applications is 1st April 2013.

excellence through diversity Errors and Corrections in Mathematics Literature Joseph F. Grcar

Table 1. Some examples of corrections.

1a. Erratum for typographical errors. Note the occasional practice of including names of authors for the corrected paper in the title of the correction. title: Errata to: “The distribution of values of L(1, χd )”, by A. Granville and K. Soun- dararajan in GAFA 13:5 (2003)” in: Geom. Funct. Anal. 14(1):245–246, 2004 (Zbl 1188.11039) 1b. Errata for substantive errors. Note different styles of titling in the same journal. title: A correction to “Bases of the contact-order filtration of derivations of Coxeter arrange- ments” in: Proc. Amer. Math. Soc. 136(7):2639, 2008 (Zbl 1141.32011) title: Erratum to “A finitely presented group with unbounded dead-end Figure 1. Better late than never. depth” in: Proc. Amer. Math. Soc. he veracity of scientific research has 136(7):2641– 2645, 2008 (Zbl 1142.20309) 1c. Addendum. become a topic of public discussion title: Addendum to “Semistable sheaves in positive recently [1]. This article examines characteristic” why mathematicians correct errors in: Ann. Math. (2), 160(3):1211–1213, 2005 at a slower pace than many other (Zbl 1080.14015) researchersT (see Figure 1). Some sociological 2. Comments and replies. aspects of mathematics research may make title: Comments on “State estimation for linear mathematicians averse to publishing corrections, systems with state equality constraints” in: Automatica 46(11):1929–1932, 2010 while mathematics journals have comparatively (Zbl 1218.93091) weak procedures to assure the correctness of the title: Authors’ reply to “Comments on ‘State literature. estimation for linear systems with state equality constraints’ ” Types of Corrections in: Automatica 46(11):1933, 2010 (Zbl 1218.93093) Corrective material appears so infrequently in 3. Other (disproving a conjecture). mathematics that readers may not be familiar with title: On the “Section Conjecture” in anabelian the terminology. The titles of corrective articles geometry can indicate whether errors are corrected or merely in: J. Reine Angew. Math. found and by whom. 588:221–235, 2005 (Zbl 1108.14021) 1. “Errata” and “corrigenda” are used mostly made by publishers and those made by authors, interchangeably for corrections of all kinds. A as in Nature. “Addenda” are further explanations. distinction (which is not implied by the Latin Strictly speaking, addenda do not correct errors, words) is sometimes drawn between mistakes although they do perform a corrective function. The corrections for other than publishers’ errors Joseph F. Grcar’s email address is [email protected]. are supplied by authors, perhaps upon prompting DOI: http://dx.doi.org/10.1090/noti988 from readers.

418 Notices of the AMS Volume 60, Number 4 2. Errors reported by readers appear in a class of Corrections 1990-2010 by Subject Area short articles with titles that begin with a formulaic “comment on”, and conversely there may be a “reply 3. Other to”. These are substantive contributions that un- 2 2. Comments dergo special forms of .1 Comments do not always challenge articles and may corroborate 1. Errata them. 3. More difficult to identify are a range of other corrective articles that disprove conjectures and 1 rule out directions of enquiry. These corrective publications (1–3) are illustrated in Table 1. Completing the lexicon are (4) “republications” due to major printing errors and (5) “retractions” As Percent of Documents in Journals due to disastrous mistakes or misbehavior. Retrac- 0 tions can be partial or full and voluntary or forced. Editors may lose sleep over retractions, but they Energy Physics Nursing Dentistry Medicine Chemistry are so rare that it is misleading to dwell on them. Veterinary All subjects Psychology Engineering Mathematics Publication rates for corrections vary widely Neuroscience Pharmacology Multidisciplinary Social Sciences

across disciplines (see Figure 2). The highest Materials Science Decision Sciences Computer Science Health Professions Arts and Humanities

rates of (1-2) corrections are in multidisciplinary Chemical Engineering Environmental Science Economics and Finance journals, core physics, and biomedical fields. For Business and Accounting Earth and Planetary Science

the most part, (1) errata correct boilerplate (author Immunology and Microbiology Physics (and no other subject) and sponsor data), typos, and substantive errors, Biochem. and Molecular Biology Engineering (and no other subject) Agricultural and Biological Sciences with the result that many of the substantive errors Mathematics (and no other subject) are found through (2) reader comments, which are especially rare in mathematics. Other corrections (3) may signify intellectual ferment in that, for Figure 2. Corrections as percent of journal example, earth science includes climatology. documents for subject areas of Scopus [2]. Three Because corrections (1-2) appear less frequently areas are shown both inclusive and exclusive of in engineering and mathematics than in many interdisciplinary work. Comments are undercounted for journals that choose to subjects, at least one alternative must be true: these identify them as “letters”. fields have lower error rates or lower correction publishing rates. There is no reason to think Mathematics literature has been shared with engineers and mathematicians make mistakes less other disciplines ever since the “scientific revolu- often than other researchers, while there is some tion” of the European Renaissance culminated in evidence that the research cultures of these fields 2 mathematics becoming the language of scientific discourage publishing corrections. These matters 3 theory, as epitomized by Isaac Newton [3]. Another are considered for mathematics: (a) proclivity to leg of science was mathematized when statistical error, (b) aversion to acknowledging errors, and methods were created to interpret experimental (c) weak editorial practice, from which follow error [5]. Scientific computation joined the group some recommendations to improve the correcting of interdisciplinary topics in the twentieth century process. [6]. All this work receives Mathematical Subject Classifications [7] over 60 and comprises slightly Proclivity for Errors over half of mathematics articles for 1990–2010 Mistakes are subjective because they have different [8]. The interdisciplinary fraction is probably larger impacts, whether in evidence or in conclusions, (see Table 2) because 03 logic and 11 number and because they are relative to a standard of theory have become interdisciplinary from their rigor. Mathematics is held to many standards in importance to computer science and information its capacity as the prototypical “meta-discipline”. security. Moreover, the core literature has some unique Errors in other fields must be considered for systemic possibilities for error. comparison and to encompass interdisciplinary mathematics. Briefly, scientific research draws 1For example, authors of a paper receiving comment do not participate in reviewing the comment. Sometimes comments confirmation from three sources: theory, simu- are misidentified as letters to the editor. Strictly construed, lation, and primarily experiment. Discrepancies letters express opinions and thus are not reviewed. are seen as paths to discovery [9], [10], with the 2Regarding engineering, corrections may be less important (the subject matter changes rapidly) and may have no venue 3Recent scholarship on mathematics and Renaissance sci- (many articles appear in conference proceedings). ence is summarized in [4].

April 2013 Notices of the AMS 419 Table 2. Percent of mathematics articles shared with any other Scopus subject area (top) and the seven largest intersection partners (bottom). Data for 1990–2010.

Other %

All Others 58 Computer Science 20 Physics 19 Engineering 16 Decision Sciences 11 Biochemistry 7 Medicine 5 Social Sciences 3 Figure 3. Errors of the scientific method.

result that research tends to be “normative” [11] from conclusions. A “heritage” [23] of important in reconciling evidence (see Figure 3). The trifecta results is supported by proofs newly written in the of evidentiary sources heightens confidence in sci- current idiom. Examples of renewal are the volumes entific conclusions and permits scientific theory to of Bourbaki [24] and the history of “Rolle’s theorem” employ its own standards of rigor for mathematics. [25], which began with a critic of infinitesimals and Philosophical studies have gone beyond “marveling now is basic to real analysis. at the applicability of mathematics to nature in In contrast to the multiple sources for scientific general” [12] to examine scientific argumentation knowledge, proofs are a single point of failure for [13]. The highest, or “mathematical”, rigor is re- garded as an obstacle to investigation [14].4 The core mathematics. Mistakes are difficult to find because mathematical arguments are abbreviated issue of appropriate rigor cannot be excised from 6 a discussion of mathematics by restricting the in two ways: (A) “Splicing” inserts results that literature, because interdisciplinary mathematics were established in earlier work. Mistakes are likely largely develops in the allied fields [18], as, for to be caught only when an article receives close example, the analysis of algorithms. scrutiny upon being “spliced” into other articles. The mathematics that is exclusive of interdisci- Scrutiny can take years, as evidenced by the low plinary work is unique among the subjects in Figure citation-impact factors for mathematics journals. 2 for being incapable of corroboration. Inspiration (B) “Skipping” leaves out parts of arguments, may have many sources [19], [20], but evidence perhaps inadvertently or to save space. Davis consists solely of tautological argument or “proof”. [26] identifies two other reasons for “skipping”: This approach developed in the nineteenth cen- boredom (“you cannot expect me to do every step”) tury.5 Despite concerns about natural language and superiority (“this step is easy for a happy few”). paradox in the twentieth century, no alternative The ambiguity in what constitutes a complete proof has been accepted: contributed to a brouhaha over the resolution of The main hope of a justification of classi- Henri Poincaré’s conjecture [29]. “Skippers”, rather cal mathematics being gone, most math- than being cavalier, may be highly functioning ematicians decided to use that system researchers: anyway…[This history] constitutes the best … he is like a clairvoyant to whom truths caution against taking the immovable rigor appear in a bright light, but mostly to him of mathematics too much for granted. alone.7 — John von Neumann [22] — Charles Hermite [30, p. 147] Variations in rigor (or in philosophies thereof) on his student Poincaré are not viewed as errors in core mathematics, because, as in the sciences, evidence is distinct 6This discussion follows Davis [26]. The rhetoric of proof is examined in [27], [28]. 4 Quinn [15, p. 36] observes that many interdisciplinary 7“Souvent Picard lui a demandé, sur des points d’une grande researchers are “several generations removed” from core importance dans ses articles des Comptes Rendus, des éclair- mathematics “and some are actually hostile to core method- cissements et des explications, sans pouvoir jamais rien ology.” For rigor in quantum theory see [12], [14], and in obtenir qu’une affirmation: ‘c’est ainsi, c’est comme cela’, de engineering see [16], [17]. sorte qu’il semble comme un voyant auquel apparaissent les 5Perspectives on origins and ramifications of using proofs vérités dans une vive lumière, mais en grande partie pour as evidence are given by [15], [21]. lui seulement.”

420 Notices of the AMS Volume 60, Number 4 Low Tolerance for Corrections Comparatively little is known about the innate talents of mathematicians [31]. Reid [32, pp. 66–67] contrasts the abilities of Bernhard Riemann and David Hilbert by arguing that creating new math- ematics is different from devising the formalist proofs which Hilbert made standard. It follows that writing proofs may use some talents found in the general population. “Perfectionism” is a personality construct ex- plained as striving for flawlessness and setting high personal standards.8 Perfectionists share characteristic tendencies, including a low toler- ance for mistakes and a preference for order [33]. Perfectionism has both adaptive and maladap- tive components: high standards coupled with organizational skills contribute to achievement, whereas concern about making mistakes coupled with self-criticism creates anxiety and depression. Figure 4. Woe to the finder of errors. To avoid the negative personality consequences, it suffices that perfectionists need only to avoid covertly. Corrections may be embedded in other learning of their failings (see Figure 4). Denying articles rather than in explicit (1) errata or (2) mistakes or trivializing their impact are defenses comments. These corrections by their nature are 9 that can be observed in mathematicians. difficult to identify (see Table 3). The relative Anecdotal evidence supports the explanation quantities found by one method suggest that that the social mores of mathematicians discour- unannounced corrections are more common in age publishing corrections. Lecat [36] recorded mathematics than in other fields. hundreds of mistakes in nineteenth-century pub- lications. Pertinent is that the authors did not Table 3. Documents with embedded corrections find and report these errors. In a subject whose as gleaned from abstracts. Percent of all journal primary research methodology is logic, there is documents in Scopus subject areas for 1990– 2010. little advantage to admit making mistakes. When Poincaré found a severe error in his prize-winning essay on dynamics, the chairman of the prize com- Subject Area % mittee, Gösta Mittag-Leffler, employed subterfuge Mathematics 0.017 to retrieve the few printed copies of the essay Computer Science 0.012 that had been distributed, including to committee Earth … Science 0.009 member Karl Weierstrass. Mittag-Leffler instructed Decision Sciences 0.008 Poincaré to acknowledge only that the essay had Econ. and Finance 0.008 been revised so as to avoid disclosing the failure Physics 0.006 of the committee [37, p. 67]. Attention is drawn to Engineering 0.004 errors in proportion to the stature of the person All Subjects 0.002 who corrects them, so correcting may be regarded The reluctance to discuss corrections is further as unsuitable work for senior mathematicians. Olga manifest in a high tolerance for unacknowledged Taussky [38, pp. 18–19] relates that David Hilbert’s errors:10 “There are a number of gray areas in collected works were “edited” by very talented and which the experts know that the proofs are not very junior temporary workers. “Hilbert’s work complete or are partly incorrect. Since they are was not free from error of all magnitudes…We experts they know exactly what you can take; in worked very hard on all this, but later even more a sense they are the happy few. This situation is errors emerged.” definitely not acceptable, and we have to fight that Like the editorial changes to Hilbert’s work, mathematicians are open to corrections made results are really available (and not by jumping over this statement, which is not true, but you 8Frost et al. [33] introduced the multidimensional view of can take the next one, and so on). In recent years, perfectionism. This discussion summarizes the survey of for many reasons [competition, complexity, and Tofaha and Ramón [34]. proliferating duties which reduce the time for 9The reluctance of authors and editors to acknowledge the need to publish a correction can be appreciated perhaps 10The quoted material is paraphrased for readability from only through experiences such as chronicled by Hill [35]. a presentation of Bourguignon [39].

April 2013 Notices of the AMS 421 careful peer review], the gray areas have been Table 4). The mathematics subject areas with growing, and if they reach a certain level, then high rates of (2) comments from readers are all mathematical development will be injured.” interdisciplinary fields (see Figure 5). Subjects 81 The most minute errors are not in mathemat- and 83 are heavily published by the Institute of 14 ical matters to be scorned.11 Physics, which invites reader comments in all — Isaac Newton [40, pp. 124–125] its journals. Subject 93 is the rare, very large referring to fluxions branch of mathematics that publishes comments in theoretical (i.e., mathematical) journals such as No Editorial Guidance Automatica. The editorial practices of journals mirror the re- Table 4. Some prominent individual journals search methods of their fields. In the sciences, as (top) and all the journals of some professional noted [9], [10], students learn to be vigilant for societies (bottom) ranked by corrections (1–3) as experimental errors (human, random, systematic) percent of documents for 1990–2010. Data from and for errors in theories (called “models” to Scopus except from Zentralblatt (*) for highlight the approximate nature). Sensitization to mathematics. “if” is the Institute for Scientific errors continues throughout scientific careers. The Information 2-year citation for Council of Science Editors expressly states, “The 2010, rounded to whole numbers. editor’s concern should be correcting the literature so the readership can rely on the information pub- Percent lished”; biomedical and multidisciplinary journals are recognized for exemplary corrective policies [41, pp. 62, 57]. These practices ensure that the

Publications 1. errata 2. cmnts 3. other if scientific literature improves continuously, like science itself. New Eng. J. Med. 3.6 5.3 0.1 53 Mathematicians take an axiomatic approach. Just J. AMA 3.1 3.4 0.3 30 as Platonic mathematics has no errors by definition, AAAS Science 3.9 0.5 0.2 31 articles emerging from peer review should be free Lancet 2.5 1.1 0.1 34 of error in principle. The editorial guidelines of Nature 2.0 0.2 0.3 36 the AMS [42] and the IMU [43] do not consider All subject areas 0.6 0.4 0.3 amending the literature—they are guidelines for All mathematics 0.6 0.2 0.4 12 an ideal world. Gray areas are not mistakes by IEEE journals 0.4 0.4 0.4 this view but rather are lacunae that would be ASME journals 0.4 0.1 0.4 filled if only the reviewers would be thorough. Such faith in an ideal process is inconsistent with its AMS journals* 0.2 0.0 0.5 implementation. Editors note that reviewers can All mathematics* 0.2 0.1 0.2 only be “reasonably certain” about the correctness SIAM journals* 0.2 0.1 0.2 of articles [45], [46]. Compounding the drawbacks of relying on peer review for correctness, editorial It Takes a Community decisions in mathematics are typically based on In summary, the mathematics literature observably a single review [47], whereas in many fields the has low correction rates, yet systemic consider- custom is two or three reviews, and the process ations suggest that mathematicians do not have may be double-blind in social sciences [48].13 lower error rates than other researchers. This The lack of editorial policies inviting correction mismatch may stem from a cultural emphasis on goes hand in hand with low rates of corrections perfection that discourages discussing mistakes. in mathematics journals. Mathematics societies The consequences are an absurdly high expectation publish vanishingly few corrections in their jour- for peer review to catch all errors and a neglect of nals compared to medical and scientific societies policies to correct the literature once published. and fewer than even engineering societies (see One might see publication practices in mathe- 11“Errores quàm minimi in rebus mathematicis non sunt matics as especially defective, but that would be contemnendi.” unfair to many other fields; see the hilarious [49]. 12Ethical guidelines urge individuals to correct errors [44], The onus is on mathematics societies, which have but where to publish corrections is moot. Some journals not led in addressing shortcomings in their publica- privately advise authors to post corrections on their per- tion processes, thereby leaving editors to fret over sonal websites, which is not consistent with the concept of an archival literature. 14See Journal of Physics A: Mathematical and Theoretical 13Anonymity strengthens the integrity of the review process and Classical and Quantum Gravity. The single journal in that articles with gray areas are less likely to be accepted with most articles indexed by Zentralblatt for the decade because a happy few authors are well known to reviewers. 2000–2009 is J. Phys. A.

422 Notices of the AMS Volume 60, Number 4 the difficulty of finding “good” (i.e., superhuman) Percent Corrections in Mathematics Subjects 1990-2010 reviewers. 0 1 A reasonable approach is to emulate the large Relativity theory 83 scientific and medical journals, whose editorial Quantum theory 81 Geophysics 86 policies welcome reader participation. Journals Systems theory, control 93 should be community forums in which new results Statistical mechanics 82 Category theory 18 evolve: Logic and foundations 03 Abstract harmonic anal. 43 1. For example, authors might prefer a stream- Algebraic geometry 14 Sequences and series 40 lined editorial process that solicits multiple Special functions 33 reviews of limited duration. Editors should Optics 78 Number theory 11 not expect to publish papers with the same Algebraic topology 55 level of refinement as Bourbaki’s ency- Astronomy 85 Game theory, economics 91 clopedia. Indeed, results become refined Classical thermodynamics 80 Topological groups 22 through discussion. Several complex variables 32 2. The criteria for acceptance could be broad- Global analysis 58 53 ened from the significance of new results Nonassociative rings 17 to corroborating recent results or to in- Manifolds, cell complexes 57 Mech. of particles and sys. 70 troducing innovative proofs. The most Field theory, polynomials 12 important contributions of some articles Dynamical systems 37 K-theory 19 are the methods rather than the conclu- Commutative rings 13 All mathematics sions, but methods are seldom explicitly Group theory 20 discussed and are not presently a criterion Measure and integration 28 Statistics 62 for publication [39]. General topology 54 3. Perhaps it should be emphasized that the Mech. of deform. solids 74 Probability theory 60 single point of failure of publishing results Integral transforms 44 Ordered alg. structures 06 only once could be addressed by inviting Linear, multilinear algebra 15 corroborating papers with alternate proofs. Combinatorics 05 Operator theory 47 In the sciences, conclusions are not ac- Information and comm. 94 cepted until duplicated by independent Functional analysis 46 Calculus of variations 49 studies [50]. Associative rings 16 4. Finally, journals should invite reader as- Real functions 26 1. Errata Fluid mechanics 76 sistance to clarify published results by Partial differential eq. 35 Potential theory 31 2. Comments establishing guidelines for comments, ac- Operations research 90 celerating review, and stating criteria for Biology 92 3. Other Ordinary differential eq. 34 acceptance. (Because editors share respon- Approximations 41 sibility for errors, they should not have Numerical analysis 65 Geometry 51 complete discretion in publishing correc- Functional equations 39 Convex & discrete geom. 52 tions.) Such practices would elevate instead Fourier analysis 42 of stigmatize corrections to the literature. One complex variable 30 Integral equations 45 Rather than rail at citation impact factors, Computer science 68 mathematics editors can be proactive to attain General alg. systems 08 high impact through reader involvement, just as Figure 5. Error rates as percent of articles in the the editors of Lancet and Nature do (see Table 4). primary Mathematics Subject Classifications [7] It should not be lost on publishers that articles in from Zentralblatt [8]. The eleven subjects in bold a journal that operates as a forum are more likely contribute 51 percent of publications. to be cited, and quickly.

Appendix: Scopus and Zentralblatt Data 2. “Comments” are documents whose titles Scopus [2] classifies documents by source and type. contain “comment* on”, “replies”, “reply”, or In Figure 2 and Table 4: “response to comment*”, except “comment*” may 1. “Errata” are documents of type “er” (errata, not be preceded by “and”, “for”, “some”, or which in Scopus includes republications and retrac- “with”. Comments are undercounted for journals tions). Searching titles for “addend*”, “correction* that identify them with only the distinguishing to”, “corrigend*”, or “errat*” finds some errata keyword “letter”. Letters are omitted because they missing from the “er” class. The asterisk matches may also be short research articles or opinion any full word. pieces.

April 2013 Notices of the AMS 423 3. “Other” corrections have titles, abstracts, [6] J. F. Grcar, John von Neumann’s analysis of Gauss- or keywords containing “contradict(s)”, “counter- ian elimination and the origins of modern numerical example to”, “disproves(s)”, “inconsistent”, “invali- analysis, SIAM Rev. 53(4):607–682, 2011. date(s)”, or “refute(s)”. [7] Amer. Math. Soc. 2010 Mathematics Subject Classification, website, 2010. The overlap among the three groups is 0.33 [8] FIZ Karlsruhe, Zentralblatt MATH–Serials Database, percent, and 97 percent have source type “j” website, 2011. (journal), so percentages are with respect to all “j” [9] H. Bassow, Interdependency and the importance of documents. The percentages may underestimate errors in chemistry, J. Chem. Educ. 68(4):273–274, corrected articles, because only 85 percent of 1991. “j” documents are of type “ar” (article) or “cp” [10] C. J. Giunta, Using history to teach scientific method: (conference paper). The balance are reviews, short The role of errors, J. Chem. Educ. 78(5):623–627, 2001. [11] T. S. Kuhn, The Structure of Scientific Revolutions, articles, editorials, errata, and letters. This other University of Chicago Press, Chicago, 2nd edition, content is numerous for the five specific journals in 1970. Table 4, but for consistency the percentages remain [12] A. Gelfert, Mathematical rigor in physics: Putting with respect to all content, thus underestimating exact results in their place, Philos. Sci. 72(5):723–738, the percent of corrections to substantive articles. In 2005. Table 3, stealth corrections are documents whose [13] S. E. Toulmin, Return to Reason, Harvard Univ. Press, Cambridge, MA, 2001. abstracts contain “we (also) correct” (not preceded [14] K. Davey, Is mathematical rigor necessary in physics? by “after”, “if”, or “when”, and not succeeded by Brit. J. Philos. Sci. 54(3):439–463, 2003. “for”) and which are not already identified as (1) [15] F. Quinn, A revolution in mathematics? What really errata or (2) comments. happened a century ago and why it matters today, The source for Figure 5 is Zentralblatt MATH Notices Amer. Math. Soc. 59(1):31–37, 2012. [8]. Corrections are found as for Scopus, except [16] W. G. Vincenti and D. Bloor, Boundaries, contingen- only titles are searched for (1) errata, and (3) cies and rigor: Thoughts on mathematics prompted by a case study in transonic aerodynamics, Soc. Stud. other omits “inconsistent”, which appears with a Sci. 33(4):469–507, 2003. technical meaning in subject class 03 logic. The [17] L. Adelman, P. E. Lehner, B. A. Cheikes, and M. F. overlap among the three groups of errors (1–3) Taylor, An empirical evaluation of structured argu- is 0.8 percent. Documents of type “j” (journal mentation using the Toulmin argument formalism, article) have 92 percent of these documents, IEEE T. Syst. Man Cyb. 37(3):340–347, 2007. while type “a” (book article) have 7 percent. In [18] J. F. Grcar, Mathematics turned inside out: The in- Figure 5 the percentages are relative to all “a” or tensive faculty versus the extensive faculty, Higher Education 61(6):693–720, 2011. “j” documents for the leading two digits of the [19] M. Atiyah et al., Responses to “Theoretical mathe- primary mathematics subject class. matics”, Bull. Amer. Math. Soc. (N.S.) 30 (2):178–207, There are considerable differences in coverage 1994. between the two databases. Zentralblatt specializes [20] D. H. Bailey and J. M. Borwein, Exploratory experi- in mathematics and finds roughly double the mentation and computation, Notices Amer. Math. Soc. number of mathematics articles for 1990–2010. 58 (10):1410–1419, 2011. Because Scopus classifies journals by subject area, [21] J. Grabiner, Is mathematical truth time-dependent? Amer. Math. Monthly 81 (4):354–365, 1974. it indexes as mathematics some articles which are [22] J. von Neumann, The mathematician, in R. B. Hey- not mathematical. wood, editor, The Works of the Mind, Univ. of Chicago Press, Chicago, IL, 1947, pp. 180–196. Acknowledgments [23] I. Grattan-Guinness, The mathematics of the past: I thank the reviewers for interesting comments Distinguishing its history from our heritage, Historia Math. 31 (2):163–185, 2004. and the editor for assembling the panel. Their [24] J. Dieudonné, The work of Nicolas Bourbaki, Amer. handiwork is clear, if only to themselves. Math. Monthly 77:134–145, 1970. [25] J. Barrow-Green, From cascades to calculus: Rolle’s References theorem, in E. Robson and J. A. Stedall, editors, The Ox- [1] G. Naik, Mistakes in scientific studies surge, The Wall ford Handbook of the History of Mathematics, Oxford Street Journal CCLVIII(34):A1, A12, Aug. 10, 2011. University Press, Oxford, UK, 2009, pp. 737–754. [2] , SciVerse Scopus, website, 2011. [26] P. J. Davis, Fidelity in mathematical discourse: Is [3] I. Newton, Philosophiae Naturalis Principia Mathe- one and one really two? Amer. Math. Monthly 79 (3): matica (Mathematical Principles of Natural Science), 252–263, 1972. William and John Innys, London, 3rd edition, 1726. [27] A. Aberdein, Mathematics and argumentation, Found. [4] J. Henry, “Mathematics made no contribution to the Sci. 14(1–2):1–8, 2009. public weal”: Why Jean Fernel (1497–1558) became a [28] A. Pease, A. Smaill, S. Colton, and J. Lee, Bridging physician, Centaurus 53(3):193–220, 2011. the gap between argumentation theory and the philos- [5] S. M. Stigler, The History of Statistics: The Measure- ophy of mathematics, Found. Sci. 14 (1–2):111–135, ment of Uncertainty before 1900, 2009. Press, Cambridge, MA, 1986.

424 Notices of the AMS Volume 60, Number 4 About the Cover [29] S. Nasar and D. Gruber, Manifold destiny, New Visualizing the impact of humans on the planet Yorker, August 28, 2006. [30] C. Hermite, Lettres de Charles Hermite à Gösta Mittag- April is Mathematics Awareness Month, and this year’s Leffler (1884–1891), Cahiers du séminaire d’histoire theme is the application of mathematics to problems des mathématiques 6:79–217, 1985. of sustainable development. One of the most inter- [31] M. Atiyah, Thoughts of a mathematician (book esting ways in which mathematics comes into this review), Brain 131 (4):1156–1160, 2008. is in the processes by which one keeps track of the [32] C. Reid, Hilbert, Springer-Verlag, ,1970. success or failure of attempts at sustainability. The [33] R. O. Frost, P. Marten, C. Lahart, and R. Rosen- most impressive projects of this kind these days are blate, The dimensions of perfectionism, Cognitive carried out from space, and largely through the Na- Ther. Res. 14 (5):449–468, 1990. [34] G. A. S. Tofaha and P. R. Ramón, Perfectionism tional Aeronautics and Space Administration (NASA). and self concept among primary school children in The cover image has been taken from the NASA Egypt, Electronic Journal of Research in Educational web page Psychology 8 (3):1099–1114, 2010. http://earthobservatory.nasa.gov/ [35] T. Hill, Hoisting the black flag (letter), Notices Amer. Features/NightLights/page2.php Math. Soc. 57 (1):7, 2010. [36] M. Lecat, Erreurs de mathématiciens des origines à and shows dramatically how civilization has extended nos jours, Castaigne, Brussels, 1935. itself over the western hemipshere. Some background [37] J. Barrow-Green, Poincaré and the three body prob- on the technology behind the night-time images can lem, Amer. Math. Soc., Providence, RI, 1997. be found at [38] M. Terrall, editor, Olga Taussky-Todd (1906–1995) Autobiography, California Institute of Technology http://www.ngdc.noaa.gov/dmsp/data/viirs_ Archives, Pasadena, CA, 1980. fire/viirs_html/viirs_ntl.html [39] J.-P. Bourguignon, The role of publications in math- ematical research: A systemic point of view, MSRI The hundreds of photographs that can be found from Workshop on Mathematics Journals, web video, 2011. links on these pages are an amazing resource. [40] I. Newton and D. T. Whiteside, The Mathematical A more detailed analysis of the effect of human Papers of Isaac Newton: 1697–1722, volume 8 of The development on the Earth can be found by searching Mathematical Papers of Sir Isaac Newton, Cambridge the NASA site for “human footprint”—perhaps coming Univ. Press, Cambridge, UK, 2008. up with the following map: [41] D. Scott-Lichter, editor, CSE’s White Paper on Promoting Integrity in Scientific Journal Publications, Council of Science Editors, Editorial Policy Committee, Wheat Ridge, CO, 2009. [42] Amer. Math. Soc., Manual for Journal Editors, web document, 2011. [43] International Math. Union, Best Current Practices for Journals, web document, 2010. [44] Amer. Math. Soc., Ethical Guidelines, web document, 2005. [45] A. J. Hildebrand, Getting Published in Mathematics, Univ. of Illinois web document, 2009. [46] P. R. Halmos, I Want to Be a Mathematician, Springer- Verlag, New York, 1985. [47] S. G. Krantz, How to write your first paper, Notices Amer. Math. Soc. 54 (11):1507–1511, 2007. [48] A. R. Young, Publishing research articles, in A. R. Young, J. Duckett, and P. Graham, editors, Publishing Each individual biome is rated on its own scale 0–100, in Politics, Political Studies Association, 2nd edition, with maximum 100. (This is cropped from the original.) 2010. [49] D. Trafimow and S. Rice, What if social scientists had The night lights images are the output of a mas- reviewed great scientific works of the past? Perspect. sive quantity of data transmitted by satellites, which Psychol. Sci. 4 (1):65–78, 2009. is then massaged for the Earth Observatory website [50] G. Naik, Scientists’ elusive goal: Reproducing study by the data visualization group at the Goddard Space results, The Wall Street Journal CCLVIII (130):A1, A16, Dec. 2, 2011. Center. We thank the group’s manager, Robert Sim- mon, for spending valuable time answering our ques- Editor’s Note: Credit for Figures 1, 3, and 4 is tions. ScienceCartoonsPlus.com. Tracking all the ways in which mathematics is applied to produce these images could be a valuable exercise. —Bill Casselman, Graphics Editor ([email protected])

April 2013 Notices of the AMS 425 Updated version available at 2012Survey-FacultySalaries-Corrected.pdf 2012-2013 Faculty Salaries Report

Richard Cleary, James W. Maxwell, and Colleen Rose

This report provides information on the distribution of 2012–2013 academic-year salaries for tenured and tenure-track faculty at four-year mathematical sciences departments in the U.S. by the departmental groupings used in the Annual Survey. (See page 431 for the definitions of the various departmental groupings.) Salaries are described separately by rank. Salaries are reported in current dollars (at time of data collection). Results reported here are based on the departments which responded to the survey with no adjustment for non-response. Departments were asked to report for each rank the number of tenured and tenure-track faculty whose 2012–2013 academic-year salaries fell within given salary intervals. Reporting salary data in this fashion ensures confidentiality of individual reponses, though it does mean that the reported quartiles are only approximations. The quartiles reported have been estimated assuming that the density over each interval is uniform. When comparing current and prior year figures, one should keep in mind that differences in the set of responding departments may be one of the most important factors in the change in the reported mean salaries. Note this report uses the new groupings of doctoral-granting mathematics departments recently adopted by the Joint Data Committee. Additional detail is provided on page 431. Math. Public Large Group Faculty Salaries Doctoral degree-granting departments of mathematics 23 responses out of 26 departments (88%) 2012–13 2011–12

ϳϬ Rank No. Reported Q1 Median Q3 Mean Mean New-Hire Asst Prof 26 78,000 83,400 90,700 83,242 80,476 ϲϱ Assistant Professor* 143 75,200 80,900 85,500 80,827 79,764 ϲϬ Associate Professor 222 78,300 86,600 94,200 87,563 86,565 Full Professor 776 105,000 124,400 151,300 131,935 127,674 ϱϱ EĞǁͲ,ŝƌĞƐƐƚ ϱϬ this is an ƐƐŝƐƚĂŶƚ ϰϱ example of a ƐƐŽĐŝĂƚĞ ϰϬ ϯϱ pull quote &Ƶůů ϯϬ Ϯϱ ϮϬ ϭϱ

Percent of Total Faculty within Rank ϭϬ ϱ  Ϭ <50 >220 50–60 60–70 70–80 80–90 90–100 210-220 200-210 190-200 110–120 120–130 100–110 130–140 140–150 150–160 160–170 170–180 180–190 2012–13 Academic-Year Salaries (in thousands of dollars) *Includes new hires.

Richard Cleary is a professor in the Department of Mathematical Sciences at Bentley University. James W. Maxwell is AMS associate executive director for special projects. Colleen A. Rose is AMS survey analyst.

426 Notices of the AMS Volume 60 Number 4 2012 Annual Survey of the Mathematical Sciences in the U.S.

Math. Public Medium Group Faculty Salaries Doctoral degree-granting departments of mathematics 36 responses out of 40 departments (90%) 2012–13 2011–12

Rank No. Reported Q1 Median Q3 Mean Mean ϳϬ New-Hire Asst Prof 44 70,300 72,700 75,900 72,284 71,892 Assistant Professor* 215 67,400 72,600 77,200 71,858 71,285 ϲϱ Associate Professor 316 70,700 77,100 84,800 78,525 76,915 ϲϬ Full Professor 685 88,700 102,100 121,800 108,387 105,851 ϱϱ ϱϬ EĞǁͲ,ŝƌĞƐƐƚ ϰϱ ƐƐŝƐƚĂŶƚ ϰϬ ƐƐŽĐŝĂƚĞ ϯϱ &Ƶůů ϯϬ Ϯϱ ϮϬ ϭϱ Percent of Total Faculty within Rank ϭϬ  ϱ Ϭ <50 >220 50–60 60–70 70–80 80–90 90–100 210-220 200-210 190-200 110–120 120–130 100–110 130–140 140–150 150–160 160–170 170–180 180–190 2012–13 Academic-Year Salaries (in thousands of dollars)

Math. Public Small Group Faculty Salaries Doctoral degree-granting departments of mathematics 54 responses out of 64 departments (84%) 2012–13 2011–12

Rank No. Reported Q1 Median Q3 Mean Mean ϳϬ New-Hire Asst Prof 58 62,500 67,500 72,400 67,043 65,012 ϲϱ Assistant Professor* 269 64,100 65,300 70,700 65,570 64,473 Associate Professor 368 65,800 71,400 77,900 72,416 72,798 ϲϬ Full Professor 618 83,000 91,800 106,100 96,630 97,712 ϱϱ EĞǁͲ,ŝƌĞƐƐƚ ϱϬ ƐƐŝƐƚĂŶƚ ϰϱ ƐƐŽĐŝĂƚĞ ϰϬ &Ƶůů ϯϱ ϯϬ Ϯϱ ϮϬ ϭϱ

Percent of Total Faculty within Rank ϭϬ ϱ  Ϭ <50 >220 50–60 60–70 70–80 80–90 90–100 210-220 200-210 190-200 110–120 120–130 100–110 130–140 140–150 150–160 160–170 170–180 180–190 2012–13 Academic-Year Salaries (in thousands of dollars) *Includes new hires.

April 2013 Notices of the AMS 427 2012 Annual Survey of the Mathematical Sciences in the U.S.

Math. Private Large Group Faculty Salaries Doctoral degree-granting departments of mathematics 17 responses out of 24 departments (71%) 2012–13 2011–12

Rank No. Reported Q1 Median Q3 Mean Mean ϳϬ New-Hire Asst Prof 20 81,800 86,500 93,100 88,080 77,800 ϲϱ Assistant Professor* 62 82,100 87,100 93,800 88,833 82,440 Associate Professor 53 87,500 97,500 108,200 98,138 97,809 ϲϬ Full Professor 272 122,000 146,100 177,700 151,314 148,074 ϱϱ EĞǁͲ,ŝƌĞƐƐƚ ϱϬ ƐƐŝƐƚĂŶƚ ϰϱ ƐƐŽĐŝĂƚĞ ϰϬ &Ƶůů ϯϱ ϯϬ Ϯϱ ϮϬ ϭϱ Percent of Total Faculty within Rank ϭϬ ϱ  Ϭ <50 >220 50–60 60–70 70–80 80–90 90–100 210-220 200-210 190-200 110–120 120–130 100–110 130–140 140–150 150–160 160–170 170–180 180–190 2012–2013 Academic-Year Salaries (in thousands of dollars)

Math. Private Small Group Faculty Salaries Doctoral degree-granting departments of mathematics 23 responses out of 28 departments (82%) 2012–13 2011–12 Rank No. Reported Q Median Q Mean Mean ϳϬ 1 3 New-Hire Asst Prof 22 66,700 77,500 86,700 75,722 75,647 ϲϱ Assistant Professor* 89 68,600 76,400 83,000 75,316 71,477 Associate Professor 119 75,900 83,300 90,400 83,289 77,184 ϲϬ Full Professor 231 99,300 114,200 136,700 120,335 117,008 ϱϱ EĞǁͲ,ŝƌĞƐƐƚ ϱϬ ƐƐŝƐƚĂŶƚ ϰϱ ƐƐŽĐŝĂƚĞ ϰϬ &Ƶůů ϯϱ ϯϬ Ϯϱ ϮϬ ϭϱ Percent of Total Faculty within Rank ϭϬ ϱ Ϭ  <50 >220 50–60 60–70 70–80 80–90 90–100 210-220 200-210 190-200 110–120 120–130 100–110 130–140 140–150 150–160 160–170 170–180 180–190 2012–13 Academic-Year Salaries (in thousands of dollars)

*Includes new hires.

428 Notices of the AMS Volume 60, Number 4 2012 Annual Survey of the Mathematical Sciences in the U.S.

Applied Mathematics Group Faculty Salaries Doctoral degree-granting departments of Applied Mathematics 19 responses out of 30 departments (63%) 2012–13 2011–12

Rank No. Reported Q1 Median Q3 Mean Mean ϳϬ New-Hire Asst Prof 14 67,500 80,800 88,300 81,234 none ϲϱ Assistant Professor* 52 62,500 74,300 87,100 75,809 69,083 Associate Professor 84 73,400 88,700 100,400 89,646 100,132 ϲϬ Full Professor 163 101,400 124,000 155,400 130,131 136,465 ϱϱ ϱϬ EĞǁͲ,ŝƌĞƐƐƚ ϰϱ ƐƐŝƐƚĂŶƚ ϰϬ ƐƐŽĐŝĂƚĞ ϯϱ &Ƶůů ϯϬ Ϯϱ ϮϬ ϭϱ ϭϬ Percent of Total Faculty within Rank ϱ  Ϭ <50 >220 50–60 60–70 70–80 80–90 90–100 210-220 200-210 190-200 110–120 120–130 100–110 130–140 140–150 150–160 160–170 170–180 180–190 2012–13 Academic-Year Salaries (in thousands of dollars)

Statistics Group Faculty Salaries** Doctoral degree-granting departments of statistics 26 responses out of 58 departments (44%) 2012–13 2011–12

Rank No. Reported Q1 Median Q3 Mean Mean ϴϱ New-Hire Asst Prof 10 80,700 82,500 84,300 77,559 83,637 ϴϬ Assistant Professor* 88 78,500 82,200 86,600 82,068 79,749 Associate Professor 109 81,800 88,700 97,100 89,850 88,937 ϳϱ Full Professor 205 108,300 134,500 155,800 137,037 137,922 ϳϬ ϲϱ EĞǁͲ,ŝƌĞƐƐƚ ϲϬ ƐƐŝƐƚĂŶƚ ϱϱ ƐƐŽĐŝĂƚĞ ϱϬ &Ƶůů ϰϱ ϰϬ ϯϱ ϯϬ Ϯϱ ϮϬ ϭϱ Percent of Total Faculty within Rank ϭϬ ϱ  Ϭ <50 >220 50–60 60–70 70–80 80–90 90–100 210-220 200-210 190-200 110–120 120–130 100–110 130–140 140–150 150–160 160–170 170–180 180–190 2012–13 Academic-Year Salaries (in thousands of dollars) *Includes new hires. **Faculty salary data provided by the American Statistical Association.

April 2013 Notices of the AMS 429 2012 Annual Survey of the Mathematical Sciences in the U.S.

Biostatistics Group Faculty Salaries** Doctoral degree-granting departments of biostatistics 20 responses out of 35 departments (56%) 2012–13 2011–12

Rank No. Reported Q1 Median Q3 Mean Mean ϳϬ New-Hire Asst Prof 13 75,800 83,300 111,700 88,362 81,398 ϲϱ Assistant Professor* 98 75,800 81,300 88,200 83,548 77,412 Associate Professor 102 90,200 96,900 107,200 100,565 95,844 ϲϬ Full Professor 187 118,200 139,500 175,000 128,519 143,059 ϱϱ ϱϬ EĞǁͲ,ŝƌĞƐƐƚ ϰϱ ƐƐŝƐƚĂŶƚ ϰϬ ƐƐŽĐŝĂƚĞ ϯϱ &Ƶůů ϯϬ Ϯϱ ϮϬ ϭϱ

Percent of Total Faculty within Rank ϭϬ ϱ  Ϭ <50 >220 50–60 60–70 70–80 80–90 90–100 210-220 200-210 190-200 110–120 120–130 100–110 130–140 140–150 150–160 160–170 170–180 180–190 2012–13 Academic-Year Salaries (in thousands of dollars)

Masters Group Faculty Salaries Master's degree-granting departments of mathematics 104 responses out of 180 departments (58%) 2012–13 2011–12

Rank No. Reported Q1 Median Q3 Mean Mean New-Hire Asst Prof 84 54,600 60,200 68,600 60,355 54,045 Assistant Professor* 429 53,400 58,700 65,700 59,606 56,886 Associate Professor 673 62,100 68,800 77,200 70,152 68,866 Full Professor 791 78,400 89,500 101,600 91,240 88,432 Percent of Total Faculty within Rank <40 >150 40–45 45–50 50–55 55–60 60–65 65–70 70–75 75–80 80–85 85–90 90–95 95–100 120–125 115–120 125–130 100–105 105–110 110–115 130–135 135–140 140–145 145–150 2012–13 Academic-Year Salaries (in thousands of dollars) *Includes new hires. **Faculty salary data provided by the American Statistical Association.

430 Notices of the AMS Volume 60, Number 4 2012 Annual Survey of the Mathematical Sciences in the U.S.

Bachelors Group Faculty Salaries Bachelor's degree-granting departments of mathematics 303 responses out of 904 departments (34%) 2012–13 2011–12

Rank No. Reported Q1 Median Q3 Mean Mean New-Hire Asst Prof 127 38,600 44,800 54,200 56,217 54,968 Assistant Professor* 731 39,600 45,200 52,900 56,359 55,686 Associate Professor 759 47,200 55,500 67,400 68,557 68,826 Full Professor 916 60,300 72,500 89,500 87,518 85,588 Percent of Total Faculty within Rank <40 >150 40–45 45–50 50–55 55–60 60–65 65–70 70–75 75–80 80–85 85–90 90–95 95–100 120–125 115–120 125–130 100–105 105–110 110–115 130–135 135–140 140–145 145–150 *Includes new hires. 2012–13 Academic-Year Salaries (in thousands of dollars)

Departmental Groupings

Starting with reports on the 2012 AMS-ASA-IMS-MAA-SIAM Annual Survey of the Mathematical Sciences, the Joint Data Committee has implemented a new method for grouping the doctorate-granting mathematics departments. These departments are first grouped into those at public institutions and those at private institutions. These groups are further subdivided based on the size of their doctoral program as reflected in the average annual number of Ph.D.’s awarded between 2000 and 2010, based on their reports to the Annual Survey during this period. Furthermore, doctorate-granting departments which self-classify their Ph.D. program as being in applied mathematics will join with the other applied mathematics departments previously in Group Va to form their own group. The former Group IV will be divided into two groups, one for departments in statistics and one for departments in biostatistics. For further details on the change in the doctoral department groupings see the article in the October 2012 issue of Notices of the AMS at http:www.ams.org/notices/201209/rtx120901262p.pdf.

Math. Public Large consists of departments with the highest annual rate of production of Ph.D.’s, with at least 7.0 per year. Math. Public Medium consists of departments with an annual rate of production of Ph.D.’s, ranging between 3.9 and 6.9 per year. Math. Public Small consists of departments with an annual rate of production of Ph.D.’s, with 3.8 per year or fewer. Math. Private Large consists of departments with an annual rate of production of Ph.D.’s, with at least 3.9 per year. Math. Private Small consists of departments with an annual rate of production of Ph.D.’s, with 3.8 per year or fewer. Applied Mathematics consists of doctoral degree granting applied mathematics departments. Statistics consists of doctoral degree granting statistics departments. Biostatistics consists of doctoral granting biostatistics departments. Group M contains U.S. departments granting a master’s degree as the highest graduate degree. Group B contains U.S. departments granting a baccalaureate degree only.

Listings of the actual departments which compose these groups are available on the AMS website at www.ams.org/annual- survey/groups.

April 2013 Notices of the AMS 431 2012 Annual Survey of the Mathematical Sciences in the U.S.

Obtain a Special Faculty Salaries Analysis

See how the salaries of your department's tenured/tenure-track faculty compare to those in similar departments. The only requirement is that your department must have responded to our latest Faculty Salary survey. Send a list of your peer institutions (a minimum of 12 institutions is required) to [email protected] along with the date the analysis is needed. (If not enough of your peer group have responded to the salary survey you'll be asked to provide additional institutions.) A minimum of two weeks is needed to complete a special analysis. The analysis produced includes a listing of your peer group institutions with along their salary survey response status, a summary table including the rank (assistant, associate, and full professor), the number reported in each rank, the 1st quartile, median, 3rd quartile, and mean salaries for each along with bar graphs.

Acknowledgments About the Annual Survey

The Annual Survey attempts to provide an accurate The Annual Survey series, begun in 1957 by appraisal and analysis of various aspects of the the American Mathematical Society, is currently academic mathematical sciences scene for the use under the direction of the Data Committee, a and benefit of the community and for filling the joint committee of the American Mathematical information needs of the professional organizations. Society, the American Statistical Association, the Every year, college and university departments in Mathematical Association of America, and the the United States are invited to respond. The Annual Society of Industrial and Applied Mathematics. Survey relies heavily on the conscientious efforts of The current members of this committee are Pam the dedicated staff members of these departments Arroway, Richard Cleary (chair), Steven R. Dunbar, for the quality of its information. On behalf of the Sue Geller, Boris Hasselblatt, Ellen Kirkman, Peter Data Committee and the Annual Survey Staff, we March, David R. Morrison, James W. Maxwell (ex thank the many secretarial and administrative staff officio), Bart S. Ng, and William Velez The committee members in the mathematical sciences departments is assisted by AMS survey analyst Colleen A. Rose. for their cooperation and assistance in responding to In addition, the Annual Survey is sponsored by the the survey questionnaires. Institute of Mathematical Statistics. Comments or suggestions regarding this Survey Report may be emailed to the committee at [email protected].

432 Notices of the AMS Volume 60, Number 4 AMERICAN MATHEMATICAL SOCIETY

NEW!

Electronic Mathematical Reviews® (eMR) Sections

You can now stay informed of new research published in your subject area by subscribing to the electronic Mathematical Reviews® (eMR) Section of your choice. Subject areas are assigned using MSC 2010. eMR Sections Provide: • Improved accessibility and portability, as eMR Section issues will be posted monthly and available as downloadable PDFs MathSciNet Mathematical Reviews • New content notifi cation via RSS feed • Perpetual access • Discounted pricing for MR Reviewers and AMS individual members • Low cost option for researchers and faculty whose institutions don’t currentlyMathSciNet subscribe to the Mathematical Reviews full database, MathSciNet

For ordering information, please visit: MathSciNet www.ams.org/bookstore/emrsections Mathematical Reviews What Is New in LATEX? VI. LATEX on an iPad. Empire

G. Grätzer

Introduction In the first part of this article, Foundation (Notices, March 2013), I hope I convinced you that the iPad is a computer. Now we proceed to discuss some LATEX implementations on this computer.

With Two Hands behind Your Back To implement LATEX on an iPad, two major man- made obstacles have to be overcome: Sandboxing and the GPL license.

Sandboxing

We discussed this topic in Foundation. Does it Richard Stallman impact LATEX implementations? You bet. For instance: The LATEX implementation Texpad on the Mac is given a single LATEX root file. It then reads through the LATEX source, gets all the included iPhone was released. Many software developers use files, and presents you with an outline of your GPL to ensure the free distribution of their software project. Sandboxing would not allow this. The (source code and executable) under reasonable handling of the auxiliary files also poses a problem. terms. Of course, these problems can be overcome by ingenious programmers. Some software developers seem not to be aware of the fact that GPL-licensed software cannot be GPL used in an app created for the iPad. Two well- Richard Stallman,1 of Emacs fame, started the GNU known developers explained to me that they use operating system in 1983. Soon after, he started GPL because their peers do. Both would like to get a nonprofit corporation called the Free Software out of it but do not know how. Foundation. How ironic: the license that was supposed Stallman wrote, with the assistance of some law to allow you to spread your free software to professors, the General Public License (GPL)—the wherever it is needed now stops you from having most widely used free software license—released in it used on the fastest-growing platform of all time. 1989. Version 3 is dated June 29, 2007, the day the The 28 Flavors… G. Grätzer is Distinguished Professor of Mathematics, re- tired, at the University of Manitoba. His email address is The iPad LATEX implementations differ, naturally, [email protected]. in almost all aspects, but we can get a crude 1Picture taken from his personal website. classification by specifying how they obtain the DOI: http://dx.doi.org/10.1090/noti963 LATEX files and how they typeset them.

434 Notices of the AMS Volume 60, Number 4 Where Are You Going? The LATEX source files, of course, can always be composed in the app, but typically you already have them. You can obtain your existing source files in two ways:

1. Using iTunes. To transfer files—one at a time— to your app from your computer using iTunes, connect your iPad to your computer and start iTunes by double-clicking on its icon. Under Devices, select the iPad from the left side of the iTunes window; see Figure 1. At the top of the iTunes window, next to Summary and Info, select Apps; see Figure 2. Figure 1. Under Devices, we selected the iPad. The lower part of the window now has File Sharing; see Figure 3. On the left you see a listing of the apps available for file transfer. Select the app; the files already in the app are then listed in the right pane. Click on the add button, and a file browser appears. Choose the file you want to transfer.

2. Via Dropbox. I assume that you have the ubiq- uitous Dropbox (the application that keeps your files safe and up to date across multiple devices Figure 2. Choose Apps. and platforms). For an introduction, go to dropbox.com/gs. In the app you sign in to Dropbox. Now the app can see the contents of your Dropbox or some part of it (at the Dropbox server) as long as you have an Internet connection.

And What Do You Wish? The app may typeset the LATEX file in the following ways:

A. On Your iPad. This is the “post-PC revolution” option: the app places a LATEX distribution on the iPad and you typeset with it. No computer or Figure 3. Select app. Internet connection is required. However, a complete LATEX distribution is about 4 GB! No app can be this big. So you only get a What Do You Get? minimal LATEX distribution and no special fonts! Originally, LATEX output was a dvi file. These days, utilizing pdftex (under GPL license) by B. On Your Computer via Dropbox. This is the most Hàn Thê´ Thành, the output is PDF. Since developers powerful option. You have all the packages and could not use GPL-d code, the output was dvi. fonts on your computer available to you. An These days, even on the iPad, PDF rules. In a more app (such as AutomaTeX by Jonathan Weisberg) perfect world, these talented developers would monitors if there is any change in the LAT X file in E not have to spend so much time reinventing GPL-d Dropbox. If there is, the file is retypeset and the wheels. PDF is made available to you via Dropbox.

C. In the Cloud. This option provides you with a Keyboard or Not to Keyboard… remote server, the Cloud; you connect to it with In Figure 4 you see editing with the iPad’s soft Wi-Fi. The server has a full LATEX implementation, keyboard (notice the extra row of LATEX keys so you miss only the special fonts. And, of course, added by the LATEX implementation Texpad) and in you must have Wi-Fi to use it. So you cannot polish Figure 5, editing with a Bluetooth keyboard (notice up your lecture on the airplane on the way to a that the extra row of LATEX keys of Texpad is still meeting. present).

April 2013 Notices of the AMS 435 Texpad Files: 2. Typesetting: A, B, C. Documentation: Excellent and detailed on the iPad interface. It is available as a help file and also at http://texpadapp.com/app-help-files/ios/help.html. Samuel Johnson famously recommended that the introduction of a book should set out the goal, explain why it cannot be done, and then proceed to accomplish it. Valletta Ventures’s Duncan P. Steele and Jawad A. Deo took this to heart. They wrote blogs explaining why there can be no LATEX on the iPad and then proceeded to implement it. Texpad is a LATEX implementation for the Mac and for the iPad. It has some interesting features, including: • Autocompletion of all common commands and autofilling \cite-s and \ref-s. • Replacement of the LATEX console with a list of errors and warnings linked to the source. • Global search, outline view, and syntax highlight. Step 1. To get started with Texpad, go to the iPad App Store and install Texpad. Step 2. Now open Texpad. Figure 6 shows Texpad at the first startup. Figure 4. Editing with soft keyboard. The Help button gets the help file. Step 3. Touch Off to turn Dropbox on. Sign up for Dropbox with the same email address and password as for your computer’s Dropbox. (If you have Dropbox installed and connected, it’s even simpler; you just have to Allow the connection.) Your File Storage now gives two options: iPad and Dropbox; see Figure 7. It is important to understand that your LATEX files will live in the Dropbox (in the Cloud, at the Dropbox server) or locally on your iPad. Step 4. The Dropbox files are now available to you by touching Dropbox under File Storage; see Figure 7. • First, create a folder for the LATEX files to be transferred. Navigate to iPad file storage. Touch the + in the bottom right and choose Folder. Name the folder. • Second, navigate to the Dropbox file system view and to the folder containing the file you want to copy. Touch Edit. Select the file to transfer. At the bottom center, touch Copy. Navigate to the folder into which you want to copy the file and touch Copy. Step 5. Typesetting will take place either on the iPad or in the Cloud. Go to the folder of a LATEX file, touch the file (on Figure 5. Editing with Bluetooth keyboard. the iPad or in the Dropbox), and typeset it on the iPad (touch Local Typeset) or in the Cloud, that is, at Valletta’s server (touch Cloud Typeset). A Two “Post-PC Revolution” LTEX Step 6. Try to visualize what is happening. Implementations 6a. If you typeset on the iPad and the file is on We now discuss the two LATEX implementations the iPad, it just typesets locally; that is it. that typeset on the iPad.

436 Notices of the AMS Volume 60, Number 4 Figure 6. Texpad first startup. Figure 8. Organizer window.

Step 8. You’ve edited and typeset your LATEX file. You want to get to another file. Touch the organize button (the folder icon in the upper left). You get the organizer window; see Figure 8. Touch the button in the upper left of the window. You get back to Dropbox, eventually to the expanded File Storage of Figure 7.

Figure 7. Expanded File Storage. **********

These eight steps should be enough to get you 6b. If you typeset on the iPad and the file is in started. Read the Help file for more information. Dropbox, then the file is transferred to the iPad, typeset, and the resulting PDF is sent back to the TeX Writer Dropbox; nothing is kept on the iPad. Files: 2. Typesetting: A. 6c. If you typeset in the Cloud and the file is Documentation: The file readme.pdf is no quick in Dropbox, the file is transferred to the Cloud, start, but it is useful for understanding how TeX typeset, and the resulting PDF is sent back to Writer works and how to customize it. Dropbox; nothing is kept in the Cloud. TeX Writer was the first to typeset on the iPad. 6d. If you typeset in the Cloud and the file is It could typeset only TeX files. Now it has LATEX and on the iPad, the file is transferred to the Cloud, the AMS packages on board. typeset, and the resulting PDF is sent back to the Step 1. When you start up TeX Writer, first link iPad; nothing is kept in the Cloud. to Dropbox. Step 7. Once you touch a LATEX file, you are ready In TeX Writer you get the display of Figure 9, to edit it. Cursor control is very important. You do showing the source file readme.tex. Pressing the it with a two-finger swipe. Of course, this is not so More icon (right-pointing arrow), you get more important if you use a Bluetooth keyboard; it has icons to read the PDF version or Air Printing cursor keys. But two-finger swipe is faster! readme.pdf.

April 2013 Notices of the AMS 437 On the left is the organize icon; touching it, you get a file listing: readme.tex and readme.pdf. At the bottom is New File; touch it to compose one. Step 2. So you are perplexed about what to do next; you’ve run out of icons. You have to know that TeX Writer accesses Dropbox in a special way. When you connect to Dropbox from TeX Writer, it creates a new folder App in Dropbox. In App it creates the subfolder TeX Writer; see Figure 10. In this subfolder you find readme.tex. Anything you put in the TeX Writer subfolder is visible in the file listing window on the iPad; anything not in this subfolder is not visible to TeX Writer. Step 3. TeX Writer gets your files from this subfolder in Dropbox. Place in there a folder with the files of your current project. These will be available to you on your iPad. Moreover, these files are fully synchronized, so the editing changes you make on your iPad show up in Dropbox. Step 4. LATEXing, you spend most of your time editing. TeX Writer’s editor has some interesting features, for instance, excellent cursor control; touch begin{}, type in the name of the environment, and the environment is placed in your article; undo, redo, search, and so on; see Figure 11. Figure 9. TeX Writer startup. Note that, when typing, you retain the editing functions you see in Figure 9 and, in addition, you get an extra row of LATEX-specific keys. You do not get them with a Bluetooth keyboard; however, the keyboard may have many of these keys. Note that the Log viewer links to error lines.

Other TEX -Related Apps The editor Textastic (see Foundations) has Dropbox connection and it is LATEX aware, so it can be viewed Figure 10. Dropbox folders. as a LATEX implementation. Now we present two LATEX implementations that are not of the “post-PC revolution”.

Tex Touch Files: 2. Typesetting: A, B. Documentation: Three videos and a quick start guide. http://www.vancapy.com/textouch/ Step 1. Sign up at TeX Cloud as instructed. Step 2. Touch Dropbox and sign in for your Dropbox account. Step 3. A window comes up to set the tex folder in Dropbox. Navigate to the folder that has your LATEX project; touch Set TeX folder. Step 4. Touch download for all the files you need. The downloaded files appear in the left pane, each with an upload button. Touch them to upload the files to the TeX Cloud. They appear in the left pane. Touch Done. Step 5. The display of Figure 12 comes back, Figure 11. Typing in TeX Writer. except that now it has a right pane listing the files in the Cloud. Touch your texfile and an important

438 Notices of the AMS Volume 60, Number 4 editor is the standard iPad text editor, so it is not suitable for editing tex files.

*************************************************** There are some more apps that use TEX and LATEX. Here are two:

iTeX In Project Gutenberg1 or arXiv.org,2 select a docu- ment, and iTeX will attempt to write the LATEX code of a nicely formatted LATEX output for the iPad’s size and resolution.

TeX Equation This app typesets math formulas with its built-in TEX engine for use in Pages and Keynote.

Figure 12. Starting up with TeX Touch. Conclusion So this is my new best friend for the road, my iPad with the Ultrathin Keyboard Cover for iPad by Logitech. How many times have you read reviews of words processors concluding with “And I wrote this review with…” Well, I did not. This article uses the Notices style, requiring the Lucida fonts, and font installations are very difficult on the iPad. So LATEXing on an iPad requires some compro- mises. Nevertheless, when not at your desk, the iPad will be nearly as functional as your MacBook Air, and it is so much easier to carry around.

Figure 13. The Dropbox interface. hint comes up: when editing, one finger swipe moves the cursor one character! Then the file comes up for editing. Touch the text anywhere, and the usual iPad keyboard comes up, augmented with a scrolling keyboard row of keys important for LATEX; these symbols are also displayed if you use a Bluetooth keyboard. Step 6. Once you are done editing, touch TeX in the upper right corner, then the TeX! in the upper left corner. The file gets uploaded to the Cloud, typeset in the Cloud, and the PDF file gets My new best friend. downloaded to the iPad. Touching PDF displays the PDF file. Exit by touching Done, then Document. iVerbTeX Files: 1. Typesetting: C. Documentation: None that I could find. iVerbTeX stores your documents locally on your 1An archive of over 40,000 public domain eBooks. iPad or in the Cloud. You edit locally. The text 2An archive of over 800,000 scientific e-prints.

April 2013 Notices of the AMS 439 New!Applied Mathematics Titles 125 YEARS from OF ADVANCING MATHEMATICS

Algebraicand Geometric Ideas in the Theoryof A Primer on MathematicalModels in Biology DiscreteOptimization Lee A. Segel and Leah Edelstein-Keshet Jesús A. De Loera, This textbook grew out of a course that the popular and Raymond Hemmecke, highly respected applied mathematician Lee Segel taught. and Matthias Köppe It introduces differential equations, biological applications, MOS-SIAM Series on Optimization 14 and simulations and emphasizes molecular events, excitable This book presents recent systems, and small protein and genetic circuits. It is intended advances in the mathematical for upper-level undergraduates in mathematics, graduate theory of discrete optimization. students in biology, and lower-level graduate students It offers several research in mathematics who would like exposure to biological technologies not yet well applications. known among practitioners of 2013 • xxvi + 424 • Softcover • 978-1-611972-49-8 discrete optimization, minimizes List $69.00 • SIAM Member $48.30 • OT129 prerequisites for learning these methods, and provides a ComputationalMathematical Modeling: transition from linear discrete An Integrated Approach Across Scales optimization to nonlinear discrete Daniela Calvetti and Erkki Somersalo optimization. Mathematical Modeling and Computation 17 2012 • xx + 322 pages • Softcover This textbook concentrates on two modeling paradigms: the 978-1-611972-43-6 • List $109.00 macroscopic, in which the authors describe phenomena in SIAM/MOS Member $76.30 • MO14 terms of time evolution via ordinary differential equations, ApproximationTheory and the microscopic, which requires knowledge of random and Approximation events and probability. The text emphasizes the development Practice of computational skills to construct predictive models and Lloyd N. Trefethen analyze the results. 2012 • xii + 222 pages • Softcover • 978-1-611972-47-4 In a book that will appeal to List $69.00 • SIAM Member $48.30 • MM17 beginners and experts alike, Oxford University’s MathematicalModels presents approximation theory for CommunicableDiseases using a fresh approach for this Fred Brauer and Carlos Castillo-Chavez established field. This textbook CBMS-NSF Regional Conference Series in Applied Mathematics 84 differs fundamentally from other works on approximation This graduate-level monograph appeals to readers interested theory in a number of ways: its in the mathematical theory of disease transmission models. emphasis is on topics close to The book provides insight into modeling cross-immunity numerical algorithms; concepts between different disease strains and the synergistic are illustrated with Chebfun; and interactions between multiple diseases; diseases transmitted each chapter is a PUBLISHable by viral agents, bacteria, and vectors; and both epidemic and Matlab® M-file, available online. endemic disease occurrences. 2012 • xiii + 305 pages • Softcover 2012 • xviii + 270 pages • Softcover • 978-1-611972-41-2 978-1-611972-39-9 • List $49.00 List $77.00 • SIAM/CBMS Member $53.90 • CB84 SIAM Member $34.30 • OT128 All prices are in US dollars.

To ORDER Please mention keycode “BNO13” when you order. Order online: www.siam.org/catalog • Use your credit card (AMEX, MasterCard, or VISA): www.ams.org Call SIAM Customer Service at +1-215-382-9800 worldwide or toll free at 800-447-SIAM in USA and Canada; Fax: +1-215-386-7999 • Send check or money order to: SIAM, Dept. BKNO13, 3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688. Society for induStrial and applied MatheMaticS 3/13 Mathematical Methods in the Study of Historical Chronology

Florin Diacu

ome ancient and medieval historical dates historian Ephorus of Cyme (ca. 405–330 BC), who we learn in school may stick in our minds: provided many dates in his work on universal 1193 BC (the fall of Troy), 753 BC (the history. His more famous predecessor Herodotus founding of Rome), AD 1 (the birth of (ca. 484–425 BC) had been far less interested in ), or AD 1066 (the battle of Hastings), placing events in time. Sextus Julius Africanus Sto mention just a few. Most of us take these dates (ca. AD 160–240), often called the first true for granted, but are they correct? Anatoli Fomenko, chronologist, is considered the father of biblical a fellow of the Russian Academy of Sciences, history, an independent field from which the finds them fictitious and promotes an alternative science of historical dating would later borrow its chronology that often differs from tradition by principles. In his work Chronologia, he put together more than a millennium [1]. dates from Hebrew, Egyptian, Greek, Persian, and The concerns about the correctness of ancient other sources by connecting various pieces of and medieval dates are not new. Isaac Newton also information, as can be noted in the example below opposed them. In his last book, which he finished [15]: preparing for publication just before his death, he If one computes backwards from the end argued that ancient Greek history should be about of captivity, there are 1,237 years. So, by 300 years shorter than claimed [9]. Since the global analysis, the same period is found to be the chronology is based on the Greek chronology, the first year of the Exodus of Israel under Moses foundation of its edifice appears to be shaky. from Egypt, as from the 55th Olympiad to It is thus natural to ask how historical events Ogygus, who founded Eleusis. And from were dated, why they are controversial, and if here we get a more notable beginning for mathematics might play any role in raising or Attic chronography. solving these problems. To lay the background The work of Julius Africanus was continued by towards throwing some light on these questions, Eusebius, Bishop of Caesarea (ca. AD 260–341), let us first make an incursion into the origins of who came up with tables that compared parallel history’s timetables. The material that follows is major events. For instance, he informs us that based on [1], the second edition of a book originally Jesus was born in the year of the 194th Greek published by Knopf Canada. Olympiad, which took place 2,010 years after the birth of Abraham, in the forty-second year A Brief History of Historical Dating of Augustus’s reign and the twenty-eighth year The first writer whom we know to have been after the Roman subjugation of Egypt, or the year interested in historical chronology is the Greek when Antony and Cleopatra died. All of Eusebius’s Florin Diacu is professor of mathematics at the Pacific In- entries start with “the year of Abraham,” i.e., 2010 stitute for the Mathematical Sciences and University of BC. Unfortunately the calculations are based on the Victoria. His email address is [email protected]. unrealistic life spans mentioned in the Bible, such DOI: http://dx.doi.org/10.1090/noti962 as Adam, 930 years; Noah, 950 years; Abraham,

April 2013 Notices of the AMS 441 175 years; or Moses 120 years, numbers upon Julian epoch to noon on Monday, January 1, 4713 which later chronologists built history. BC, which thus became a convenient reference Among the medieval researchers who followed point for all his computations [13]. A method he this tradition were Martin Luther and Gerardus then devised, based on the twelfth-century work Mercator. But in spite of their contributions, of Roger of Hereford and used extensively by chronology remained a gathering of disparate Scaliger’s follower Dionysius Petavius, was that dates used mainly for religious purposes, such of combined cycles. This method employs the as making sense of the biblical stories or decid- numbers 19, 28, and 15 in terms of congruences: ing Easter Day. The current consensus among 19 stays for the lunar cycle, i.e., the number of researchers, as expressed by Denys Hay, a profes- integer years the moon takes to complete an integer sor at the University of Edinburgh, is that “[i]n number of orbits (235) around Earth; 28 stays for classical antiquity there was virtually no system of the solar cycle, i.e., the minimum number of years chronology available to historians” [7]. after which the Gregorian calendar repeats itself; The breakthrough came in 1583, when the French and 15 stays for the Roman indiction, a taxation scholar Joseph Justus Scaliger published a seven- cycle established in AD 313 and used as late as volume work titled De Emendatione Temporum, the sixteenth century in some places. This method which provided dates for the main historical events assigns to each date in history its Julian count of humankind. This colossal enterprise treats in (the number of years since the Julian epoch). Every detail the astronomical bases of more than fifty Julian count up to 7980 has a unique triplet of calendars. To achieve this feat, Scaliger had taught numbers resulting from the remainder obtained himself thirteen languages, Hebrew and Arabic when dividing the Julian count by 19, 28, and 15. So among them, becoming one of the most respected if a certain event can be associated with the lunar philologists of his time. He embarked on his cycle, the solar cycle, and the Roman indiction, its chronology work aiming to understand the flow year can be determined. of ideas and who had influenced whom in ancient But Scaliger’s most important contributions literature. He ended up sorting out the problems are with understanding calendars, most of which of history, an achievement for which he received were long forgotten during his time. Such studies in 1593 a research professorship at the University are multidisciplinary. To reach the point when of Leiden, where he worked until his death in mathematics can be of any use, he had to first 1609. His second and last treatise was Thesaurus unravel the calendar’s language and the deeper Temporum, a collection and arrangement of all meaning of the nomenclature. Scaliger started available ancient chronological sources. almost from scratch. An example of the difficulties Despite their deficiencies, his historical time- he encountered are made evident in the lines he tables survived and were later used to build all wrote in 1568 prompted by some third century AD Eastern chronologies, including those of India, statements he disagreed with [5]: China, and Japan. Anthony Grafton, Professor of European History at Princeton University, remarked I do not see how the month of April can recently that “[t]he few modern historians who derive its name from aperio [to open, to mentioned Scaliger described him as a brilliant discover]. First of all, since the year initially innovator who created a discipline in the teeth of had only ten months, they must have always ferocious opposition” [6]. No doubt, this was the wandered and had no fixed position in the new science on which history would rest. year…. [In fact] aprilis comes from aper, which is boar. Early Use of Mathematics Once the calendrical language was clear, many Apart from simple arithmetic operations aimed at other difficulties occurred. The rules of the old understanding various calendars, the early use of Roman (pre-Julian) calendars, for instance, often mathematics in chronology was mostly reduced changed according to the interests of the political to the computation of eclipses. Scaliger based leaders. These alterations are not only hard to trace his method on the astronomical and calendrical but are also detrimental in chronology studies information he gathered from ancient sources, because they can lead to misinterpretations that aiming to fix some historical landmarks, which he may give rise to large errors in time. Apart from could then connect to other events. Since total solar such subjective issues, another problem Scaliger eclipses, for instance, could be computed with had to deal with was the type of calendar he tried reasonable accuracy, he interpreted the ancient to decipher: lunar, solar, lunisolar, or of some descriptions of such celestial phenomena to place arbitrary type, most of which required not only them in time. solid knowledge of astronomy, but some algo- Scaliger also intuitively applied congruences rithmic sophistication as well. The computational and the Chinese Remainder Theorem to fix the difficulties are easy to overcome today, so it’s no

442 Notices of the AMS Volume 60, Number 4 wonder that calendrical calculations now belong to the first volume of a book that considered the the realm of computer science [14], but Scaliger had issue by looking at historical solar eclipses [11]. to approach them without proper mathematical Five years later, he came up with a second volume, training, thus having at his disposal only some which approached the problem from the point of rudimentary tools. view of lunar observations. His conclusion was that the behavior of D00 could be explained only Modern Developments by factoring in some unknown forces [12]. Scaliger’s work was attacked from the very begin- Fomenko found the idea of unknown forces ning, but apart from being subject to fine tuning, it outlandish, so he used his own chronology to redate survived unscathed. Historians based their theories Robert Newton’s astronomical records, which led on it, and today every student of history takes it for to the conclusion that D00 was almost constant granted. Astronomers all did their best to support in time [2]. His result was in agreement with it with observational and computational data, and Newtonian gravitation, according to which the other scholars studied the match between dates and the information that can be gathered from rotation of the Earth around its axis slows down 00 ancient horoscopes (more about them later). when D decreases. Among the detractors of Scaliger’s chronology Robert Newton either ignored Fomenko’s results are a few scientists and mathematicians, such or never learned about them. For the rest of his as Isaac Newton and Anatoli Fomenko, but also life, he continued to present evidence for the many cranks, the most famous being Immanuel unpredictable changes of the moon’s acceleration. Velikovsky, who made a fortune from the books he Among the potential factors that change the values wrote on this topic in the 1950s [1]. The latter give of D00, he suggested the Earth’s magnetic force, a bad name to anyone who opposes the traditional the tidal friction between water and sea bottoms, view, so neither Newton nor Fomenko have been the growth of the Earth’s core, and the withdrawal free of harsh criticism. Fomenko, especially, has of the ice caps, but he offered no computations many opponents, who often find good ammunition towards proving their influence on the behavior of against his conclusions. Nevertheless, he and D00. his collaborators introduced new mathematical Fomenko and Newton approached the problem methods in the modern study of chronology. We from opposite points of view. The former doubted will further present some of them, leaving it to the the date of every eclipse, whereas the latter reader to pass judgment about their merit. accepted them all, going as far as to disagree The Moon’s Acceleration with the descriptions of the ancient observers, trusting only a few of the 370 cases he studied: Understanding the moon’s orbit around Earth is a difficult mathematical problem. Isaac Newton “We have found too many instances of an eclipse was the first to consider it, and it took more than that could not possibly have been total but that two centuries until the American mathematician was so recorded, sometimes in a quite picturesque found a suitable framework in manner,” he wrote in the second volume of his which to address this question. Still, the moon’s book. Fomenko, instead, trusted the word of orbit is not fully explained today, and Fomenko the observers, refusing to accept the existence was dealing with some details at the end of the of mysterious forces. A closer look at the data, 1970s. however, shows that Fomenko’s graph after AD 00 He was concerned with the acceleration, D , of 900 is similar to Newton’s. In the middle period, for the moon’s elongation, which is the angle between which Newton found a sharp drop of D00, Fomenko the moon and the sun as viewed from Earth. This obtained results he deemed unreliable. The most 00 acceleration D is computable from observations, ancient period vanished, because he shifted the and its past behavior can be determined from chronology forward in time. records of eclipses. Its values vary between −18 So, if we ignore the period before AD 900, there and +2 seconds of arc per century squared. Also, is not much difference between the results of D00 is slightly above zero and almost constant from Newton and Fomenko. The change of chronology about 700 BC to AD 500, but it drops significantly for the next five centuries, to settle at around −18 has not led to a straight line starting with antiquity after AD 1000. Unfortunately this variation cannot but has only eliminated the data before AD 500 and be explained from gravitation, which requires the cast a doubt upon the information between 500 and graph to be a horizontal line. 900. Newton’s results can be interpreted similarly: Among the other experts in celestial mechanics if we exclude the possibility of mysterious forces, who attacked this problem was Robert Newton from his graph puts traditional ancient and medieval Johns Hopkins University. In 1979, he published chronology in doubt.

April 2013 Notices of the AMS 443 Calendar Reform and the Council of Nicaea on it. Indeed, the Christian calendar has a rigid The Christian calendar has its origins in Rome. and a flexible part. The former is the old-style In 46 BC, Julius Caesar established the length solar Julian system, with its fixed celebrations, of months at thirty or thirty-one days, except whereas the latter is the lunar Easter Book, which for February, for which he introduced the leap determines the variable feasts and festivals of the year. Subsequently the emperors Augustus and Christian church. With no exception, all religious Constantine the Great slightly amended it. services are based on these two systems. The solar and the Julian year differ by only Tradition claims that the difficulty of combining a few minutes, but this little discrepancy led the lunar and the solar calendars has confronted to significant errors after enough time passed. theologians since the second century AD, when Towards the end of the sixteenth century, for the church first celebrated Easter. The Easter instance, the spring equinox fell in early March. Book, canonized by the First Council of Nicaea In 1582, this anomaly prompted Pope Gregory in AD 325, provides the rules on which day this XIII to issue a papal bull, according to which that celebration should occur. But the dating system year’s month of October was shortened by ten in the Easter Book is confusing, since the original days and the day of February 29 was cancelled in text of the Nicaean Creed has not survived. We the end-of-the-century years, except for years that know how to compute the date of Easter only from are multiples of 400. the message of Constantine to the bishops who But Gleb Nosovski, a close collaborator of were absent from the council, and this document Anatoli Fomenko, disagreed with the computations doesn’t ask for Easter to take place after the spring that led to Pope Gregory’s changes. He specifically equinox. By about AD 1330, the medieval scholar Matthew Vlastar wrote the following about how to referred to the following passage from the papal determine the anniversary of Christ’s resurrection bull: in the Collection of Rules of the Holy Fathers of the Our care was not only to reinstate the Church: equinox in its long ago nominated place The rule on Easter has two restrictions: not from which it has deviated since the Council to celebrate together with the Israelites and of Nicaea by approximately ten days, and to to celebrate after the spring equinox. Two return the 14th Moon [full Moon] to its place, more were added by necessity: to have the from which it has deviated by four and five festival after the very first full Moon after days, but also to settle such modes and the equinox and not on any day but on the rules according to which future equinoxes first Sunday after the full Moon. All the and the 14th Moon would never move off restrictions except the last one have been their places. . . . Therefore, to return the kept firmly until now, but now we often equinox to its proper place established by change for a later Sunday. We always count the Church fathers of the Council of Nicaea two days after the Passover [full Moon] and on the 12th day before the April calends then turn to the following Sunday. This [March 21], we prescribe and order relative happened not by ignorance or inability of to October of the current year, 1582, that the Church fathers who confirmed the rules, ten days, from the third day before nonas but because of the lunar motion. [October 5] to the eve of the ides [October In Vlastar’s time, the last condition of Easter 14] inclusive, be deleted. was violated: if the first Sunday took place within Nosovski found two errors in this quote. The two days after the full moon, the celebration of first has to do with the time difference between Easter was postponed until the next weekend. This the full moon and the spring equinox, an interval change was necessary because of the difference the bull wants to keep constant. But this cannot be between the real full moon and the one computed done, because the cycle of full moons and the date in the Easter Book. The error, of which Vlastar of the equinox shift at different rates. Nosovski knew, is twenty-four hours in 304 years. Therefore concluded that this mistake likely belonged to the Easter Book must have been written around AD those who wrote the bull, for no astronomer could 722. Had Vlastar been aware of the Easter Book’s have fallen into this trap. AD 325 canonization, he would have noticed the The second mistake, however, proves to be three-day gap that had accumulated between the essential, and it relates to the determination of dates of the real and the computed full moon in the dates of the spring equinox and the full moon. more than 1,000 years. So he either was unaware To understand it, we should mention that Scaliger of the Easter Book or knew the correct date when dated the First Council of Nicaea at AD 325. This it was written, which could not be near AD 325. year is crucial for the accuracy of the Gregorian Nosovski used Gauss’s Easter formula to calcu- reform, because the ten-day correction depends late the Julian dates of all spring full moons from

444 Notices of the AMS Volume 60, Number 4 the first century AD up to our time and compared In The Crime of Claudius Ptolemy [10], a book them with the Easter dates obtained from the published in 1977, Robert Newton accused the Easter Book. He thus concluded that three of the ancient astronomer of fabricating evidence. Newton four conditions imposed by the First Council of argued that many of the coordinates presented Nicaea were violated until 784, whereas Vlastar in the Almagest as observations are nothing but had noted that “all the restrictions except the last fraud. Fomenko disagreed, so he took on the task one have been kept firmly until now.” Scaliger had of dating the book. no chance of detecting this fault when proposing His first attempt was based on the fact that every the year 325, because in the sixteenth century the star has a proper motion that is unrelated to the full moon calculations for the distant past couldn’t apparent one due to precession. The discovery of be performed with high accuracy. this phenomenon is attributed to Edmund Halley, There is another reason against the validity who described it at the beginning of the eighteenth of AD 325: the 532-year periodicity of the Easter century. Ptolemy had also asked if stars moved dates. The last cycle started in 1941. The previous independently of each other, but he missed the ones were 1409 to 1940, 877 to 1408, and 345 to correct answer. 876. So it appears strange that the council met in The motion of stars can be detected only by AD 325 and started the Easter cycle 20 years later. hundreds of years of precise observations of Therefore Nosovski thought that the First Coun- their tangential components. Using the relative cil of Nicaea had taken place in AD 876 or 877, positions given in the Almagest and comparing since the latter is the starting year of the first them with the present ones, Fomenko wanted to Easter cycle after AD 784, when Nosovski believed find out when the book had been written. But the Easter Book was probably compiled. This con- that goal was not easy to achieve. One hurdle clusion also agreed with his full moon calculations, was the use of Ptolemy’s catalog for tracing the which showed that the computed and the real full motion of some stars. If the catalog’s dating was moons occurred on the same day only between AD incorrect, the computed speeds of these stars 700 and 1000. From 1000 on, the real full moons were also wrong. Fomenko had, therefore, to trace occurred more than a day after the computed ones, the history of those determinations and eliminate whereas before 700 the order was reversed. The from his analysis the stars related to the Almagest. years 784 and 877 also matched the traditional But the most difficult process was to identify the claim that about a century had passed between the compilation and the canonization of the Easter cataloged stars, a problem that had preoccupied Book. many astronomers starting with the sixteenth Unfortunately, this conclusion generated no century. reaction from historians. Nosovski’s mathematical In ancient and medieval times the shapes of reasoning seems plausible, but it would be inter- constellations were not standardized, and their esting to know if the historical aspects he invokes description was often vague. Therefore, telling hold water. which star from the catalog corresponds to the one we see in the night sky is difficult. Ptolemy The Almagest, Probabilities, and the Method provided positions and magnitudes. For bright of Least Squares objects identification is easier because there are The moon’s acceleration was only one disagreement few to choose from, but with faint stars, things between Robert Newton and . get complicated: in the Almagest their coordinates, They also strongly differed on the Almagest, the and also their magnitudes, are often incorrect. most influential astronomy book ever written. Research done on this problem assumed that the Claudius Ptolemy, one of the greatest scientists of observations were made in the second century AD, antiquity, wrote it in Alexandria during the reign of a fact that influenced the identification of the stars. the Roman emperor Antoninus Pius, traditionally The outcome changes for different suppositions. set from AD 138 to 161. Any firm evidence for a This leads to a circular argument. Fortunately, different dating of this treatise would affect the identification is easier for the stars of zodiacal chronology of Rome and consequently most ancient constellations because they have been studied history. This opus touches on the main problems of more carefully for astrological purposes and there astronomy, from the nature of the universe to lunar is more historical information about them. Of and planetary motion, and contains detailed star the 350 zodiacal stars recorded in the Almagest, catalogs and records of eclipses, occultations, and Fomenko chose to focus on the very fast ones, with equinoxes, all of which are prone to mathematical an individual motion of at least one arc second per dating. The original version of the Almagest has year, because slower objects could have traveled been lost, but in its many translations the work distances that were less than those resulting from has been in circulation since ancient times. Ptolemy’s observational errors.

April 2013 Notices of the AMS 445 Fomenko then applied the method of least In the 1990s, Fomenko and his collaborators squares. He took the distance between the position worked on deciphering and dating some Egyptian of a star as recorded in the Almagest and its images, which they interpreted as horoscopes, such real position in a given year, as determined by as the Denderah stones, the Esna bas-reliefs, the computations. He then summed up the distances paintings at Athribis, the Petosiris tomb of Dakhla, for all stars and repeated the procedure for all and the murals found in the burial chambers of the years within some interval long enough to avoid pharaohs Rameses VI and VII. The difficult part was bias, from 500 BC to AD 1800. Finally, he compared that of finding the correct interpretation of each the results and chose the year corresponding to symbol. Based on these findings, the team obtained the minimum sum. Estimates for each century mostly dates from the middle ages, results they pointed out that the only interval in which the claimed in support of their shorter chronology. errors were smaller than Ptolemy’s ten-arc-minute But these conclusions don’t agree with work precision was from AD 600 to 1300, with the done in the late 1950s by Otto Neugebauer and highest probability around AD 800. Henry Bartlett Van Hoesen, who published a This conclusion depends on several assump- study of some 200 horoscopes, mostly Greek, tions, and Fomenko checked the reliability of his but also Egyptian and Arabic [8]. All of them result. His estimate showed a very small, but occur explicitly, not in symbolic form, so whenever nonzero, probability that the Almagest had been the text is complete, the interpretation is certain. written outside this interval. With admissible (but Unlike Fomenko and his team, Neugebauer and unrealistic) changes in the parameters, the interval Van Hoesen didn’t take every horoscope seriously. could have been extended as far back in time as AD They found a few impossible configurations, such 350, a date still two centuries after the traditional as one in which Venus opposes the sun, but most dating. The good news was that the outcome didn’t are plausible from the astronomical point of view. change when slightly varying the data. To gain They also restricted their study to an ancient more confidence in this procedure, he also tested interval, ignoring possible dates closer to our time. star catalogs from the sixteenth and seventeenth Their results are statistically meaningful: the Greek centuries, as well as some computer-generated dates range from 71 BC to AD 621, clustering ones. The results proved more than satisfactory: around AD 100; the Egyptian dates fall between he recovered the known dates within a ten-year 37 BC and AD 93; and most of the Arabic dates margin of error. are from around AD 800. Their book provides the The Almagest contains other resources, such as necessary information for further investigations, occultations and lunar eclipses, phenomena that as there are many more unstudied horoscopes in are prone to independent dating. Fomenko and the papyrological literature, which comprises tens his collaborators devised methods to check the of thousands of texts. dates, which then led them to the time when the Almagest was written. The estimates they obtained Even if all of Fomenko’s solutions were correct, were consistent with the previous dating of the the number of cases he has studied is too small to Almagest to about AD 800 [3], [4]. justify drawing any conclusion from them. Histori- So far, historians have ignored these studies, ans can easily dismiss them as irrelevant because of which are published in a mathematics journal that the uncertainty surrounding the interpretations of has a reasonably good ranking. the symbols. Nevertheless, the study of horoscopes is an important method, whose potential has not Horoscopes been exhausted yet. A horoscope depicts the positions of the sun, the moon, and the planets Mercury, Venus, Mars, Empirical-Statistical Methods Jupiter, and Saturn among the standard twelve One of Fomenko’s empirical-statistical methods zodiacal constellations at a given time. Except for aims to identify various chronicles that seem Mercury and Venus, which are never too far from different but describe the same historical period, the sun, with a span of three constellations for the even if they appear in different languages, call their former and five for the latter, the other planets may characters differently, and use different geographic show up anywhere. As a result, there are 3, 732, 480 names. He started from the premises that a person, possible configurations for these celestial bodies. deity, country, or city can be known by more Because of the planets’ fast motion, horoscopes than one name: Charlemagne is also known as change almost daily. They may repeat themselves Carol the Great, God as Allah, Finland as Suomi, after hundreds of thousands of years, or as early as and Bratislava as Pressburg. It is reasonable to within a few decades. The tedious calculations for think that lack of communication allowed name finding the possible dates of a particular horoscope variations to be common in the past, so such are easily performed by computers today. chronicles are likely to exist.

446 Notices of the AMS Volume 60, Number 4 Fomenko designed his method as follows. Take is so used to its current position. But Fomenko’s two texts describing several historical events that frequency analysis suggest that it belongs near have a relative but not an absolute dating. Collect the prophecies of the Old Testament. His new various data, such as the number of words used ordering moves Revelation into the same period to chronicle a period or the number of times a as the books of Isaiah, Jeremiah, Ezekiel, Daniel, name occurs in a certain interval of time. Then Exodus, and Leviticus. Fomenko himself did not compare the homolog information. If the numbers find this placement surprising, because St. John’s you obtain are very different, the periods are Revelation reminded him of the apocalyptic nature probably unrelated. If they are close, continue the of Daniel’s prophecies in the Old Testament. investigation with the help of various statistical Can this empirico-statistical analysis change our tools. For consistent results, the two chronicles understanding of the Bible? So far, biblical scholars are likely to describe the same events. Tested seem to have ignored Fomenko’s conclusions. But on specific texts, such as the Russian Supras’l it would be good to see studies that either refute and Nikiforov chronicles, both referring to events these ideas or use them to better understand occurring in the period AD 850–1256, the method Christian theology. gave similar statistical results. A related problem is that of ordering several The Dating of Maps writings that contain many historical characters, Fomenko also came up with a method for dating some of whom appear in more than one document. maps, in which he used the following assumptions. For that, divide the texts into generations, i.e., Once an error is corrected on a map, it does not chapters spanning twenty-four to thirty-three appear on subsequent maps, and all the accurate years of history. In any given chapter, only names features are maintained. So, for a region with a from the past or present show up. Introduce a long cartographic history, the fewer the number quantitative measure: compare the occurrence of mistakes a map contains, the more recent the of names from previous generations with those map. Given a sequence of maps where the dates in the investigated chapters and write down the when they were drawn is unknown, one can order ratios. Since parents are better remembered than them chronologically by mutually comparing them grandparents, more distant generations are ideally and finding the changes that occur. Many criteria less frequently mentioned. In the end, order the that involve some mathematics must be taken into chronicles so that all mutual frequencies are close account, including the type of map (globe, flat); the to ideal. This principle yielded good results when kind of projection (conical, cylindrical, azimuthal); tested on reliable documents of the past few centuries. orientation; the arrangements of poles, equator, Fomenko applied these methods to the Old and and tropics; the representation of climatic zones; New Testaments. According to tradition, the Bible and so on. describes distinct events, except for the well-known This idea is known to historians. Sir Flinders overlaps between the four books of Samuel and Petrie, the father of modern archeology, used a Kings and the two books of the Chronicles. But similar (but nonmathematical) technique at the Fomenko’s conclusion was different. To reach it, beginning of the twentieth century, after noticing he first divided the Scriptures into 218 chapters, the stylistic differences between the articles of one for each generation that occurs from the total pottery found in various graves. By charting those of about 2,000 characters. For instance, Genesis changes, he determined the relative chronology of was split into seventy-three chapters: Genesis 1–3 the graves. (Adam, Eve), Genesis 4:1–4:16 (Cain, Abel), and so Applying this method, Fomenko found that on, whereas Exodus formed only one chapter. The cartography developed very slowly. The maps of Old Testament consisted of parts 1 to 191, and the third and fourth centuries AD were simple the New Testament consisted of parts 192 to 218. sketches, very different from what they depicted. To check the validity of this division, Fomenko Then their quality improved, with the occurrence tested it on the already-known biblical overlaps of the first fairly accurate globes and planar maps and confirmed them easily. appearing in the 1500s. But in spite of having He then ordered all the chapters according to Earth’s main features present in the latter drawings, the above principles and concluded that the Old their proportions were still poor. Several famous and New Testaments describe interwoven events maps attracted Fomenko’s attention, such as the and are not separated by several centuries, as globe of Crates, from the second century BC, previously thought. For instance, the Revelation Tabula Peutingeriana, thought to have originated of St. John the Divine, the last book in the Bible, in the time of the Emperor Augustus (27 BC–AD belongs to the New Testament. Placing it anywhere 14), and one attributed to Claudius Ptolemy in the else would look strange at best since everybody second century AD. Using his method, Fomenko

April 2013 Notices of the AMS 447 concluded that all of them were in fact produced conditions such as temperature or humidity do not about a millennium later. affect the decay of radiocarbon; the dated samples Disregarding these conclusions for the moment, are not contaminated, so the ratio of carbon-14 the principles of Fomenko’s method meet the to carbon-12 is not affected by external factors. same standards historians apply when dealing These hypotheses were often criticized in the early with evolution patterns, as Sir Flinders Petrie did days of the method, when large errors occurred, in his analysis of pottery. Moreover, from the mainly because the measuring techniques were not mathematical point of view, there is nothing wrong refined enough. But starting with the 1980s, the new with these techniques; they use the standard tools accelerator mass spectroscopy technique (which is of mathematical statistics, which nobody questions. as different from the original radiocarbon method Therefore, as in the case of applying empirical as digital photography is from film) led to very statistics to texts, this idea seems worth pursuing. good measurements that have small approximation But can these methods withstand criticism on errors. The results obtained in this way are often other fronts? The discipline that deals with the tested with the help of dendrochronology, the interpretation of such data is known as applied science of determining dates from tree rings. statistics, and its experts are aware of its traps, Thermoluminescence is based on the light that such as failing to ensure that the pool of data is is emitted, in addition to the usual glow, when a relevant or not assessing how many experiments crystalline material reaches a temperature of about fail. These issues are overlooked in Fomenko’s 500o C. When pottery, which contains minerals work. (feldspars, calcite, quartz) with high emissions, Although he claims to obtain absolute dates, it breaks and the shards are buried, the process of is doubtful that he can achieve that without relying building up this energy starts all over again. The on nonmathematical results. As in the case of quantity of thermoluminescence found in these texts, he might be able to order the charts, but how fragments indicates their age. Instead of decay, as can he tell if the earliest maps are from the third happens with carbon-14, this process is described century BC or the ninth century AD? It appears that, by a differential equation that expresses growth. in his haste to support his previous astronomical Fission tracking is based on particle physics. findings, he jumped to conclusions too early, thus If the atoms of an element prone to fission, like risking making a yet unsettled method look weak. uranium, are trapped inside a crystal structure, the released radiation “scratches” the inside of Scientific Methods the structure. An electron microscope can detect Starting with the twentieth century, the progress the marks, whose number provides the age of of physics, chemistry, biology, and other branches the sample, according to probability theory. If the of science led to several new dating methods, material is manufactured glass, the heating used which established themselves, but not without in the production erases previous traces, allowing encountering resistance. Some benefit from math- an evaluation of when the sample was made. ematical support as well. Among them, the best Finally, archeomagnetic dating establishes the known are the radiocarbon method, dendrochronol- age of objects by comparing their magnetic infor- ogy, thermoluminescence, fission tracking, and mation with changes in Earth’s magnetic field. As archeomagnetic dating. with all the other dating methods, the techniques From the mathematical point of view, the radio- used in this approach are continually improving. carbon method uses a simple linear differential As time progresses, the scientific dating methods equation that describes radioactive decay. It is physically based on the disintegration of carbon-14, have a greater impact on historical research, and a chemical element with a half-life of about 5,370 someday historians might use them more heavily years that exists in all living organisms. Once an for a critical assessment of traditional chronology. organism dies, the decay of this element starts, and the method provides the time of death by com- Closing Remarks puting the variable ratio between carbon-14 and At the end of the sixteenth century, when historical the element carbon-12, which is stable, remaining chronology became a science, most mathematical constant in time. In living organisms this ratio is methods used to determine ancient and medieval about 10−12. dates were based on celestial mechanics. Things The radiocarbon method is based on several changed in the twentieth century with the introduc- hypotheses: the concentration of radiocarbon in tion of several scientific dating methods. In parallel, the atmosphere is constant and hasn’t changed some mathematicians tried new approaches, with throughout history; the proportion of radiocarbon some degree of success. These methods provide in all living beings is the same as in the atmosphere insight into history but don’t seem to be taken and independent of location; physical and chemical very seriously if regarded alone.

448 Notices of the AMS Volume 60, Number 4 Indeed, historians usually take the results of [11] , The Moon’s Acceleration and Its Physical Origin, these methods into account only when the conclu- Vol. 1, As Deduced from Solar Eclipses, Johns Hopkins sions agree with their chronological expectations. Univ. Press, 1979. In other words, they give priority to traditional [12] , The Moon’s Acceleration and Its Physical Origin, Vol. 2, As Deduced from General Lunar Observations, chronology over the above scientific and mathe- Johns Hopkins Univ. Press, 1984. matical methods. This attitude is not surprising. [13] R. L. Reese, S. M. Everett, and E. D. Craun, The origin After all, they built the traditional chronology with of the Julian Period: An application of congruences much effort over a significant interval of time. and the Chinese Remainder Theorem, Amer. J. Phys. Nevertheless, it is clear that the more distant 49 (1981), 658–661. historical dates go into the past, the less reliable [14] E. M. Reingold and N. Dershowitz, Calendrical Cal- they are. So, next time we hear that Rome was culations: The Millennium Edition, Cambridge Univ. Press, 2001. founded in 753 BC, we should take the claim with a [15] J. Repcheck, The Man Who Found Time: James Hut- grain of salt. We may actually never know for sure ton and the Discovery of Earth’s Antiquity, Perseus, exactly when that happened, but we need to hope Cambridge, Mass., 2003. that the ordering of historical events is correct. [16] A. A. Zalyzniak, Linguistics according to A. T. The mathematical methods described above are Fomenko, Russian Math. Surveys 55 (2000), 369–404. a first step towards providing historians with more than their original chronological tools and help them keep regarding the past critically. But it is a hasty move to jump to conclusions about the cor- rect dating of ancient events without corroborating many pieces of evidence. Mathematics, however, seems to have infiltrated the study of history, as it did so successfully in other disciplines. After the dust of controversy settles, there is hope that historians, scientists, and mathemati- cians will together revisit the existing methods and devise new tools, following the same spirit of cooperation that occurs in so many other fields of human endeavor. But we should not expect too much too soon. This is a field in which progress has been slow, and there are no signs that things will take a sudden turn in the near future.

References [1] F. Diacu, The Lost Millennium: History’s Timetables under Siege, 2nd ed., Johns Hopkins Univ. Press, 2011. [2] A. T. Fomenko, The jump of the second derivative of the moon’s elongation, Cel. Mech. 25 (1981), 33–40. [3] A. T. Fomenko, V. V. Kalashnikov, and G. V. Nosov- ski, When was Ptolemy’s star catalogue in Almagest compiled in reality? Statistical analysis, Acta Appl. Math. 17 (1989), 203–229. [4] , The dating of Ptolemy’s Almagest based on the covering of the stars and on lunar eclipses, Acta Appl. Math. 29 (1992), 281–298. [5] A. Grafton, Joseph Scaliger: A Study in the History of Classical Scholarship, vol. 2, Historical Chronology, Clarendon Press, Oxford, 1993. [6] , A premature autobiography?, unpublished manuscript, 2002. [7] D. Hay, Annalists and Historians: Western Historiog- raphy from the Eighth to the Eighteenth Centuries, Methuen, London, 1977. [8] O. Neugebauer and H. B. Van Hosen, Greek Horo- scopes, Amer. Philos. Soc., Philadelphia, 1959. [9] I. Newton, The Chronology of Ancient Kingdoms Amended, J. Tonson, J. Osborn, and T. Longman, London, 1728. [10] R. Newton, The Crime of Claudius Ptolemy, Johns Hopkins Univ. Press, 1977.

April 2013 Notices of the AMS 449 Newton’s Laws and Coin Tossing

Don Ornstein

bstract ergodic theory is the study of This paper is narrowly focused and is not a sur- 1-1 (invertible) measure-preserving vey. In particular, I do not discuss the first deep transformations T on a measure isomorphism theorem, which is due to Adler and space X of total measure 1 or a Weiss [AW67] (or the directions it initiated: Markov one-parameter family of such trans- partitions, finitary codings, etc.), even though it was Aformations T where T (T (x)) = T (x) (x ∈ X). t t2 t1 t1+t2 an inspiration for so much later work. Nor do I dis- (We call the latter a flow.) cuss the Keane-Smorodinsky finitary isomorphism There has been a recent explosion of interest theorem for Bernoulli shifts. in abstract ergodic theory, due mainly to its un- The bibliography is very incomplete. One crite- expected applications to number theory. It may rion I used is that the title should give some idea therefore be of interest to revisit an early chapter in of the main result, so that reading the bibliography its history. will add to the picture described in the text. I am going to tell the story1 of Bernoulli shifts, the abstract version of coin tossing. A central piece Isomorphism of this story is the existence of an abstract flow, the Bernoulli flow, or Bt , which pastes together the To pin down the objects we are talking about, we Bernoulli shifts. need to say when two abstract transformations or Bt is the “most random abstract flow possible”. flows are the same or isomorphic. A priori, this statement has no real meaning, and to (Tt ,X) and (Tbt , X)b are isomorphic if there is a 1-1 the extent that it does, one would expect a host of invertible measure-preserving map ψ from X to Xb −1 competitors for “most random”. We will, however, such that Tbt = ψ Tt ψ. (We have the same definition state several properties of Bt which, when taken to- for transformations.) gether, can be interpreted as “Bt is the most random For flows there is another weaker sense in which abstract flow possible”, even though we do not give two flows could be the same. We say that Tbt is “essen- a precise definition of “random”.2 tially the same” as Tt if Tbt = Tct for some constant B is the cause of all randomness in dynamical sys- t c. tems (for a precise statement, see “Positive Entropy and Factors”). There Are Two Contexts In Which Abstract There are classical dynamical systems, governed Ergodic Theory Arose: Concrete or Classical by Newton’s Laws, that are the same as Bt at our level of abstraction. We will discuss the concrete implica- Dynamical Systems (Like the Time Evolution tions of this abstract equivalence and the relevance of a Gas Confined to a Box) and Stationary to chaos theory. Processes Classical Dynamical Systems Don Ornstein is emeritus professor of mathematics Here, we try to abstract-out the statistical properties at Stanford University. His email address is ornstein@math. of the mechanism governing the time evolution. In stanford.edu. particular, we ignore events of probability zero. 1We will also point out certain questions which at the time The state or configuration of the system is rep- were overlooked. resented by a point x in a smooth manifold M, the 2See the section “Positive Entropy and Factors”. phase space. Newton’s laws determine where x in M DOI: http://dx.doi.org/10.1090/noti974 will be at time t; denote this by Tt (x).

450 Notices of the AMS Volume 60, Number 4 We will give an alternate description of how, in this case, we go from a stationary process to a measure-preserving transformation. Let π be a set with k elements ei with probabili- Pk ties pi, i=1 pi = 1, and x in X is a doubly infinite sequence of the elements in π. x is a realization of the process. A cylinder set is the set of x where we fix the ei at a finite set of coordinates (or times). Its measure is the product of those pi. This extends to a measure on X. T is the transformation that shifts these sequences (the n coordinates of T (x) in the Figure 1. Sinai billiards: the motion of a billiard n + 1 coordinate x). We will call the transformation ball on a frictionless table with convex obstacles. above B(p1, . . . , pk). We get the model for the independent process by a function F on X where F(x) is the zero component of x. F(T i(x)) is a realization of the process. There is a smooth invariant probability measure We will now describe a continuous time analog µ with the same sets of measure 0 as Lebesgue of the Bernoulli shifts. For simplicity, we will take measure (Liouville’s theorem). µ(E) models the 1 1 B( 2 , 2 ). probability that the state of the system is in E. We need to introduce a probability structure, because The Coin Tossing Flow Newton’s laws do not respect the direction of time We start with the Continuous Time Coin Tossing Pro- (and can’t alone imply the increase of entropy or cess. Pick t and t such that t /t is irrational. Toss keep a pond from ejecting a stone). 1 2 1 2 a fair coin. If we get heads, output 0 for time t . If At the level of abstract ergodic theory, we regard 1 we get tails, output 1 for time t . Toss again. (M, µ) as an abstract measure space (X, µ) (i.e., we 2 ignore sets of measure 0 and the geometry of M). We can visualize the flow Tt that we get from the above process as follows: Start with B( 1 , 1 ), defined We get back the geometry by a function that maps a 2 2 point in X to the corresponding point in M. on X. Tt is defined on a subset, Y , of X × R, pictured A good example is Sinai billiards. Here we fix a below: speed, and the points in M are the position and direction of motion of the ball. µ is 3-dimensional Lebesgue measure. See Figure 1. T Stationary Processes A stationary process is a sequence or a one- parameter family of random variables, Xn or Xt , 0 1 defined on the same measure space ( , µ). (For ω in Ω Figure 2. Coin Tossing Flow Ω, Xn(ω) or Xt (ω) is a realization of the process.) We define a transformation in Ω: A point x in the base flows straight up for time t1 T {Xn} = {Xn+1} and Tt {Xt } = {Xt+t }. (or t2) and returns to the base at B 1 1 (x). ( 2 , 2 ) Stationarity means that the joint distribution of Although this is a continuous time analog of a {X0,X1,...,Xn} is the same as the joint distribution Bernoulli shift, the connection is much deeper. For {Xk,X1+k,...,Xn+k} for all k, and the joint distribu- example, the isomorphism theory will show that by discretizing time, we can get B( 1 , 1 ). tion of {Xt1 ,...,Xtn } is the same as the joint distri- 2 2 bution of {Xt1+t ,...,Xtn+t } for all t. This is the same We can visualize the model for continuous time as saying that T (or Tt ) is measure preserving. coin tossing by a function, F, that labels the points We get the Kolmogorov model for a stationary above 1 by 1 and those above 0 by 0. F(Tt (y)) are process by starting with (T , Ω, µ) or (Tt , Ω, µ), and the realization of our process. n ∞ a function F on Ω, F(ω) = X0(ω), {F(T (ω))}−∞ ∞ or {F(Tt (ω))}−∞ are the realizations we started out with. We will denote this process by (T , Ω,F) or (Tt , Ω,F), or by (T , X, F) or (Tt ,X,F). The Kolmogorov point of view puts dynamical sys- tems and stationary processes into the same mathe- Bernoulli Shifts matical framework: a measure-preserving transfor- These are the transformations that come from inde- mation or flow, together with a function that gives pendent processes. the concrete structure.

April 2013 Notices of the AMS 451 Entropy and Isomorphism 1. Bernoulli shifts of the same entropy are iso- Our story of Bernoulli shifts begins with the long- morphic. (The entropy of the transformation

standing question: Are all Bernoulli shifts isomor- Tt0 is proportional to t0.) phic? 2. We can complete our understanding of Kolmogorov [Kol59] was able to show that not Bernoulli shifts by representing them as the all Bernoulli shifts were isomorphic by introduc- discretizations of the Bernoulli flow, Bt , or ing an invariant, which he called “the entropy of the time of one transformation of the Bct . the transformation”. This revolutionized abstract Thus, returning to Kolmogorov, even though ergodic theory. the Bernoulli shifts are not all isomorphic, To define the entropy of a transformation, we they are “essentially” the same in the sense start with the Shannon entropy of a stationary that they differ only by a rescaling of time. process with a finite (or countable) number of out- 3. A Bernoulli shift can be embedded in a flow puts. Define the entropy to be the infimum of the α and in particular it has roots of all order (it such that, given , there is an N() and the number was not previously known that the Bernoulli αn 1 1 of outputs of length n (n > N()) is less than 2 shift 2 , 2 had a square root). after removing a subset of these outputs of measure We can now return to some of the concrete systems ≤ . that gave birth to abstract ergodic theory The entropy of a transformation (T , X) is the sup of the entropies of the stationary process (T , X, F) An Important Aspect of the Proof of the that come from (T , X). Isomorphism Theorem Is Giving Criteria for Kolmogorov’s key result is that the entropy of a Proving Bernoulli Bernoulli shift is the same as the entropy of the in- The criteria are somewhat technical and we post- dependent, identically distributed process used to pone their description. We will start with two define it. It is not hard to see that if the outputs of applications of the Bernoulli criteria. this process have probability Pi, then the entropy of P P 3 this process is − Pi log Pi (where − Pi log Pi could Theorem. The coin tossing flow and Sinai billiards be finite or infinite). are both isomorphic to Bt . After Kolmogorov, the problem became: Are all This theorem links the two contexts in which ab- Bernoulli shifts of the same entropy isomorphic? An stract ergodic theory arose. answer was given twelve years later. It came as a surprise that continuous time coin Theorem ([Orn70b], [Orn70d]). Bernoulli shifts of the tossing and a classical dynamical system could be same entropy are isomorphic. the same as abstract flows. The isomorphism between Sinai billiards and the Thus, when we go to the level of abstract ergodic coin tossing flow implies that we can realize con- theory, we get a very simple picture. tinuous time coin tossing by a fixed (nonrandom) Independent processes are the least predictable function on the phase space of Sinai billiards and and in this sense the most random possible, and its time evolution could be derived from Newton’s Bernoulli shifts are in this sense the most random laws. transformations. The proof of the theorem above is the basis for Some Bernoulli Examples the stronger and deeper theorem below. • Our first example is a generalization of the con- tinuous time coin tossing flow where we replace Bernoulli Isomorphism Theorem 1 ([Orn70b], coin tossing by an N-step Markov process. [Orn70d], [Orn70a], [Orn73b]). There exists an We replace the continuous time coin tossing abstract finite entropy flow, B , with the following t process by a process that we will call a ct (con- properties: tinuous time) Markovian process. This process (a) If we discretize time at multiples of t0, then has a finite number of outputs Oi. Oi lasts for

for all t0 the resulting transformation, Bt0 , is time ti. The next output is determined by a a Bernoulli shift of finite entropy. roulette wheel4 whose probabilities depend on i (b) Every Bernoulli shift of finite entropy is iso- and the previous N outputs. morphic to B for some t . t0 0 We can visualize the flow as a “flow built under a (c) Uniqueness. For any flow T , if for some t t 0 function” where the transformation on the base T is isomorphic to a finite entropy Bernoulli t0 is an N-step Markovian process. shift, then Tt as a flow is isomorphic to Bct for some constant c. • Sinai billiards [GO74].

∞ 3 We also have an analogous Bernoulli flow, Bt , of Deep results of Sinai about Sinai billiards are needed to infinite entropy. check our criteria. Three consequences of the theorem above are: 4Or many coin tosses.

452 Notices of the AMS Volume 60, Number 4 • Geodesic flow on a manifold, M, of negative flow, then given ε we can change F to (F)b on a set of curvature (the flow is really on the tangent measure ε; and if we regard M as an abstract mea- bundle of M) [OW73]. sure space, then (Tt ,M, F)b is the model for a ct N-step This was the first concrete flow that was Markovian process on the same Oi. shown to be Bernoulli and showed how to use hyperbolic structure to verify our criteria (this A good example: display Sinai billiards on a TV was Weiss’s insight and the beginning of a screen. This is a (Bt ,M,F) where the range of F is long collaboration). Almost all of the concrete the states of the TV screen. If we fix a set of posi- flows that were shown to be Bernoulli follow tions and directions of measure < ε and force the this format. TV to misread the position when in these states in • The following deep theorem of Pesin gives a a fixed nonrandom way, then the display on the TV general class of Bernoulli flows [Pes77]: screen becomes the model for a ct N-step Markovian process. Thus, the evolution of the slightly altered TV Let T be a smooth ergodic5 flow in C1+ε on t movie that we get by looking at a fixed billiard ball a compact 3-dimensional manifold with a orbit is determined by the outcome of a sequence of smooth invariant measure. If T has positive t coin tosses. entropy, T is either B or B × R (R is a t ct ct t t The discussion above holds for any classical sys- rotation of the circle). (This also applies to tem that is isomorphic to B where F is any coarse the ergodic components of positive entropy. t graining of the phase space. These have positive measure. A nice example The theorems above have implications for chaos is the double pendulum. It is not known if “theory”, because they show that there are “super this has positive entropy.) chaotic systems”, which, at an abstract level, are • We get an infinite entropy example by replac- the most random possible (see “Positive Entropy and ing the two shift by a Bernoulli shift of infinite Factors”) and, at a concrete level, are essentially entropy in “flow built under a function” (see indistinguishable from systems that are driven by Figure 2). coin tossing and where the coin tossing, after many • Other infinite entropy examples are the flows tosses, wipes out any memory of the past. we get from Brownian motion on the unit inter- The cornerstone of chaos “theory” is Lorenz’s ob- val with reflecting barriers, Poisson processes, servation that for certain classical systems, a small and diffusion processes. change in the initial position grows exponentially Some discrete time examples are: fast. (Sinai billiards are an example.) This is usually • Mixing Markov (or multistep Markov). referred to as “sensitivity to initial conditions” or • Ergodic automorphisms of compact groups. the “butterfly effect”. The conclusion is that the be- This result is due to Lind [Lin77] and, havior of these systems is unpredictable or random. independently, Miles and Thomas [MT78]. “Super chaos” is the ultimate strengthening of this • One of the deepest results in abstract ergodic conclusion. In particular, the lack of predictability theory is due to Rudolph: mixing compact due to sensitivity to initial conditions is qualitatively extensions of a Bernoulli shift are Bernoulli different from the lack of predictability due to coin [Rud78a], [Rud78b]. tossing. The titles of some of the papers in the bibliogra- The results above also mean that the distinction phy extend this sampling. between long-term determinism of Newtonian me- chanics6 and the long-term randomness introduced Another Aspect of the Proof of the Isomor- by coin flipping is not as sharp as one might expect. phism Theorem Is That in Certain Cases the Isomorphism Preserves the Concrete Statistical Stability Structure This is a more special case of isomorphisms be- Proving that a dynamical system is Bernoulli identi- tween concrete systems where the proof of the fies the system if we ignore the concrete structure isomorphism theorem allows us to preserve the given by a function on X. geometry. Theorem ([OW91]). Let F be geodesic flow on a man- The Concrete Structures of All Bernoulli Systems t ifold or surface M of negative curvature (the flow is Have a Certain Randomness Property T really on the tangent bundle M ), and define Fbt by a Recall the definition of a ct N-step Markovian pro- C2 change in the Riemannian structure of M that is cess. small in the C2-metric.

Theorem ([OW91]). Fix (Ft ,M,F), where the range of F is a finite set (Oi). If Ft is isomorphic to the Bernoulli 6It is critical that we are talking about the determinism of Newtonian mechanics, because any abstract system is 5Ergodic means that the only invariant sets have measure 0 deterministic in the sense that the starting point x in X or 1. determines the entire orbit.

April 2013 Notices of the AMS 453 Then, given ε, if the change is small enough, Fbt lump these points together, we get a new transfor- will be isomorphic to Fct ,(|c − 1| < ε) by a map ψ mation or flow, which we call a factor of (T , X, µ) or T of M to itself that moves all but ε of the points by (Tt , X, µ). a distance < ε. (In particular, orbits map to orbits, A factor can also be described as restricting the preserving time.) measurable sets to belong to a sub-σ -algebra, which is invariant under T (or Tt ). We would get the same conclusion if, instead of changing the Riemannian structure, we made a small Bernoulli Isomorphism Theorem 2 ([Orn70b], variable speed change along the orbits of Ft . [Orn70d], [Orn70c], [Orn70a], [Orn73b]). We would also get the same conclusion if we (a) The only factors of Bct are the Bαt for α ≤ c. made a small change in the obstacles in Sinai (b) If Tt has finite positive entropy, then Tt has Bct billiards [Elo88]. as a factor of the same or smaller entropy. The theorems above are examples of what we call (There exists an abstract flow of infinite entropy, statistical stability. B∞, with the analogous properties.) The isomorphisms above give a clean picture of t (a) and (b) are also true for Bernoulli shifts. Sinai the potentially messy long-term consequences of the proved (b) for Bernoulli shifts before the isomor- small perturbations above. phism theorems in this article. Even though Sinai’s method was entirely different, knowledge of this Comparison with the Structural Stability (of result was of major importance. Peixoto, Anosov, Smale, Mané, etc.) The main consequences of the theorem above are: In the case of geodesic flow, if we define F as com- bt • There exists a unique abstract flow of finite ing from the above change in Riemannian structure, (infinite) entropy that is the “most random” then structural stability asserts that there is a home- flow possible: omorphism of ψ of MT onto itself that takes the One meaning comes from characterizing orbits of F onto those of F (without preserving time) t bt the randomness of a flow, (T ,X), by the and moves all points by < ε. t randomness of Tt ,X,F, the stationary pro- In statistical stability, we lose the continuity of ψ, cesses that come from the flow. (b) implies and instead of moving all points by < , there is a that any flow of positive (infinite) entropy set of measure  where points may be moved by a that is not Bt gives rise to more station- large distance. ary processes and includes processes that However, statistical stability corrects the following are not VWB (see “Very Weak Bernoulli problems with structural stability in continuous time. (VWB) Systems”) and are therefore less 1. It is possible that ψ maps a set of probability unpredictable. one onto a set of probability zero. Another sense of “most random” comes 2. Since ψ does not preserve the speed along from its realization as the flow we get from orbits, sets that correspond under ψ at time continuous time coin tossing. 0 may not correspond to each other at time Yet another sense comes from its t ≠ 0. discretizations being the most random If our perturbation were a variable speed change transformations. along the same orbits, ψ would be the identity, • (b) means that Tt can be realized as a skew whereas ψ must scramble orbits. product with base Bct , where Bct has full en- Sinai billiards is statistically stable (we make a tropy and the action on the fibers does not small change in the curvature of the obstacles) but contribute to the entropy. In this sense, Bt or ∞ not structurally stable [Elo88]. Bt is the cause of all randomness in flows The systems that are structurally stable are the (flows of zero entropy are not considered axiom A attractors. These are Bernoulli with respect random). to the SBR measure, which is considered to be the • (a) says that the flow we get from any measure that is physically relevant, and we can ask Bt ,X,F is still Bct ; i.e., the processes that if they are statistically stable. The only obstruction give a Bernoulli flow are the same as the comes from the existence of eigenfunctions. A pre- processes that we get from a Bernoulli flow. cise statement can be found in [OW91]. We thus have a nice class of processes which we call Bernoulli. Positive Entropy and Factors and the Bernoulli Criterion The Entropy Hierarchy and the Limits of the We now return to abstract ergodic theory. Isomorphism Theory An arbitrary F defines a measure on the Our story began with Kolmogorov’s introduction of n ∞ ∞ {F(T (x))}−∞ or {F(Tt (x))}−∞. This is the model entropy as an invariant for Bernoulli shifts. Soon for a stationary process, except that many x can after, Sina˘ı [Sin59] modified Kolmogorov’s defini- n ∞ ∞ give the same {F(T (x))}−∞ or {F(Tt (x))}−∞. If we tion to give an invariant for any measure-preserving

454 Notices of the AMS Volume 60, Number 4 transformation or flow. This led to a hierarchy of conjecture is true. Every flow of positive entropy is transformations and flows and to the question: the direct product of Bt and a rotation (see item 4 What is the role of Bernoulli shifts (and later Bt ) in above for an exact statement). this hierarchy? In general, the Pinsker conjecture is false [Orn73a], 1. Zero Entropy. A discrete time process with a [Orn73c]. finite (or countable) number of outputs has Open Problem: Is every transformation the direct zero entropy if the infinite past determines product of a transformation of arbitrarily small en- the infinite future. tropy and a Bernoulli shift? These are called weak A transformation or flow has zero en- Pinsker transformations. tropy if all of the discrete time processes The first smooth non-Bernoulli K-automorphism that come from it have zero entropy. is due to Katok [Kat80]. Rudolph gave the first 2. Positive Entropy. A transformation or flow example of a smooth non-Bernoulli K-flow. The has positive entropy if some discrete time first “natural” example of a non-Bernoulli K- process that comes from it has nonzero en- automorphism is due to Kalikow [Kal82]. It’s tropy. easy to describe—it is a random walk on a ran- 3. K. A transformation or flow is K if no dis- dom environment—but difficult to show that it is crete time process that comes from it has non-Bernoulli. zero entropy. The next question is: What kinds of non-Bernoulli This is equivalent to having no factors of behavior can we get without introducing any deter- zero entropy. ministic elements? 4. The Bernoulli shifts in discrete time or Bt We now have a zoo of K-automorphisms with ∞ qualitatively different behaviors [Hof99b], [Hof99d], (Bt ) in continuous time (Bernoulli implies K). [Hof99c], [OS73], [Rud78a], [Rud76], [Rud77]. A small sampling is: The isomorphism theorem shows that Bernoulli shifts provide a simple top to the entropy hierarchy • K-automorphisms that are not isomorphic to in discrete time. their inverses. ∞ The existence of Bt (Bt ) shows that there is a single • Two K-automorphisms that are not isomor- top to the entropy hierarchy in continuous time. phic, but all of their powers are isomorphic Furthermore, all entropy comes from Bernoulli [Rud77]. shifts or the Bernoulli flow. • K-automorphisms that have no square root Kolmogorov introduced the K property around [Cla72]. the same time as the introduction of the Kolmogorov- Hoffman, using work of Rudolph, showed that the Sina˘ı entropy. The two main questions about behaviors of finite permutations could be lifted to Bernoulli shifts at the time were: Are Bernoulli behaviors of K-automorphisms [Hof99c], [Hof99b], shifts of the same entropy isomorphic, and are [Rud79]. Thus, instead of isolated examples, we have K-transformations Bernoulli shifts? [KAT07] a systematic method for producing examples. The answer to the second question is no [Orn73a]. At the time these counterexamples were pro- In fact, there are uncountably many nonisomorphic duced, the main focus was on the discrete time K-automorphisms of the same entropy [OS73] and un- hierarchy, and the issue of embedding the examples countably many nonisomorphic K-flows of the same in a flow was largely ignored, leaving a large gap in entropy [Smo73]. the theory. Sinai and his school are responsible for the first applications of hard abstract results to classical dy- Factors of a Bernoulli Shift namical systems, proving the time one transforma- tions to be K. The proofs rested on analyzing the We end by tying together the proofs and results of hyperbolic structure of these systems, the structure the two main strands of the abstract part of our that makes the distance between nearby points grow story: the isomorphism theorem and the entropy hi- exponentially fast. Later, in some cases, this analysis erarchy, together with the counterexamples. also allowed the application of the Bernoulli criteria The results concern the relationship of a factor (see “Some Bernoulli Examples”), proving the time to the whole Bernoulli shift. We say that two fac- one transformation to be Bernoulli, thus narrowing tors of T , A, and B sit the same way if there is an the class to a single flow. automorphism of T that takes A onto B. Pinsker [Pin60] conjectured that every trans- We say that T is Bernoulli relative to a factor A if formation of positive entropy was the direct T = A × B. product of a transformation of zero entropy and a K-automorphism. Thouvenot’s Relative Isomorphism Theory A theorem of Pesin shows that for flows on The starting point is Thouvenot’s relative isomor- 3-dimensional manifolds (transformations on 2- phism theory, which tells us when T is relatively dimensional surfaces), a strong form of the Pinsker Bernoulli with respect to A. (In Thouvenot’s theory

April 2013 Notices of the AMS 455 there is no restriction on A. In particular, A does not characterizations of B processes where F is finite have to be a factor of a Bernoulli shift.) valued. The theory gives the relative analog of the Bernoulli criteria in the appendix (relatively finitely The d Metric determined, relatively VWB). Here is the analog of We will describe a distance between (T , X, F) and the theorem that factors of Bernoulli shifts are (T, X, F) where F and F have the same range. Bernoulli. Both processes put a measure on sequences of Theorem (Thouvenot). If T = A × B where B is a length n which can be realized by functions f Bernoulli shift, then any factor of T that includes A and f from nondiscrete measure spaces Y and Y also has the form A × B, where B is a Bernoulli shift. of total measure 1 to the sequences of length n. (In general, B is not a factor of B.) Denote these by Y , f , and Y, f . We say that these measures are closer than α in d if there is a 1-1 Fieldsteel extended the relative isomorphism the- measure preserving map between Y and Y where orem to flows. the corresponding sequences differ in less than αn places except for a set in Y (or Y ) of measure < α. The Relative Entropy Hierarchy d((T , X, F), (T, X, F)) < α if the measure they put We now restrict to factors of a Bernoulli shift. on sequences of length n for all n are closer than α. Recall that T is Bernoulli relative to a factor A if there is a factor B such that T = A × B. We say that Finitely Determined (FD) Systems T is K relative to a factor A if any B > A has greater (T , F) is said to be finitely determined if, given ε, entropy. There exists a factor with respect to which there is an n and δ > 0 such that if T, F satisfies T is relatively K but not relatively Bernoulli [Orn75]. Hoffman [Hof99a], by a subtle modification of this  n n  _ i _ i example, produced uncountably many such factors d F(T (x)), F(t (x)) < δ that do not sit the same way. i=0 i=0 T has positive entropy relative to A if the entropy and the entropies of T,F, and T, F are closer than δ, of T is greater than the entropy of A. T has zero then d((T , F), (T, F)) < ε. entropy relative to a factor of full entropy. FD is the crucial idea behind our proofs, since it Rudolph proved that there are only a finite allows us to control infinite behavior by finite con- number of ways that a factor whose fibers have k structions. We prove: points can sit and that these are classified by an algebraic structure on the symmetric group on k Theorem. FD processes of the same entropy are iso- points [Rud78b]. morphic. Hoffman, using results of Rudolph, showed that the behaviors of permutations could be lifted to We then show that a specific process (e.g., an in- the ways that factors of Bernoulli shifts could sit dependent process) is FD. [Hof99a]. Feldman made a subtle modification of FD that works for flows. He used this to give a proof of the Proofs. We get the factors by taking a skew prod- technically more difficult isomorphism theorem for uct over a Bernoulli base with a counterexample. We flows that parallel the discrete time proof [Fel80]. use a criteria for Bernoulli to prove that the skew The original proof used a sequence of finer and finer product is Bernoulli. The base is the factor we are discretizations. interested in. Very Weak Bernoulli (VWB) Systems These results show how far we have come since the time of Kolmogorov. Before Kolmogorov, it T,F is said to be “very weak Bernoulli” (VWB) if, for every ε > 0, there is an n such that for all m and was not known whether B 1 , 2 had any nontrivial −1 3 3 n ≥ n and all but ε of the atoms in W F(T i(x)), factors. After Kolmogorov, essentially all that was i=−m the conditional distribution of Wn F(T i(x)) is known was that a Bernoulli shift had a factor of any i=0 closer than ε in d to the unconditional distribution. smaller entropy. It was not known whether or not a A process is FD if—and only if —it is VWB [OW74]. Bernoulli shift had a nontrivial factor of full entropy. VWB is usually the easiest to check. We now know that the relative classification of fac- tors of a Bernoulli shift mimics, to a large extent, the classification of general transformations. The mixing hierarchy (discrete time). Factors of a Bernoulli flow form another largely VWB allows us to view the relationship between neglected area. K and Bernoulli in terms of the dependence of the future on the past. VWB is such a condition; Appendix: Criteria for Bernoulli K can be described by an analogous but weaker, We will call a stationary process (T , X, F), where T condition, where we condition F(T n(x)) rather than Wn i is a Bernoulli shift, a B process. We will now give i=0 F(T (x)) on the past.

456 Notices of the AMS Volume 60, Number 4 VWB completes the traditional mixing hierarchy: tions, volume 567 of Contemp. Math., pages ergodicity, weak mixing, mild mixing, mixing, K, 67–78. Amer. Math. Soc., Providence, RI, 2012. Bernoulli. [BR88] Vitaly Bergelson and Daniel J. Rudolph, Weakly mixing actions of F∞ have infinite sub- Postscript 7 group actions which are Bernoulli, in Dynamical Systems (College Park, MD, 1986–87), volume Most of the isomorphism theory of Bernoulli sys- 1342 of Lecture Notes in Math., pages 7–22. tems was carried over to actions of unimodular Springer, Berlin, 1988. amenable groups in [OW87]. At that time we be- [BS97] Robert M. Burton and Jeffrey E. Steif, Cou- lieved that there was no reasonable entropy-like pling surfaces and weak Bernoulli in one and theory for nonamenable groups and in particular higher dimensions, Adv. Math., 132(1):1–23, for the free group on two generators where one can 1997. define a factor map from the 2-shift onto the 4-shift. [Bur78] Robert Main Burton Jr., A non-Bernoulli skew It came as quite a surprise when Lewis Bowen in product, ProQuest LLC, Ann Arbor, MI, 1978, Thesis (Ph.D.), Stanford University. [Bow10b] proved the analogue of Kolmogorov’s [Cla72] Jack Elman Clark, A Kolmogorov shift with theorem for Bernoulli shifts defined over the free no roots, ProQuest LLC, Ann Arbor, MI, 1972, group, namely, that if they are isomorphic then Thesis (Ph.D.), Stanford University. they must have the same base entropy. He went on [dHKSS03] F. den Hollander, M. S. Keane, J. Serafin, to extend this new theory to a very wide class of and J. E. Steif, Weak Bernoullicity of random groups called sofic groups (no countable group has walk in random scenery, Japan. J. Math. (N.S.), yet to be shown to be nonsofic) in [Bow10a]. This 29(2):389–406, 2003. new theory has sparked a great deal of activity in [dHS97] Frank den Hollander and Jeffrey E. Steif, the ergodic theory of actions of general groups. New On K-automorphisms, Bernoulli shifts and Markov random fields. Ergodic Theory Dynam. phenomena appear here, as was shown a few years Systems, 17(2):405–415, 1997. ago by Sorin Popa, who gave examples of factors [dHS00] , On the equivalence of certain ergodic of Bernoulli shifts over many nonamenable groups properties for Gibbs states, Ergodic Theory that were not isomorphic to Bernoulli shifts [Pop06]. Dynam. Systems, 20(1):231–239, 2000. Finally, one should mention Bowen’s recent exten- [Elo88] Kari Vaino Eloranta, Alpha congruence for sion of the isomorphism theorem to all groups (if billiards and Markov processes, ProQuest LLC, the distributions are not supported on just two ele- Ann Arbor, MI, 1988, Thesis (Ph.D.), Stanford ments) in [Bow12]. It is remarkable that the proof of University. this result makes essential use of the relative theory [Fel76] J. Feldman, New K-automorphisms and a prob- that was developed by Thouvenot for the classical lem of Kakutani, Israel J. Math., 24(1):16–38, 1976. Bernoulli shift. [Fel80] Jacob Feldman, r-entropy, equipartition, and Ornstein’s isomorphism theorem in Rn, Israel References J. Math., 36(3-4):321–345, 1980. [Fie81] Adam Fieldsteel, The relative isomorphism [AS93] Scot Adams and Jeffrey E. Steif, An applica- theorem for Bernoulli flows, Israel J. Math., tion of the very weak Bernoulli condition for 40(3-4):197–216 (1982), 1981. amenable groups, Pacific J. Math., 159(1):1–17, [FRM80] J. Feldman, D. J. Rudolph, and C. C. Moore, 1993. Affine extensions of a Bernoulli shift, Trans. [AW67] R. L. Adler and B. Weiss, Entropy, a com- Amer. Math. Soc., 257(1):171–191, 1980. plete metric invariant for automorphisms of the [GO74] Giovanni Gallavotti and Donald S. Orn- torus, Proc. Nat. Acad. Sci. U.S.A., 57:1573–1576, stein, Billiards and Bernoulli schemes, Comm. 1967. Math. Phys., 38:83–101, 1974. [Aze70] R. Azencotte, Diffeomorphismes d’Anosov et [Hof] Christopher Hoffman, Endomorphisms schémas de Bernoulli, C. R. Acad. Sci. Paris, which are measurably isomorphic to Bernoulli 270:A1105–A1107, 1970. shifts, in Information and Randomness, pages [BFK81] M. I. Brin, J. Feldman, and A. Katok, Bernoulli 1–39. Collection Travaux en Cours, Hermann diffeomorphisms and group extensions of dy- Éditeurs des Sciences et des Arts, Paris. namical systems with nonzero characteristic [Hof99a] , The behavior of Bernoulli shifts relative exponents, Ann. of Math. (2), 113(1):159–179, to their factors, Ergodic Theory Dynam. Systems, 1981. 19(5):1255–1280, 1999. [Bow10a] Lewis Bowen, Measure conjugacy invariants [Hof99b] ,A K counterexample machine, Trans. for actions of countable sofic groups, J. Amer. Amer. Math. Soc., 351(10):4263–4280, 1999. Math. Soc., 23(1):217–245, 2010. [Hof99c] , A loosely Bernoulli counterexample [Bow10b] , A measure-conjugacy invariant for free machine, Israel J. Math., 112:237–247, 1999. group actions, Ann. of Math. (2), 171(2):1387– [Hof99d] , A Markov random field which is K 1400, 2010. but not Bernoulli, Israel J. Math., 112:249–269, [Bow12] , Every countably infinite group is almost 1999. Ornstein, in Dynamical Systems and Group Ac- [Hof04a] , An endomorphism whose square is Bernoulli, Ergodic Theory Dynam. Systems, 7Contributed by B. Weiss. 24(2):477–494, 2004.

April 2013 Notices of the AMS 457 [Hof04b] , A family of nonisomorphic Markov [Orn70a] D. S. Ornstein, Imbedding Bernoulli shifts in random fields, Israel J. Math., 142:345–366, flows, in Contributions to Ergodic Theory and 2004. Probability (Proc. Conf., Ohio State Univ., Colum- [HR02] Christopher Hoffman and Daniel Rudolph, bus, Ohio, 1970), pages 178–218. Springer, Uniform endomorphisms which are isomorphic Berlin, 1970. to a Bernoulli shift, Ann. of Math. (2), 156(1):79– [Orn70b] , Bernoulli shifts with the same entropy 101, 2002. are isomorphic, Advances in Math., 4:337–352 [Kal82] Steven Arthur Kalikow, T,T −1 transforma- (1970), 1970. tion is not loosely Bernoulli, Ann. of Math. (2), [Orn70c] , Factors of Bernoulli shifts are Bernoulli 115(2):393–409, 1982. shifts, Advances in Math., 5:349–364 (1970), [Kat71] Yitzhak Katznelson, Ergodic automorphisms 1970. of T n are Bernoulli shifts, Israel J. Math., 10:186– [Orn70d] , Two Bernoulli shifts with infinite en- 195, 1971. tropy are isomorphic, Advances in Math., [Kat80] A. Katok, Smooth non-Bernoulli K- 5:339–348 (1970), 1970. automorphisms, Invent. Math., 61(3):291–299, [Orn73a] , An example of a Kolmogorov automor- 1980. phism that is not a Bernoulli shift, Advances in [KKW92] Steven Kalikow, Yitzhak Katznelson, and Math., 10:49–62, 1973. Benjamin Weiss, Finitarily deterministic gener- [Orn73b] , The isomorphism theorem for Bernoulli ators for zero entropy systems, Israel J. Math., flows, Advances in Math., 10:124–142, 1973. 79(1):33–45, 1992. [Orn73c] , A mixing transformation for which [Kol59] A. N. Kolmogorov, Entropy per unit time as a Pinsker’s conjecture fails, Advances in Math., metric invariant of automorphisms, Dokl. Akad. 10:103–123, 1973. Nauk SSSR, 124:754–755, 1959. [Orn75] , Factors of Bernoulli shifts, Israel J. [KR02] Janet Whalen Kammeyer and Daniel J. Math., 21(2-3):145–153, 1975. Conference on Rudolph, Restricted Orbit Equivalence for Ac- Ergodic Theory and Topological Dynamics tions of Discrete Amenable Groups, volume (Kibbutz Lavi, 1974). 146 of Cambridge Tracts in Mathematics, [Orn78] , Guessing the next output of a stationary Cambridge University Press, Cambridge, 2002. process, Israel J. Math., 30(3):292–296, 1978. [KS77] M. Keane and M. Smorodinsky, A class of [ORW82] Donald S. Ornstein, Daniel J. Rudolph, and finitary codes, Israel J. Math., 26(3-4):352–371, Benjamin Weiss, Equivalence of measure pre- 1977. serving transformations, Mem. Amer. Math. [KS79] Michael Keane and Meir Smorodinsky, Soc., 37(262):xii+116, 1982. Bernoulli schemes of the same entropy [OS73] Donald S. Ornstein and Paul C. Shields, are finitarily isomorphic, Ann. of Math. (2), An uncountable family of K-automorphisms, 109(2):397–406, 1979. Advances in Math., 10:63–88, 1973. [Led84] F. Ledrappier, Propriétés ergodiques des [OW73] Donald S. Ornstein and Benjamin Weiss, mesures de Sinaï, Inst. Hautes Études Sci. Publ. Geodesic flows are Bernoullian, Israel J. Math., Math., (59):163–188, 1984. 14:184–198, 1973. [Lin73] Douglas Allen Lind, Locally-compact measure [OW74] , Finitely determined implies very weak preserving flows, ProQuest LLC, Ann Arbor, MI, Bernoulli, Israel J. Math., 17:94–104, 1974. 1973. Thesis (Ph.D.), Stanford University. [OW87] , Entropy and isomorphism theorems for [Lin74] , Ergodic automorphisms of the infinite actions of amenable groups, J. Analyse Math., torus are Bernoulli, Israel J. Math., 17:162–168, 48:1–141, 1987. 1974. [OW91] , Statistical properties of chaotic systems, [Lin75] , Locally compact measure preserving Bull. Amer. Math. Soc. (N.S.), 24(1):11–116, 1991, flows, Advances in Math., 15:175–193, 1975. with an appendix by David Fried. [Lin77] , The structure of skew products with [Pes77] Ja. B. Pesin, Characteristic Ljapunov exponents, ergodic group automorphisms, Israel J. Math., and smooth ergodic theory, Uspehi Mat. Nauk, 28(3):205–248, 1977. 32(4 (196)):55–112, 287, 1977. [LS94] Douglas Lind and Klaus Schmidt, Bernoul- [Pin60] M. S. Pinsker, Dynamical systems with com- licity of solenoidal automorphisms and global pletely positive or zero entropy, Dokl. Akad. fields, Israel J. Math., 87(1-3):33–35, 1994. Nauk SSSR, 133:1025–1026, 1960. [LS03] Russell Lyons and Jeffrey E. Steif, Station- [Pol74] Stephen Henry Polit, Weakly isomorphic ary determinantal processes: phase multiplicity, transformations need not be isomorphic, Pro- Bernoullicity, entropy, and domination, Duke Quest LLC, Ann Arbor, MI, 1974, Thesis (Ph.D.), Math. J., 120(3):515–575, 2003. Stanford University. [MT78] G. Miles and R. K. Thomas, Generalized torus [Pop06] Sorin Popa, Some computations of 1-coho- automorphisms are Bernoullian, in Studies in mology groups and construction of nonorbit- Probability and Ergodic Theory, volume 2 of equivalent actions, J. Inst. Math. Jussieu, Adv. in Math. Suppl. Stud., pages 231–249. 5(2):309–332, 2006. Academic Press, New York, 1978. [Rah76] Maurice Hampton Rahe, Relatively finitely de- [MYG96] Gusztáv Morvai, Sidney Yakowitz, and termined implies relatively very weak Bernoulli, László Györfi, Nonparametric inference for ProQuest LLC, Ann Arbor, MI, 1976, Thesis ergodic, stationary time series, Ann. Statist., (Ph.D.), Stanford University. 24(1):370–379, 1996.

458 Notices of the AMS Volume 60, Number 4 [Rat74] M. Ratner, Anosov flows with Gibbs measures spin systems, Ann. Probab., 19(2):609–635, are also Bernoullian, Israel J. Math., 17:380–391, 1991. 1974. [Tho75a] Jean-Paul Thouvenot, Quelques propriétés [Rat78] , Bernoulli flow over maps of the interval, des systèmes dynamiques qui se décomposent Israel J. Math., 31(3-4):298–314, 1978. en un produit de deux systèmes dont l’un est un [RS95] Daniel J. Rudolph and Klaus Schmidt, Al- schéma de Bernoulli, Israel J. Math., 21(2-3):177– most block independence and Bernoullicity 207, 1975, Conference on Ergodic Theory and of Zd -actions by automorphisms of compact Topological Dynamics (Kibbutz, Lavi, 1974). abelian groups, Invent. Math., 120(3):455–488, [Tho75b] , Une classe de systèmes pour lesquels la 1995. conjecture de Pinsker est vraie, Israel J. Math., [Rud76] Daniel J. Rudolph, Two nonisomorphic K- 21(2-3):208–214, 1975, Conference on Ergodic automorphisms with isomorphic squares, Israel Theory and Topological Dynamics (Kibbutz J. Math., 23(3-4):274–287, 1976. Lavi, 1974). [Rud77] , Two nonisomorphic K-automorphisms [Tsa97] Anna M. Tsalenko, Stochastic stability of all of whose powers beyond one are isomorphic, Bernoulli flows, ProQuest LLC, Ann Arbor, MI, Israel J. Math., 27(3–4):277–298, 1977. 1997, Thesis (Ph.D.), Stanford University. [Rud78a] , Classifying the isometric extensions of [Yak97] Zohar H. Yakhini, Billiard approximations for a Bernoulli shift, J. Analyse Math., 34:36–60 Brownian motion, ProQuest LLC, Ann Arbor, MI, (1979), 1978. 1997, Thesis (Ph.D.), Stanford University. [Rud78b] , Counting the relatively finite factors of a Bernoulli shift, Israel J. Math., 30(3):255–263, Books and Expositions 1978. [Rud78c] , If a finite extension of a Bernoulli shift [Kat07] Anatole Katok, Fifty years of entropy in dy- has no finite rotation factors, it is Bernoulli, namics: 1958–2007, J. Mod. Dyn., 1:545–596, Israel J. Math., 30(3):193–206, 1978. 2007. [Rud78d] , If a two-point extension of a Bernoulli [KM10] Steven Kalikow and Randall McCutcheon, shift has an ergodic square, then it is Bernoulli, An Outline of Ergodic Theory, volume 122 of Israel J. Math., 30(1-2):159–180, 1978. Cambridge Studies in Advanced Mathematics, [Rud79] , An example of a measure preserv- Cambridge University Press, Cambridge, 2010. ing map with minimal self-joinings, and [Orn74] Donald S. Ornstein, Ergodic Theory, Ran- applications, J. Analyse Math., 35:97–122, 1979. domness, and Dynamical Systems, James K. [Rud82] , Ergodic behaviour of Sullivan’s geomet- Whittemore Lectures in Mathematics given at Yale ric measure on a geometrically finite hyperbolic University, Yale Mathematical Monographs, No. 5, manifold, Ergodic Theory Dynam. Systems, Press, New Haven, Conn., 1974,. 2(3-4):491–512 (1983), 1982. [Shi73] Paul Shields, The Theory of Bernoulli Shifts, Uni- [Rud83] , An isomorphism theory for Bernoulli versity of Chicago Press, Chicago, Ill.-London, free Z-skew-compact group actions, Adv. in 1973. Math., 47(3):241–257, 1983. [Smo71] Meir Smorodinsky, Ergodic Theory, Entropy, [Rud85] , Restricted orbit equivalence, Mem. Lecture Notes in Mathematics, vol. 214. Springer- Amer. Math. Soc., 54(323):v+150, 1985. Verlag, Berlin, 1971. [Sin59] Ja. Sina˘ı, On the concept of entropy for [Smo00] , Information, entropy and Bernoulli sys- a dynamic system, Dokl. Akad. Nauk SSSR, tems, in Development of Mathematics, 1950–2000, 124:768–771, 1959. pages 993–1012, Birkhäuser, Basel, 2000. [Sin62] , A weak isomorphism of transforma- [Tho02] Jean-Paul Thouvenot, Entropy, isomorphism tions with invariant measure, Dokl. Akad. Nauk and equivalence in ergodic theory, in Handbook SSSR, 147:797–800, 1962. of Dynamical Systems, Vol. 1A, pages 205–238. [Sin70] , Dynamical systems with elastic reflec- North-Holland, Amsterdam, 2002. tions. Ergodic properties of dispersing billiards, Uspehi Mat. Nauk, 25(2 (152)):141–192, 1970. [Smo73] M. Smorodinsky, β-automorphisms are Bernoulli shifts, Acta Math. Acad. Sci. Hungar., 24:273–278, 1973. [ST75] Paul Shields and J.-P. Thouvenot, Entropy zero × Bernoulli processes are closed in the d¯-metric, Ann. Probability, 3(4):732–736, 1975. [Ste88] Jeffrey Edward Steif, The ergodic structure of interacting particle systems, ProQuest LLC, Ann Arbor, MI, 1988, Thesis (Ph.D.), Stanford University. [Ste91a] , d-convergence to equilibrium and space-time Bernoullicity for spin systems in the M <  case, Ergodic Theory Dynam. Systems, 11(3):547–575, 1991. [Ste91b] , Space-time Bernoullicity of the lower and upper stationary processes for attractive

April 2013 Notices of the AMS 459 MAA-AMS Ad Feb2013_MAA 2/12/13 11:37 AM Page 1 New From the Mathematical Association of America New Horizons in Geometry By Tom M. Apostol & Mamikon A. Mnatsakanian In a remarkable display of mathematical versatility and imagination, the authors present us with a wealth of geometrical gems. These beautiful and often surprising results deal with a multitude of geometric forms, their interrelationships, and in many cases, their connection with patterns underlying the laws of nature. —Don Chakerian New Horizons in Geometry represents the fruits of 15 years of work in geometry by a remarkable team of prize-winning authors—Tom Apostol and Mamikon Mnatsakanian. It serves as a capstone to an amazing collaboration. Apostol and Mamikon provide fresh and powerful insights into geometry that requires only a modest background in mathematics. Using new and intuitively rich methods, they give beautifully illustrated proofs of results, Catalog Code: the majority of which are new, and frequently develop extensions of familiar theorems that DOL-47 are often surprising and sometimes astounding. It is mathematical exposition of the highest order. The hundreds of full color illustrations by Mamikon are visually enticing and provide 520 pp., Hardbound, 2013 great motivation to read further and savor the wonderful results. Lengths, areas, and volumes 978-0-88385-354-2 of curves, surfaces, and solids are explored from a visually captivating perspective. It is an List: $75.00 understatement to say that Apostol and Mamikon have breathed new life into geometry. MAA Member: $60.00

Six Sources of Collapse Charles Hadlock Six Sources of Collapse is a wonderful book in numerous ways. Chance, group behavior, evolutionary processes, instability, nonlinearity, and networks are adroitly brought under the same roof and applied to a stunning range of important examples, from the collapse of ancient civilizations to the collapse of financial markets. Lucid engaging primers in relevant areas of mathematics—including non-linear differential equations, network theory, and extreme value statistics—are presented with an unpretentious informality attainable only by those with the deepest command. In effect, Hadlock offers both the call to arms and the armamentarium for a unified theory of collapse. An important scientific and pedagogical contribution. —Professor Joshua M. Epstein Director of the Center for Advanced Modeling in the Social, Behaviorial, and Health Sciences, Johns Hopkins University, and External Professor, Santa Fe Institute Catalog Code: Beginning with one of the most remarkable ecological collapses of recent time, that SSC of the passenger pigeon, Hadlock goes on to survey collapse processes across the entire spectrum of the natural and man-made world. He takes us through extreme weather 296 pp., Hardbound, 2012 events, technological disasters, evolutionary processes, crashing markets and companies, the chaotic nature of Earth’s orbit, revolutionary political change, the spread and elimination ISBN 978-0-88385-579-9 of disease, and many other fascinating cases. His key thesis is that one or more of six List: $50.00 fundamental dynamics consistently show up across this wide range. These “six sources MAA Member: $40.00 of collapse” can all be best described and investigated using fundamental mathematical concepts. They include low probability events, group dynamics, evolutionary games, instability, nonlinearity, and network effects, all of which are explained in readily understandable terms.

To order call 1-800-331-1622 or online at: http://maa-store.hostedbywebstore.com/ Book Review

A Mathematician’s Lament Reviewed by William Schmidt

A Mathematician’s Lament of providing “a creative Paul Lockhart and rewarding process Bellevue Library Press, 2009 of invention and dis- US$12.95, 140 pages covery.” He suggests ISBN-13: 978-1934137178 that, by concentrating on the what and leav- A Mathematician’s Lament by Paul Lockhart pro- ing out the why, “math- vides a searing and pointed critique of mathemat- ematics is reduced to ics education in the United States. The book is built an empty shell.” on the assumption that mathematics is art. But, un- The author further like what we traditionally consider the arts, where asserts that reform ef- the patterns depend on properties of the physical forts are often ill-con- world, mathematics is about making patterns of ceived and do more ideas. The book is a lament because the author harm than good. For believes that school mathematics essentially de- example, he suggests stroys the creative aspects of mathematics. Since that attempts to make mathematics is art, he argues, it is more than the math interesting and relevant to kids’ lives often memorization of definitions, formulas, and algo- end up appearing forced and contrived. He be- rithms and should therefore be taught in a way lieves that the “glory” of mathematics is that it that encourages exploration and discovery. is irrelevant to our lives, and this is what makes My reading of this book leaves me with two dis- mathematics fun. Algebra is not about daily life, tinctly different reactions—that the lament is both he says, but rather about numbers and symmetry. realistic and unrealistic. First, the author describes He also criticizes attempts that resort to what he the characteristics that led him to conclude that calls “cutesiness” in order to reduce mathematics school mathematics is a “senseless tragedy”. I find anxiety. his characterizations to be realistic criticisms of Another point the author makes, which I think is schooling in America and consistent with my own particularly important, is that mathematics is one research. On the other hand, he makes unrealistic of the few school subjects that is typically taught assumptions about the typical student’s apprecia- without reference to its history and philosophical tion of the beauty of mathematics as an art form, underpinnings. He also criticizes the way school and this leads him to an unrealistic perspective on mathematics is considered by many—including the implications for school mathematics. parents, teachers, and students—to be a race in Let me first focus on the author’s criticisms which some students are ahead of others. He then about school mathematics in the United States. asks the crucial question: Where does this race School mathematics, he claims, stunts the imagina- lead? This is an important criticism, because treat- tions of children through a “sterile set of facts to be ing mathematics instruction as a race facilitates memorized and procedures to be followed” instead poor educational practices, such as tracking in the middle grades. William Schmidt is University Distinguished Professor in The author reserves some of his strongest criti- the Department of Counseling, Educational Psychology, cism for high school geometry, which he considers and Special Education at Michigan State University and an “instrument of the devil.” In fact, his criticism co-director of MSU’s Educational Policy Center. His email is even more strongly stated: “A geometry class is address is [email protected]. by far the most mentally and emotionally destruc- DOI: http://dx.doi.org/10.1090/noti981 tive component of the entire K–12 mathematics

April 2013 Notices of the AMS 461 curriculum.” Forcing students to follow a rigid and Given his view of school mathematics, I am dogmatic format in laying out their proofs, he ar- not surprised by the author’s rejection of what gues, destroys the very essence of what geometric he calls the “ladder myth”, which is the idea that proofs should be and undermines the students’ mathematics can be arranged as a sequence of intuition. His view is that a proof is like a work topics, with each being more advanced than the of fiction or a poem in that its goal is to satisfy. other. He believes the path through such a curricu- He actually goes on to say, “A well-written, well- lum prevents teachers and students from seeing crafted argument should feel like a splash of cool mathematics as an organic whole. He then goes water, and be a beacon of light—it should refresh on to advise mathematics teachers to play around the spirit and illuminate the mind. And it should in “Mathematical Reality” and “throw the stupid be charming.” curriculum and textbook out the window!” Math- I share in much of the author’s lament. It cor- ematics is, in general, hierarchical; I am therefore rectly characterizes much of mathematics instruc- confused about what the implications would be tion as found, unfortunately, in both the interna- for the organization of curricula. Is he suggesting tional and national research studies in which I mathematics teaching should be more episodic? have been involved. On the other hand, I believe The “ladder myth” is pretty universally present the same research suggests that the author’s as- in school mathematics in most of the countries sumption about the typical student’s comfort level that I have studied. Without some structure, I fear with mathematics is unrealistic and therefore can- many, if not most, students would find an episodic not serve as a sound basis for public schooling in journey through “Mathematical Reality” to be simi- math. In the last section he describes an imaginary lar to Alice’s adventures in Wonderland. Such an place he calls “Mathematical Reality”, which by im- approach also could exacerbate the inequalities in plication is where school mathematics should take mathematics education that are currently rampant students. In the chapter titled “Exultation”, the in U.S. schools (see Inequality for All: The Chal- author takes the reader on several explorations of lenge of Unequal Opportunity in American Schools, Schmidt & McKnight, 2012). number and shape in “Mathematical Reality”. One All in all, I found this short book an easy, example refers to the seventeenth-century inven- thoughtful, and entertaining read. The author tion of projective geometry as a means for expand- clearly is passionate about mathematics as an art, ing Euclidean geometry by removing parallelism. as a creative process. In reading this book, one can In sharing this example, the author’s words cited easily get the impression that mathematics instruc- below make very clear his view of what “doing” tion should be more like an unfettered journey into mathematics should look like. a jungle where an individual can make his or her The point is that there is no reality to own way through that terrain. I do not disagree any of this, so there are no rules or re- with his view of mathematics as a special place for strictions other than the ones we care one’s imagination. Using this vision as a guiding to impose. And the aesthetic here is principle for structuring mathematics education very clear, both historically and philo- would enrich the study of mathematics for some. sophically: if a pattern is interesting But it would desert many others. and attractive, then it’s good. (And if This brings us back to where I started this re- it means having to work hard to bend view. The author provides an accurate character- your mind around a new idea, so much ization of mathematics instruction in the United the better.) Make up anything you want, States. His passion makes the critique compel- so long as it isn’t boring. Of course ling. He persuasively rails against textbooks. But, this is a matter of taste, and tastes other than the suggestion to explore the land of change and evolve. Welcome to art his- “Mathematical Reality” and a clever and enlighten- tory! Being a mathematician is not so ing discussion of several examples, what school much about being clever (although lord mathematics should look like in a more specific knows that helps); it’s about being aes- sense is left to the reader’s imagination. I have a thetically sensitive and having refined hunch this is what the author would like. I am left and exquisite taste. with two questions: (1) What would school math- ematics look like—not at the vision level, but at I found this part of the book fun and an interest- the day-to-day instructional level—if explorations ing read. It posits a very different vision, although of “Mathematical Reality” formed the basis for one that is very abstract, of what mathematics teaching in classrooms around the world? (2) Who education should look like, one I personally could would end up being excluded from playing in the enjoy, as I am sure many others would too. But unique place known as “Mathematical Reality?” If I wonder whether the world of “Mathematical truly accessible to all, it would be a wonderful place Reality” would be accessible and interesting to all in which to learn mathematics. students.

462 Notices of the AMS Volume 60, Number 4 Quantitative Approaches to Sustainability Seminars Rachel Levy

This article appears in celebration of Mathematics Awareness Month, an event sponsored annually in April by the AMS, the American Statistical Association, the Mathematical Association of America, and the Society for Industrial and Applied Mathematics. For 2013 the MAM theme is Mathematics and Sustainability. To find out more about MAM, visit the website http://www.mathaware.org/.

How can mathematicians contribute to education Hot Air (SEWTHA), could be used in a seminar to about sustainability? Mathematicians study cli- introduce students to principles of sustainability. mate change, energy-related technologies, models of energy availability, production and consump- Why Should Mathematicians Provide a tion, and even the political and social aspects of Sustainability Seminar? sustainable legislation and practices. However, at We want students to be savvy consumers of this point, few courses on sustainability can be quantitative information about sustainability in found in mathematics department offerings. When the popular and technical press. We also want we consider problems that our current and future them to apply their mathematical skills to model students will face, energy sustainability certainly energy problems and solutions. To do so we must seems important. But how many of these ideas help them avoid common misconceptions such reach our classrooms? as “million=big” and “every little bit counts” that We can incorporate sustainability into our persist when communicators, policymakers, and existing courses through problem sets, projects, readers fail to comprehend which magnitudes capstones, or independent studies. We can also really matter. We also want mathematicians and reach a broad audience through the topical semi- their students who pursue technical research to nars offered at many colleges and universities. be able to communicate how their particular focus These seminars, sometimes specially designed of study (such as a climate change model) fits into for first-year students, tend to be characterized the big sustainability picture. While the concepts by small class size, discussion, and project-based in SEWTHA may differ from those in traditional learning, and perhaps by speakers and field trips. mathematics curricula, MacKay’s text has success- When students of all majors enter these courses fully provided students of varying mathematical backgrounds with a quantitative conceptual foun- they bring a wide range of mathematical prepara- dation regarding sustainability energy practices. tion, confidence, and enthusiasm, which can make mathematical versions of these seminars challeng- A Free Online Text: Sustainable Energy ing to design. A seminar in sustainability offered Without the Hot Air by mathematicians can provide students with an Using only elementary mathematics, SEWTHA attractive path toward quantitative literacy. In takes a quantitative perspective to give the reader this article I will explore how David MacKay’s free context for further technical exploration of sus- online textbook, Sustainable Energy Without the tainability. The primary mathematical content in the energy calculations includes scaling, conver- Rachel Levy is associate professor of mathematics at Har- sions between units, and estimation. Algebraic vey Mudd College. Her email address is [email protected]. expressions appear in energy estimates but do not DOI: http://dx.doi.org/10.1090/noti979 require much manipulation. Thus the text does not

April 2013 Notices of the AMS 463 discuss traditional university mathematics topics. Throughout the book, SEWTHA focuses on Instead it: calculations for the UK, which provides an inter- •demonstrates that any feasible plan for sus- esting opportunity for students to pursue similar tainability has to “add up”—that is, supply and calculations for a region of interest. Because many demand must be balanced; issues of feasibility relate to population density, •bases simple and memorable quantitative esti- students performing the same types of calcula- mates on underlying physical principles; tions might reach conclusions different from those •promotes comprehension and retention through in SEWTHA. The text explicitly does not deal with analogies between hard-to-grasp and well- some important aspects of sustainability, such as understood quantities; and economic feasibility and social acceptability, and •probes myths and misconceptions about energy only touches on the issue of environmental impact. and replaces them with factual rather than ethi- These questions are put aside in favor of a quanti- cal assertions. tative view, which can be seen as a prerequisite for The first third of the text, “Numbers Not Adjec- tackling the other aspects of the problem. tives”, creates a two-column account balancing energy supply and demand. Each chapter adds to Assignments for a Sustainability Seminar the balance sheet by expressing a type of produc- To envision a sustainability seminar using SEWTHA, tion or consumption in terms of a common unit: I have designed assignments in four categories: energy in kilowatt-hours (kWh) or power in kilo- calculation, discussion, communication, and inven- watt-hours/day (where 40 watts = 1 kWh/d). Major tion. As MacKay notes, convictions are stronger if sources of demand include transport, heating and they are self-generated rather than taught, so each cooling, lighting, information systems and other of the projects is structured to help students form gadgets, food, and manufacturing. The sustainable and develop ideas about sustainability rather than supply includes wind, solar, hydroelectric, wave, simply receive them. tidal, geothermal, and possibly nuclear energy Calculation: Students can perform similar calcula- sources. Most calculations boil down to the equa- tions in a new context. Completing the following tion energy = power × time (with the analogy to types of studies and comparisons can provide the equation volume = flow × time) and sometimes students with a meaningful sense of the scale of kinetic energy = 1/2mv 2. Technical questions are the energy issues and sustainable solutions. addressed as asides or at the end of the chapters 1. Redo calculations for a geographic region of in a question-and-answer format, along with an interest. For example, calculate the cost of air con- annotated bibliography. ditioning/heating in different parts of the United The second third of the book focuses on the States using average daily temperatures. idea that “Every Big Helps”. That is, unlike the 2. Estimate each student’s yearly energy con- conventional wisdom that “every little bit counts”, sumption. the mathematics of sustainability demonstrates 3. Compute the energy per trip required for the unfortunate truth that if everyone only does a some car/bus/train trip the student has made reg- little bit to help, the sum of the efforts will address ularly. Compare this energy to some other activity. only a little bit of the problem. If we want to alter 4. Calculate the area required for energy produc- our behavior to improve energy sustainability, tion installations based on some physical dimen- we need to understand which changes will make sion the student knows well (home acreage, soccer a significant dent in energy consumption. Then, field, campus). through a series of calculations and comparisons, 5. Choose a vacation destination that would MacKay explains how we might significantly in- require air transport. Choose a favorite electronic crease supply or reduce demand. For example, gadget. Compare the energy cost between the we might shift to electric vehicles with batteries transportation and the powering of the gadget. charged by wind or solar. 6. Figure out which gadgets in the dorm or The last part of the book, called “Technical college “suck” the most energy. Buy a meter to Chapters”, revisits the items on the balance sheet determine which devices (sometimes known as from the first chapters and reviews how the tech- vampires) require the most energy when they are nologies might be improved. The mathematical “off”. details are still fairly simple but discuss energy 7. Conduct a lifecycle assessment of both energy concepts such as force and efficiency in addition to requirement and use for a product or system. power. For example, in heating a structure, power 8. Compare the lifecycle assessments for mul- used = (average temperature difference × leakiness tiple ways to accomplish the same task (such as of the building)/(efficiency of heating system). More landscaping a yard). advanced undergraduate topics such as a partial Discussion: Through discussion and debate, stu- differential equation (the heat equation) appear as dents can wrestle with the choices required to cre- supplementary information. ate and achieve sustainability goals. Discussions

464 Notices of the AMS Volume 60, Number 4 can be facilitated by groups of students who, in ad- course, accessible to many students, using SEWTHA vance, prepare ways to stimulate the conversation. as a starting point. I hope to hear from those of 1. Discuss a chapter from the text or a related you who choose to do the same. (and perhaps more mathematically focused) ar- ticle. Acknowledgments 2. Identify and discuss both factual and ethi- I would like to thank David Keffer of the University cal assertions in popular (and perhaps scientific) of Tennessee, Leidy Klotz of Clemson University, articles. Erin Byrne of Olin College, and Ellen Swanson of 3. Debate the pros/cons and viability of various Centre College for helpful conversations. energy options, such as solar versus wind. Mathematical Reading and Resources Communication: Student projects can focus on Many syllabi, articles, datasets, and other resources communicating the messages of MacKay’s book, can easily be found online. The SEWTHA website or their own ideas about mathematics and sus- itself contains materials for teachers. References tainability. for a few additional resources are included here. 1. Choose a quantitative myth about sustain- 1. The National Renewable Energy Lab (NREL) ability and develop a way to communicate the website contains datasets, publications, and con- misconception and truth to the public (such as a tact information for potential speakers, http:// memorable cartoon, graphic, or video). www.nrel.gov/. 2. Extract/write/share the underlying principles 2. Free online course materials developed by and take-home messages of a particular chapter. Leidy Klotz at Clemson are available at https:// 3. Design an outreach activity (perhaps a game) canvas.instructure.com/courses/593350. that would help kids explore some ideas from the 3. Online resource for mathematical model- text about mathematics and sustainability. ing and sustainability, http://home.comcast. 4. Identify a misleading idea communicated in a net/~benfusaro/Book/FrontMatter.html. popular article. Write a press release or advertise- 4. Mathematical Modeling of Earth’s Dynamical ment refuting the misconception and promoting Systems: A Primer, by Rudy Slingerland and Lee the facts. Address how the misconception could Kump, Princeton University Press, 2011. affect public opinion in a harmful way. 5. United Nations report on sustainability, Invention: Students can investigate and develop http://www.un.org/Depts/dhl/resguide/ ways for current businesses, common practices, specenv.htm#commission. and systems to be made more sustainable. Projects 6. A Joint Initiative of the North American Math- that involve real clients can motivate hard work ematical Institutions: Climate Change, Sustainabil- and innovation. ity and the Mathematical Sciences, http://www. 1. Learn how a local company performs energy mathinstitutes.org/climate-initiative/. audits. Then perform an energy audit for a local business and make some recommendations to significantly improve sustainability practices of the business. 2. Consider a common practice at the college or university and determine a feasible plan to make the practice more sustainable. Examples include food usage, garbage/recycling practices, energy use, transportation. 3. Create a new way to take advantage of an existing or new technology. Many more ideas for discussion topics and projects can be found online. An example of a nice companion to MacKay’s book has been developed as a freely available online course through Clemson (see reference below). As a mathematician, when I first read SEWTHA, I was hoping to find material that I could directly incorporate into my existing courses. Instead, I gained a new quantitative perspective on sustain- ability to keep in mind when considering technical information in the future. If I decide to pursue a new avenue of research related to sustainability, I will make my choice with a better understanding of the potential impact of my work on society. At the same time, I am inspired to offer a quantitative

April 2013 Notices of the AMS 465 Remembering Steve Rallis

Jim Cogdell and Dihua Jiang, Coordinating Editors

Steve Rallis was born in and their L-functions. His work was very original Bennington, Vermont, and has left a lasting impact on number theory in 1942. He was an un- and representation theory. dergraduate at Harvard, receiving his BA in 1964. Michael Harris From there he moved Steve Rallis used to apologize before explaining his to MIT where he was a new ideas. It was one of his most endearing habits, student of Bert Kostant, and it was always a cue to pay close attention. receiving his Ph.D. in Number theorists have been especially attentive 1968. His early work to his work of the 1980s and 1990s in connection was on invariant the- with what is called the Rallis inner product formula, ory. After graduating which calculates the inner product of a pair of from MIT, Rallis spent Steve Rallis theta functions as a special value or residue of a two years at the IAS in Langlands L-function. As developed and refined in Princeton and two years at SUNY StonyBrook, an extended series of papers with Piatetski-Shapiro, followed by a number of visiting positions at Stras- this formula provides an effective classification of bourg, Texas, Notre Dame, and Princeton. Along poles of L-functions of classical groups. Combined the way, the primary focus of Steve’s research with Shimura’s explicit local calculations (which shifted to the theory of automorphic forms and inexplicably fail to refer to Rallis’s work), Rallis’s representation theory, but always keeping invari- ant theory as one of his powerful techniques. In formula provides the most natural starting point 1970 Steve married Michele Kaufmann, who had for the study of p-adic L-functions of classical received her Ph.D. in astronomy from Harvard in groups; a striking application is the proof by the same year that Steve graduated from MIT. In Skinner and Urban of the Iwasawa-Greenberg main 1977 he became a visiting assistant professor at conjecture for elliptic modular forms. Kudla and his Ohio State University. He became a tenure-track collaborators have adapted the formula to analyze assistant professor the following year and moved the central derivatives of L-functions in the setting up through the ranks, staying at OSU throughout of a vast program, combining representation theory the rest of his career. But he always kept a busy with Arakelov theory, to generalize the Gross-Zagier traveling schedule, making regular visits to both formula. Strasbourg and Israel, where he had active col- Number theory has yet to absorb all the lessons laborations for many years. He became professor of Rallis’s more recent work, but the process is emeritus at OSU in 2008 and passed away in 2012. beginning. His series of papers with Ginzburg and The primary focus of Steve’s research was Soudry provided the basis for the results of Jiang automorphic forms, automorphic representations, and Soudry on the local Langlands correspondence for generic representations. His papers on the Jim Cogdell is professor of mathematics at Ohio State Uni- relative trace formula, mostly with Jacquet, have versity. His email address is [email protected]. been of crucial importance in recent work of Wei edu. Dihua Jiang is professor of mathematics at University of Michael Harris is professor of mathematics at Uni- Minnesota. His email address is [email protected]. versité Paris-Diderot Paris 7. His email address is DOI: http://dx.doi.org/10.1090/noti964 [email protected].

466 Notices of the AMS Volume 60, Number 4 Zhang and others on the global and arithmetic to rewrite a short note of his which complemented Gross-Prasad conjectures.  one of his old papers. I wish I could still hear him Steve was always a collaborative mathematician. say, “Hi, Gerard. Let’s start a new project. What are Of the ninety-four citations listed on MathSciNet, all you up to….”  but seven are joint papers. Among his seventeen collaborators were Gerard Schiffmann, with whom Hervé Jacquet he collaborated off and on for forty years, as well as I wrote a few papers with Steve Rallis. Since Steve Hervé Jacquet and part of his “Israeli team” David had many collaborators at the same time, this was Ginzburg and David Soudry. somewhat of a challenge because, at times, one had to struggle to get his attention. At a conference, for Gerard Schiffmann instance, one had to wait patiently in line to get a “Hi, Gerard. What’s up? I found this paper; it might chance to talk to Steve. Moreover, it was difficult to be useful for us!” This was a typical start of a day keep track of the ongoing work of Steve with other working with Steve. Never short of ideas, always mathematicians (work which could be relevant to enthusiastic, very tenacious—once convinced that my work with Steve). On the other hand, once one something should work, he would never give up could get Steve’s attention, working with him was until he succeeded. a pleasure. I still have lengthy unpublished notes I first met Steve in Princeton. Both of us were from our collaborations. spending the academic year 1969–1970 at the IAS. Steve’s erudition was astonishing. Rather than We started to work on the oscillator representation. looking up a reference in the mathematical litera- It took us several years, two of which Steve spent ture, it was much easier to call Steve on the phone in Strasbourg. It was perhaps at that time that to find out the facts. he adopted his amazing work schedule: six or Of the papers we wrote together the most interesting ones are also the most tentative. There seven days a week from early in the morning are papers where we conjecture or simply suggest to late in the evening! Steve went back to the new relative trace formulas. Steve’s erudition was States and eventually settled in Columbus. Also of paramount importance in formulating these that year in Princeton I met Michele for the first conjectures. time, just a few months before their wedding.  Later during their stay in Strasbourg my wife David Ginzburg and and I had many occasions to visit with them. Early in their relationship Michele recognized David Soudry Steve’s gift for mathematics and his devotion to “Hey, the first team is here!” We still hear Steve’s research. Remarkably, throughout Steve’s career voice, greeting us happily, each time the three she managed to maintain her career in astronomy of us met. Indeed, we were a team, and Steve while facilitating his life in mathematical research. was our captain. Steve used to come to Tel Aviv I met Steve again in 1987. He had invited me and visit Piatetski-Shapiro, already in the early to Columbus, the first of a long series of visits. 1980s. Each one of us was a graduate student Mathematically, he had matured and within a few in those days. Steve had a keen interest in our years he had become a leading expert in his field, work, listened to questions, gave his advice, and due in part to his collaboration with Piatetski- shared his vast knowledge and his personal ideas Shapiro. Feeling less pressure, he relaxed and generously. Steve became our mentor, and as the enjoyed more of the social life of the university. He years went by, we teamed up and worked together, did his best to play the absent-minded professor, visiting each other frequently. Working with Steve hardly visible among the piles of papers in his was a great experience for us, still very vivid in our office, but I remember many discussions with minds: Steve’s great passion for mathematics, his him on topics outside mathematics. He remained enthusiasm and devotion; times of breakthrough, highly focused on his work but was very perceptive moments of frustration; meeting together the next of everything around him. He had a sharp mind, morning and sharing our separate thoughts of the sometimes a sharp tongue, but above all he was a night before; taking a break and getting coffee very generous person with his collaborators and or ice cream, playing pool, or bowling. Together his students. with Steve, we did our best mathematical work. The last years were sad. Until the end, he tried Hervé Jacquet is professor of mathematics at Columbia Uni- to go on with his work, but he was less and less versity. His email address is [email protected]. able to do so. The last thing we did, in 2011, was David Ginzburg is professor of mathematics at Tel Aviv Uni- Gerard Schiffmann is professor of mathematics at versity. His email address is [email protected]. the Université de Strasbourg. His email address is David Soudry is professor of mathematics at Tel Aviv [email protected]. University. His email address is [email protected].

April 2013 Notices of the AMS 467 a hub of activity and attracted speakers from all over the U.S. and beyond on a regular basis. It became one of Steve’s trademarks as is reflected by his email address: [email protected]. Mathematics was a top priority for Steve. He pursued it tirelessly and was extremely effective in multitasking. At the same time, Steve loved his daily routine and never seemed in a hurry. Steve cared a lot about the development of his current and former postdocs and took a keen interest in their mathematical upbringing. He will be remembered as a great mathematician who largely shaped the field of automorphic forms.  Steve never had that many students. In his entire career at OSU he had five students total, beginning Rallis and his coworkers. with Dihua Jiang, who later became a collaborator. Steve was a pretty demanding advisor, probably because he also treated his students as if they were collaborators. This could be quite daunting in the We feel very privileged to have known Steve, to beginning, but if one survived, the experience was have learned from him and worked with him. quite rewarding. Among Steve’s later students was Working with Steve was accompanied by many Ameya Pitale. conversations about life, personal memories and experiences, dreams, hopes, and fears; history, politics, literature, art, films, and so much more. Ameya Pitale He was a true friend. Steve will always live in our I was very fortunate to have Steve as my thesis minds: Steve the great mathematician and Steve advisor. He was very generous with his time and ideas. I recall one summer when I used to meet our great beloved friend.  with Steve and Cary Rader every day for a couple of Steve was known as an excellent postdoc mentor. hours. This was before I had obtained any results This is probably because when he brought post- towards my thesis, and it was very frustrating to docs to OSU, he expected to collaborate with them me when every angle of approach seemed to fail. I and treated them the same way he treated his col- still remember Steve’s comment that I was actually laborators, as equal partners in their endeavors. learning to be a mathematics researcher—you get He was eager to share ideas and engage mathe- to a good idea only after hitting a million dead matically with them. All of his postdocs left OSU ends, and the key is to keep working. This advice as stronger mathematicians than when they ar- has stayed with me since then and has indeed rived, and they continued to collaborate with Steve helped to shape me into the mathematician that I long after they left. Among his postdocs were David am today. Ginzburg, above, and Erez Lapid.  Steve was a definite presence in the OSU mathe- Erez Lapid matics department. He would come in every morn- ing and start making his rounds. He would check Steve Rallis was a devoted mathematician, full of in with the staff, visit with his colleagues, talk with ideas and very generous in sharing them. I’ve known his visitors and his post-docs, meet with his students Steve from his frequent visits to Israel in the 1990s (usually in the lounge with others present), and still to work with his close collaborators David Ginzburg manage to teach his classes. He seemed to be ev- and David Soudry. Typically, he would come in erywhere, except his office; there was no point in December to avoid the long, idle winter vacation in trying to call him during the day. But he came back the U.S. I became closer to him as a postdoc at the to the department after dinner to work and kept Ohio State University during the years 1999–2002. the same schedule on the weekends; then you had This was the most productive period of my career, a better chance of finding him in his office working, and I couldn’t have asked for a more supportive, but still ready to talk mathematics. His work ethic dedicated mentor. The “harmonic analysis and was an inspiration to all. automorphic representations” seminar (known as Steve could also be a polarizing presence in the the “HAAR seminar”) that Steve organized became department. Steve was deeply committed to the OSU Erez Lapid is professor of mathematics at the Hebrew Ameya Pitale is professor of mathematics at the University University of Jerusalem and Weizmann Institute of Science. of Oklahoma. His email address is [email protected]. His email address is [email protected].

468 Notices of the AMS Volume 60, Number 4 mathematics department and had definite ideas about the direction that the OSU department should take in terms of hiring and growth…and he had no qualms about sharing these. Two of his long-time colleagues in the department were Paul Ponomarev and Bob Stanton. Paul Ponomarev and Bob Stanton Historical sources depict Socrates as a “gadfly” The Association for Women in who strolled the agora and challenged the “wise” is a professional men with simple questions—of course, eventually, Mathematics (AWM) it cost him his life. In our department Steve was an society whose goal is to encourage women incarnation of his ethnic ancestor. The agora—all and girls to pursue careers in the the floors of the department, most importantly mathematical sciences, and to promote the chairman’s suite; the “wise” men—colleagues equal opportunity and equal treatment of who sounded off authoritatively. It was our good women in the mathematical sciences. fortune to have Steve come by our offices three to five times a day, but few of our chairs were AWM sponsors a wide variety of activities enlightened enough to appreciate his several daily for women at all levels, from middle school visits to them. Was he more than a “gadfly”? to university faculty. AWM programs During his time with us he was a positive force in include research conferences and workshops, the department, a presence that could never be lecture series and prizes, travel grants, an ignored. He tried to stop bad appointments but was not always successful, and he championed essay contest, and day-long workshops for excellent appointments; he ridiculed, perhaps middle and high school girls. not always within collegial bounds, misguided policies advocated by certain “leaders” in the OSU AWM is currently accepting nominations department. He left his mark on the department, for two awards (deadline April 30): but with time any mark wears off, and his leadership • The Gweneth Humphreys Award for has been sorely missed since the onset of his illness. His presence in the department gave it a Mentorship of Undergraduate visibility not seen since, whether it be through Women in Mathematics the conferences he helped organize at OSU, the • The Louise Hay Award for speakers he brought to the HAAR seminar, the Contributions to Mathematics collaborators and postdocs who came for long Education visits—from Israel, the United States, , and Asia. In preparation for our fall election, we also He was also a friend—faithful when in need and with a ready amusing story when glum. In strongly encourage nominations for the return, we had ample opportunity to respond in following AWM officers: kind. We used to kid him about his purported • President resemblance to Jerry Garcia until the day on High • Clerk Street when visitors to the university stopped • Four Members-at-Large us to ask for Jerry’s autograph. Until the end, we could tease him about his classes: the world expert on automorphic forms who became the Please send your suggestions to awm@ prime expositor of real analysis to mathematics awm-math.org education majors, acquiring the reputation among them as the “teddy bear” of the department. We Are you interested in helping more girls can’t kid him anymore, but we will never stop discover a love of mathematics? Would you telling stories about him.  like to see more women build successful Paul Ponomarev is professor of mathematics at Ohio careers in the field? Join AWM now and State University. His email address is ponomar@math. get involved! ohio-state.edu. Bob Stanton is professor of mathematics at Ohio http://awm-math.org State University. His email address is stanton@math. ohio-state.edu.

April 2013 Notices of the AMS 469 Doceamu s doceamus . . . let us teach

An Update on Jobs for Doctorates in Mathematics Education at Institutions of Higher Education in the United States Robert Reys, Barbara Reys, and Anne Estapa

For more than two decades, institutions of higher What has been the impact of these reductions of education have reported shortages for doctorates revenue on jobs for doctorates in mathematics in mathematics education; i.e., there have been education in institutions of higher education? This more jobs announced for doctorates in mathemat- study reports results from a survey of national ics education than qualified people to fill them [1], searches by institutions of higher education for [2], [3]. During the last five years there has been new faculty members in mathematics education. a major downturn in the national economy in the United States. The bleak economic conditions have Survey Procedures resulted in declines in state revenues, which have Beginning in September 2011 and continuing to created decreases in state expenditures. Conse- April 2012, announcements of job searches for quently, nearly all of the state legislatures have re- tenure-track positions in mathematics education duced funding for institutions of higher education. at institutions of higher education were compiled. Over one hundred institutions of higher educa- Robert Reys is Curators’ Professor Emeritus of mathemat- tion were initially identified as searching for at ics education in the College of Education at the University least one new tenure-track faculty member in of Missouri. His email address is [email protected]. mathematics education. Job announcements from Barbara Reys is Curators’ Professor and Lois Knowles Faculty Fellow in the College of Education at the University of Missouri. Her email address is [email protected]. The senior author acknowledges IRD support from the National Science Foundation which allowed this work to Anne Estapa is assistant professor of mathematics edu- be conducted. However, the content and opinions in this cation at Iowa State University. Her email address is [email protected]. report are the authors’ only and do not necessarily reflect any position of the NSF. Members of the Editorial Board for Doceamus are: David Bressoud, Roger Howe, Karen King, William McCallum, The authors gratefully acknowledge the chairs of the and Mark Saul. search committees of institutions for their willingness to take the time to complete the survey and contribute the DOI: http://dx.doi.org/10.1090/noti976 information on which this report is based.

470 Notices of the AMS Volume 60, Number 4 institutions outside the U.S., and postdocs or hires to teach a wider array of upper-division nonregular appointments were eliminated. In April mathematics courses, and a few (about 20 percent) 2012, a personal email message was sent to the expected the new hires to teach graduate courses search chair at each institution on the remaining in mathematics. search list (N =104). Twelve respondents indicated New faculty hires in departments/colleges of that, due to budget reductions and some major education were typically expected to teach under- policy changes, their search had been abandoned. graduate courses focusing on methods for teaching The chairs of the search committees of the remain- mathematics (elementary, middle, and secondary ing ninety-two institutions were asked to complete levels) and student-teaching supervision. The an online survey on the status and success of their graduate teaching expectations in colleges/depart- job searches. Subsequent emails to nonrespon- ments of education varied greatly and ranged from dents together with personal phone calls resulted leading seminars on current issues to teaching spe- in responses from eighty-two institutions (89 cific courses, including curriculum, assessment, percent return rate). learning theory, technology, and research, each Job Expectations with a specific focus on mathematics education. Research expectations varied greatly across Over 90 percent of the positions announced were at the assistant professor level, with the institutions but tended to be closely aligned with others indicating their rank was open depending the Carnegie Classification shown in Table 1. Thus, on the qualifications of the applicants. Teaching while some research was encouraged by baccalau- expectations for the posted positions were very reate and master’s institutions hiring new faculty, similar to the profiles previously reported [3]. their primary focus was on teaching, and this was More specifically, the teaching responsibilities reflected in teaching loads of 12 to 15 credit-hours for faculty joining mathematics departments typi- per semester. Doctoral-granting institutions had cally included teaching a range of undergraduate lower teaching loads (typically 3–6 credit-hours per mathematics courses (e.g., content courses for semester), along with a clearly stated priority for future elementary/middle/secondary teachers, the applicant to establish a research agenda and including calculus and linear algebra). Some in- develop a productive record of scholarship while stitutions (about 40 percent) looked to these new working toward tenure and promotion.

Table 1. Number of positions (and hires) in mathematics education by type of Carnegie institution. Carnegie Classification Total # of Total # of Math Education Joint Percent of positions institutions Department successful searches Bac/Diverse: Baccalaureate 1 1 1 Colleges—diverse fields (0) 0 Bac/A&S: Baccalaureate Colleges— 2 2 1 1 Arts & Sciences (0) (0) 50% Master’s S: Master’s Colleges and 2 2 2 Universities (smaller programs) (2) 100% Master’s M: Master’s Colleges and 1 1 1 Universities (medium programs) (0) 0 Master’s L: Master’s Colleges and 27 24 22 5 Universities (larger programs) (16) (5) 78% DRU: Doctoral/Research 5 4 3 2 Universities (3) (2) 100% RU/H: Research University (high 26 23 8 16 2 research activity) (6) (10) (1) 65% RU/VH: Research University (very 22 17 3 17 2 high research) (3) (16) (2) 95% Other unidentified institutions 8 8 3 5 (2) (5) Total 94 82 44 46 4 (32) (38) (3) Percent of successful searches 72% 82% 75% 77%

Note: There were a total of 94 positions and 73 total hires. (n) reports the number of hires in each cell.

April 2013 Notices of the AMS 471 Table 2. Percent of positions* reporting salaries within a range. of the mathematics department positions were in the range of US$40,000–$50,000. Salary Range Education Mathematics Joint (US$) Department/ Department Appointment Further examination revealed the institu- College tions offering the lower salary range were in the Carnegie baccalaureate and master’s $40,000–50,000 2 9 0 classifications, and these institutions were $50,001–60,000 20 25 3 least likely to make a hire, suggesting the $60,001–70,000 23 6 1 importance of having a more competitive $70,001–80,000 4 5 1 salary. The other discrepancy was in the salary range of US$60,001–$70,000, where Above $80,000 2 0 0 the percent of positions in education (23 *Based on 84 responses to the salary range question. percent ) was nearly four times greater than in mathematics departments (6 percent). Information about the Job Opportunities Respondents were also asked, “What best de- and Success in Hiring scribes the start-up package associated with your Table 1 shows the Carnegie Classification of each of position?” The statement went on to say that a the eighty-two institutions1 included in the study. start-up package might include items such as a Their current classification was confirmed at http:// computer, software, travel, and moving expenses. classifications.carnegiefoundation. Start-up packages ranged from none (about 6 org/lookup_listings/institution.php. The percent of the positions) to more than US$20,000 eighty-two searching institutions reported a total (provided for about 8 percent of the positions). of 94 positions, and these positions were about About one-third of the institutions offered between equally distributed between mathematics depart- US$1,000 and US$5,000, and about one-eighth ments (44), education (46), and joint appointments of the institutions offered between US$10,000 (4). Table 1 shows that nearly three-fourths (72 and US$20,000 for start-up. There were no major percent) of the positions in the mathematics de- differences between the positions in start-up partments and about 82 percent of the positions packages between mathematics departments and in education were filled. The baccalaureate- and education. In addition, a majority of institutions master’s-level institutions had the most difficulty offered new faculty members a reduction in teach- in successfully making a hire. ing load their first year. The searches resulted in 73 hires, of which 65 had either a Ph.D. or Ed.D. in mathematics educa- Some Implications tion, 6 had a Ph.D. in mathematics, and two had This study documents that, while there is not the doctorates in other fields. In each case the institu- excess of jobs for doctorates in mathematics in tion hiring a Ph.D. in mathematics noted that their institutions of higher education that there has been hire had either done some academic work in math- in the past [2], [3], there are still jobs for doctorates ematics education or expressed a keen interest in in mathematics education that were unfilled. This mathematics education. Of the 73 hires reported, study showed that of the 70 percent of the insti- 37 were new doctorates in mathematics education tutions that began their job searches early in the and 36 were experienced faculty members moving fall, more than 25 percent filled their positions by from one institution to another. Thus nearly one- December or January. Thus candidates interested half of the new hires were moving from one institu- in these positions are well advised to get their ap- tion to another, thereby creating a new opening in plication papers ready early so they are prepared mathematics education at the previous institution. when job announcements are forthcoming. Institu- This seemingly high rate of mobility among faculty tions that wait until late in the academic year to members in mathematics education is consistent announce a position in mathematics education are with results reported in 2006 [3]. at a disadvantage, as many potential applicants Salaries and Start-up Packages have already been hired. In addition to the unfilled positions for Respondents were asked, “What range best de- 2012–13, over 25 percent of the institutions in- scribes your projected salary for this position?” dicated that they will be searching for another Table 2 summarizes the responses. The salary ranges of searches in mathematics and education doctorate in mathematics education for 2013–14. departments/colleges are similar for several ranges So, even in the current austere financial environ- but differ in the lowest range, where 9 percent ment, jobs in mathematics education exist. And when you consider the fact that about one-half of 1Eight of the institutions did not include their name but these announced positions were filled by math- completed the survey. Their Carnegie classification was ematics educators moving from another institu- not known, so their data are included in the “Other” tion, thereby creating another position, it is likely category. that the job opportunities within mathematics

472 Notices of the AMS Volume 60, Number 4 education for doctorates in mathematics education will remain strong. In closing, it should be noted that this survey THE CHINESE UNIVERSITY OF HONG KONG focused only on nationally announced job searches that were distributed in primary outlets. Many Applications are invited for:- regional institutions do not conduct national Department of Mathematics searches, and this study does not reflect many Research Assistant Professor regional postings. In fact, the majority of the (Ref. 1213/104(576)/2) (Closing date: May 15, 2013) Applications are invited for a Research Assistant Professorship in positions identified in this study were in doctoral- any area of mathematics. Applicants should have a relevant PhD granting institutions, even though there are many degree and good potential for research and teaching. Appointment more baccalaureate or master’s-level institutions. will initially be made on contract basis for up to two years Furthermore, there were no associate degree commencing August 2013, renewable subject to mutual agreement. awarding institutions, even though more than Salary and Fringe Benefits Salary will be highly competitive, commensurate with qualifications 1,700 junior colleges exist. So it is likely that this and experience. The University offers a comprehensive fringe report significantly underestimates the number of benefit package, including medical care, a contract-end gratuity for jobs available in institutions of higher education an appointment of two years, and housing benefits for eligible for doctorates in mathematics education. appointee. Further information about the University and the general terms of service for appointments is available at http://www.per.cuhk.edu.hk. References The terms mentioned herein are for reference only and are subject to revision by the University. [1] J. Hiebert, J. Kilpatrick, and M. Lindquist (2001), Im- proving U.S. doctoral programs in mathematics edu- Application Procedure Please send full resume, copies of academic credentials, a cation, In R. E. Reys and J. Kilpatrick (eds.), One Field, publication list and/or abstracts of selected published papers, Many Paths: U. S. Doctoral Programs in Mathematics together with names, addresses and fax numbers/e-mail addresses of Education (pp. 153–162). Washington, DC: American three referees to whom the applicants’ consent has been given for Mathematical Society/Mathematical Association of their providing references (unless otherwise specified), to the America. Personnel Office, The Chinese University of Hong Kong, Shatin, [2] R. E. Reys (2000), Doctorates in mathematics educa- N.T., Hong Kong (Fax: (852) 3943 1462) by the closing date. The Personal Information Collection Statement will be provided upon tion: An acute shortage, Notices of the Amer. Math. request. Please quote the reference number and mark ‘Application Soc. 47, 1267–1270. - Confidential’ on cover. [3] ______(2006), Jobs in mathematics education in institutions of higher education in the United States, Notices of the Amer. Math. Soc. 55, 676–680.

AMERICAN MATHEMATICAL SOCIETY

NEW! AMS Titles Now Available Electronically on Google Play

e AMS o ers over 2,500 pure and applied mathematics titles in eBook form via Google Play.

Your favorite texts (even the out-of-print ones!) are now fully searchable and easily The AMS o ers over 2,500 pure readable on web-browsing devices. and applied mathematics titles in eBook form via Google Play. Start reading today at: https://play.google. com/store. Your favorite texts (even the out-of-print ones!) are now fully searchable and easily readable on web-browsing devices.

Start reading today at: play.google.com/store

April 2013 Notices of the AMS 473 Aschbacher and Caffarelli Awarded 2012 Wolf Prize

and tightly embedded subgroups and brought the theory of groups of odd characteristic type close to completion. Turning next to groups of characteristic type, Aschbacher handled all of the most difficult cases, notably the Thin Group case, the p-Uniqueness Case, and finally the Quasithin Case. This last result, contained in two massive monographs written jointly with S. D. Smith, completed the clas- sification of finite simple groups. In the process, he significantly advanced the theories of GF (2) representations, Thompson factorization, and The 2012 Wolf Prize in Mathematics has been pushing up. awarded to: Also worthy of mention are Aschbacher’s work Michael Aschbacher, California Institute of Technology, for his work in classifying finite on maximal subgroups of finite simple groups, simple groups. his joint work with Y. Segev on the uniqueness Luis Caffarelli, University of Texas, Austin, of the sporadic groups, and his joint work with for his work on partial differential equations. S. D. Smith on the Quillen conjecture. The prize of US$100,000 was divided equally Caffarelli has repeatedly made very deep break- between the prizewinners. The list of previous throughs. His early work on free boundary prob- recipients of the Wolf Prize in Mathematics is lems was the first place where his extraordinary available on the website of the Wolf Foundation, talent and intuition began to show. Free boundary http://www.wolffund.org.il. problems are about finding both the solutions to an equation and the region where the equation Description of the Prizewinners’ Work holds. In a series of pioneering papers, Caffarelli The following descriptions of the prizewinners’ put forward a novel methodology which eventu- work were prepared by the Wolf Foundation. ally leads, after several truly amazing technical Michael Aschbacher and John Thompson (Wolf estimates that step by step improve the regularity Prize laureate of 1992) are the two great modern of the solutions and the boundary, to full regularity masters of the theory of finite groups in an era that under very mild assumptions. Although the theory brought to fruition a line of research going back is complicated, the arguments are elementary and to Galois in the 1830s. The breadth and depth of full of beautiful geometric intuition and mastery Aschbacher’s understanding of finite groups in of analytic technique. general, and finite simple groups in particular, and A second fundamental contribution by Caf- the power he brought to bear on their analysis are farelli is the study of fully nonlinear elliptic par- astonishing. tial differential equations (including the famous Aschbacher astounded the finite group theory Monge-Ampère equation), which he revolutionized. community with a series of papers that raised the The upshot is that, although the equations are classification project for finite simple groups from nonlinear, they behave for purposes of regularity a distant dream to the reality it is today. as if they were linear. (Work of Nirenberg, Spruck, In a series of papers in the 1970s Aschbacher Evans, Krylov, and others also played a significant developed the theory of standard components role here.) Another fundamental contribution by Caffarelli DOI: http://dx.doi.org/10.1090/noti980 is his joint work with Kohn and Nirenberg on partial

474 Notices of the AMS Volume 60, Number 4 regularity of solutions of the incompressible (2011), and the Leroy P. Steele Prize for Mathemati- Navier-Stokes equation in 3 space dimensions. cal Exposition (2012). He has been elected to the Although the full regularity of solutions is still National Academy of Sciences (1990) and is a fel- unknown and likely very hard, Caffarelli-Kohn- low of the American Academy of Arts and Sciences Nirenberg showed that the singular set must have (1992) and of the American Mathematical Society parabolic Hausdorff dimension strictly less than one. In particular, singular fibers cannot occur. (2012). (V. Scheffer also deserves partial credit.) Luis Caffarelli was born in 1948 in Buenos Aires, Caffarelli has also produced deep work on Argentina. He received his (1968) homogenization and on equations with nonlocal and Ph.D. (1972) degrees from the University of dissipation. The list could be continued. Caffarelli Buenos Aires. He has taught at the University of is the world’s leading expert on regularity of solu- Minnesota, the University of Chicago, the Courant tions of partial differential equations. Institute of Mathematical Sciences at New York Biographical Sketches University, and the Institute for Advanced Study. Michael Aschbacher was born in 1944 in Little He is currently the Sid W. Richardson Foundation Rock, Arkansas. He received his B.S. from the Regents Chair in Mathematics at the University California Institute of Technology in 1966 and his of Texas at Austin. He has received the Bôcher Ph.D. from the University of Wisconsin, Madison, Memorial Prize (1984), the Rolf Schock Prize in 1969. He joined the faculty of the California (2005), and the Leroy P. Steele Prize for Lifetime Institute of Technology in 1970 and is currently Shaler Arthur Hanisch Professor of Mathematics at Achievement (2009). He is a member of the Na- Caltech. He was a visiting scholar at the Institute tional Academy of Sciences (1991) and a fellow for Advanced Study in 1978–1979. He was awarded of the American Mathematical Society (2012). the Cole Prize in Algebra (1980), the Rolf Schock Prize of the Royal Swedish Academy of Sciences — Elaine Kehoe Platonism Is the Law of the Land David A. Edwards

It is a well-established principle of our legal tradi- is not patentable. In fact the patent office itself tion that mathematical formulas, laws of nature, has argued repeatedly against the patentability of and natural phenomena are not patentable subject software because it would be flooded with applica- matter. It is the purpose of this essay to argue that tions.1 This implies a very active software industry this tradition should be reversed. Our argument much desirous of patent protection. is that it is and always has been in our society’s At present, only those things which are made by economic self-interest to grant very broad intellec- man are patentable. Thus, the courts have allowed tual property rights. This is particularly important new forms of bacteria which have been engineered in our present Information Age where “software to have useful properties using recombinant DNA dominates.” techniques to be patented but would not allow The patent system is designed to promote the such a bacterium to be patented if it were naturally flow of technological innovation, a topic of much occurring even if it were newly discovered. This current concern in America. One often sees refer- is the basis for the nonpatentability of computer ences to the relative decline in American vs. for- programs. They are algorithms, which are essen- eign patents as evidence of a decline in American tially mathematical formulas, which—as everyone innovation. But the main frontier of current tech- knows—are “eternal” and hence discovered by nological innovation is in software, and software man and not created by him. This argument which, David A. Edwards is associate professor of mathematics 1Harold T. Johnson Jr., Comment. “Computer program (retired) at the University of Georgia. His email address patentability —­ the CCPA refuses to follow the lead of the is [email protected]. Supreme Court in Parker v. Flook”, North Carolina Law DOI: http://dx.doi.org/10.1090/noti975 Review 58(1980), 319.

April 2013 Notices of the AMS 475 regularity of solutions of the incompressible (2011), and the Leroy P. Steele Prize for Mathemati- Navier-Stokes equation in 3 space dimensions. cal Exposition (2012). He has been elected to the Although the full regularity of solutions is still National Academy of Sciences (1990) and is a fel- unknown and likely very hard, Caffarelli-Kohn- low of the American Academy of Arts and Sciences Nirenberg showed that the singular set must have (1992) and of the American Mathematical Society parabolic Hausdorff dimension strictly less than one. In particular, singular fibers cannot occur. (2012). (V. Scheffer also deserves partial credit.) Luis Caffarelli was born in 1948 in Buenos Aires, Caffarelli has also produced deep work on Argentina. He received his Master of Science (1968) homogenization and on equations with nonlocal and Ph.D. (1972) degrees from the University of dissipation. The list could be continued. Caffarelli Buenos Aires. He has taught at the University of is the world’s leading expert on regularity of solu- Minnesota, the University of Chicago, the Courant tions of partial differential equations. Institute of Mathematical Sciences at New York Biographical Sketches University, and the Institute for Advanced Study. Michael Aschbacher was born in 1944 in Little He is currently the Sid W. Richardson Foundation Rock, Arkansas. He received his B.S. from the Regents Chair in Mathematics at the University California Institute of Technology in 1966 and his of Texas at Austin. He has received the Bôcher Ph.D. from the University of Wisconsin, Madison, Memorial Prize (1984), the Rolf Schock Prize in 1969. He joined the faculty of the California (2005), and the Leroy P. Steele Prize for Lifetime Institute of Technology in 1970 and is currently Shaler Arthur Hanisch Professor of Mathematics at Achievement (2009). He is a member of the Na- Caltech. He was a visiting scholar at the Institute tional Academy of Sciences (1991) and a fellow for Advanced Study in 1978–1979. He was awarded of the American Mathematical Society (2012). the Cole Prize in Algebra (1980), the Rolf Schock Prize of the Royal Swedish Academy of Sciences — Elaine Kehoe Platonism Is the Law of the Land David A. Edwards

It is a well-established principle of our legal tradi- is not patentable. In fact the patent office itself tion that mathematical formulas, laws of nature, has argued repeatedly against the patentability of and natural phenomena are not patentable subject software because it would be flooded with applica- matter. It is the purpose of this essay to argue that tions.1 This implies a very active software industry this tradition should be reversed. Our argument much desirous of patent protection. is that it is and always has been in our society’s At present, only those things which are made by economic self-interest to grant very broad intellec- man are patentable. Thus, the courts have allowed tual property rights. This is particularly important new forms of bacteria which have been engineered in our present Information Age where “software to have useful properties using recombinant DNA dominates.” techniques to be patented but would not allow The patent system is designed to promote the such a bacterium to be patented if it were naturally flow of technological innovation, a topic of much occurring even if it were newly discovered. This current concern in America. One often sees refer- is the basis for the nonpatentability of computer ences to the relative decline in American vs. for- programs. They are algorithms, which are essen- eign patents as evidence of a decline in American tially mathematical formulas, which—as everyone innovation. But the main frontier of current tech- knows—are “eternal” and hence discovered by nological innovation is in software, and software man and not created by him. This argument which, David A. Edwards is associate professor of mathematics 1Harold T. Johnson Jr., Comment. “Computer program (retired) at the University of Georgia. His email address patentability —­ the CCPA refuses to follow the lead of the is [email protected]. Supreme Court in Parker v. Flook”, North Carolina Law DOI: http://dx.doi.org/10.1090/noti975 Review 58(1980), 319.

April 2013 Notices of the AMS 475 to say the least, is philosophically controversial, allowance by the King, by his agents, commission, leads to our present unfortunate policy. From an or otherwise, to any person or persons, bodies poli- economic point of view, there is no rationale for tic or corporate, of or for the sole buying, selling, distinguishing between discovery and invention, making, working, or using anything; whereby any and we would advocate dropping entirely any person or persons, bodies politic or corporate, are subject matter restrictions whatsoever on what sought to be restrained of any freedom or liberty can be patented. One should be able to patent that they had before, or hindered in their lawful anything not previously known to man. In fact, a trade.”5 If such a restraint was to be legal, good good economic case can be made2 for allowing the reasons had to be forthcoming. Unfortunately, patenting of many things that are well known but good reasons weren’t always forthcoming. Abuses are not being commercially produced. led to tensions between Parliament and the Crown, The remainder of this essay is divided into which culminated during the reign of Charles I. An three sections. In the first we discuss the historical acceptable good reason for a temporary monopoly background of the patent system. In the second we was to promote new manufacturing. In this case discuss the cases and laws which bear on subject the freedom that was constrained was not one that matter patentability. In the last we summarize the persons were considered to have had before. our position and provide further arguments in One should note in particular that this included its favor. not only forms of manufacturing that were entirely new to mankind, but also forms that were only The Origins of Our Patent System new to the realm. All societies regulate economic behavior. This Based upon the long experience in England and ranges from such extremes of regulation as found in the separate colonies, the framers of the U.S. in ancient Egypt and modern on the one Constitution unanimously adopted Article I, Sec- hand to such relatively free-trade societies as tion 8: “The Congress shall have the Power… To Imperial Rome and nineteenth-century England promote the Progress of Science and useful Arts, by on the other hand. However, there seems to be no securing for limited Times to Authors and Inven- evidence of intellectual property rights (patents, tors the exclusive Right to their respective Writings copyrights, and trademarks) in the ancient world. and Discoveries.” The Constitution does not have Such rights seem to have been systematically any subject matter restrictions. In fact, up to that time no patent had been declared invalid because established for the first time during the late fif- of subject matter.6 But this almost happened (the teenth century in Venice.3 court split 2-2) in the famous case of Boulton v. In England, from which our traditions most Bull (1795) to which we now turn. immediately derive, the patent for new inventions evolved slowly out of a much more general policy Subject Matter Patentability of regulating and encouraging trade. As early as The case of Bolton and Watt against Bull seems to 1331, King Edward III was issuing letters patent be the first case in which the validity of a patent (i.e., open, public) for the express purpose of was questioned because of its subject matter. The encouraging new industry (in this case textiles) first commercially successful steam engine was into the realm.4 These patents allowed foreign developed by Newcomen in the early eighteenth workers, with their knowledge of techniques not century.7 In the middle of the century Smeaton then known in England, to enter the realm and performed a series of painstaking Baconian stud- bypass the usual guild restrictions which applied ies on the Newcomen engine and managed to to domestic workers. These policies were intensi- roughly double its efficiency. Watt’s approach fied during the reign of Queen Elizabeth as Lord was much more theoretical and resulted not in Burleigh, the secretary of state, sought to make small evolutionary changes, but instead in the England self-sufficient, especially in those areas revolutionary change of using two cylinders in- related to defense. stead of one. Watt’s patent starts as follows: “My The English common law was strongly for free method of lessening the consumption of steam, trade except for those circumstances which could and consequently fuel in fire engines, consists of be clearly shown to be in the general interest (e.g., the following principles.” He goes on to describe defense). This tradition goes back at least as far his new principles, which would today be called as the Magna Carta. The famous Elizabethan ju- the laws of thermodynamics. These principles rist, Chief Justice Coke, in his influential treatise were first abstractly treated by Sadi Carnot several Institutes, defined a monopoly as “an institution or decades later.

2Edmund W. Kitch, “The nature and function of the patent 5Co. 3 Inst. 181, C. 85. Quoted in Fox, p. 8. system”, The Journal of Law and Economics 265. 6Fox, p. 234. 3 B. W. Bugbee, Genesis of American Patent and Copyright 7The development of the steam engine is very clearly Law, Public Affairs Press, 1967. discussed in D. S. L. Cardwell, Turning Points in Western 4Harold G. Fox, Monopolies and Patents, 1947. Technology, 1972.

476 Notices of the AMS Volume 60, Number 4 Watt was asking for a patent not merely for in Gottshalk v. Benson.12 Until 1990 software had a new steam engine with a specific new design, only copyright protection. This is adequate protec- but instead for a whole new class of possible new tion only for software containing no new ideas, as steam engines whose designs were not described copyright protects only the expression in a work explicitly but were instead described by Watt’s and not the ideas in it. (See Baker v. Selden.13) higher-level principles. He was not asking for a Around 1990 the courts “found” a way to patent on these principles themselves in the sense protect software via the patent system. They that anyone desiring to use or describe these also allowed business methods and many areas principles would have to get his permission and of bio-tech. But more recently the issue has re- possibly pay him royalties, but only for a patent on surfaced in Bilski. There is no economic basis for steam engines designed according to these princi- the distinction between discovery and invention. ples. All four judges agreed that one couldn’t get a If we're going to have a general patent system, patent for the principles themselves. However, the then algorithms should be as patentable as la- judges split 2-2 on the question of Watt’s patent sers. For example, general relativity is used in GPS itself; two of them felt that its scope was much too (see http://en.wikipedia.org/wiki/Global_ broad, while the other two felt that the principles Positioning_System). My colleague Carl Pomer- were acceptably being used to adequately describe ance developed fast primality testing algorithms Watt’s innovations in steam engine design. in the late 1970s but couldn't patent them. My Similar situations arose throughout the nine- colleague Victor Wickerhauser developed the teenth century. (See, for instance, O’Reilly v. fast wavelet transform in the early 1990s and Morse.8) A twentieth-century case in which the was able to patent it as a software patent. I issues are particularly clearly drawn is the case believe the recent rulings (see http://www. of MacKay Radio and Tel. Co. v. Radio Corp. of nytimes.com/2010/03/31/nyregion/31about. America.9 Radio Corporation of America (RCA) html?sq=inventions%20and%20patents& had a patent for an antenna whose structure was st=cse&scp=2&pagewanted=print and described using a mathematical formula. The court http://www.nytimes.com/2010/10/30/ held the patent valid as not being on the formula business/30drug.html?_r=1&hp=& itself but instead on the antenna. The court stated pagewanted=print) were correct current law and that “[w]hile a scientific truth, or the mathematical will end up also invalidating software patents. If we expression of it, is not a patentable invention, a want these things to be patentable, then Congress novel and useful structure created with the aid of needs to change the law. knowledge of scientific truth may be.” Computer software presents a problem area Conclusions where, as the Supreme Court has recently put it, For the past two hundred years all our courts and “[t]he line between a patentable ‘process’ and an legislatures have unanimously ruled that math- unpatentable ‘principle’ does not always shimmer ematical formulas, laws of nature, and natural with clarity.”10 In In re Bernhart,11 the Court of phenomena are not patentable subject matter. A Customs and Patent Appeals (C.C.P.A.) stated: typical rationale for these rulings is the following [I]f a machine is programmed in a certain new footnote which occurs in the Supreme Court’s re- and unobvious way, it is physically different from cent opinion in the case of Parker v. Flook:14 the machine without that program; its memory ele- The underlying notion is that a scientific prin- ments are differently arranged. The fact that these ciple, such as that expressed in respondent’s al- physical changes are invisible to the eye should not gorithm, reveals a relationship that has always tempt us to conclude that the machine has not been existed. changed. If a new machine has not been invented, “An example of such a discovery [of a scientific certainly a “new and useful improvement” of the principle] was Newton’s formulation of the law unprogrammed machine has been, and Congress of universal gravitation, relating the force of at- has said in 35 U.S.C. #101 that such improvements traction between two bodies, F, to their masses, m are statutory subject matter for a patent. and m', and the square of the distance, d, between This argument of the C.C.P.A. would have their centers, according to the equation F=mm'/d 2. extended patent protection to all uses of algo- But this relationship always existed —even before rithms in computers, but not to other uses of al- Newton announced his celebrated law. Such ‘mere’ gorithms (such as human use). Unfortunately, this recognition of a theretofore existing phenomenon argument was overturned by the Supreme Court or relationship carries with it no rights to exclude others from its enjoyment…. Patentable subject matter must be new (novel); not merely heretofore 856 U.S. (15 How.) 62 (1853). 9 306 U.S. 86 (1939). 12409 U.S. 63 (1972). 10437 U.S. 584 (1978). 13101 U.S. 99 (1879). 11417 F. 2d 1395 (C.C.P.A. 1969). 14437 U.S. 584 (1978).15437 U.S. 584 (1978).

April 2013 Notices of the AMS 477 MATHEMATICS AT THE N ATIONAL S ECURITY A GENCY unknown. There is a compelling reason for this rule. The reason is founded upon the proposition that in granting patent rights, the public must not Far from Ordinary be deprived of any rights that it theretofore freely enjoyed.” P. Rosenberg, Patent Law Fundamentals, #4, at 13(1975). This “very compelling reason” goes back to Cooke (see quote in #1) but is taken totally out of context. Cooke would not have considered the public to have been enjoying rights to unknown things. This line of argument could be pushed to void all patents. Up until recently, the economic consequences of these restrictions in intellectual property rights have probably been quite slight. Similarly, the eco- nomic consequences of allowing patents for new inventions were also probably quite slight up to about 1800. Until then, patents were mainly import franchises. After 1800 the economic consequences of allowing patents for new inventions became im- mense as our society moved from a predominately agricultural stage into a predominately industrial stage. Since the end of World War II,our society has The U.S. is under cyber attack, every minute of every day. That’s been moving into an information stage, and it is why cyberspace has become today’s new front line. What you becoming more and more important to have prop- know can make a difference at the National Security Agency. erty rights appropriate to this stage. We believe Whether it’s collecting foreign intelligence or preventing foreign that this would best be accomplished by Congress adversaries from accessing U.S. secrets, you can protect the amending the patent laws to allow anything not previously known to man to be patented. nation by putting your intelligence to work. Explore technology More specifically, the distinction between dis- that’s years ahead of the private sector. Plus exciting career covery and invention should be eliminated. This fields, paid internships, co-op and scholarship opportunities. See would allow the patent incentive to motivate how you can be a part of our tradition of excellence and help the exploration for previously unknown useful forms nation stay a step ahead of the cyber threat. of bacteria, plants, animals, materials, molecules, atoms, particles, etc. Previously unknown math- KNOWINGMATTERS ematical formulas and laws of nature should also be patentable. Since patents only give control over the commercial applications of his or her discovery or invention to the patentee, granting patents on mathematical formulas, laws of nature, and natural phenomena would have no negative side effects on pure science. The economic stimulation of pure Excellent Career science that would be provided by such patents Opportunities for is particularly important today as the traditional Experts in the Following: economic support of pure science, namely univer- sity faculty positions and government grants, are n Number Theory in decline. For the society as a whole, the positive n Probability Theory economic effects of such extended intellectual n Group Theory property rights would be quite substantial. Today’s n Finite Field Theory technology depends upon yesterday’s science. n Combinatorics In conclusion, it is time for Congress to update the patent laws and provide for the appropriate n Linear Algebra intellectual property protection which our age >> Plus other demands. opportunities where intelligence goes to work ® Additional References 1. D. S. Chisum, Intellectual Property: Copyright, Patent and Trademark Law, Mathew Bender, N.Y., 1980. Search NSA to Download 2. E. W. Kitch and H. S. Perlman, Legal Regulation of U.S. citizenship is required. NSA is an Equal Opportunity Employer. the Competitive Process, The Foundation Press, Inc., N.Y., 1979.

478 Notices of the AMs VoluMe 60, NuMber 4 REFRESH_10BWNS-02_3.25x9.5.indd 1 8/22/12 9:20 AM

2013 Steele Prizes

The 2013 AMS Leroy P. Steele Prizes were presented few research groups were applying center mani- at the 119th Annual Meeting of the AMS in San Diego, fold reduction to physical problems, computing California, in January 2013. The Steele Prizes were normal forms, and unfolding bifurcations, but the awarded to John Guckenheimer and Philip Holmes Guckenheimer and Holmes work was the first text- for Mathematical Exposition, to for book to lay out clearly the theory for dissipative a Seminal Contribution to Research, and to Yakov dynamical systems, to separate rigorous results Sinai for Lifetime Achievement. from speculation, to begin to reunite perturba- tion methods with the geometric and topological Mathematical Exposition: John ideas of global analysis, and to provide analyses Guckenheimer and Philip Holmes of practical problems. Citation Both theoretical and applied dynamical systems remain strong research areas, with theoretical The 2013 Leroy P. Steele Prize for Mathematical Ex- research appearing in physics, engineering, and position is awarded to John Guckenheimer and Philip applied mathematics departments and applied Holmes in recognition of their book, Nonlinear Oscil- work being produced by researchers in mathemat- lations, Dynamical Systems, and Bifurcations of Vector ics departments—a healthy trend that was given Fields, Applied Mathematical Sciences, 42, Springer- substantial help by the publication of Nonlinear Verlag, New York, 1983; reprinted with revisions and Oscillations, Dynamical Systems, and Bifurcations corrections, 1990. of Vector Fields. Dynamical systems underwent a rebirth in the 1960s and 1970s with the work of mathematicians Biographical Sketches such as (in alphabetical order) Anosov, Arnold, Kol- John Guckenheimer was born in Baton Rouge, mogorov, Moser, Ruelle, Sinai, Smale, Takens, Thom, Louisiana, in 1945. He received his undergraduate and many others on the theoretical side and engi- degree from Harvard neers and experimental physicists such as Lorenz, Swinney, Gollub, and many others on the applied in 1966 and his Ph.D. side. Not surprisingly, it was difficult for the two com- from the University munities to know about each other’s work until the of California Berke- publication of the now-classic text by Guckenheimer ley in 1970 under the and Holmes. Thirty years later this book remains in direction of Stephen wide use as a standard text for graduate-level courses Smale. He held posi- in mathematics departments and throughout the tions at IMPA (1969), sciences and engineering, and Chinese and Russian the University of War- translations have appeared. wick (1969–1970), the In the late 1970s dynamical systems theory was Institute for Advanced still largely the preserve of mathematicians, at Study (1970–1972), least in Europe and the Americas (the and the Massachusetts had maintained somewhat stronger links among Institute of Technology John Guckenheimer mathematical scientists and physicists, chemists, (1972–1973) before and engineers). Excitement was growing over chaos joining the faculty of the University of California and sensitive dependence (the butterfly effect) and Santa Cruz (1973–1985). Since 1985 he has been on bifurcation and unfolding theories. Physicists such the faculty of , where he is now as Swinney and Gollub were generating experimental the A. R. Bullis Professor of Mathematics. data on constrained fluid systems, but the fundamen- During the past fifteen years, his research has tal work of Smale and his students was appearing in investigated dynamical systems with multiple journals unknown to many researchers who could time scales and associated numerical methods. most benefit from them, beyond and even within the He has also continued to investigate the use of mathematics community. dynamical systems theory in diverse areas, notably Research monographs were beginning to appear: in neuroscience and animal locomotion. He was Abraham and Marsden’s Foundations of Mechanics a 1984 Guggenheim Fellow and is a fellow of the (1967) focused on Hamiltonian systems and clas- AMS, the American Academy of Arts and Sciences, sical mechanics; Marsden and McCracken edited a the American Association for Advancement of collection of papers on Hopf bifurcation (1976). A Science, and the Society for Industrial and Applied Mathematics (SIAM). He served as president of DOI: http://dx.doi.org/10.1090/noti972 SIAM in 1997–1998.

480 Notices of the AMS Volume 60, Number 4 Phil visited UC Berkeley and David visited UC Santa Philip Holmes was Cruz. We discussed continuing the writing as a born in Lincolnshire, three-author team, but ultimately David decided England, in 1945 and to withdraw while the two of us proceeded. We was educated in engi- completed the first draft in approximately nine neering science at the months, starting with notes for applied dynamical Universities of Oxford systems courses that we taught at UC Santa Cruz and Southampton. He and Cornell and spending an hour or two discuss- taught in the depart- ing our differences on the phone every Friday. We ments of theoretical received enthusiastic support from Jerry Marsden and applied mechan- and others. Walter Kaufmann-Buehler at Springer- ics and mathematics Verlag was willing to take a risk in pricing the book at Cornell University at a level that would encourage individuals to buy from 1977 to 1994. it. Our handwritten manuscript was painstakingly Philip Holmes In 1994 he moved to typed by Dolores Pendell at Cornell’s Center for Princeton University, Applied Mathematics, and the diagrams were pro- where he is Eugene Higgins Professor of Mechani- duced by Barbara Boettcher. cal and Aerospace Engineering, professor of ap- The book’s success has been extraordinarily plied and computational mathematics, associated gratifying, especially when younger (than us!) faculty in the Department of Mathematics, and scientists tell us that they studied it carefully and a member of Princeton’s Neuroscience Institute. keep returning to take it from their bookshelves. Much of his research has been in dynamical sys- We tried hard to explain mathematical concepts tems and their applications in engineering and the and arguments in their simplest manifestations physical sciences, but in the past fifteen years he while relying on as little formal training as seemed has increasingly turned to biology. He currently feasible. It helped that we came to the interface works on the neuromechanics of animal locomo- between mathematics and the physical sciences tion and neurodynamics of decision making. He is from opposite sides. With our different back- a member of the American Academy of Arts and grounds, we sought to bring alive how dynamical Sciences, an honorary member of the Hungarian systems theory has been enriched repeatedly by Academy of Sciences, and a fellow of the AMS, of questions from the “real world”, while, at the same the American Physical Society, and of SIAM. He has time, demonstrating the power of mathematical also published four collections of poems (Anvil thinking and abstraction to unify the sciences. We Press, London). thank the Committee for recognizing our efforts to present significant results to a broad scientific Joint Response from John Guckenheimer audience that stretches far beyond the boundaries and Philip Holmes of mathematics. We are honored and delighted to receive the Steele Prize for Mathematical Exposition. We come from Seminal Contribution to Research: Saharon very different places (Baton Rouge, Louisiana, and Shelah Brigg, Lincolnshire, ) and very dif- ferent training: John from Ph.D. studies with Steve Citation Smale and Phil from applied mechanics. We first The 2013 Leroy P. Steele Prize for Seminal Contri- met in 1976 at a dynamical systems conference bution to Research is awarded to Saharon Shelah in Southampton coorganized by David Rand. At for his book Classification Theory and the Number that time rapid advances in dynamical systems of Nonisomorphic Models, Studies in Logic and the theory were stimulating experimental work that Foundations of Mathematics, 92, North-Holland demonstrated the usefulness of the theory in ex- Publishing Co., Amster- plaining empirical phenomena across the sciences dam–New York, 1978; and engineering. We saw a real need for a book 2nd edition, 1990. that made the new mathematics accessible to a Before Shelah’s broad audience, including mature scientists and work, the great theo- students. The excitement of the period was cap- rem of pure model tured vividly by James Gleick in his book, Chaos: theory was Morley’s Making a New Science, Penguin Books, New York, theorem on categoric- 1987, which received the first JPBM Communica- ity. It concerned a class tions Award in 1988. of theories whose un- In the late 1970s Phil and David Rand, after countable models are working together on nonlinear oscillators, began completely determined teaching courses in dynamical systems and as- by their cardinality. sembling notes toward a book. Independently, Shelah visualized a John also began planning a book. In spring 1981 vast extension of the Saharon Shelah

April 2013 Notices of the AMS 481 problem to a classification of arbitrary first-order welcome such applications and interactions. It is theories, based on the number of models they may a happy day for me that this line of thought has have of a given uncountable size. Solving this problem received such honourable recognition. Thank you. required some twenty years that made model theory into a mature field, completely transform- Lifetime Achievement: ing its aims, methods, and ability to connect to Citation algebra and geometry. Shelah isolated the class of stable theories, The 2013 Steele Prize for Lifetime Achievement is where finitely generated extensions admit, in a awarded to Yakov Sinai for his pivotal role in shap- certain local sense, finitary descriptions. He was ing the theory of dynamical systems and for his able to show, on the other hand, that any unstable groundbreaking contributions to ergodic theory, theory has the maximum set-theoretically permis- probability theory, statistical mechanics, and sible number of models. All theories of modules mathematical physics. are stable. Among the stable theories, he isolated Sinai’s research ex- the superstable theories, analogous to Noetherian hibits a unique com- rings, and again found many models if this condi- bination of brilliant tion fails. These were the first two of a series of analytic technique, outstanding geomet- dividing lines, characterized by a deep theorem ric intuition, and pro- on either side. On the stable side, he was able to found understanding define a canonical tensor product of extensions of of underlying physi- structures and made it into an incisive tool for the cal phenomena. His decomposition of structures. An arsenal of notions work highlights deep became available to the previously bare-handed and unexpected con- model theorist: algebraic closure, canonical bases, nections between dy- imaginary sorts, domination, forking, regular namical systems and types. These concepts proved useful beyond the statistical mechanics. Yakov G. Sinai stable framework and led to substantial applica- Sinai has opened up tions when investigated in algebraic settings. The new directions, including Kolmogorov–Sinai en- problem of the number of models was solved in the tropy, Markov partitions, and Sinai–Ruelle–Bowen second edition of his monograph, but the ideas of measures in the hyperbolic theory of dynamical the solution remained central and proved critical systems; dispersing billiards, a rigorous theory for many others. It would be impossible to imagine of phase transitions in statistical mechanics and model theory today without them. space-time chaos. In addition, Sinai has made Biographical Sketch seminal contributions in the theory of Schrödinger operators with quasi-periodic potentials, random Saharon Shelah earned his B.Sc. from Tel Aviv walks in random environments, renormalization University, his M.Sc. from the Hebrew University theory, and statistical hydrodynamics for Burgers under the supervision of Professor H. Gaifman, and and Navier–Stokes equations. his Ph.D. from the Hebrew University under the Sinai pioneered the study of dispersing billiards: supervision of Professor M. Rabin. He has taught dynamical systems which model the motion of at the Hebrew University and Rutgers University, molecules in a gas. The simplest example of such a among others. He is a member of the Israel Acad- billiard table, a square with a disk removed from its emy of Sciences and Humanities and the American center, is called “Sinai’s billiard”. Studying billiard Academy of Arts and Sciences. motions within the framework of hyperbolic the- ory, Sinai discovered that they exhibit deep ergodic Response and statistical properties (such as the central limit I am grateful for this great honour. theorem). Owing to Sinai’s work, some key laws of While it is great to find full understanding of statistical mechanics for the Lorentz gas can be that for which we have considerable knowledge, I established with mathematical rigor. In particular, have been attracted to trying to find some order in Sinai made the first steps towards justification of the darkness; more specifically, finding meaningful Boltzmann’s famous ergodic hypothesis, proposed dividing lines among general families of structures. in the end of the nineteenth century: “For large sys- This means that there are meaningful things to be tems of interacting particles in equilibrium, time said on both sides of the divide: characteristically, averages are close to the ensemble average.” Sinai understanding the tame ones and giving evidence returned to this subject several times in the period of being complicated for the chaotic ones. It is 1970–1990 with various coauthors, including his expected that this will eventually help in under- students Bunimovich and Chernov. standing even specific classes and even specific Together with his student Pirogov, Sinai created structures. Some others see this as the aim of a general theory of low-temperature phase transi- model theory; not so for me. Still I expect and tions for statistical mechanics systems with a finite

482 Notices of the AMS Volume 60, Number 4 number of ground states. Pirogov–Sinai theory Science, the Hungarian Academy of Science, the forms essentially the basis for modern equilibrium Polish Academy of Science, and Academia Europea. statistical mechanics in a low-temperature regime. Among his other recognitions are the Wolf Prize Sinai made seminal contributions to the theory in Mathematics, the Nemmers Prize, the Lagrange of random walks in a random environment. With Prize, the Boltzmann Medal, the Dirac Medal, and his model, known nowadays as “Sinai’s random the Poincaré Prize. walk”, he obtained remarkable results about its asymptotic behavior. With his student Khanin, Response Sinai pioneered applications of the renormaliza- It is a great honor to be awarded the Steele Prize tion group method to multifractal analysis of the for Lifetime Achievement from the American Math- Feigenbaum attractor and to the Kolmogorov– ematical Society. I worked in several directions in Arnold–Moser theory on invariant tori of Hamil- mathematics, including the theory of dynamical tonian systems. systems, statistical and mathematical physics, and In the past fifteen years Sinai has brought novel probability theory. tools and insights from dynamical systems and My mentors who had a big influence on me mathematical physics to statistical hydrodynam- were A. N. Kolmogorov, V. A. Rokhlin, and E. B. ics, obtaining new results for the Navier–Stokes Dynkin. I also benefited a lot from many contacts systems. Specifically, along with D. Li, Sinai devised with my colleagues. I was very fortunate to have a new renormalization scheme which allows the talented students, many of whom became strong proof of existence of finite time singularities for and famous mathematicians. Unfortunately, it is complex solutions of the Navier–Stokes system in not possible to list the names of all of them here. dimension three. I thank my family and friends for their encourage- Sinai’s mathematical influence is overwhelming. ment and support. Finally, I thank the selection During the past half-century he has written more committee for its work. than 250 research papers and a number of books. About the Prize Sinai’s famous monograph, Ergodic Theory (with The Steele Prizes were established in 1970 in Cornfeld and Fomin), has been an introduction to honor of , William Fogg the subject for several generations, and it remains Osgood, and William Caspar Graustein. Osgood a classic. was president of the AMS during 1905–1906, and Sinai supervised more than fifty Ph.D. students, Birkhoff served in that capacity during 1925–1926. many of whom have become leaders in their own The prizes are endowed under the terms of a right. Sinai’s work is impressive for its breadth. In bequest from Leroy P. Steele. Up to three prizes addition to its long-lasting impact on pure math- are awarded each year in the following catego- ematics, it has played a crucial role in the creation ries: (1) Lifetime Achievement: for the cumulative of a concept of dynamical chaos which has been influence of the total mathematical work of the extremely important for the development of phys- recipient, high level of research over a period of ics and nonlinear science over the past thirty-five time, particular influence on the development of a years. The Steele Prize for Lifetime Achievement field, and influence on mathematics through Ph.D. is awarded to Sinai in recognition of all these students; (2) Mathematical Exposition: for a book achievements. or substantial survey or expository research paper; Biographical Sketch (3) Seminal Contribution to Research: for a paper, whether recent or not, that has proved to be of Yakov G. Sinai was born in 1935 in Moscow, Soviet fundamental or lasting importance in its field or Union, now Russia. He received his Ph.D. degree a model of important research. Each Steele Prize (called a Candidate of Science in Russia) and then carries a cash award of US$5,000. his doctorate degree (Doctor of Science) from Beginning with the 1994 prize, there has been a . For several years, he held five-year cycle of fields for the Seminal Contribu- combined positions at Moscow State University and tion to Research Award. For the 2013 prize, the the Landau Institute of Theoretical Physics of the field was logic. The Steele Prizes are awarded by Russian Academy of Sciences. Since 1993, he has the AMS Council acting on the recommendation been a professor in the mathematics department of a selection committee. For the 2013 prizes, the of Princeton University. members of the selection committee were Yakov Sinai has received various honors recognizing Eliashberg, John E. Fornæss, Irene M. Gamba, his contributions. He was elected as a foreign as- Barbara L. Keyfitz, Tomasz S. Mrowka, Gang sociate of the National Academy of Sciences and Tian, , Lai-Sang Young, and a foreign member of the Academy of Arts and Efim I. Zelmanov. The list of previous recipients of Sciences. He is a full member of the Russian Acad- the Steele Prize may be found on the AMS website emy of Sciences, and he was recently elected as a at http://www.ams.org/prizes-awards. foreign member of the Royal Society in London. He is also a member of the Brazilian Academy of —Elaine Kehoe

April 2013 Notices of the AMS 483 2013 Conant Prize

John Baez and John Huerta received the 2013 plex vector space V. More precisely, particles are Levi L. Conant Prize at the 119th Annual Meeting of the basis vectors of the associated irreducible the AMS in San Diego, California, in January 2013. representations on V. Ideas coming from the era of grand unification persist into the twenty-first Citation century and are still important today. The article The 2013 Levi L. Conant Prize is awarded to John focuses upon the group representation theory Baez and John Huerta for their article, “The algebra that describes particles in the Standard Model. of grand unified theories”, Bulletin of the The authors eschew symmetry breaking (particle American Mathematical Society 47 (2010), interaction) so they can limit the discussion to no. 3, 483–552. algebraic ideas that mathematicians not coming The Standard Model of particle physics from theoretical physics will find familiar. Their is one of the central theoretical constructs goal is to introduce the unfamiliar ways in which of twentieth-century physics. It attempts physicists use these algebraic ideas. to describe all particles and all the forces Even with the simplifications inherent in the of nature except gravity. Although the restricted range of the Standard Model alluded Standard Model seems complicated and to above, it is a daunting task to convey decades somewhat arbitrary, it has been very suc- of work in one relatively short article. A grand cessful in describing mathematically what unified theory is based on different versions of we see in reality. Physicists are not fully extending the Standard Model. The gauge group of satisfied with it because it does leave out the Standard Model is GSM = U(1) × SU(2) × SU(3). John Baez gravity, described by Einstein’s general An extension of this gauge group, together with theory of relativity, which has not been an appropriate representation, composes a grand reconciled with the Standard Model, and unified theory. The article deals with three such it may not be able to account for dark extensions whose gauge groups are SU(2) × SU(2) matter. Looking beyond the Standard × SU(4), SU(5) and Spin(10). It transpires that the Model leads to , loop quan- third theory can be viewed as an extension of the tum gravity, and theories based on non- first two, thereby uniting the points of view. The commutative geometry. article finishes with some intriguing theorems that In the second half of the twenti- lead to interesting speculations about the nature eth century, a program of “grand uni- of matter and force in the universe. Along the fication” commenced whose aim was way, we are treated to a running account of the to unify particles and the forces, Standard Model, how it began with the Heisenberg save gravity, within the confines model of the proton and neutron, and how it has John Huerta of the Standard Model using Lie evolved with grand unification. Such wonderful groups, Lie algebras, and their representa- physical notions as color, spin, handedness, anti- tions. Such a theory features a compact Lie matter, and the like make their appearance along group G, called the gauge group. Particles lie in a the way. At the same time, concrete theory of the representation of G on a finite-dimensional com- aforementioned Lie groups and their representa- tions are related to these physical notions, and DOI: http://dx.doi.org/10.1090/noti967 the group theory is shown to lead to interesting

484 Notices of the AMS Volume 60, Number 4 physical conclusions and puzzles. The authors do best mathematics blog. But I am very happy to an excellent job of keeping the mathematics within receive this prize for a more traditional form of reach of even a well-educated, advanced graduate mathematics exposition. student and alternate nicely between the physics From John Huerta: It is a great honor to receive and the mathematics. this award and to share it with my mentor, John The article is a very well-written piece about an Baez. I know there were many other worthy con- interesting and important topic. Too often the ex- tenders for this award, which only adds to my pository aspect of an article is limited to the initial gratitude. I hope, if nothing else, the attention sections, so that once a reader is “hooked”, he or this honor brings leads more people to learn this she has to continue through much drier material. wonderful story we did our best to tell. This was That is not the case in this article. The authors are our first paper together, and my first publication. able to maintain an engaging tone throughout the I remain deeply proud of it, but it belongs to more seventy pages it takes to tell this marvelous tale. people than I can name. I’d like to thank the AMS And as with the stories in the Thousand and One for their top-notch editors, UC Riverside Mathemat- Nights, we are left hoping for more. ics for having me as a student, my parents and my friends for their love and support. But most of all, I Biographical Sketches want to thank John, for being tough and for being John Baez is professor of mathematics at the Uni- kind, and for teaching me how to write. versity of California Riverside. Until recently he About the Prize worked on higher category theory and quantum gravity. His Internet column “This Week’s Finds” The Conant Prize is awarded annually to recog- dates back to 1993 and is sometimes called the nize an outstanding expository paper published world’s first blog. In 2010, concerned about climate in either the Notices of the AMS or the Bulletin of change and the future of the planet, he switched to the AMS in the preceding five years. Established working on more practical topics and started the in 2001, the prize honors the memory of Levi L. Azimuth Project, an international collaboration to Conant (1857–1916), who was a mathematician at create a focal point for scientists and engineers Worcester Polytechnic Institute. The prize carries interested in saving the planet. a cash award of US$1,000. The Conant Prize is awarded by the AMS Coun- John Huerta is starting a postdoctoral fel- cil acting on the recommendation of a selection lowship at Instituto Superior Técnico in Lisbon, committee. For the 2013 prize, the members of where he will work with Roger Picken on higher the selection committee were Thomas F. Banchoff, gauge theory and its role in string theory. Before Jerry L. Bona, and J. Brian Conrey. that he was a postdoc at the Australian National Previous recipients of the Conant Prize are: University, where he studied the smallest excep- Carl Pomerance (2001); Elliott Lieb and Jakob tional and its relationship with a rolling Yngvason (2002); Nicholas Katz and Peter Sarnak ball. He was a Ph.D. student of John Baez, study- (2003); Noam D. Elkies (2004); Allen Knutson and ing supersymmetry, higher gauge theory, and (2005); Ronald M. Solomon (2006); Jef- their connection to division algebras. Though he frey Weeks (2007); J. Brian Conrey, Shlomo Hoory, studied mathematics, he’s always loved physics Nathan Linial, and Avi Wigderson (2008); John W. and derives great inspiration from it. His interest Morgan (2009); Bryna Kra (2010); began with a popular astronomy book that his sister gave to him as a child. This evolved into (2011); and Persi Diaconis (2012). a fascination with stars, with physical laws, and eventually the underlying mathematics and its —Elaine Kehoe conceptual interplay.

Responses From John Baez: I put a lot of energy into explain- ing math and physics online. Blogging is no sub- stitute for more formal writing about academic subjects, but it fills a gap, especially for the millions who don’t live near a good research university. Socrates complained that “writing is unfortunately like painting, for the creations of the painter have the attitude of life, yet if you ask them a question they preserve a solemn silence.” This is no longer true with blogs: the author is there to answer your questions! So, I am hoping that eventually blogs will be taken seriously by academia, and the AMS will have an award for the

April 2013 Notices of the AMS 485 2013 Morgan Prize

Fan Wei received the 2013 AMS-MAA-SIAM Frank internship at Microsoft Research New England with and Brennie Morgan Prize for Outstanding Re- Henry Cohn, she went to Cambridge University, where search in Mathematics by an Undergraduate Stu- she is studying Part III mathematics for a master’s dent at the 119th Annual Meeting of the AMS in degree. She plans to return to the United States to San Diego, California, in January 2013. Receiving obtain a Ph.D. honorable mentions were Dhruv Ranganathan and Jonathan Schneider. Response I am very honored and grateful to receive the Frank Citation and Brennie Morgan Prize. It is a great encourage- Fan Wei is awarded the 2013 AMS-MAA-SIAM ment for me and I would like to thank the AMS, Frank and Brennie Morgan Prize for Outstand- MAA, and SIAM for selecting me for this award. ing Research in Mathematics by an First and foremost, I want to thank my parents Undergraduate Student for her wide for their constant love, understanding, and toler- range of scholarly contributions. As ance. My home has always been, and will continue an undergraduate at the Massachu- to be, my motivation. My gratitude goes to my setts Institute of Technology, Wei research mentors, class lecturers, and nominators, authored or coauthored five papers Richard Dudley and Richard Stanley. Furthermore, in fields as diverse as number theory, I want to express my gratitude to Henry Cohn, my combinatorics, statistics, and tropi- mentor at Microsoft Research; to the hosts of the cal geometry, and she is recognized, UMN REU—Gregg Musiker, Victor Reiner, and Pavlo in particular, for her single-authored Pylyavskyy; and to the hosts of Williams College paper that solves a separable permu- SMALL REU, especially Allison Pacelli, for provid- tations problem posed by Dr. Richard ing me with two memorable summers. I am also Stanley. Her work has been described grateful to the MIT mathematics department and as “impressive and ingenious” and as the many people, including the staff members and Fan Wei “enthusiastically received by other professors, such as , Alan Edelman, mathematicians.” Ju-Lee Kim, Gigliola Staffilani, Daniel Stroock, and Wei has attended REUs at Williams College and other analysis and combinatorics professors, for the University of Minnesota-Twin Cities and has their great help, patience, and support. Lastly, I participated in many research projects at MIT. She want to thank all my friends for giving me a second has presented her results at two conferences in family. I am lucky to know you all. 2010: the Young Mathematician’s Conference and Permutation Patterns. Citation for Honorable Mention: Dhruv Wei was part of a Meritorious Winner Team Ranganathan for the 2010 Mathematical Contest in Modeling, a Honorable mention goes to Dhruv Ranganathan for mentor of the Girl’s Angle Math Club in Cambridge, the 2013 AMS-MAA-SIAM Frank and Brennie Mor- and has served on the board of MIT’s Society of gan Prize for Outstanding Research in Mathemat- Women Engineers. Additionally, Wei won the 2012 ics by an Undergraduate Student. Ranganathan Alice T. Schafer Prize of the Association for Women is recognized for his research in Gromov-Witten in Mathematics (AWM). theory and toric geometry and, in particular, for 3 Biographical Sketch his article “Toric symmetry of CP ”. Letter writers remarked on his ability to “master the formidable Fan Wei is from Beijing, China, where she finished prerequisite material” and commented on his “key high school and became interested in mathematics. insights” and “surprising” results. In 2012, Fan Wei received her bachelor’s degree from Ranganathan has presented his results at the the Massachusetts Institute of Technology, where she majored in mathematics and became more com- Algebra, Number Theory and Combinatorics Semi- mitted to the subject. Her interests in mathematics nar of the Claremont Colleges, the 2011 Joint are in analysis and combinatorics. After a summer Mathematics Meetings, and the Western Algebraic Geometry Seminar. Ranganathan is a 2012 gradu- ate of Harvey Mudd College. DOI: http://dx.doi.org/10.1090/noti968

486 Notices of the AMS Volume 60, Number 4 Biographical Sketch computer science. During his freshman summer, Dhruv Ranganathan grew up in , India, Jon was introduced to the world of mathemat- and Johannesburg, South Africa. Despite early ics research through MIT’s Summer Program in aspirations to be a professional cricket player, Undergraduate Research, where he studied the his interest in mathematics and physics quickly combinatorics of chip-firing and hyperplane ar- took over. It was at the age of nine that he began rangements. The next summer, Jon did research learning advanced mathematics outside of school, with Richard Stanley, solving an open problem under the private tutelage of A. V. Ramamoorthy regarding the coefficients of chromatic polynomi- in Chennai. At Harvey Mudd College, he worked als of certain graphs. Jon presented his solution to closely with Dagan Karp for over two years, study- this problem at the 2011 Joint Mathematics Meet- ing Gromov-Witten theory and toric geometry. ings, for which he was recognized with an MAA This project shaped his current research interests, Undergraduate Poster Prize. Next year Jon intends which include algebraic geometry, toric and tropi- to attend graduate school in either mathematics cal geometry, and high-energy physics. Rangana- or computer science. than is a winner of the Giovanni Borrelli Fellowship Response from Jon Schneider and was awarded the Chavin Prize for his Harvey I would like to thank the AMS, MAA, and SIAM for Mudd College senior thesis entitled “Gromov–Wit- selecting me for this honorable mention. ten theory of blowups of toric threefolds”. He is I am especially grateful to Professor Richard currently a first-year graduate student at Yale Stanley for introducing me to this problem and University. When not studying mathematics, Ran- for advising me over the course of my research. ganathan can be found playing the violin, finding I am further thankful to all of my math professors innovative ways to cook for himself, or honing his and classmates at MIT, from whom I have learned a flying trapeze skills. great deal. Finally, I would like to thank my parents Response from Dhruv Ranganathan for their incredible love and support. I am deeply humbled and honored to be selected About the Prize for Honorable Mention for the 2013 Morgan Prize. The Morgan Prize is awarded annually for out- I thank the AMS, MAA, and SIAM for selecting me. standing research in mathematics by an under- I owe a debt of gratitude to my advisor Dagan graduate student (or students having submitted Karp for his friendship, guidance, and patience. I joint work). Students in Canada, Mexico, or the also express my gratitude to my teachers, Alfonso United States or its possessions are eligible for Castro, Jon Jacobsen, Michael Orrison, and A. V. Ra- consideration for the prize. Established in 1995, mamoorthy, all of whom have inspired me to study the prize was endowed by Mrs. Frank (Brennie) mathematics. Finally, I owe a massive thank you to Morgan of Allentown, Pennsylvania, and carries my family and friends for their ever-present belief, the name of her late husband. The prize is given unquestioning support, and unbounded love. jointly by the AMS, the Mathematical Association Citation for Honorable Mention: Jonathan of America (MAA), and the Society for Industrial and Applied Mathematics (SIAM) and carries a cash Schneider award of US$1,200. Honorable mention goes to Jonathan Schneider Recipients of the Morgan Prize are chosen by for the 2013 AMS-MAA-SIAM Frank and Brennie a joint AMS-MAA-SIAM selection committee. For Morgan Prize for Outstanding Research in Math- the 2013 prize, the members of the selection com- ematics by an Undergraduate Student. Schneider mittee were Colin C. Adams, Jill Dietz, Kathleen R. is recognized for his solution to a problem posed Fowler, Reza Malek Madani, Kannan Soundarara- by Dr. Richard Stanley concerning polynomial jan, and . sequences of binomial type. One letter writer de- Previous recipients of the Morgan Prize are scribes Schneider’s “ingenious insights and argu- Kannan Soundararajan (1995), ments” as having “vast potential for applications (1996), Jade Vinson (1997), Daniel Biss (1998), and generalizations.” Jon is currently a senior at Sean McLaughlin (1999), Jacob Lurie (2000), Ciprian the Massachusetts Institute of Technology. Manolescu (2001), Joshua Greene (2002), Mela- nie Wood (2003), Reid Barton (2005), Jacob Fox Biographical Sketch (2006), Daniel Kane (2007), Nathan Kaplan (2008), Jon Schneider was born in Montreal, Canada, and Aaron Pixton (2009), Scott Duke Kominers (2010), grew up in Toronto. His high school mathematics Maria Monks (2011), and John Pardon (2012). teachers at the University of Toronto Schools en- couraged Jon’s passion for math, and while at high — Elaine Kehoe school Jon represented Canada three times at the International Mathematics Olympiad. Jon then enrolled at MIT, where he is currently concentrating in mathematics and theoretical

April 2013 Notices of the AMS 487 2013 E. H. Moore Prize

Michael J. Larsen and Richard versity of Missouri (1997–1998) before coming to Pink received the 2013 E. H. Moore Indiana University, where he is now Distinguished Research Article Prize at the 119th An- Professor of Mathematics. nual Meeting of the AMS in San Diego, Richard Pink received his Ph.D. in mathematics California, in January 2013. from Bonn University in 1989 under the supervi- sion of Professor Dr. G. Harder. An invited lecturer Citation to the 2002 ICM in Beijing, China, he has taught The 2013 E. H. Moore Research at Bonn University, Harvard University, the Max- Prize is awarded to Michael J. Planck-Institut für Mathematik, and Mannheim Larsen and Richard Pink for their University. Currently, he is teaching at ETH Zürich, article “Finite subgroups of alge- where he served as chairman of the Department braic groups”, Journal of the Ameri- of Mathematics from 2004–2006. His research can Mathematical Society 24 (2011), interests include number theory and arithmetic Michael J. Larsen no. 4, 1105–1158. The article provides geometry, specifically the arithmetic of Shimura a conceptual proof of an approxima- varieties, the topological-geometric nature of the tion to the classification of finite Lefschetz trace formula, Drinfeld modules and simple groups, using methods that their generalizations, and motives over function are independent of the classification. fields, including the arithmetic of the associ- Specifically, the article generalizes a ated Galois representations. In 1996, after some fundamental result of Camille Jordan preliminary sketches, he commissioned a master (1878) that concerns finite subgroups carpenter to design and build a fancy modular

of GLn over a field of characteristic cabinet for his house (http://www.math.ethz. 0; Jordan’s theorem states that each ch/~pink/ModularCabinet/cabinet.html); the such subgroup has a normal abelian design is based upon the action of the modular subgroup of index bounded by a con- group on the Poincaré half-plane, an important stant that depends only on n. Larsen idea that underlies much of his work. and Pink work with arbitrary fields; they allow positive characteristic. Joint Response from Michael J. Larsen and Richard Pink Their situation is visibly more compli- Richard Pink cated than that studied by Jordan: con- We would like to express our deep appreciation sider, for example, what happens over a finite field, to the American Mathematical Society for this where the ambient group GLn is finite and close to unexpected honor. At the time that we wrote the being simple. The authors prove, in particular, that original draft of our paper (1998), the quasi-thin finite simple subgroups of GLn are either small (in case of the classification of finite simple groups terms of n) or are of Lie type. remained open. We wanted a weak version of the Experts spoke of the impact of this article, classification theorem that could be used for sub- praising the methods of the article as well as its groups of GLn , for instance for Galois representa- conclusions. One report ended, “This was a tour tions associated to Drinfeld modules or for strong de force, and I cannot think of a better choice for approximation in the spirit of Boris Weisfeiler’s the E. H. Moore Prize.” work. We were inspired by the beautiful paper of

Madhav Nori on subgroups of GLn (Fp) and were Biographical Sketches very glad to succeed using only the comparatively Michael Larsen graduated from Harvard College soft methods of algebraic geometry. in 1984 and received a Ph.D. in mathematics from Princeton University in 1988. He has worked at the About the Prize Institute for Advanced Study (1988–1990), the Uni- The Moore Prize is awarded every three years for versity of Pennsylvania (1990–1997), and the Uni- an outstanding research article that appeared in one of the primary AMS research journals: Journal DOI: http://dx.doi.org/10.1090/noti966 of the AMS, Proceedings of the AMS, Transactions of

488 Notices of the AMS Volume 60, Number 4

!!Not Supplied!! !!Not Supplied!! Notices of the AMS 1 the AMS, AMS Memoirs, Mathematics of Computa- The Moore Prize is awarded by the AMS Coun- tion, Electronic Journal of Conformal Geometry and cil acting on the recommendation of a selection Dynamics, or Electronic Journal of Representation committee. For the 2013 prize, the members of Theory. The article must have appeared during the selection committee were: Sergiu Klainerman, the six calendar years ending a full year before the Howard Masur, Bjorn Poonen, Kenneth A. Ribet, meeting at which the prize is awarded. The prize and Ulrike L. Tillman. carries a cash award of US$5,000. The previous recipients of the Moore Prize are The prize honors the extensive contributions of E. H. Moore (1862–1932) to the AMS. Moore Mark Haiman (2004), Ivan Shestakov and Ualbai founded the Chicago section of the AMS, served Umirbaev (2007), and Sorin Popa (2010). as the Society’s sixth president (1901–1902), deliv- ered the Colloquium Lectures in 1906, and founded —Elaine Kehoe and nurtured the Transactions of the AMS.

2013 Robbins Prize

Alexander Razborov received the David P. Rob- was done by ingenuity and the trial- bins Prize at the 119th Annual Meeting of the AMS and-error method. Remarkably, the in San Diego, California, in January 2013. work of Razborov gives a systematic approach to these arguments. His flag Citation algebra calculus provides a formalism The 2013 David P. Robbins Prize is awarded to through which the problem of finding Alexander Razborov of the University of Chicago relations between subgraph densi- for his paper “On the minimal density of triangles ties can be reduced to a semidefinite in graphs”, Combinatorics, Probability and Comput- programming (SDP) problem. This in ing 17 (2008), no. 4, 603–618, and for introducing turn enables the use of computers to a new powerful method, flag algebras, to solve find solutions, with rigorous proofs, problems in extremal combinatorics. to problems in extremal combina- Razborov solves an old extremal problem about torics. This method already had a the minimum possible number of triangles in a great impact on the area, and it has graph with n vertices and m edges. The origin of been used to settle a number of long- Alexander Razborov this problem goes back more than one hundred standing open problems in extremal years to one of the oldest results in extremal com- graph theory. binatorics, by Mantel, who proved that any such graph with more than n2/4 edges must have a Biographical Sketch triangle. This leads to the natural question of how Alexander Razborov was born in 1963 in the small many such triangles (as a function of the number Siberian town of Belovo. He received his B.Sc. in of edges) should exist. Although the problem has mathematics from Moscow State University and been studied by leading combinatorialists for de- his Ph.D. from the Steklov Mathematical Institute cades, it remained open until its recent solution (Moscow). Currently, he is an Andrew MacLeish by Razborov. Distinguished Service Professor at the Department The paper by Razborov not only settled a long- of Computer Science at the University of Chicago, standing open problem; much more importantly, with part-time appointments at the Steklov Math- it introduced a new method, called flag algebra ematical Institute and Toyota Technological Insti- calculus, for attacking a large class of extremal tute at Chicago. He received the Rolf Nevanlinna questions. This method was originally invented Prize in 1990 and the Gödel Prize in 2007, was an by Razborov to study the triangle density problem invited speaker at the ICM in Berkeley (1986), and and was developed in full generality in his closely was elected as a corresponding member of the related paper, “Flag algebras”, Journal of Symbolic Russian Academy of Sciences in 2000. His research Logic 72 (2007), no. 4, 1239–1282. spans several areas in theoretical computer sci- The solution of many extremal problems re- ence, including computational complexity, proof quires finding inequalities involving densities of complexity, , and computa- small subgraphs of large graphs. Until recently this tional complexity, as well as related mathematical areas, notably discrete mathematics and combina- DOI: http://dx.doi.org/10.1090/noti969 torial group theory.

April 2013 Notices of the AMS 489 the AMS, AMS Memoirs, Mathematics of Computa- The Moore Prize is awarded by the AMS Coun- tion, Electronic Journal of Conformal Geometry and cil acting on the recommendation of a selection Dynamics, or Electronic Journal of Representation committee. For the 2013 prize, the members of Theory. The article must have appeared during the selection committee were: Sergiu Klainerman, the six calendar years ending a full year before the Howard Masur, Bjorn Poonen, Kenneth A. Ribet, meeting at which the prize is awarded. The prize and Ulrike L. Tillman. carries a cash award of US$5,000. The previous recipients of the Moore Prize are The prize honors the extensive contributions of E. H. Moore (1862–1932) to the AMS. Moore Mark Haiman (2004), Ivan Shestakov and Ualbai founded the Chicago section of the AMS, served Umirbaev (2007), and Sorin Popa (2010). as the Society’s sixth president (1901–1902), deliv- ered the Colloquium Lectures in 1906, and founded —Elaine Kehoe and nurtured the Transactions of the AMS.

2013 Robbins Prize

Alexander Razborov received the David P. Rob- was done by ingenuity and the trial- bins Prize at the 119th Annual Meeting of the AMS and-error method. Remarkably, the in San Diego, California, in January 2013. work of Razborov gives a systematic approach to these arguments. His flag Citation algebra calculus provides a formalism The 2013 David P. Robbins Prize is awarded to through which the problem of finding Alexander Razborov of the University of Chicago relations between subgraph densi- for his paper “On the minimal density of triangles ties can be reduced to a semidefinite in graphs”, Combinatorics, Probability and Comput- programming (SDP) problem. This in ing 17 (2008), no. 4, 603–618, and for introducing turn enables the use of computers to a new powerful method, flag algebras, to solve find solutions, with rigorous proofs, problems in extremal combinatorics. to problems in extremal combina- Razborov solves an old extremal problem about torics. This method already had a the minimum possible number of triangles in a great impact on the area, and it has graph with n vertices and m edges. The origin of been used to settle a number of long- Alexander Razborov this problem goes back more than one hundred standing open problems in extremal years to one of the oldest results in extremal com- graph theory. binatorics, by Mantel, who proved that any such graph with more than n2/4 edges must have a Biographical Sketch triangle. This leads to the natural question of how Alexander Razborov was born in 1963 in the small many such triangles (as a function of the number Siberian town of Belovo. He received his B.Sc. in of edges) should exist. Although the problem has mathematics from Moscow State University and been studied by leading combinatorialists for de- his Ph.D. from the Steklov Mathematical Institute cades, it remained open until its recent solution (Moscow). Currently, he is an Andrew MacLeish by Razborov. Distinguished Service Professor at the Department The paper by Razborov not only settled a long- of Computer Science at the University of Chicago, standing open problem; much more importantly, with part-time appointments at the Steklov Math- it introduced a new method, called flag algebra ematical Institute and Toyota Technological Insti- calculus, for attacking a large class of extremal tute at Chicago. He received the Rolf Nevanlinna questions. This method was originally invented Prize in 1990 and the Gödel Prize in 2007, was an by Razborov to study the triangle density problem invited speaker at the ICM in Berkeley (1986), and and was developed in full generality in his closely was elected as a corresponding member of the related paper, “Flag algebras”, Journal of Symbolic Russian Academy of Sciences in 2000. His research Logic 72 (2007), no. 4, 1239–1282. spans several areas in theoretical computer sci- The solution of many extremal problems re- ence, including computational complexity, proof quires finding inequalities involving densities of complexity, quantum computing, and computa- small subgraphs of large graphs. Until recently this tional complexity, as well as related mathematical areas, notably discrete mathematics and combina- DOI: http://dx.doi.org/10.1090/noti969 torial group theory.

April 2013 Notices of the AMS 489 Response developing this theory. Last but not the least my I am truly grateful and honored to receive this special thanks go to my wife, Iren, and my children, prize, and I am equally delighted to receive it for Andrew and Maria, for humorously bearing with this particular topic, which is very close to my the half-absence of their husband and daddy even heart. when he appears to be fully present physically. All my professional career so far has been spent on the interface between computer science and About the Prize mathematics, and I genuinely believe that certain The Robbins Prize was established in 2005 in amusing cultural differences between the two memory of David P. Robbins by members of his communities look really insignificant when com- family. Robbins, who died in 2003, received his pared to the amount of inspiration and fresh and Ph.D. in 1970 from the Massachusetts Institute novel ideas their interaction brings to both disci- of Technology. He was a long-time member of the plines. In a sense, the work I am being awarded Institute for Defense Analysis Center for Commu- for is a quintessence of this philosophy. In order nications Research and a prolific mathematician to be able to do something really computational whose work (much of it classified) was in discrete (computer-aided theorem proving in extremal mathematics. combinatorics using packages for semidefinite The prize is given for a paper that (1) reports programming), one has to be able to reveal and on novel research in algebra, combinatorics, or understand fundamental, albeit somewhat simple discrete mathematics; (2) has a significant ex- by mathematical standards, algebraic and analyti- perimental component; (3) is on a topic broadly cal structures behind this activity and be guided accessible; and (4) provides a simple statement by them. Thank you again, both for the recognition of the problem and clear exposition of the work. of my own contribution and for promoting this The US$5,000 prize is awarded every three years. important interdisciplinary ideology! The Robbins Prize is awarded by the AMS Coun- I would like to use this opportunity and thank cil acting on the recommendation of a selection all institutions I have been fortunate to be af- committee. For the 2013 prize, the members of filiated with (Steklov Mathematical Institute, IAS, the selection committee were: Daniel J. Allcock, University of Chicago, Toyota Technological Insti- J. P. Buhler, Maria Chudnovsky, Bernd Sturmfels, tute) for the stimulating intellectual environment and Benjamin Sudakov. that encouraged work on difficult and interesting The previous recipients of the Robbins problems. I am very grateful to my own collabora- Prize are Samuel Ferguson and Thomas C. tors on the project (Hamed Hatami, Jan Hladky, Hales (2007) and Ileana Streinu (2010). Daniel Kral, Sergei Norin, Oleg Pikhurko), as well as to many other young “flag algebraists” for —Elaine Kehoe 2013 Satter Prize

Maryam Mirzakhani received the 2013 AMS Ruth boundary components. That there exists a formula Lyttle Satter Prize in Mathematics at the 119th An- of this nature was itself surprising, but more sur- nual Meeting of the AMS in San Diego, California, prising were the results she was able to extract in January 2013. from it: a new proof of the celebrated conjecture of Witten on the intersection numbers of tautol- Citation ogy classes on and, in a completely The 2013 Ruth Lyttle Satter Prize in Mathematics different direction, an asymptotic formula for the is awarded to for her deep lengths of simple closed geodesics on a compact contributions to the theory of moduli spaces of hyperbolic surface. Riemann surfaces. Much of her work subsequent to this has fo- Her earliest work, the topic of her thesis, was a cused on the Teichmüller dynamics of moduli volume formula for the moduli space of bordered space. In particular, she was able to construct a Riemann surfaces of genus g with n geodesic measure-preserving conjugacy between Thurston’s boundary components, a formula that expresses earthquake flow on Teichmüller space and horocy- this volume as a polynomial in the lengths of the cle flow on the associated space of quadratic differ- entials and as an immediate and long sought-after DOI: http://dx.doi.org/10.1090/noti969 consequence of this to prove that earthquake flow

490 Notices of the AMS Volume 60, Number 4 Response developing this theory. Last but not the least my I am truly grateful and honored to receive this special thanks go to my wife, Iren, and my children, prize, and I am equally delighted to receive it for Andrew and Maria, for humorously bearing with this particular topic, which is very close to my the half-absence of their husband and daddy even heart. when he appears to be fully present physically. All my professional career so far has been spent on the interface between computer science and About the Prize mathematics, and I genuinely believe that certain The Robbins Prize was established in 2005 in amusing cultural differences between the two memory of David P. Robbins by members of his communities look really insignificant when com- family. Robbins, who died in 2003, received his pared to the amount of inspiration and fresh and Ph.D. in 1970 from the Massachusetts Institute novel ideas their interaction brings to both disci- of Technology. He was a long-time member of the plines. In a sense, the work I am being awarded Institute for Defense Analysis Center for Commu- for is a quintessence of this philosophy. In order nications Research and a prolific mathematician to be able to do something really computational whose work (much of it classified) was in discrete (computer-aided theorem proving in extremal mathematics. combinatorics using packages for semidefinite The prize is given for a paper that (1) reports programming), one has to be able to reveal and on novel research in algebra, combinatorics, or understand fundamental, albeit somewhat simple discrete mathematics; (2) has a significant ex- by mathematical standards, algebraic and analyti- perimental component; (3) is on a topic broadly cal structures behind this activity and be guided accessible; and (4) provides a simple statement by them. Thank you again, both for the recognition of the problem and clear exposition of the work. of my own contribution and for promoting this The US$5,000 prize is awarded every three years. important interdisciplinary ideology! The Robbins Prize is awarded by the AMS Coun- I would like to use this opportunity and thank cil acting on the recommendation of a selection all institutions I have been fortunate to be af- committee. For the 2013 prize, the members of filiated with (Steklov Mathematical Institute, IAS, the selection committee were: Daniel J. Allcock, University of Chicago, Toyota Technological Insti- J. P. Buhler, Maria Chudnovsky, Bernd Sturmfels, tute) for the stimulating intellectual environment and Benjamin Sudakov. that encouraged work on difficult and interesting The previous recipients of the Robbins problems. I am very grateful to my own collabora- Prize are Samuel Ferguson and Thomas C. tors on the project (Hamed Hatami, Jan Hladky, Hales (2007) and Ileana Streinu (2010). Daniel Kral, Sergei Norin, Oleg Pikhurko), as well as to many other young “flag algebraists” for —Elaine Kehoe 2013 Satter Prize

Maryam Mirzakhani received the 2013 AMS Ruth boundary components. That there exists a formula Lyttle Satter Prize in Mathematics at the 119th An- of this nature was itself surprising, but more sur- nual Meeting of the AMS in San Diego, California, prising were the results she was able to extract in January 2013. from it: a new proof of the celebrated conjecture of Witten on the intersection numbers of tautol- Citation ogy classes on moduli space and, in a completely The 2013 Ruth Lyttle Satter Prize in Mathematics different direction, an asymptotic formula for the is awarded to Maryam Mirzakhani for her deep lengths of simple closed geodesics on a compact contributions to the theory of moduli spaces of hyperbolic surface. Riemann surfaces. Much of her work subsequent to this has fo- Her earliest work, the topic of her thesis, was a cused on the Teichmüller dynamics of moduli volume formula for the moduli space of bordered space. In particular, she was able to construct a Riemann surfaces of genus g with n geodesic measure-preserving conjugacy between Thurston’s boundary components, a formula that expresses earthquake flow on Teichmüller space and horocy- this volume as a polynomial in the lengths of the cle flow on the associated space of quadratic differ- entials and as an immediate and long sought-after DOI: http://dx.doi.org/10.1090/noti969 consequence of this to prove that earthquake flow

490 Notices of the AMS Volume 60, Number 4 is ergodic. In another grateful to my husband, Jan, for being my best vein, her recent work friend and companion and for encouraging me with Eskin establishes when I need it the most. I would like to thank my striking analogues for parents, who always believed in me and let me be the Teichmüller flow who I am. They have been my inspiration through- and the mapping class out my life. group of Selberg’s classical “Prime Geo- About the Prize desic Theorem” for the The Satter Prize is awarded every two years to modular surface and recognize an outstanding contribution to math- the modular group. ematics research by a woman in the previous six Moreover, in a work years. Established in 1990 with funds donated by in progress, they have Joan S. Birman, the prize honors the memory of unearthed some unex- Birman’s sister, Ruth Lyttle Satter. Satter earned a Maryam Mirzakhani pected and intriguing bachelor’s degree in mathematics and then joined analogues, in this Teichmüller setting, of the Rat- the research staff at AT&T Bell Laboratories during ner unipotent rigidity theorems in homogeneous World War II. After raising a family, she received dynamics. a Ph.D. in botany at the age of forty-three from the University of Connecticut at Storrs, where she Biographical Sketch later became a faculty member. Her research on Maryam Mirzakhani grew up in Tehran, Iran. She the biological clocks in plants earned her recogni- obtained her B.Sc. in mathematics (1999) from the tion in the United States and abroad. Birman re- Sharif University of Technology. She holds a Ph.D. quested that the prize be established to honor her from Harvard University (2004), where her advi- sister’s commitment to research and to encourage sor was Curtis McMullen. From 2004 to 2008 she women in science. The prize carries a cash award was a Clay Mathematics Institute Research Fellow of US$5,000. and an assistant professor at Princeton University. The Satter Prize is awarded by the AMS Coun- She is currently a professor at Stanford University. cil acting on the recommendation of a selection committee. For the 2013 prize, the members of Her research interests include Teichmüller theory, the selection committee were Victor W. Guillemin, hyperbolic geometry, and ergodic theory. Svetlana Y. Jitomirskaya, and Raman Parimala. Response Previous recipients of the Satter Prize are: Dusa McDuff (1991), Lai-Sang Young (1993), Sun-Yung I am deeply honored to receive the Ruth Lyttle Alice Chang (1995), Ingrid Daubechies (1997), Satter Prize. This would not have been possible Bernadette Perrin-Riou (1999), Karen E. Smith without many people who helped me along. I am (2001), Sijue Wu (2001), Abigail Thompson (2003), grateful to my collaborators and colleagues who Svetlana Jitomirskaya (2005), Claire Voisin (2007), helped me all these years. I would like to thank Laure Saint-Raymond (2009), and Amie Wilkinson my great teachers in Iran, both in high school and (2011). at Sharif University, for providing a stimulating —Elaine Kehoe environment for their students. All these oppor- tunities and the people who made them possible, regardless of the difficulties of the times, deserve my sincere gratitude. I am also grateful to my Ph.D. advisor, Curt McMullen, for his unceasing support and for introducing me to fascinating areas of mathematics. I have enjoyed a pleasant and supportive en- vironment during my time at Harvard, Princeton, and Stanford. Still, in my opinion, the situation of women in math is far from ideal. The social bar- riers for girls who are interested in mathematical sciences might not be lower now than they were when I grew up. And balancing career and family remains a big challenge. It makes most women face difficult decisions which usually compro- mise their work. However, there has been a lot of progress over the years, and I am sure this trend will continue. Finally, I would like to thank my friends who have been like my family away from home. I am

April 2013 Notices of the AMS 491 2013 Wiener Prize

Andrew J. Majda received the 2013 AMS-SIAM and Sciences and has received numerous honors Prize in Applied Mathematics at and awards, including the National Academy of the 119th Annual Meeting of the AMS in San Diego, Science Prize in Applied Mathematics, the John California, in January 2013. von Neumann Prize of the Society of Industrial and Applied Mathematics, and the Gibbs Prize of Citation the AMS. He has been awarded the Medal of the The 2013 Norbert Wiener Prize in Applied Math- Collège de France twice, and he is a Fellow of the ematics is awarded to Andrew J. Majda for his Japan Society for the Promotion of Science. Majda groundbreaking work in theoretical has received three honorary doctorates, including fluid mechanics and its application to one from his undergraduate alma mater, Purdue problems in atmospheric science and University. He has given plenary one-hour lectures oceanography. His many outstanding at both the ICM (Kyoto 1990) and the first ICIAM contributions to the field include his (Paris 1987) and is a fellow of both AMS and SIAM. work on vortex dynamics, turbulent In his years at the Courant Institute, Majda has diffusion, concentration phenomena created the Center for Atmosphere Ocean Science for the Euler equations, multidimen- with seven multidisciplinary faculty to promote sional shock fronts, and absorbing cross-disciplinary research with modern applied boundary conditions for wave propa- mathematics in climate modeling and prediction. gation. Mathematicians and geophysi- cists alike have embraced Majda’s Response pioneering advances on important I am honored and delighted to receive the 2013 and recalcitrant issues arising in cli- Norbert Wiener Prize from the American Math- Andrew J. Majda mate modeling and prediction. This ematical Society and the Society for Industrial work includes the development and and Applied Mathematics. I believe in modern exploitation of the methods of statistical physics applied mathematics with the broadest possible in geophysical problems, as well as the multiscale toolkit blending asymptotic methods, numerical analysis of moist fluid dynamics in the atmo- methods, and rigorous mathematical analysis sphere. with physical reasoning used to attack the most difficult and pressing scientific problems, such Biographical Sketch as climate modeling or turbulence. In this way, Andrew J. Majda is Morse Professor of Arts and both a scientific discipline and mathematics can Sciences at the Courant Institute of New York Uni- enrich each other in amazing ways. As a young versity. He was born in East Chicago, Indiana, on scientist, I found I truly loved the serendipity January 30, 1949. He received a B.S. degree from between applied mathematics and complex physi- in 1970 and a Ph.D. degree from cal phenomena, and I would like to thank Peter Stanford University in 1973. Majda began his scien- Lax and Joe Keller for being role models, in very tific career as a Courant Instructor at the Courant different ways, during my early days at Courant. Institute from 1973–1975. Prior to returning to the I would like to thank my many collaborators, Ph.D. Courant Institute in 1994, he held professorships students, and postdoctoral fellows for all of their at Princeton University (1984–1994), the Univer- insight, help, and friendship. For my cited earlier sity of California Berkeley (1978–1984), and the works, I would like to mention especially Tom University of California (1976–1978). Beale, Bjorn Engquist, and Ron DiPerna, a truly Majda is a member of the National Academy brilliant mathematician who left us too soon. For of Sciences and the American Academy of Arts my cited work in climate atmosphere ocean science and applied mathematics, I would like to acknowl- DOI: http://dx.doi.org/10.1090/noti971 edge Eric Vanden-Eijnden, Rupert Klein, Boualem

492 Notices of the AMS Volume 60, Number 4 2013 San Diego, CA, Khouider, Sam Stechmann, and Dimitri Giannakis. Finally, I would like to thank my many friends Joint Mathematics Meetings and colleagues at the Courant Institute, especially Photo Key for pages 390–391 David McLaughlin, for the wonderful, supportive atmosphere for genuine interdisciplinary research with mathematics and applications. 1 2 14 15 16 About the Prize 17 The Wiener Prize is awarded every three years 3 5 18 to recognize outstanding contributions to applied 4 19 mathematics in the highest and broadest sense 6 8 20 (until 2001, the prize was awarded every five 7 21 22 years). Established in 1967 in honor of Norbert 10 23 Wiener (1894–1964), the prize was endowed by the 9 Department of Mathematics of the Massachusetts 26 Institute of Technology. The prize is given jointly 11 12 13 24 25 by the AMS and the Society for Industrial and Ap- plied Mathematics (SIAM). The recipient must be a member of one of these societies and a resident of the United States, Canada, or Mexico. The prize 1. AMS Booth in the Exhibits area. carries a cash award of US$5,000. 2. AMS Booth in the Exhibits area. The recipient of the Wiener Prize is chosen by a 3. Contestants in Who Wants to Be a Mathematician joint AMS-SIAM selection committee. For the 2013 National Game. prize, the members of the selection committee 4. Coming down the escalator in the San Diego were: Jerry L. Bona, Roger Temam, and Horng- Convention Center. Tzer Yau. 5. Meeting and greeting in the AMS Booth. The previous recipients of the Wiener Prize are: 6. Listening to a talk. Richard E. Bellman (1970), Peter D. Lax (1975), 7. Mathematical Art Exhibit. Tosio Kato (1980), Gerald B. Whitham (1980), 8. In the Exhibits area. Clifford S. Gardner (1985), Michael Aizenman 9. One of the winners, Mathematical Art Exhibit. (1990), Jerrold E. Marsden (1990), Hermann 10. Email Center. Flaschka (1995), Ciprian Foias (1995), Alexan- 11. In the Exhibits area. dre J. Chorin (2000), Arthur T. Winfree (2000), 12. AMS Colloquium Lecture speaker Alice Guionnet. James Sethian (2004), Craig Tracy and Harold 13. In the Exhibits area. Widom (2007), and David L. Donoho (2010). 14. Attentive audience. 15. Calvin Deng, winner of the San Diego Who Wants — Elaine Kehoe to Be a Mathematician Game. 16. Grad Student Fair. 17. Outside the San Diego Convention Center. 18. Undergraduate Student Poster Session. 19. Fellow students cheer on a Who Wants to Be a Mathematician contestant. 20. Invited Address audience. 21. I Love Math buttons. 22. Cédric Villani, AMS Josiah Willard Gibbs Lecture speaker. 23. Networking. 24. MAA Invited Address speaker Tony DeRose. 25. AMS-MAA Invited Address speaker Emily Shuckburgh. 26. AMS 125th Anniversary Gala, Saturday, January 12.

April 2013 Notices of the AMS 493 2013 Veblen Prize

Ian Agol and Daniel Wise received the 2013 quasiconvex subgroup of a hyperbolic group is Prize in Geometry at the 119th An- separable. nual Meeting of the AMS in San Diego, California, In April 2012 Agol posted an article to the arXiv in January 2013. announcing a proof of a conjecture of Daniel Wise implying both Waldhausen’s virtual Haken conjec- Citation for ture and Thurston’s virtual fibering conjecture. The 2013 Oswald Veblen Prize in Geometry is The proof involves a generalization of the methods awarded to Ian Agol for his many fundamen- of these papers, clever new ideas, and it makes tal contributions to hyperbolic crucial use of results of Wise and joint work of geometry, 3-manifold topology, Haglund and Wise. The resolution of these conjec- and geometric group theory. In tures is a milestone achievement in the theory of particular, Agol is cited for the 3-dimensional manifolds. following papers: (with P. Storm and W. P. Thur- Biographical Sketch ston) “Lower bounds on volumes Ian Agol was born in Hollywood, California, in 1970 of hyperbolic Haken 3-manifolds” and received his Ph.D. at the University of Califor- with an appendix by Nathan Dun- nia San Diego in 1998 under the supervision of field, Journal of the American . He held postdoctoral positions Mathematical Society 20 (2007), at UC Davis and the University of Melbourne before no. 4, 1053–1077. teaching at the University of Illinois at Chicago. “Criteria for virtual fibering”, He moved to UC Berkeley in 2007. He received a Journal of Topology 1 (2008), Guggenheim fellowship in 2005 and spoke at the Ian Agol no. 2, 269–284. International Congress of Mathematicians (ICM) in (with D. Groves and J. F. Man- Madrid (2006). ning) “Residual finiteness, QCERF and fillings of He shared the 2009 Clay Research Award with hyperbolic groups”, Geometry and Topology 13 Danny Calegari and for the solution (2009), no. 2, 1043–1073. of the Marden tameness conjecture. He received In 2004 Agol (and independently Calegari and the 2012 Senior Berwick Prize for his cited paper, Gabai) proved the Marden tameness conjecture, “Criteria for virtual fibering”. which led to many important results in hyper- bolic geometry, among them the Ahlfors measure Response conjecture. It is an honor to share the 2013 Oswald Veblen Agol subsequently proved a number of impor- Prize in Geometry with Dani Wise. tant results on volumes of hyperbolic 3-manifolds, I owe a debt of gratitude to my family for their including an important inequality, found with support, especially my mother, who made sure Nathan Dunfield in the first cited paper, relating that I had a top-notch education and encouraged the volume of a closed hyperbolic 3-manifold with me to pursue a career in a field I am passionate that of a drilled manifold and the tube radius of about, and my wife, Michelle, for her support. I also the drilled geodesic. This played an important role share this award in spirit with my collaborators, in the proof (by Gabai, Meyerhoff, and Milley) that especially Nathan Dunfield, Peter Storm, Daniel the Weeks manifold is the unique lowest volume Groves, Jason Manning, and the late Bill Thurston, closed orientable hyperbolic 3-manifold. with whom I collaborated on the papers mentioned In the second paper Agol found a key group in the citation. My research on the virtual Haken theoretic criterion for a closed irreducible 3-mani- conjecture was influenced heavily by the work of fold to have a finite sheeted covering space that Dani Wise and his collaborators Nicolas Bergeron, fibers over the circle. With Daniel Groves and Jason Tim Hsu, Michah Sageev, and especially Frédéric Manning, he showed in the third paper that if all Haglund. Without Wise’s insights, this approach to hyperbolic groups are residually finite, then any the conjecture would have been impossible, and his vision completely changed my perspective on DOI: http://dx.doi.org/10.1090/noti973 the problem. Another key contribution was made

494 Notices of the AMS Volume 60, Number 4 by Jeremy Kahn and Vlad Markovic on the surface (with F. Haglund) “Spe- subgroup problem. The subject of hyperbolic cial cube complexes”, groups and CAT(0) cube complexes, introduced by Geometric and Func- Misha Gromov, has reached a mature state, which tional Analysis 17 (2008), I think is now made evident by the first Veblen no. 5, 1551–1620. Prize given to a geometric group theorist after (with F. Haglund) “A com- Gromov himself. bination theorem for special In the 1980s, Bill Thurston laid out his vision cube complexes”, Annals of for three-dimensional topology and Kleinian Mathematics 176 (2012), no. groups. Over the past thirty years, his vision has 3, 1427–1482. In the first of been validated, with such milestones achieved as these papers, Wise and Ha- the geometrization theorem, the ending lamina- glund introduced the concept tion theorem, the tameness theorem, the density of a “special cube complex”. theorem, and now the virtual fibering theorem. The The fundamental groups of Daniel Wise virtual Haken theorem (originating as a question such complexes have various of Waldhausen) is a purely topological statement; remarkable properties. For example, they embed however, its resolution is in some sense mostly into right-angled Artin groups and their quasi- geometrical and depends on many developments convex subgroups are separable. Generalizing the from outside the field, including PDEs, Riemannian methods of the two papers of Wise cited above, and Alexandrov geometry, dynamics, representa- Wise and Haglund show that a group obtained tion theory, and geometric group theory. from two virtually special hyperbolic groups that I hope that 3-manifolds and Kleinian groups are amalgamated along quasiconvex malnormal will give back equally to these and other subjects subgroups is virtually special. In subsequent work in the future. Moreover, I think the most impor- Wise used this to prove that the fundamental tant project for the future is to make connections groups of Haken hyperbolic 3-manifolds are virtu- between this highly developed area of geometric ally special. 3-manifold topology and other approaches, such Beyond what is cited in these papers, Wise made as various flavors of Floer homology, topological the far-reaching conjecture that a Gromov hyper- quantum field theory, knot theory and 4-mani- bolic group that acts cocompactly and properly on folds, and other types of geometric structures on a CAT(0) cube complex is the fundamental group of 3-manifolds. virtually special cube complexes. Ian Agol proved Citation for Daniel Wise this conjecture making essential use of the technol- The 2013 Veblen Prize in Geometry is awarded to ogy of Wise and his collaborators. Combined with Daniel Wise for his deep work establishing sub- the work of Jeremy Kahn and Vladimir Markovic, group separability (LERF) for a wide class of groups and with the work of Michah Sageev, this gives a and for introducing and developing with Frédéric positive solution to the long-standing Waldhausen Haglund the theory of special cube complexes, virtual Haken conjecture. which are of fundamental importance for the to- Biographical Sketch pology of 3-dimensional manifolds. In particular Wise is cited for the following papers: Daniel Wise was born in 1971 and grew up in the “Subgroup separability of graphs of free groups area. He received a B.A. in 1991 from with cyclic edge groups”, Quarterly Journal of Yeshiva College and a Ph.D. in 1996 from Princeton Mathematics 51 (2000), no. 1, 107–129. A sub- University. After stints at UC Berkeley, Cornell Uni- group is separable if it is the intersection of all versity, and Brandeis University, he joined McGill the finite index subgroups that contain it. This University, where he has been teaching since 2001. is an important condition in 3-manifold topology Dani lives in Montreal with his wife, Yael, and their since up to homotopy an immersed surface in a four children. Their house is full of music, art, closed irreducible 3-manifold lifts to an embedded laughter, and sleep deprivation. surface in some finite sheeted cover if the immer- sion induces an isomorphism of the fundamental Response group of the surface with a separable subgroup of I am grateful to be the recipient, together with the fundamental group of the 3-manifold. Ian Agol, of the Oswald Veblen Prize of the AMS. “Residual finiteness of negatively curved poly- I share my part of this recognition with my coau- gons of finite groups”, Inventiones Mathematicae thors Nicolas Bergeron, Chris Hruska, Tim Hsu, 149 (2002), no. 3, 579–617. This paper is represen- Michah Sageev, and especially Frédéric Haglund. tative of a long arc of work in which Wise describes This is a welcome opportunity to thank Jonathan powerful and ingenious criteria for showing that a Sondow and Sylvain Cappell, who launched me into group is residually finite or, more generally, that mathematics, Martin Bridson, who initiated my all of its quasiconvex subgroups are separable. interest in geometric group theory, my colleagues

April 2013 Notices of the AMS 495 and students at McGill, my friends and family, and committee. For the 2010 prize, the members of my wife, Yael. the selection committee were Tobias H. Colding, I was attracted to residual finiteness at the very William G. Dwyer, and Michael J. Hopkins. beginning of my career—and latched on to the Previous recipients of the Veblen Prize are: idea of a “clean complex” in 1993. From that point Christos D. Papakyriakopoulos (1964), Raoul H. onward I progressively nurtured a belief that a vari- Bott (1964), Stephen Smale (1966), ant of this was a key toward understanding basic and Barry Mazur (1966), Robion C. Kirby (1971), properties of fundamental groups of 3-manifolds. I Dennis P. Sullivan (1971), William P. Thurston made progress in 1997 toward understanding that (1976), James Simons (1976), Mikhael Gromov many prime alternating link groups are virtually (1981), Shing-Tung Yau (1981), Michael H. Freed- fundamental groups of clean complexes. In 2000, man (1986), (1991), Clifford H. I realized that most small-cancellation groups had Taubes (1991), Richard Hamilton (1996), Gang codimension-one subgroups, made the connec- Tian (1996), (2001), Yakov Eliash- tion to Michah Sageev’s work on dual CAT(0) cube berg (2001), Michael J. Hopkins (2001), David complexes, and then began a crusade to “cubulate” Gabai (2004), Peter Kronheimer, , all groups in sight. To my great fortune, Frédéric Peter Ozsváth, and Zoltán Szabó (2007), Tobias H. Haglund visited in 2002, and we developed the Colding and William P. Minicozzi (2010), and Paul notion of a “special cube complex”. These “general- Seidel (2010). ized graphs” improved upon clean complexes in a manner that I had only dreamed about beforehand. — Elaine Kehoe My previous work was readapted in this more elegant context with no 2-dimensional limitation, and this began to catch the attention of the geo- metric group theory community. There was now something concrete to aim for—and at the 2004 Spring Topology and Dynamics conference, I laid out a plan for understanding groups: first cubu- late and then find a special cover. I explained that my conjecture, that each hyperbolic 3-manifold M is virtually special, is equivalent to the pair of

well-known conjectures that π1M has separable quasiconvex subgroups and that π1M has suffi- ciently many quasiconvex surface subgroups (now a recent accomplishment of Kahn and Markovic). Though some topologists were reluctant to believe in the cubical route, by 2005 I had mapped out a strategy for proving a significant case: the virtual specialness of “hyperbolic groups with a quasiconvex hierarchy”. The various parts of this plan were accomplished with the great help of my collaborators, and during my 2008–2009 sabbati- cal at the Hebrew University, I was able to write up details of this work. I remain deeply grateful to Zlil Sela for his feedback and encouragement during that time. It brings tremendous satisfaction to have seen Ian Agol bring this program to completion among his many great achievements.

About the Prize The Veblen Prize is awarded every three years for a notable research memoir in geometry or topology that has appeared during the previous five years in a recognized North American journal (until 2001 the prize was usually awarded every five years). Established in 1964, the prize honors the memory of Oswald Veblen (1880–1960), who served as president of the AMS during 1923–1924. It carries a cash award of US$5,000. The Veblen Prize is awarded by the AMS Coun- cil acting on the recommendation of a selection

496 Notices of the AMS Volume 60, Number 4 Mathematics People

1987, Journal of the AMS, 1988, and Journal of Differential Jerison and Lee Awarded 2012 Geometry, 1989), Jerison and Lee established this result Bergman Prize for the case n bigger than or equal to 2, with M not lo- cally CR equivalent to the sphere. (The remaining cases David Jerison of the Massachusetts Institute of Technol- n=1 and M conformally flat were subsequently settled by ogy and John M. Lee of the University of Washington have Gamara, Journal of the European Math Society, 2001, and been awarded the 2012 Stefan Bergman Prize. Established Gamara-Yacoub, Pacific Journal of Math, 2001). In settling in 1988, the prize recognizes mathematical accomplish- a large part of this problem, Jerison and Lee developed a ments in the areas of research in which Stefan Bergman myriad of important tools in the subject, such as casting worked. Jerison and Lee will each receive US$12,494, which the problem as a nonlinear subelliptic partial differential is half of the 2012 income from the prize fund. equation for the conformal factor, finding the best con- The previous Bergman Prize winners are: David W. stant in Sobolev inequalities in the Heisenberg group, and Catlin (1989), Steven R. Bell and Ewa Ligocka (1991), constructing suitable normal CR coordinates on M. (1992), Yum Tong Siu (1993), John Erik Fornæss (1994), Harold P. Boas and Emil J. Straube (1995), Biographical Sketch: David Jerison David E. Barrett and Michael Christ (1997), John P. David Jerison was born in Lafayette, Indiana, in 1953. He D’Angelo (1999), Masatake Kuranishi (2000), László received his B.A. from Harvard College in 1975 and then Lempert and Sidney Webster (2001), M. Salah Baouendi spent a year at the University of Paris and Linda Preiss Rothschild (2003), Joseph J. Kohn at Orsay on a Rotary Fellowship. He (2004), Elias M. Stein (2005), Kengo Hirachi (2006), Alex- received his Ph.D. from Princeton ander Nagel and Stephen Wainger (2007–2008), Ngaiming University in 1980, writing a thesis Mok and Duong H. Phong (2009), and Gennadi Henkin under the direction of Elias M. Stein. (2011). On the selection committee for the 2012 prize After a year as a National Science were Harold P. Boas, Carlos E. Kenig, and Alexander Nagel. Foundation Postdoctoral Fellow at the University of Chicago, he joined Citation the faculty of MIT in 1981, where The Bergman prize for 2012 is awarded jointly to Profes- he is professor of mathematics and sor David S. Jerison of MIT and Professor John M. Lee David Jerison MacVicar Faculty Fellow. He has held of the University of Washington for their pioneering visiting positions at the University works on the CR Yamabe problem, which led to finding of Paris at Orsay, the Mathematical Sciences Research canonical metrics in a given conformal class, for strictly Institute in Berkeley, the Institute for Advanced Study pseudo-convex manifolds. This problem was introduced in Princeton, and Harvard University. He has served in by Jerison and Lee in their landmark paper in the Journal various administrative positions in the MIT mathematics of Differential Geometry, 1987. It consists of finding, for department: undergraduate chair, graduate chair, chair of a compact, strictly pseudo-convex, hypersurface-type the Pure Mathematics Committee, and director of SPUR, CR manifold M of dimension 2n+1, a contact form for M the summer research program for undergraduates in whose Webster-Tanaka scalar curvature is constant, i.e., to mathematics at MIT. He served on the editorial boards of change the pseudo-Hermitian structure to obtain constant several journals, including the Transactions of the AMS, scalar curvature. The problem is a striking analog of the the Notices of the AMS, and Inventiones. He is currently an Yamabe problem in Riemannian geometry. In their funda- associate editor of the Duke Mathematical Journal and an mental series of papers (Journal of Differential Geometry, editor of the Cambridge Journal of Mathematics.

April 2013 Notices of the AMS 497 Mathematics People

In 1985 Jerison received an A. P. Sloan Foundation Fel- of partial differential equations of elliptic type with atten- lowship and a Presidential Young Investigator Award. He tion to Bergman’s operator method. delivered an invited address at the AMS summer meeting in Salt Lake City in 1987 and an invited address in the —Allyn Jackson Partial Differential Equations Section of the International Congress of Mathematicians in Zurich in 1994. In 1999 he was elected to the American Academy of Arts and Sciences. 2012 Poincaré Prizes Awarded Biographical Sketch: John M. Lee The International Association of Mathematical Physics John M. (Jack) Lee is a professor of mathematics at the Uni- (IAMP) has awarded the 2012 Henri Poincaré Prizes for versity of Washington in Seattle. Born and raised in Phila- mathematical physics to Nalini Anantharaman, Uni- delphia, he received his bachelor’s degree from Princeton versité Paris-Sud; Freeman Dyson, Institute for Advanced University in 1972. After spending five years trying out a Study; Sylvia Serfaty, Université Pierre et Marie Curie few different career paths (computer Paris 6; and Barry Simon, California Institute of Technol- systems programmer, high school ogy. Anantharaman was honored “for her original con- teacher, vagabond), he settled on tributions to the area of quantum chaos, dynamical sys- mathematics and started graduate tems and Schrödinger equations, including a remarkable school at Tufts University. He trans- advance in the problem of quantum unique ergodicity.” ferred a year later to MIT, where he Dyson was recognized “for his many decisive contribu- received his Ph.D. in 1982 under the tions to physics and mathematical physics, including, in direction of Richard Melrose. After particular, the study of QED, the stability of matter and MIT he spent five years as a Benjamin random matrix theory.” Serfaty was honored “for her Peirce Assistant Professor at Harvard outstanding work on the theory of Ginzburg-Landau equa- John M. Lee University. While there, he struck up tions, including remarkable progress towards the rigorous a fruitful collaboration with David proof of the onset of the Abrikosov lattice in the theory Jerison, which lasted more than seven years and resulted of superconductivity.” Simon was recognized “for his im- in the work that has been cited for the Bergman award. In pact on many areas of mathematical physics including, in 1987 he moved to the University of Washington and has particular, the spectral theory of Schrödinger operators, been on the faculty there ever since. In addition to his work for his mentoring of generations of young scientists, and with Jerison on the CR Yamabe problem, Lee has published for his lucid and inspirational books.” research on asymptotically hyperbolic Einstein manifolds, The Poincaré Prize, which is sponsored by the Daniel low-dimensional CR structures, and the constraint equa- Iagolnitzer Foundation, recognizes outstanding contribu- tions in general relativity. He is the author of a popular tions that lay the groundwork for novel developments in series of graduate textbooks on manifolds and differential mathematical physics. It recognizes and supports young geometry and of a new AMS undergraduate textbook on people of exceptional promise who have already made axiomatic geometry. Previous awards include a National outstanding contributions to the field of mathematical Science Foundation Postdoctoral Research Fellowship, an physics. The prize is awarded every three years at the AMS Centennial Research Fellowship, and the Joseph R. International Mathematical Physics Congress. Levenson Prize for excellence in teaching at Harvard. —From an IAMP announcement About the Prize The Bergman Prize honors the memory of Stefan Bergman, best known for his research in several complex variables, as well as the Bergman projection and the Bergman ker- Simon Awarded Sacks Prize nel function that bear his name. A native of Poland, he Pierre Simon of the Hebrew University of Jerusalem has taught at Stanford University for many years and died in been awarded the Gerald Sacks Prize of the Association for 1977 at the age of eighty-two. He was an AMS member for Symbolic Logic (ASL). He received his Ph.D. in 2011 from thirty-five years. When his wife died, the terms of her will the Université Paris-Sud, and, according to the prize cita- stipulated that funds should go toward a special prize in tion, his thesis “provides new model theoretic tools and her husband’s honor. concepts for the study of NIP structures, a generalization The AMS was asked by Wells Fargo Bank of California, of stability that also encompasses o-minimal structures, the managers of the Bergman Trust, to assemble a commit- algebraically closed valued fields, and the p-adics. The tee to select recipients of the prize. In addition, the Society thesis makes substantial progress at the full generality assisted Wells Fargo in interpreting the terms of the will of NIP theories, and at the same time obtains new results to assure sufficient breadth in the mathematical areas in and new proofs in classical settings. It stands out for its which the prize may be given. Awards are made every one depth, originality, and elegance in seeking the appropriate or two years in the following areas: (1) the theory of the tools and dividing lines for the subject.” kernel function and its applications in real and complex analysis; and (2) function-theoretic methods in the theory —From an ASL announcement

498 Notices of the AMS Volume 60, Number 4 Mathematics People

this time he moved to Vermont and taught at Bennington Mawhin Awarded Schauder College for one year. Prize Adler wrote fifty-six trade books, all except one on top- ics of science and mathematics. He wrote thirty additional Jean Mawhin of the Université Catholique de Louvain, books with his wife, Ruth, for the Reason Why series. Ruth Belgium, has been awarded the first Juliusz Schauder illustrated most of his books, and his daughter, Peggy, an Prize “in recognition of his enormous contributions artist, illustrated some as well. Their books were published to nonlinear analysis and outstanding achievements in in thirty-one countries in nineteen different languages, and this area.” The Schauder Prize is awarded to individuals some were cited as outstanding science books for children for significant achievements related to topological methods by the Children’s Book Council and the National Science in nonlinear analysis. Teachers Association. In the 1960s Adler became interested in connections —Wojciech Kryszewski between plant growth and the Fibonacci numbers and the Director, Juliusz Schauder Center for Nonlinear Studies Golden Ratio. He published several papers and books on the subject, including his eighty-seventh book, Solving the Riddle of Phyllotaxis: Why the Fibonacci Numbers and Prizes of the Accademia Golden Ratio Occur on Plants, published posthumously five days after his death. Nazionale dei Lincei Adler was active throughout his adult life as a trade The Accademia Nazionale dei Lincei in Rome, Italy, has unionist and in the peace and civil rights movements. With awarded a number of prizes in mathematics. Christopher his wife, Ruth, he was a principal organizer of the Southern Hacon of the University of Utah has been awarded the Vermont contingent of the 1963 March on Washington 2011 Antonio Feltrinelli Prize in Mathematics, Mechan- for Jobs and Freedom. In 2009 he received the Lifetime ics, and Applications for his work in algebraic geometry. Achievement Award from the American Civil Liberties Mauro Fabrizio of the University of Bologna was honored Union of Vermont. He also received honorary doctorates from Vermont’s St. Michael’s College (1990) and the City with the 2012 Lyneus Prize for Mathematics in Mechanics College of New York (2002). Adler was an AMS member and Applications. Guido de Philippis of Scuola Normale for fifty-three years. Superiore, Pisa, was awarded the 2012 Joachim Iapichino In addition to his daughter, Peggy Adler, a police com- Prize for mathematical analysis. Founded in 1603, the Ac- missioner in Clinton, Connecticut, Irving Adler is survived cademia is the oldest academy in the world, and its goal by his son, Stephen L. Adler, a theoretical particle physicist is to promote, coordinate, integrate, and disseminate the and emeritus professor at the Institute for Advanced Study highest expression of scientific knowledge in the context in Princeton, New Jersey. of unity and universality of culture.

—Peggy Adler —From an Accademia dei Lincei announcement

Irving Adler (1913–2012) Irving Adler, a mathematician known for his popular books about mathematics and science and for his political activ- ism, died on September 22, 2012, at the age of ninety-nine. Adler was born on April 27, 1913, in New York City. He received his B.S. degree from the City College of New York in 1931 at the age of eighteen. He then began teaching mathematics in the New York City public high schools. During the McCarthy era, Adler refused to answer ques- tions about his association with groups the U.S. govern- ment considered to be subversive. As a result, he was suspended from his teaching position and department chairmanship in 1952 and dismissed in 1954. In the well- known U.S. Supreme Court case Adler v. Board of Educa- tion of the City of New York, his dismissal was upheld until its reversal in 1967, when that same Court declared unconstitutional the law that had led to his dismissal in the first place. Adler was then reinstated, with his pension rights restored, in 1977. From 1957 until 1960, Adler taught at Columbia Uni- versity. Under the direction of Ellis Kolchin, he received a Ph.D. in mathematics from Columbia in 1961. Around

April 2013 Notices of the AMS 499 Mathematics Opportunities

applications for the scholarship program are April 15, Math in Moscow Scholarship 2013, for the fall 2013 semester and September 15, 2013, Program for the spring 2014 semester. Information and application forms for Math in Mos- The Math in Moscow program at the Independent Univer- cow are available on the Web at http://www.mccme.ru/ sity of Moscow (IUM) was created in 2001 to provide for- mathinmoscow, or by writing to: Math in Moscow, P.O. Box eign students (primarily from the United States, Canada, 524, Wynnewood, PA 19096; fax: +7095-291-65-01; email: and Europe) with a semester-long, mathematically inten- [email protected]. Information and application forms for sive program of study in the Russian tradition of teaching the AMS scholarships are available on the AMS website mathematics, the main feature of which has always been at http://www.ams.org/programs/travel-grants/ the development of a creative approach to studying math- mimoscow or by writing to: Math in Moscow Program, Mem- ematics from the outset—the emphasis being on problem bership and Programs Department, American Mathemati- solving rather than memorizing theorems. Indeed, discov- cal Society, 201 Charles Street, Providence RI 02904-2294; ering mathematics under the guidance of an experienced email: [email protected]. teacher is the central principle of the IUM, and the Math in Moscow program emphasizes in-depth understanding —AMS Membership and Programs Department of carefully selected material rather than broad surveys of large quantities of material. Even in the treatment of the most traditional subjects, students are helped to explore NSF-CBMS Regional significant connections with contemporary research top- ics. The IUM is a small, elite institution of higher learning Conferences 2013 focusing primarily on mathematics that was founded in With funding from the National Science Foundation (NSF), 1991 at the initiative of a group of well-known Russian the Conference Board of the Mathematical Sciences (CBMS) research mathematicians, who now compose the Aca- will hold four NSF-CBMS Regional Research Conferences demic Council of the university. Today, the IUM is one of during the summer of 2013. These conferences are in- the leading mathematical centers in Russia. Most of the tended to stimulate interest and activity in mathematical Math in Moscow program’s teachers are internationally research. Each five-day conference features a distin- recognized research mathematicians, and all of them have guished lecturer who delivers ten lectures on a topic of considerable teaching experience in English, typically in important current research in one sharply focused area of the United States or Canada. All instruction is in English. the mathematical sciences. The lecturer subsequently pre- With funding from the National Science Foundation pares an expository monograph based on these lectures. (NSF), the AMS awards five US$9,000 scholarships each Support for about thirty participants will be provided semester to U.S. students to attend the Math in Moscow for each conference. Both established researchers and program. To be eligible for the scholarships, students interested newcomers, including postdoctoral fellows must be either U.S. citizens or enrolled at a U.S. institu- and graduate students, are invited to attend. Information tion at the time they attend the Math in Moscow program. about an individual conference may be obtained by con- Students must apply separately to the IUM’s Math in tacting the conference organizer. The conferences to be Moscow program and to the AMS Math in Moscow Schol- held in 2013 follow. arship program. Undergraduate or graduate mathematics May 20–24, 2013: Solitons in Two-Dimensional Water or computer science majors may apply. The deadlines for Waves and Applications to Tsunami. Yuji Kodama,

500 Notices of the AMS Volume 60, Number 4 Mathematics Opportunities lecturer. University of Texas, Pan American. Organizers: Kenichi Maruno, 956-665-3536, [email protected]; and NSF Scholarships in Science, Virgil Pierce, 956-665-3535, [email protected]. Confer- Technology, Engineering, and ence website: faculty.utpa.edu/kmaruno/nsfcbms- tsunami.html. Mathematics June 17–21, 2013: The Global Behavior of Solutions to The NSF Scholarships in Science, Technology, Engineering, Critical Nonlinear Wave Equations. Carlos Kenig, lecturer. and Mathematics (S-STEM) program provides institutions Kansas State University. Organizers: Virginia Naibo, 785- with funds for student scholarships to encourage and enable academically talented students demonstrating 532-0554, [email protected]; and Diego Maldonado, financial need to enter the STEM workforce or STEM 785-532-0581, [email protected]. Conference graduate school following completion of an associate, website: www.math.ksu.edu/events/conference/ baccalaureate, or graduate degree in fields of science, cbms2013/. technology, engineering, or mathematics. Students to be August 12–16, 2013: Uncertainty Principles in Harmonic awarded scholarships must demonstrate academic tal- Analysis: Gap and Type Problems. Alexei Poltoratski, lec- ent and financial need. S-STEM grants may be made for turer. Clemson University. Organizers: Mishko Mitkovski, up to five years and provide individual scholarships of up to US$10,000 per year, depending on financial need. 864-656-0411, [email protected]; Constanze Liaw, Proposals must be submitted by institutions, which are 254-710-3859, [email protected]; and Brett responsible for selecting the scholarship recipients. The Wick, 404-894-4311, [email protected]. Confer- deadline for full proposals is August 13, 2013. For more ence website: people.clemson.edu/~mmitkov/cbms/ information, see the website http://www.nsf.gov/ index.html. pubs/2012/nsf12529/nsf12529.htm. August 19–23, 2013: Analysis of Stochastic Partial Differential Equations. Davar Khoshnevisan, lecturer. —From an NSF announcement Michigan State University. Organizers: Yimin Xiao, 517- 432-5411, [email protected]; and V. S. Mandrekar, 517- 353-7172, [email protected]. Conference website: Project NExT 2013–2014 www.stt.msu.edu/CBMS2013. Project NExT (New Experiences in Teaching) is a profes- sional development program for new and recent Ph.D.’s —From a CBMS announcement in the mathematical sciences (including pure and ap- plied mathematics, statistics, operations research, and mathematics education). It addresses all aspects of an academic career: improving the teaching and learning of DMS Workforce Program in the mathematics, engaging in research and scholarship, and participating in professional activities. It also provides the Mathematical Sciences participants with a network of peers and mentors as they The Division of Mathematical Sciences (DMS) of the Na- assume these responsibilities. In 2013 about eighty fac- ulty members from colleges and universities throughout tional Science Foundation (NSF) welcomes proposals for the country will be selected to participate in a workshop the Workforce Program in the Mathematical Sciences. preceding the Mathematical Association of America (MAA) The long-range goal of the program is increasing the summer meeting, in activities during the summer MAA number of well-prepared U.S. citizens, nationals, and meetings in 2013 and 2014 and the Joint Mathematics permanent residents who successfully pursue careers in Meetings in January 2014, and in an electronic discussion the mathematical sciences and in other NSF-supported network. Faculty for whom the 2013–2014 academic year disciplines. Of primary interest are activities centered on will be the first or second year of full-time teaching (post- Ph.D.) at the college or university level are invited to apply education that broaden participation in the mathematical to become Project NExT Fellows. sciences through research involvement for trainees at the Applications are invited for the fellowship year 2013– undergraduate through postdoctoral educational levels. 2014, the twentieth year of Project NExT. The deadline The program is particularly interested in activities that for applications is April 12, 2013. For more information, improve recruitment and retention, educational breadth, see the Project NExT website, http://archives.math. and professional development. utk.edu/projnext/ or contact Aparna Higgins, direc- The submission period for unsolicited proposals is tor, at [email protected]. Project NExT is a program of the MAA. It receives major funding from May 15–June 15, 2013. For more information and a list of the Mary P. Dolciani Halloran Foundation and additional cognizant program directors, see the website http://www. funding from the Educational Advancement Founda- nsf.gov/funding/pgm_summ.jsp?pims_id=503233. tion, the American Mathematical Society, the American Statistical Association, the National Council of Teachers —From a DSM announcement of Mathematics, the Association for Symbolic Logic, the

April 2013 Notices of the AMS 501 Mathematics Opportunities

W. H. Freeman Publishing Company, John Wiley & Sons, by spending an academic year immersed in the thematic MAA Sections, and the Greater MAA Fund. program at the IMA, where they learn new mathematics and applications, connect their research with important —Aparna Higgins, Director problems, and establish new contacts and collaborations. Applications for professorships during the 2013–2014 (Scientific and Engineering Applications of Algebraic To- AWM Gweneth Humphreys pology) and 2014–2015 (Discrete Structures: Analysis and Applications) thematic programs are still being accepted. Award Please visit www.ima.umn.edu/new-directions for more information as well as for the online application. The Association for Women in Mathematics (AWM) spon- Applications for New Directions Short Course: Applied sors the Gweneth Humphreys Award to recognize out- Statistics and Machine Learning. From June 17–28, 2013, standing mentorship activities. This prize will be awarded the IMA will hold its New Directions Short Course, Ap- annually to a mathematics teacher (female or male) who plied Statistics and Machine Learning. Organized by Bin has encouraged female undergraduate students to pursue Yu (University of California Berkeley) and David Madigan mathematical careers and/or the study of mathematics at (), the two-week course will intro- the graduate level. The recipient will receive a cash prize duce participants to a broad array of modern statistical and honorary plaque and will be featured in an article in concepts and techniques with a focus on critical thinking the AWM newsletter. The award is open to all regardless and practical data analysis. The statistical software R will of nationality and citizenship. Nominees must be living be used extensively, and students are expected to have at at the time of their nomination. least rudimentary knowledge of R prior to the course. The The deadline for nominations is April 30, 2013. For course will cover exploratory data analysis (visualization, details, see www.awm-math.org, telephone: 703-934-0163, dimension reduction, clustering), statistical modeling (lin- or email: [email protected]. ear models, generalized linear models, logistic regression, graphical models), and statistical computation (Monte —From an AWM announcement Carlo, Markov chain Monte Carlo, convex optimization). The course will also cover regularized and large-scale modeling techniques. The deadline for applications is News from the IMA April 15, 2013. More information and an online appli- cation are available online at http://www.ima.umn. The Institute for Mathematics and its Applications (IMA), edu/2012-2013/ND6.17-28.13. located on the University of Minnesota campus, will soon launch its new Annual Thematic Program, running from —IMA announcement September 2013 to June 2014. The theme, Scientific and Engineering Applications of Algebraic Topology, will gather researchers from topology, computational geom- etry, networking, statistics, biology, and other fields to address methods for qualitative analysis and recognition problems in contemporary contexts, including data (finite metric spaces as samples from experiments, surveys, or sensors), networks (Internet traffic, gene regulation, coordinated robotics, communications), and dynamics (systems equipped with only finite resolution or which are stochastic). The six workshops planned for the year are designed to be truly interdisciplinary. More information about this year’s workshops and the thematic program are available online at www.ima.umn.edu/programs. Call for Proposals for Hot Topics Workshops. The IMA’s Hot Topics Workshops cover rapidly developing areas of interest, focusing on a specific problem or area of excep- tional contemporary significance. These workshops are often cosponsored by a participating institution, corpo- ration, government funding agency, or an NSF-focused research group. More information on submitting a work- shop proposal is available online at www.ima.umn.edu/ solicit/hot-topics-guidelines.html. New Directions Research Professorships. New Direc- tions Research Professorships provide an extraordinary opportunity for established mathematicians—typically mid-career faculty at U.S. universities—to branch into new directions and increase the impact of their research

502 Notices of the AMS Volume 60, Number 4 Inside the AMS

AMS Hosts Congressional From the AMS Public Briefing Awareness Office The American Mathematical Society hosted a congres- •Selected Highlights of the 2013 Joint Mathematics sional briefing on December 4, 2012, titled “Chaos and Meetings. Over 6,700 mathematicians, exhibitors, and Avalanches in Science and Socio-Political Systems”. This students came to the 2013 Joint Mathematics Meetings Capitol Hill presentation was given by James A. Yorke, of the American Mathematical Society (AMS) and the Distinguished University Professor of Mathematics and Mathematical Association of America (MAA) in San Diego, Physics at the University of Maryland. California, January 9–12. Read about some of the many highlights and see slideshows of the prizes and awards, Professor Yorke talked about how the science of chaos invited addresses, sessions, Mathematical Art Exhibition, has completely changed the understanding of physical exhibits, AMS 125th anniversary events, national Who processes in the last thirty years. From astronomy to Wants to Be a Mathematician game, and more: http:// weather prediction, chaos has become a way to understand www.ams.org/meetings/national/jmm13-highlights. physical systems. Avalanches are a particularly sudden •Mathematics Awareness Month April 2013. The type of chaos. This presentation demonstrated how politi- theme for Mathematics Awareness Month 2013 is “Math- cal upheavals have much in common with avalanches and ematics of Sustainability”. See the website for the theme earthquakes. The AMS hosts an annual congressional brief- announcement, essay, related resources, and poster: ing as a way to communicate information to policymakers http://www.mathaware.org/. about the importance of mathematics research. •Mathematics Events at the 2013 AAAS Meeting. Read about some of the events related to mathematics that —AMS Washington Office took place at the 2013 annual meeting of the American Association for the Advancement of Science in Boston, Massachusetts, February 14–18, including Who Wants to National Who Wants to Be a Be a Mathematician: http://www.ams.org/meetings/ aaas2013. Mathematician Competition •Celebrating . This poster celebrates the 125th birthday of Ramanujan and links to Calvin Deng, a senior at the North Carolina School of Sci- the series of articles in the De- ence and Mathematics and a gold medalist at the 2012 cember 2012 and January 2013 International Mathematical Olympiad, won the national issues of Notices. The poster Who Wants to Be a Mathematician competition at the is available free upon request. Joint Mathematics Meetings in San Diego. Calvin edged Email paoffice@ams. out Shyam Narayanan, last year’s champion, on the last org with subject line question of the finals to win $5,000 for himself and $5,000 “Ramanujan poster - Notices”. for the NCSSM math department. —Annette Emerson and Mike —AMS Public Awareness Office Breen AMS Public Awareness Officers [email protected]

April 2013 Notices of the AMS 503 Reference and Book List

The Reference section of the Notices Steven Ferrucci, email: ams-simons@ in Moscow Program, Membership is intended to provide the reader ams.org, telephone: 800-321-4267, and Programs Department, American with frequently sought information in ext. 4113. Mathematical Society, 201 Charles an easily accessible manner. New March 31, 2013: Applications Street, Providence, RI 02904-2294; information is printed as it becomes for IPAM graduate summer school email: [email protected]. available and is referenced after the on computer vision. See www.ipam. April 15, 2013: Applications for first printing. As soon as information ucla.edu. IMA New Directions Short Course: is updated or otherwise changed, it April 1, 2013: Letters of intent for Applied Statistics and Machine Learn- will be noted in this section. proposals for one-semester programs ing. See “Mathematics Opportunities” Contacting the Notices at the Bernoulli Center (CIB). See the in this issue. website http://cib.epfl.ch/. April 30, 2013: Nominations for The preferred method for contacting April 12, 2013: Applications for AWM Gweneth Humphreys Award. the Notices is electronic mail. The Project NExT fellowships. See “Math- See “Mathematics Opportunities” in editor is the person to whom to send ematics Opportunities” in this issue. this issue. articles and letters for consideration. April 15, 2013: Applications for fall Articles include feature articles, me- May 1, 2013: Applications for 2013 semester of Math in Moscow. See morial articles, communications, May review for National Academies opinion pieces, and book reviews. The http://www.mccme.ru/mathinmos- Research Associateship Programs. editor is also the person to whom to cow or write to: Math in Moscow, P.O. See the website http://sites. send news of unusual interest about Box 524, Wynnewood, PA 19096; fax: nationalacademies.org/PGA/RAP/ other people’s mathematics research. +7095-291-65-01; email: mim@mccme. PGA_050491 or contact Research The managing editor is the person ru. Information and application forms Associateship Programs, National to whom to send items for “Math- for the AMS scholarships are available Research Council, Keck 568, 500 Fifth ematics People”, “Mathematics Op- on the AMS website at http://www. Street, NW, Washington, DC 20001; portunities”, “For Your Information”, ams.org/programs/travel-grants/ telephone: 202-334-2760; fax: 202- “Reference and Book List”, and “Math- mimoscow or by writing to: Math 334-2759; email: [email protected]. ematics Calendar”. Requests for permissions, as well as all other Where to Find It inquiries, go to the managing editor. A brief index to information that appears in this and previous issues of the Notices. The electronic-mail addresses are AMS Bylaws—January 2012, p. 73 [email protected] in the case of the editor and [email protected] AMS Email Addresses—February 2013, p. 249 in the case of the managing editor. AMS Ethical Guidelines—June/July 2006, p. 701 The fax numbers are 314-935-6839 AMS Officers 2010 and 2011 Updates—May 2012, p. 708 for the editor and 401-331-3842 for AMS Officers and Committee Members—October 2012, p. 1290 the managing editor. Postal addresses may be found in the masthead. Contact Information for Mathematical Institutes—August 2012, p. 979 Upcoming Deadlines Conference Board of the Mathematical Sciences—September 2012, March 18, 2013: Registration for CRM p. 1128 International School and Research IMU Executive Committee—December 2011, p. 1606 Workshop on Complex Systems. Information for Notices Authors—June/July 2012, p. 851 Seethe website http://www.crm. cat/en/Activities/Pages/Activ- Mathematics Research Institutes Contact Information—August 2012, ityDescriptions/International- p. 979 School-and-Research-Workshop- National Science Board—January 2013, p. 109 on-Complex-systems.aspx. NRC Board on Mathematical Sciences and Their Applications—March March 31, 2013: Nominations for 2013, p. 350 CMS Graham Wright Award for Dis- NSF Mathematical and Physical Sciences Advisory Committee—February tinguished Service. See the website 2013, p. 252 http://cms.math.ca/Prizes/dis- nom. Program Officers for Federal Funding Agencies—October 2012, March 31, 2013: Applications for p. 1284 (DoD, DoE); December 2012, p. 1585 (NSF Mathematics Education) AMS-Simons Travel Grants.See Program Officers for NSF Division of Mathematical Sciences—Novem- www.ams.org/programs/travel- ber 2012, p. 1469 grants/AMS-SimonsTG or contact

504 Notices of the AMS Volume 60, Number 4 Reference and Book List

May 1, 2013: Applications for Institutional Transformation Catalyst The Big Questions: Mathematics, AWM Travel Grants and Mathematics awards. See http://www.nsf.gov/ by Tony Crilly. Quercus, April 2011. Education Research Travel Grants. pubs/2012/nsf12584/nsf12584. ISBN-13: 978-18491-624-01. (Re- See https://sites.google.com/ htm?WT.mc_id=USNSF_36&WT.mc_ viewed October 2012.) site/awmmath/programs/travel- ev=click. Calculating Curves: The Mathemat- grants; or telephone: 703-934-0163; November 1, 2013: Applications ics, History, and Aesthetic Appeal of email: [email protected]; or contact for November review for National T. H. Gronwall’s Nomographic Work, Association for Women in Mathemat- Academies Research Associateship by Thomas Hakon Gronwall, with ics, 11240 Waples Mill Road, Suite Programs. See the website http:// contributions by Ron Doerfler and 200, Fairfax, VA 22030. sites.nationalacademies.org/ Alan Gluchoff, translation by Paul May 15–June 15, 2013: Proposals PGA/RAP/PGA_050491 or contact Hamburg, and bibliography by Scott for NSF DMS Workforce Program in Research Associateship Programs, Guthery. Docent Press, April 2012. the Mathematical Sciences. See “Math- National Research Council, Keck 568, ISBN-13: 978-09837-004-32. ematics Opportunities” in this issue. 500 Fifth Street, NW, Washington, DC Classic Problems of Probability, August 1, 2013: Applications for 20001; telephone: 202-334-2760; fax: by Prakash Gorroochurn. Wiley, May August review for National Acad- 202-334-2759; email: [email protected]. 2012. ISBN-13: 978-1-1180-6325-5. emies Research Associateship Pro- November 12, 2013: Full pro- The Crossing of Heaven: Mem- grams. See the website http:// posals for NSF Program ADVANCE oirs of a Mathematician, by Karl sites.nationalacademies.org/ Institutional Transformation and Gustafson. Springer, January 2012. PGA/RAP/PGA_050491 or contact Institutional Transformation Catalyst ISBN-13: 978-36422-255-74. Research Associateship Programs, awards. See http://www.nsf.gov/ Elliptic Tales: Curves, Counting, and National Research Council, Keck 568, pubs/2012/nsf12584/nsf12584. Number Theory, by Avner Ash and 500 Fifth Street, NW, Washington, DC htm?WT.mc_id=USNSF_36&WT.mc_ Robert Gross. Princeton University 20001; telephone: 202-334-2760; fax: ev=click. Press, March 2012. ISBN-13: 978- 202-334-2759; email: [email protected]. 06911-511-99. Book List August 13, 2013: Full proposals The Foundations of Geometry And for NSF Scholarships in Science, Tech- The Book List highlights recent books Religion from an Abstract Standpoint, nology, Engineering, and Mathemat- that have mathematical themes and by Salilesh Mukhopadhyay. Outskirts ics (S-STEM) program. See “Mathemat- are aimed at a broad audience po- Press, July 2012. ISBN-13: 978-1- tentially including mathematicians, ics Opportunities” in this issue. 4327-9424-8. students, and the general public. Sug- September 15, 2013: Applications The Fractalist: Memoir of a Scien- gestions for books to include on the list for spring 2014 semester of Math in tific Maverick, by Benoit Mandelbrot. may be sent to notices-booklist@ Moscow. See http://www.mccme. Pantheon, October 2012. ISBN-13: ru/mathinmoscow or write to: Math ams.org. 978-03073-773-57. in Moscow, P.O. Box 524, Wynnewood, *Added to “Book List” since the *Fueling Innovation and Discovery: PA 19096; fax: +7095-291-65-01; list’s last appearance. The Mathematical Sciences in the 21st email: [email protected]. Information Algorithmic Puzzles, by Anany Century, by the National Research and application forms for the AMS Levitin and Maria Levitin. Oxford Council. National Academies Press, scholarships are available on the University Press, October 2011. ISBN- 2012. ISBN-13: 978-0-309-25473-1. AMS website at http://www.ams. 13: 978-01997-404-44. Galileo’s Muse: Renaissance Math- org/programs/travel-grants/ American Mathematicians as Edu- ematics and the Arts, by Mark Austin- mimoscow or by writing to: Math cators, 1893–1923: Historical Roots Peterson. Harvard University Press, in Moscow Program, Membership of the “Math Wars”, by David Lindsay October 2011. ISBN-13: 978-06740- and Programs Department, American Roberts. Docent Press, July 2012, Mathematical Society, 201 Charles ISBN-13: 978-09837-004-49. 597-26. (Reviewed November 2012.) Street, Providence, RI 02904-2294; *Assessing the Reliability of Com- Game Theory and the Humanities: email: [email protected]. plex Models: Mathematical and Sta- Bridging Two Worlds, by Steven J. October 1, 2013: Applications for tistical Foundations of Verification, Brams. MIT Press, September 2012. AWM Travel Grants and Mathematics Validation, and Uncertainty Quan- ISBN-13: 978-02625-182-53. Education Research Travel Grants. tification, by the National Research Games and Mathematics: Subtle See https://sites.google.com/ Council. National Academies Press, Connections, by David Wells. Cam- site/awmmath/programs/travel- 2012. ISBN: 978-0-309-25634-6. bridge University Press, November grants; or telephone: 703-934-0163; The Best Writing on Mathematics 2012. ISBN-13: 978-11076-909-12. email: [email protected]; or contact 2012, edited by Mircea Pitici. Prince- Gösta Mittag-Leffler: A Man of Con- Association for Women in Mathemat- ton University Press, November 2012. viction, by Arild Stubhaug (translated ics, 11240 Waples Mill Road, Suite ISBN-13: 978-06911-565-52. by Tiina Nunnally). Springer, Novem- 200, Fairfax, VA 22030. Bibliography of Raymond Clare ber 2010. ISBN-13: 978-36421-167-11. October 4, 2013: Letters of in- Archibald, by Scott Guthery. Docent Guesstimation 2.0: Solving Today’s tent for NSF Program ADVANCE Press, April 2012. ISBN-13: 978- Problems on the Back of a Napkin, Institutional Transformation and 0983700425. by Lawrence Weinstein. Princeton

April 2013 Notices of the AMS 505 Reference and Book List

University Press, September 2012. and Harrie Willems. Springer, March Association of America, December ISBN-13: 978-06911-508-02. 2012. ISBN NTSC: 978-3-642-24482-7; 2011. ISBN-13: 978-08838-557-82. Henri Poincaré: Impatient Genius, ISBN PAL: 978-3-642-24522-0. (Re- *A Mathematician’s Lament: How by Ferdinand Verhulst. Springer, Au- viewed January 2013.) School Cheats Us Out of Our Most Fas- gust 2012. ISBN-13: 978-14614-240- Lemmata: A Short Mathematical cinating and Imaginative Art Form, by 62. Thriller, by Sam Peng. CreateSpace, Paul Lockhart. Bellvue Literary Press, Henri Poincaré: A Scientific Bi- December 2011. ISBN-13: 978-14681- April 2009. ISBN-13: 978-1-934137- ography by Jeremy Gray. Princeton 442-39. 17-8. (Reviewed in this issue.) University Press, November 2012. *Levels of Infinity: Selected Writings Mathematicians in Bologna 1861- ISBN-13: 978-06911-527-14. on Mathematics and Philosophy, by 1960, edited by Salvatore Coen. ISBN- *How to Study as a Mathematics Hermann Weyl. Edited by Peter Pesic. 13: 978-30348-022-60. Major, by Lara Alcock. Oxford Uni- Dover Publications, February 2013. Mathematics in Popular Culture: versity Press, March 2013. ISBN-13: ISBN-13: 978-0486489032. Essays on Appearances in Film, Fic- 978-0199661312. The Logician and the Engineer: How tion, Games, Television and Other I Died for Beauty: Dorothy Wrinch George Boole and Cre- Media, edited by Jessica K. Sklar and and the Cultures of Science, by ated the Information Age, by Paul J. Na- Elizabeth S. Sklar. McFarland, Febru- Marjorie Senechal. Oxford University hin, Princeton University Press, Octo- ary 2012. ISBN-13: 978-07864-497-81. Press, December 2012. ISBN-13:978- ber 2012. ISBN-13: 978-06911-510-07. Mathematics in Victorian Britain, 01997-325-93. Magical Mathematics: The Math- by Raymond Flood, Adrian Rice, and *Ibn al-Haytham’s Theory of ematical Ideas That Animate Great Robin Wilson. Oxford University Conics, Geometrical Constructions Magic Tricks, by Persi Diaconis and Ron Press, October 2011. ISBN-13: 978- and Practical Geometry, by Roshdi Graham. Princeton University Press, 019-960139-4. Rashed. Routledge, February 2013. November 2011. ISBN-13: 978-06911- Meaning in Mathematics, edited by ISBN: 978-0-415-58215-5. 516-49. (Reviewed August 2012.) John Polkinghorne. Oxford University In Pursuit of the Traveling Sales- The Man of Numbers: Fibonacci’s Press, July 2011. ISBN-13: 978-01996- man: Mathematics at the Limits of Arithmetic Revolution, by Keith Devlin. 050-57. Computation, by William J. Cook. Walker and Company, July 2011. ISBN- Measurement, by Paul Lockhart. Princeton University Press, December 13: 978-08027-781-23. (Reviewed May Belknap Press of Harvard University 2011. ISBN-13: 978-06911-527-07. 2012.) Press, September 2012. ISBN-13: 978- In Pursuit of the Unknown: 17 *Manifold Mirrors: The Crossing 06740-575-55. Equations That Changed the World, Paths of the Arts and Mathematics, Nine Algorithms That Changed the by Ian Stewart. Basic Books, March by Felipe Cucker. Cambridge Uni- Future: The Ingenious Ideas That Drive 2012. ISBN-13: 978-04650-297-30. versity Press, April 2013. ISBN-13: Today’s Computers, by John MacCor- (Reviewed December 2012.) 978-0521728768. mick. Princeton University Press, Infinity: New Research Frontiers, Math Girls, by Hiroshi Yuki (trans- December 2011. ISBN-13: 978-06911- edited by Michael Heller and W. Hugh lated from the Japanese by Tony 471-47. Woodin. Cambridge University Press, Gonzalez). Bento Books, November Number-Crunching: Taming Un- February 2011. ISBN-13: 978-11070- 2011. ISBN-13: 978-09839-513-15. ruly Computational Problems from 038-73. (Reviewed August 2012.) Mathematical Physics to Science The Infinity Puzzle: Quantum Field Math Goes to the Movies, by Fiction, by Paul J. Nahin. Princeton Theory and the Hunt for an Orderly Burkard Polster and Marty Ross. University Press, August 2011. ISBN- Universe, by Frank Close. Basic Books, Johns Hopkins University Press, July 13: 978-06911-442-52. November 2011. ISBN-13: 978-04650- 2012. ISBN-13: 978-14214-048-44. On the Formal Elements of the 214-44. (Reviewed September 2012.) Math Is Murder, by Robert C. Absolute Algebra, by Ernst Schröder Introduction to Mathematical Think- Brigham and James B. Reed. Universe, (translated and with additional ing, by Keith Devlin. Keith Devlin, July March 2012. ISBN-13: 978-14697- material by Davide Bondoni; with 2012. ISBN-13: 978-06156-536-31. 972-81. German parallel text). LED Edizioni The Irrationals: A Story of the Mathematical Excursions to the Universitarie, 2012. ISBN-13: 978-88- Numbers You Can’t Count On, by World’s Great Buildings, by Alex- 7916-516-7. Julian Havil. Princeton University ander J. Hahn. Princeton University Our Days Are Numbered: How Press, June 2012. ISBN-13: 978- Press, July 2012. ISBN-13: 978-06911- Mathematics Orders Our Lives, by 0691143422. 452-04. Jason Brown. Emblem Editions, April The Joy of x: A Guided Tour of The Mathematical Writings of 2010. ISBN-13: 978-07710-169-74. Math, from One to Infinity, by Steven Évariste Galois, edited by Peter M. (Reviewed October 2012.) Strogatz. Eamon Dolan/Houghton Neumann. European Mathematical Paradoxes in Probability Theory, by Mifflin Harcourt, October 2012. ISBN- Society, October 2011. ISBN-13: 978- William Eckhardt. Springer, Septem- 13: 978-05475-176-50. 3-03719-104-0. (Reviewed December ber 2012. ISBN-13: 978-94007-513-92. Late Style: Yuri I. Manin Looking 2012.) (Reviewed March 2013.) Back on a Life in Mathematics. A DVD A Mathematician Comes of Age, Pricing the Future: Finance, Phys- documentary by Agnes Handwerk by Steven G. Krantz. Mathematical ics, and the 300-Year Journey to the

506 Notices of the AMS Volume 60, Number 4 Reference and Book List

Black-Scholes Equation, by George G. Platform, January 2012. ISBN-13: 978- Szpiro. Basic Books, November 2011. 14699-123-32. ISBN-13: 978-04650-224-89. Transcending Tradition: Jewish Proving Darwin: Making Biology Mathematicians in German Speaking Mathematical, by Gregory Chaitin. Academic Culture, edited by Birgit Pantheon, May 2012. ISBN-13: 978- Bergmann, Moritz Epple, and Ruti 03754-231-47. Ungar. Springer, January 2012. ISBN- The Search for Certainty: A Jour- 13: 978-36422-246-38. (Reviewed ney through the History of Mathemat- February 2013.) ics, 1800-2000, edited by Frank J. Turbulent Times in Mathematics: Swetz. Dover Publications, September The Life of J. C. Fields and the History 2012. ISBN-13: 978-04864-744-27. of the , by Elaine McKin- Secrets of Triangles: A Mathemati- non Riehm and Frances Hoffman. cal Journey, by Alfred S. Posamentier AMS, November 2011. ISBN-13: 978- and Ingmar Lehman. Prometheus 0-8218-6914-7. Books, August 2012. ISBN-13: 978- Uneducated Guesses: Using Evi- 16161-458-73. dence to Uncover Misguided Educa- Seduced by Logic: Emilie Du Châe- tion Policies, by Howard Wainer. let, Mary Somerville and the Newto- Princeton University Press, August nian Revolution, by Robyn Arianrhod. 2011. ISBN-13: 978-06911-492-88. Oxford University Press, September (Reviewed June/July 2012.) 2012. ISBN-13: 978-01999-316-13. The Universe in Zero Words: The Selected Papers: Volume II: On Story of Mathematics as Told through Algebraic Geometry, including cor- Equations, by Dana Mackenzie. respondence with Grothendieck, by Princeton University Press, April David Mumford. Edited by Amnon 2012. ISBN-13: 978-06911-528-20. Neeman, Ching-Li Chai, and Takahiro (Reviewed in this issue.) Shiota. Springer, July 2010. ISBN-13: A Wealth of Numbers: An Anthology 978-03877-249-11. (Reviewed Febru- of 500 Years of Popular Mathematics ary 2013.) Writing, edited by Benjamin Ward- The Signal and the Noise: Why haugh. Princeton University Press, So Many Predictions Fail—but Some April 2012. ISBN-13: 978-06911-477- Don’t, by Nate Silver. Penguin Press, 58. (Reviewed March 2013.) September 2012. ISBN-13:978-15942- Who’s #1?: The Science of Rating 041-11. and Ranking, by Amy N. Langville and Carl D. Meyer. Princeton University Simon: The Genius in My Base- Press, February 2012. ISBN-13: 978- ment, by Alexander Masters. Dela- 06911-542-20. (Reviewed January corte Press, February 2012. ISBN-13: 2013.) 978-03853-410-80. Why Cats Land on Their Feet (and Sources in the Development of 76 Other Physical Paradoxes and Mathematics: Series and Products Puzzles), by Mark Levi. Princeton from the Fifteenth to the Twenty-first University Press, May 2012. ISBN-13: Century, by Ranjan Roy. Cambridge 978-0691148540. University Press, June 2011. ISBN-13: 978-05211-147-07. A Strange Wilderness: The Lives of the Great Mathematicians, by Amir D. Aczel. Sterling, October 2011. ISBN- The AMS provides down- 13: 978-14027-858-49. The Theory That Would Not Die: loadable pdf files of posters How Bayes’ Rule Cracked the Enigma promoting awareness of Code, Hunted Down Russian Sub- mathematics, programs, and marines, and Emerged Triumphant services to post in common from Two Centuries of Controversy, areas, classrooms, and offices. by Sharon Bertsch McGrayne. Yale University Press, April 2011. ISBN- 13: 978-03001-696-90. (Reviewed Visit ams.org/posters for May 2012.) more information. Thinking Statistically, by Uri Bram. CreateSpace Independent Publishing

April 2013 Notices of the AMS 507 AMERICAN MATHEMATICAL SOCIETY

The selection committees for these prizes request nominations for consideration for the 2014 awards, which will be presented at the Joint Mathematics Meetings in Baltimore, MD, in January 2014. Information about these prizes may be found at www.ams.org/profession/prizes-awards/prizes or in the January 2012 issue of the Notices, pp. 79-100.

BÔCHER MEMORIAL PRIZE

The Bôcher Prize, awarded for a notable paper in analysis published during the preceding six years, is awarded every three years. To be eligible, papers must be either authored by an AMS member or published ATIONS in a recognized North American journal.

FRANK NELSON COLE PRIZE IN NUMBER THEORY

The Cole Prize in Number Theory, which recognizes a notable paper in MI N number theory published during the preceding six years, is awarded every three years. To be eligible, papers must be either authored by an AMS member or published in a recognized North American journal. NO LEVI L. CONANT PRIZE

The Levi L. Conant Prize is presented annually for an outstanding expos-

Call f or itory paper published in either the Notices or the Bulletin of the American

Mathematical Society during the preceding five years.

Nominations with supporting information should be submitted using the online form available here: www.ams.org/profession/prizes-awards/nominations. Include a short description of the work that is the basis of the nominations, including complete bibliographic citations. A brief curriculum vitae for the nominee should be included. Those who prefer to submit by postal mail may send nominations to AMS Secretary, Carla Savage, Box 8206, Computer Science Department, North Carolina State University, Raleigh, NC 27695-8206. The nominations will be forwarded by the secretary to the appropriate prize selection committee, which will make final decisions on the awarding of these prizes.

Deadline for nominations is June 30, 2013. Deadline for nominations is June 30, 2013. 30, June is nominations for Deadline these prizes. by to the appropriate thesecretary prize selection committee, whichwillmake finaldecisions on the awarding of Science Department, Carolina North State University, Raleigh, NC 27695-8206. The nominations beforwardedwill nominee should be included. Those who prefer to submit by postal mail may send nominations to AMS Secretary, Carla Savage, Box 8206, Computer descriptionoftheworkInclude ashort thatisthebasisofnominations, includingcompletebibliographic citations. A briefcurriculumvitae for the information should be submitted using the online formNominations with supporting available here: www.ams.org/profession/prizes-awards/nominations. Call for NO MIN ATIONS T A preceding five years. five preceding the during profession mathematician mathematics research the to contribution a distinguished a recognizes made has years, who two every made is which award, This DISTINGUISHED PUBLICSERVICE AWARD years. six way.novel awardedis prize The threeprecedingevery the in published work a for years a in mathematics modern exploiting theory physical a of development the for or physics in developments modern by inspired mathematics to contribution a for given be might prize the example, together.for closer Thus, physics and mathematics brings that works of group or work a honors Physics and Mathematics for Prize Eisenbud Leonard The LEONARD EISENBUD PRIZE FOR MATHEMATICS AND PHYSICS publishers. by or tees, of members committee, selection the of members members, AMS commit- editorial AMS by nominated be may Books nominated. is it which in year precedingthe years calendar prize is awarded every three years and the book must have been published within the six The area. its in impact long-term and deep a have to promises and exposition, research of standards highest the reflects literature,research the to contribution seminal a makes that book research outstanding recent, relatively single, a recognizes Prize Doob The JOSEPH L. DOOBPRIZE the 79-100. pp. Notices, of at issue found 2012 be January may the prizes in or these www.ams.org/profession/prizes-awards/prizes about Information 2014. in January Meetings in Mathematics MD, Joint the Baltimore, at presented be consideration will for which nominations awards, request 2014 prizes the these for for committees selection he MERICAN M ATHEMATICAL S OCIETY The prize is awarded each year to an undergraduate student (or students having submitted joint work) for outstanding research in mathematics. Any student who is an undergraduate in a college or university in the United States or its possessions, or Canada or Mexico, is eligible to be considered for this prize.

The prize recipient’s research need not be confined to a single paper; it may be contained in several papers. However, the paper (or papers) to be considered for the prize must be submitted while the student is an undergraduate; they cannot be submitted after the student’s graduation. The research paper (or papers) may be submitted for consideration by the student or a nominator. All submissions for the prize must include at least one letter of support from a person, usually a faculty member, familiar with the student’s research. Publication of research is not required.

The recipients of the prize are to be selected by a standing joint committee of the AMS, MAA, and SIAM. The decisions of this committee are final. The 2014 prize will be awarded for papers submitted for consideration no later than June 30, 2013, by (or on behalf of) students who were undergraduates in December 2012.

Questions may be directed to: Nominations and submissions Barbara T. Faires should be sent to: Secretary Morgan Prize Committee Mathematical Association of America c/o Carla Savage, Secretary Westminster College Box 8206 New Wilmington, PA 16172 Computer Science Department North Carolina State University Telephone: 724-946-6268 Raleigh, NC 27695-8206 Fax: 724-946-6857 Email: [email protected] Mathematics Calendar

Please submit conference information for the Mathematics Calendar through the Mathemat- ics Calendar submission form at http://www.ams.org/cgi-bin/mathcal-submit.pl. The most comprehensive and up-to-date Mathematics Calendar information is available on the AMS website at http://www.ams.org/mathcal/.

April 2013 Funding: Is available to cover accommodation and travel for up to 30 participants. We expect to give all of them the opportunity to * 1–June 14 Research Term on Real Harmonic Analysis and Appli- contribute either with a talk or a poster. cations to PDE, ICMAT: Instituto de Ciencias Matemáticas, located Information: For more information contact qfgi2013@notting- at the campus of the U.A.M., Universidad Autonoma de Madrid. ham.ac.uk or visit our website: http://qfgi2013.weebly. Description: This is a trimester called “Research Term on Real Har- com/. monic Analysis and Applications to PDE”. More than 40 researchers have agreed to participate in the trimester. Each of them will deliver * 4–6 44th Interface Symposium, Chapman University, Orange, Cali- specialized lectures on their topics. Four minicourses on modern fornia. harmonic analysis will be delivered by S. Hofmann, T. Hytönen, A. Description: Interface is a membership society of computational McIntosh and A. Volberg. Also there will be a workshop during the scientists, statisticians, mathematicians and individuals from related discipline areas interested in the interface between computing sci- week of May 27–31. ence and statistics. Information: http://www.icmat.es/NTHA/RT-HAPDE/. Topics: Interests include topics such as computational statistics, * 3–5 Quantum Fields, Gravity and Information, University of Not- statistical software, exploratory data analysis, mathematical mod- tingham, Nottingham, United Kingdom. eling, data mining, pattern recognition, scientific visualization and Description: We are pleased to invite postgraduate researchers, from related fields. inside and outside the UK, to exchange ideas on the mathematical Cooperating Organizations: American Statistical Association aspects of relativistic quantum physics. A special focus will be given (ASA), Classification Society of North America (CSNA), Eastern North to relativistic quantum information, a relatively young field which American Region (of the International Biometric Society) (ENAR), combines techniques from quantum field theory, quantum gravity, International Association of Statistical Computing (IASC), Institute quantum information and quantum optics. of Mathematical Statistics (IMS), Institute for Operations Research Invited speakers: Our invited speakers reflect the diversity of areas and Management Science (INFORMS), Society for Industrial and Ap- covered: Prof. Bob Coecke (Oxford), Dr. Ivette Fuentes (Nottingham), plied Mathematics (SIAM), Western North American Region (of the Dr. Etera Livine (Lyon) and Prof. Tim Ralph (Brisbane). International Biometric Society) (WNAR).

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

April 2013 Notices of the AMS 511 Mathematics Calendar

Information: http://www.chapman.edu/events/ * 6–17 The Topology of 3-dimensional Manifolds, Centre de recher- interface-2013/index.aspx. ches mathématiques, Montréal, Canada. * 8–13 Joint CRM-Imperial College School and Workshop in Com- Description: Recent advances in our understanding of the topol- plex Systems, Centre de Recerca Matemàtica, Bellaterra, Barcelona. ogy of 3-dimensional manifolds have led to a conjectural picture Description: The overall theme is the dynamics of complex systems, which indicates surprising restrictions on their topology. We will with an emphasis on emergent phenomena such as spatio-temporal explore the major elements of this picture and what it says about patterns. Complex systems are characterized by different phenom- basic classification and decision problems. The main themes to be ena occurring at different time scales as well as different length addressed are the study of finite covers and virtual properties of scales. Specific mathematical techniques are available to analyze 3-manifolds, the study of the set of all 3-manifolds and how they how phenomena occurring at one scale are related to phenomena relate to each other, and the algebraic and geometric properties of at different scales. Typically, one moves from microscopic, through 3-manifold fundamental groups and their character varieties. This mesoscopic, to macroscopic levels of description. New interactions event will consist of introductory mini-courses held during the first and new behaviors emerge at each level. week, a workshop held during the second, and a series of Aisenstadt Information: http://www.crm.cat/2013/ACComplexSystems. Chair lectures to be delivered by David Gabai (Princeton) which will provide a bridge between them. * 8–20 Cimpa-Unesco-Mesr-Mineco-Morocco Research School on Information: http://www.crm.umontreal.ca/ Statistical Methods and Applications in Insurance and Finance, 2013/Manifolds13/index_e.php. Cadi Ayyad University, Faculty of Sciences and Techniques, Mar- rakesh, Morocco. * 13–15 Spring School 7th Montreal Scientific Computing Days, Description: A first purpose of this school is to offer to Ph.D. stu- Centre de recherches mathématiques, Montréal, Canada. dents and young researchers from Morocco and other countries of Description: To foster scientific exchanges within the scientific the region and other countries of the world, advanced courses about computing community; to train senior undergraduate and graduate recent research in the area of statistics, probability and related ap- students, post-doctoral fellows, and young researchers in the form of plications in finance and insurance, and provide a platform for them three minicourses given by world recognized experts in the general to discus some of the latest trends in these fields. A second pur- areas of scientific computing in science, engineering and medicine; pose consists in establishing research links between international to maximize interactions between the students, the senior partici- researchers and African ones. Finally, the school wants to encourage pants, and the main speakers by reserving up to half of the time young researchers to work in these areas. This is important from for student presentations; to encourage the participation of non- scientific and professional point of view. Success of the school will academic (private or public sector) research or other organizations. definitely have a positive scientific and professional impact, both Information: http://www.crm.umontreal.ca/ in the North-South cooperation and for our young researchers, in Comp13/index_e.shtml. important domains of modern economy. * 20–24 Probability and PDEs, Centro di Ricerca Matematica Ennio Information: http://cimpa2013.uca.ma/. De Giorgi, Palazzo Puteano, Piazza dei Cavalieri 3, 56100, Pisa, Italy. * 18–22 A tribute to S. Ramanujan and Professor S. Chandrasekhar Description: This workshop aims to bring together researchers in- (Life and Work), Department of Mathematics, T.D.P.G. College, Jaun- terested in the interaction between probability and PDEs. Particular pur-222002 (U.P.), India. interest will be devoted to the regularizing effect that comes from Description: Respected Sir/ Madam, in order to motivate students both: randomization of initial data and/or introduction of noise. towards Mathematical Sciences (Mathematics/Physics), a workshop Host: This workshop is hosted by the Centro di Ricerca Matematica on “A tribute to S. Ramanujan and Professor S. Chandrasekhar (Life and it is also supported by: ERC Starting grant Dis- and Work)”, has been organized in the Department of Mathemat- peq (agreement number 257293); FIRB 2012 “Dinamiche dispersive: ics, T.D.P.G. College, Jaunpur, during April 18-22, 2013. You are Analisi di Fourier e Metodi Variazionali”; PRIN 2012 “Problemi dif- requested to deliver talk (popular/technical)/present paper during ferenziali di evoluzione: approcci deterministici e stocastici e loro the above mentioned workshop. interazioni”. Support: We shall provide local hospitalities and AC IInd class fare Registration: There is no registration fee, but preliminary registra- for senior scientists and teachers. For research scholars local hospi- tion is required. For registration please link to http://www.crm. talities and AC IIIrd class fare will be given. Kindly send the topic of sns.it/event/284/registration.html. talk/ abstract of the paper at your earliest. In order to make proper Support: Some funds are available to offer financial support to a arrangement for your comfortable stay we require your travel pro- number of selected young researchers and students. Applications gramme soon. In anticipation of your early reply. can be made at http://crm.sns.it/event/284/financial. Organizing secretary: Dr. Satya Prakash Singh, Department of Math- html. ematics, T.D.P.G. College, Jaunpur-222002, (U.P.) India. Deadline: For applications is March 31, 2013. Information: http://workshoptdcollege.hpage.com/. Information: http://www.crm.sns.it/event/284/. May 2013 * 20–30 Cube complexes and 3-manifolds, University of Illinois at * 6–10 Instructional Workshop on Applied Mathematics, South Chicago, Chicago, Illinois. Asian University, New Delhi, India. Description: This meeting will focus on the recent results on vir- Description: The workshop aims to bring together the research- tual questions, bringing together the geometric group theory and ers studying the topics numerical analysis, differential equations, 3-manifold communities to discuss the impact and future directions. mathematical modeling, stochastic modeling, cryptography, graph The format will be a week long workshop, followed by a conference. theory, fractal geometry, coding theory, applied algebra, dynamical Deadline: We strongly encourage graduate students to participate; systems. The academic programme will consist of series of instruc- register by March 15, 2013, to receive full consideration for confer- tional lectures by eminent mathematicians and scientists in the iden- ence funding. tified topics, invited talks to acquaint the participants in the recent Workshop: May 20–24, 2013. development in the area and contributory talks/paper presentations Main speakers: Ian Agol, UC Berkeley; Dani Wise, McGill Univer- by the participants, if they wish so. sity; Alan Reid. Information: http://mathworkshop.sau.ac.in. UT Austin Conference: May 27–30, 2013. See website for speakers.

512 Notices of the AMS Volume 60, Number 4 Mathematics Calendar

Information: http://www.math.uic.edu/conferences/ working in automata theory, as well as those working on the inter- cubes_man. face, to report on recent research results, to learn from tutorials, to brainstorm on open problems, to explore emerging connections, * 20–31 Advanced School and Workshop in Real and Complex Dy- and to identify new challenges at the intersection of the two fields. namics, The Abdus Salam International Centre for Theoretical Phys- Information: http://www.pims.math.ca/ ics (ICTP), Trieste, Italy. scientific-event/130603-atsdw. Description: Dynamical Systems is a very broad field with many ap- plications and with lots of interest in many developing countries. * 3–28 Rational Points, Rational Curves and Entire Holomorphic This activity will focus on some advanced topics in dynamics and so Curves on Projective Varieties, Centre de recherches mathéma- will be suitable for mathematicians who have a strong background tiques, Montréal, Canada. and are ambitious and interested in developing their research in Description: During the first three weeks, there will be a dozen mini- dynamical systems in a solid and rigorous direction. The first week courses given on three objects of interests in an algebraic variety: ra- of the activity is devoted to a school giving an overview of current tional curves, rational points and holomorphic curves. Key intercon- research topics in real and complex dynamics. nections between these objects will be explored throughout. During Courses: Four intensive courses will be given. the last week of June, there will be a workshop featuring new and Topics: One-dimensional real dynamics (S. Van Strien, Imperial exciting developments in these fields and in their interconnections. College), two-dimensional real dynamics (M. Martens, Stony Brook), Information: http://www.crm/umontreal.ca/ one-dimensional complex dynamics (M. Lyubich, Stony Brook), two- 2013/Integral13/index_e.php. dimensional complex dynamics (J. Smillie, Cornell). The second * 17–20 Young Researchers in Mathematics (YRM2013), Edinburgh, week will be an international workshop, where leading experts will Scotland. report on the newest developments in the field of Real and Com- Description: This is an annual conference organized this year by plex Dynamics. graduate students at Edinburgh, Heriot-Watt, and Glasgow Univer- Information: http://agenda.ictp.it/smr.php?2460. sity. The meeting is principally aimed at UK based mathematics Ph.D. * 21–25 Great Plains Operator Theory Symposium (GPOTS2013), students and postdocs. Previous conferences in the series have taken University of California, Berkeley, California. place in Cambridge, Warwick, and Bristol. Theme: The conference will be centered on the legacy in the fields of Information: http://www.maths.gla.ac.uk/YRM2013/Home. operator theory and operator algebras of the late William B. Arveson. html. Plenary Speakers (50 minute talks): Kennith Davidson (Univ. Wa- * 17–21 NSF-CBMS Regional Research Conference in Mathematical terloo), Ronald Douglas (Texas A & M Univ.), Ruy Exel (Federal Univ. Sciences: The Global Behavior of Solutions to Critical Nonlinear of Santes Caterina, Brazil), Richard Kadison (Univ. Pennsylvania), Wave Equations, Kansas State University, Manhattan, Kansas. Paul Muhly (Univ. Iowa), Vern Paulsen (Univ. Houston). Judith Packer Description: The conference will feature ten lectures by Profes- (Univ. Colorado), Gilles Pisier (Texas A&M Univ.), Robert Powers (Univ. sor Carlos Kenig (University of Chicago) on ‘The Global Behavior of Pennsylvania), Sarah Reznikoff (Kansas State Univ.) Solutions to Critical Nonlinear Wave Equations’. The main topic of Contributed talks (15 minutes each): The schedule allows for up the lectures will be about issues of local and global well-posedness, to 200 contributed talks. scattering, finite-time blow-up and soliton resolution for a class of Information: http://math.berkeley.edu/gpots2013/. non-linear dispersive equations, namely the focusing, energy-critical * 27–31 Advanced Course on Compactifying Moduli Spaces, Centre nonlinear wave equation and the related energy-critical co-rotational de Recerca Matemàtica, Bellaterra, Barcelona. wave maps into the sphere. Description: The Advanced School “Compactifying Moduli Spaces” Invited speakers: In addition to Carlos Kenig’s ten lectures, there deals with various instances of moduli spaces in algebraic geometry, will be eight 50-minute talks by invited speakers: Ioan Bejenaru (Uni- with particular attention to questions related to compactification. versity of California San Diego), Thomas Duyckaerts (Université Paris Goals: It has three goals: To introduce a new generation of students 13), Justin Holmer (Brown University), Andrea Nahmod (University and researchers to these subjects, to collect and survey recent devel- of Massachusetts Amherst), Natasa Pavlovic (University of Texas at opment in the theory, to formulate and disseminate new problems Austin), Gustavo Ponce (University of California Santa Barbara), Gi- and directions of research. gliola Staffilani (Massachusetts Institute of Technology), Luis Vega Information: http://www.crm.cat/2013/ACModuliSpaces. (Universidad del País Vasco). Information: http://www.math.ksu.edu/events/ * 30–June 1 GAP 2013: Geometry and Physics, Centre de Recherches conference/cbms2013/. Mathématiques in Montréal, Québec, Canada. Description: Every year, the format of the GAP conference combines * 17–21 String-Math 2013, SCGP, Stony Brook, New York. three separate but related themes in geometry and physics. This Description: The Simons Center for Geometry and Physics is hosting approach results in a very interesting mixture of talks. The aim of the third annual meeting of String-Math series of conferences in June the conference is to present some of the exciting new developments 2013. The main goal of the conference is to bring together mathema- in the areas of the chosen themes, as well as to expose local area ticians and physicists who work on ideas related to string theory. graduate students and postdocs to these ideas. In fact, the previ- String theory, as well as quantum field theory, has contributed a se- ous conferences involved heavy participation by graduate students ries of profound ideas which gave rise to entirely new mathematical and postdoctoral fellows (over one third of all participants) from fields and revitalized older ones. By now there is a large and rapidly the universities of Toronto, Queen’s, Waterloo, McMaster, and West- growing number of both mathematicians and physicists working at ern Ontario, the Perimeter Institute, as well as other institutions in the string-theoretic interface between the two academic fields. The Canada and around the world. influence flows in both directions, with mathematical techniques and ideas contributing crucially to major advances in string theory. Information: http://www.math.uwaterloo.ca/~gap/. Information: http://scgp.stonybrook.edu/events/ June 2013 event-pages/string-math-2013. * 3–7 PIMS/EQINOCS Automata Theory and Symbolic Dynamics * 17–21 XVII International Conference on Waves and Stability in Workshop, University of British Columbia, Vancouver, BC, Canada. Continuous Media, Bellavista Hotel, Levico terme (TN), Italy. Description: The aim of this workshop is to bring together math- Description: The International Conference on Waves and Stabil- ematicians working in symbolic dynamics and computer scientists ity in Continuous Media (WASCOM), now in its XVII edition, is a

April 2013 Notices of the AMS 513 Mathematics Calendar

biennial international conference on Mathematical Physics. The Information: http://csgt13.upjs.sk. meeting will encompass different fields including discontinuity and * 15–August 2 School and Workshop on Geometric Measure Theory shock waves, linear and nonlinear stability in fluid dynamics and and Optimal Transport, International Centre for Theoretical Phys- solid mechanics, small parameter problems, kinetic theories and ics (ICTP), Trieste, Italy. comparison with continuum models, wave propagation and non- Organizer: The Abdus Salam International Centre for Theoretical equilibrium thermodynamics, diffusion processes in biology and Physics (ICTP). in continuum mechanics, group analysis and reduction techniques, Director: It will be directed by Luigi Ambrosio (Scuola Normale Su- numerical and technical applications. periore di Pisa), Claudio Arezzo (ICTP), (MIT) and Information: http://www.dmi.unict.it/wascom2013. Camillo De Lellis (University of Zurich). * 24–28 Constructive Mathematics: Foundations and Practice, Uni- Topics: Regularity for area-minimizing currents, optimal Transpor- versity of Nisˇ (Faculty of Mechanical Engineering), Nisˇ, Serbia. tation, Monge-Ampére equations, Riemannian Ricci bounds. Aim: The aims of the meeting are: (i) To offer mini-series of lectures Information: http://agenda.ictp.it/smr.php?2459/. presented by experts and designed to lead mathematicians (includ- * 22–26 Planetary Motions, Satellite Dynamics, and Spaceship Or- ing graduate students) from the basic elements of constructive bits, Centre de recherches mathématiques, Montréal, Canada. mathematics to a level where they can appreciate recent develop- Description: The initial goal of Celestial Mechanics was to explain ments in the area. Each mini-series will comprise between 2 and 4 the motion of the Sun, the Moon and planets. Nowadays the math- one-hour lectures. ematical methods of Celestial Mechanics find several different ap- Subjects: The subjects covered will cover the foundations of con- plications, including the determination of the dynamics of planets, structive mathematics (logic, type theory, and set theory); the prac- asteroids, comets, artificial satellites, and the design of orbits for tice of constructive analysis, algebra, and topology; and construc- interplanetary travels. The discovery in the ‘90s of the Kuiper belt tive reverse mathematics. (ii) To enable selected speakers to present and of extrasolar planetary systems gave a new impulse to Celestial one-hour seminars on their recent research in aspects of construc- Mechanics as a mean to understand the birth, evolution, and the tive mathematics. future of the planetary system around the Sun as well as of those Information: http://www.masfak.ni.ac.rs/cmfp2013/. around other stars. Also, the observational campaigns of small bod- * 24–28 Riemann and Klein Surfaces, Symmetries and Moduli ies of the solar system and the increase of the number of artificial Spaces, Linkoping University, Linkoping, Sweden. satellites around our planet led to a new field of application of Ce- Description: The conference will be held in honour of Professor lestial Mechanics: the safeguard of planet Earth. Dismissed satellites Emilio Bujalance on the occasion of his 60th birthday. It will be de- and space rockets constitute dangerous space debris around the voted to the mathematics Emilio Bujalance has worked with, with Earth, which must be carefully monitored and eventually removed. special focus on Riemann and Klein surfaces, i.e., complex curves, Information: http://www.crm.umontreal.ca/2013/ automorphisms of surfaces (real and complex), group actions on Satellites13/index_e.php. surfaces, Grothendieck theory of dessins d’enfants and topological * 22–26 Centenary Conference, , Poland properties of moduli spaces of complex curves. The conference will Description: The Conference will be held on the old campus of the have a computational/combinatorial flavour, with focus on group University of Warsaw, Alma Mater of Samuel Eilenberg. actions on Riemann surfaces, Klein surfaces and related structures Organizer: By the University of Warsaw, Polish Mathematical Soci- such as complex surfaces or hyperbolic manifolds. ety and the Institute of Mathematics of the Polish Academy of Sci- Topics: Among others, the following topics will be covered: Real and ences in collaboration with the American Mathematical Society and complex algebraic curves and surfaces automorphisms of Riemann Columbia University. and Klein surfaces, Dessins d’Enfants, combinatorics and graph Scientific Program: Confirmed plenary lectures will be delivered by theory, geodesics in hyperbolic surfaces, Teichmüller theory and Paul Balmer (University of California at Los Angeles), Andre Joyal moduli spaces of algebraic curves. (Université du Qu{é}bec, Montr{é}al), Martin Hyland (University of Information: http://www.mai.liu.se/surfaces/. Cambridge), Ieke Moerdijk (Radboud University Nijmegen), Brooke E * 24–28 Sz.-Nagy Centennial Conference, Szeged, Hungary. Shipley (University of Illinois at Chicago), Bernard Toen (Universit{é} Description: In the summer of 2013 we commemorate the 100th an- de Montpellier), Shmuel Weinberger (University of Chicago), Mari- niversary of Béla Szo˝kefalvi-Nagy who made deep impact on operator usz Wodzicki (University of California at Berkeley). We plan also to theory over a long period of time. On this occasion the Bolyai Insti- organize parallel sessions of contributed talks and poster session tute of the University of Szeged organizes a conference devoted to which will cover the areas of Eilenberg’s contributions: Algebraic to- recent advances in operator theory and related fields with emphasis pology, category theory and computability. Sectional talks will last on areas related to research carried out by B. Sz.-Nagy. 30-45 minutes. Proposals of talks can be submitted electronically Information: http://www.math.u-szeged.hu/SzNagy100/. after registration for the conference. Proceedings on the occasion * 27–28 A singular life – in honour of Eduard Looijenga, Utrecht of the conference, a special volume of Fundamenta Mathematicae University, Utrecht, The Netherlands. will appear, the journal in which Eilenberg published many papers, Description: The conference “A singular life” marks the retire- including his first (in 1934) and the last (in 1988) topological pa- ment of Professor Eduard Looijenga and will be a celebration of his pers. Participants are welcome to submit their papers which all will achievements and influence in Dutch and international mathematics. be refereed. Information: http://www.uu.nl/singularlife. Financial support: Grants will be offered to a limited number of students and postdocs, towards covering their local expenses in July 2013 Warsaw. Applications can be submitted via the registration website. * 7–13 Seventh Czech-Slovak International Symposium on Graph Information: http://eilenberg100.ptm.org.pl. Theory, Combinatorics, Algorithms and Applications, Kosˇice, Slo- vakia. August 2013 Description: The symposium is a scientific event celebrating the first * 1–3 15th IMS New Researchers Conference, Centre de recherches Czech-Slovak international symposium on graph theory, combinator- mathématiques, Montréal, Canada. ics, algorithms and applications, which took place at the Smolenice Description: The purpose of the conference is to promote interaction Castle fifty years ago. and networking among new researchers in probability and statistics.

514 Notices of the AMS Volume 60, Number 4 Mathematics Calendar

Information: http://www.math.mcgill.ca/nrc2013/index. * 21–27 “Wavelets and Related Multiscale Methods” within IC- html. NAAM2013: 11th International Conference of Numerical Analysis & Applied Mathematics, Rodos Palace Hotel, Rhodes, Greece. * 18–24 International Conference “Differential Equations. Function Aim: To bring together specialists in both theory and application of Spaces. Approximation Theory” dedicated to the 105th anniver- multiscale harmonic analysis. sary of the birthday of S.L. Sobolev, Sobolev Institute of Mathemat- Topics: The topics include but are not restricted to: Wavelet bases ics, Novosibirsk, Russia. and frames (construction, properties, etc.), applications of wavelet Description: October 6, 2013, will be the 105th anniversary of the and other multiscale decompositions to computational problems, birthday of Sergei L’vovich Sobolev (1908–1989), an outstanding modeling multiscale (fractal) structures and anomalous (multiscale mathematician of the 20th century. The Sobolev Institute of Math- self-similar) kinetics, mathematical modeling and analysis biophysi- ematics of the Siberian Branch of the Russian Academy of Sciences cal (e.g., in acoustics, oscillating chemical reactions, neuroscience, jointly with Novosibirsk State University are organizing the Inter- etc.) signals with non-stationary multifrequent periodicity. Thus, national Conference “Differential Equations. Function Spaces. Ap- presentations of researchers developing mathematical basics of proximation Theory” dedicated to this event. multiscale analysis as well as those who apply these methods for Topics: The range of topics that are within the scope of the confer- practical computational applications are welcomed. ence includes (but is not limited to): ordinary differential equations, Information: http://www.icnaam.org/sessions_minisym- partial differential equations, equations of mathematical physics, posia.htm. operator theory, spectral theory, function spaces, embedding theo- rems, approximation theory, cubature formulas. November 2013 Information: http://www.math.nsc.ru/conference/sobo- * 12–14 2013 IEEE International Conference on Technologies for lev/105/english/. Homeland Security (IEEE HST.13), The Westin, Waltham, Boston, * 19–23 Fifth Montreal Problem Solving Workshop, A CRM-Mprime 70 Third Avenue, Waltham, Massachusetts 02451. Event, Centre de Recherches Mathématiques, Montréal, Canada. Description: The meeting will bring together innovators from lead- Description: The goal of the workshop is to gather industry rep- ing academic, industry, business, homeland security centers of ex- resentatives, academic researchers, graduate students and post- cellence, and government programs to provide a forum to discuss doctoral fellows to work on concrete problems proposed by the ideas, concepts, and experimental results. Produced by IEEE with technical support from DHS S&T, IEEE Boston Section, and IEEE-USA industry. The workshop is organized by the Centre de recherches and organizational support from MIT Lincoln Laboratory, Raytheon, mathématiques, along with GERAD, the CIRRELT (Interuniversity Battelle, and MITRE, this year’s event will once again showcase se- Research Centre on Enterprise Networks, Logistics and Transporta- lected technical paper and posters highlighting emerging technolo- tion), and ncm2, and is sponsored by the Mprime network of centres gies in the areas of cyber security, attack and disaster preparation, of excellence. The participants will work in teams, and each team recovery, and response, land and maritime border security and bio- will analyze a problem supplied by a company or a public sector metrics & forensics. institution. The workshop will provide companies and institutions Information: http://ieee-hst.org. with mathematical tools for solving problems, and will enable aca- demic researchers and students in applied mathematics to work on real-world problems. Information: http://www.crm.umontreal.ca/ probindustriels2013/index_e.php. * 26–30 AIM Workshop: Rigorous computation for infinite dimen- sional nonlinear dynamics, American Institute of Mathematics, Palo Alto, California. Description: This workshop, sponsored by AIM and the NSF, will focus on the extension of the rigorous computational tools used in fi- nite dimensional dynamical systems to the infinite dimensional case. Information: http://www.aimath.org/ARCC/workshops/ computenonlinear.html. September 2013 * 9–11 S.Co. 2013 - Complex Data Modeling and Computationally Intensive Statistical Methods for Estimation and Prediction, Po- litecnico di Milano, Milano, Italy. Description: The conference provides a forum for the discussion of new developments and applications of statistical models and computational methods for the analysis of complex and high di- mensional data. Information: http://mox.polimi.it/sco2013/. * 11–14 The Sixth International Workshop on Differential Equations and Applications, Izmir University of Economics, Izmir, Turkey. Description: The scope of the conference is to bring together mem- bers of the mathematical community whose interest lies in applied mathematics to assess new developments, ideas and methods. The conference will cover a wide range of topics of differential equations, difference equations, dynamic equations and stochastic differential equations. Information: http://dm.ieu.edu.tr/wdea2013.

April 2013 Notices of the AMS 515 New Publications Offered by the AMS To subscribe to email notification of new AMS publications, please go to http://www.ams.org/bookstore-email.

Algebra and Algebraic Geometry and graph clustering algorithms are: • What are the communities within an (online) social network? • How do I speed up a numerical simulation by mapping it efficiently onto a parallel computer? The Reductive • How must components be organized on a computer chip such that they can communicate efficiently with each other? Subgroups of F4 • What are the segments of a digital image? • Which functions are certain genes (most likely) responsible David I. Stewart, New College, for? Oxford, United Kingdom The 10th DIMACS Implementation Challenge Workshop was devoted to determining realistic performance of algorithms where worst Contents: Introduction; Overview; General case analysis is overly pessimistic and probabilistic models are too Theory; Reductive subgroups of F4; unrealistic. Articles in the volume describe and analyze various Appendices; Bibliography. experimental data with the goal of getting insight into realistic algorithm performance in situations where analysis fails. This item will also be of interest to those working in discrete mathematics and combinatorics. Memoirs of the American Mathematical Society, Volume 223, Number 1049 Contents: P. Sanders and C. Schulz, High quality graph partitioning; B. O. Fagginger Auer and R. H. Bisseling, Abusing a hypergraph April 2013, 88 pages, Softcover, ISBN: 978-0-8218-8332-7, 2010 partitioner for unweighted graph partitioning; S. Rajamanickam Mathematics Subject Classification: 20G07, 20G10, 20G41, 20G30, and E. G. Boman, Parallel partitioning with Zoltan: Is hypergraph 18G50, Individual member US$41.40, List US$69, Institutional partitioning worth it?; U. V. Çatalyürek, M. Deveci, K. Kaya, member US$55.20, Order code MEMO/223/1049 and K. Uçar, UMPa: A multi-objective, multi-level partitioner for communication minimization; H. Meyerhenke, Shape optimizing load balancing for MPI-parallel adaptive numerical simulations; A. Buluç and K. Madduri, Graph partitioning for scalable Applications distributed graph computations; H. Djidjev and M. Onus, Using graph partitioning for efficient network modularity optimization; D. Aloise, G. Caporossi, P. Hansen, L. Liberti, S. Perron, and M. Ruiz, Modularity maximization in networks by variable Graph Partitioning and neighborhood search; A. Verma and S. Butenko, Network clustering Graph Clustering via clique relaxations: A community based approach; S. Srinivasan, T. Chakraborty, and S. Bhowmick, Identifying base clusters and their David A. Bader, Georgia Institute application to maximizing modularity; M. Hamann, T. Hartmann, of Technology, Atlanta, GA, and and D. Wagner, Complete hierarchical cut-clustering: A case study on expansion and modularity; U. V. Çatalyürek, K. Kaya, J. Langguth, Henning Meyerhenke, Peter and B. Uçar, A partitioning-based divisive clustering technique for Sanders, and Dorothea Wagner, maximizing the modularity; M. Ovelgönne and A. Geyer-Schulz, Karlsruhe Institute of Technology, An ensemble learning strategy for graph clustering; E. J. Riedy, Germany, Editors H. Meyerhenke, D. Ediger, and D. A. Bader, Parallel community detection for massive graphs; B. O. Fagginger Auer and R. H. Graph partitioning and graph clustering Bisseling, Graph coarsening and clustering on the GPU. are ubiquitous subtasks in many applications where graphs play Contemporary Mathematics, Volume 588 an important role. Generally speaking, both techniques aim at the identification of vertex subsets with many internal and few external March 2013, 240 pages, Softcover, ISBN: 978-0-8218-9038-7, 2010 edges. To name only a few, problems addressed by graph partitioning Mathematics Subject Classification: 05C85, 68W05, 05C82, 68W10, 68R05, 68R10, 05C05, 05C65, AMS members US$71.20, List US$89, Order code CONM/588

516 Notices of the AMS Volume 60, Number 4 New Publications Offered by the AMS

Mathematics in Image properties; Uniqueness of solutions; Boundary data in H1(∂V ); Processing Concluding remarks; Bibliography. Memoirs of the American Mathematical Society, Volume 223, Hongkai Zhao, University of Number 1051 California, Irvine, CA, Editor April 2013, 106 pages, Softcover, ISBN: 978-0-8218-8740-0, 2010 Mathematics Subject Classification: 35J25; 31A25, Individual mem- The theme of the 2010 PCMI Summer School ber US$43.20, List US$72, Institutional member US$57.60, Order was Mathematics in Image Processing in a broad sense, including mathematical code MEMO/223/1051 theory, analysis, computation algorithms and applications. In image processing, information needs to be processed, Global Regularity for extracted and analyzed from visual content, such as photographs or the Yang-Mills Equations videos. These demands include standard tasks such as compression and denoising, as well as high-level understanding and analysis, on High Dimensional such as recognition and classification. Centered on the theme of mathematics in image processing, the summer school covered Minkowski Space quite a wide spectrum of topics in this field. The summer school Joachim Krieger, University of is particularly timely and exciting due to the very recent advances and developments in the mathematical theory and computational Pennsylvania, Philadelphia, PA, methods for sparse representation. and Jacob Sterbenz, University of This volume collects three self-contained lecture series. The topics are California, San Diego, La Jolla, CA multi-resolution based wavelet frames and applications to image This item will also be of interest to those working in mathematical processing, sparse and redundant representation modeling of images physics. and simulation of elasticity, biomechanics, and virtual surgery. Recent advances in image processing, compressed sensing and sparse Contents: Introduction; Some gauge-theoretic preliminaries; representation are discussed. Reduction to the “main a-priori estimate”; Some analytic preliminaries; Proof of the main a-priori estimate; Reduction to Titles in this series are co-published with the Institute for Advanced approximate half-wave operators; Construction of the half-wave Study/Park City Mathematics Institute. Members of the Mathematical operators; Fixed time L2 estimates for the parametrix; The dispersive Association of America (MAA) and the National Council of Teachers of estimate; Decomposable function spaces and some applications; Mathematics (NCTM) receive a 20% discount from list price. Completion of the proof; Bibliography. Contents: H. Zhao, Introduction; B. Dong and Z. Shen, MRA-based Memoirs of the American Mathematical Society, Volume 223, wavelet frames and applications; M. Elad, Five lectures on sparse Number 1047 and redundant representations modelling of images; J. M. Teran, J. L. Hellrung, Jr., and J. Hegemann, Simulation of elasticity, April 2013, 99 pages, Softcover, ISBN: 978-0-8218-4489-2, 2010 biomechanics, and virtual surgery. Mathematics Subject Classification: 35L70; 70S15, Individual member IAS/Park City Mathematics Series, Volume 19 US$41.40, List US$69, Institutional member US$55.20, Order code MEMO/223/1047 May 2013, approximately 251 pages, Hardcover, ISBN: 978-0-8218- 9841-3, 2010 Mathematics Subject Classification: 35-XX, 42-XX, 49-XX, 65-XX, 94-XX, AMS members US$51.20, List US$64, Order code PCMS/19 Discrete Mathematics and Combinatorics Differential Equations The Poset of k-Shapes and Branching Rules for Elliptic Partial k-Schur Functions Differential Equations Thomas Lam, University of with Almost-Real Michigan, Ann Arbor, MI, Luc Coefficients Lapointe, Universidad de Talca, Chile, Jennifer Morse, Drexel Ariel Barton, University of University, Philadelphia, PA, Minnesota, Minneapolis, MN and Mark Shimozono, Virginia Contents: Introduction; Definitions and Polytechnic Institute and State the main theorem; Useful theorems; The University, Blacksburg, VA fundamental solution; Properties of layer potentials; Boundedness of layer potentials; Contents: Introduction; The poset of k-shapes; Equivalence of paths Invertibility of layer potentials and other in the poset of k-shapes; Strips and tableaux for k-shapes; Pushout of

April 2013 Notices of the AMS 517 New AMS-Distributed Publications strips and row moves; Pushout of strips and column moves; Pushout sequences; Pushouts of equivalent paths are equivalent; Pullbacks; Appendix A. Tables of branching polynomials; Bibliography. New AMS-Distributed Memoirs of the American Mathematical Society, Volume 223, Number 1050 Publications April 2013, 101 pages, Softcover, ISBN: 978-0-8218-7294-9, 2010 Mathematics Subject Classification: 05E05, 05E10; 14N10, Individual member US$43.20, List US$72, Institutional member US$57.60, Order code MEMO/223/1050 Algebra and Algebraic Geometry Geometry and Topology Sur Les Conjectures de Gross et Prasad I Characterization and Wee Teck Gan, University of the Topological Rigidity California at San Diego, CA, of Nöbeling Manifolds Benedict H. Gross, Harvard University, Cambridge, MA, Andrzej Nagórko, University of , Tata Institute of Warsaw, Poland Fundamental Research, , India, and Jean-Loup Waldspurger, Contents: Introduction and preliminaries: Institut de Mathématiques de Introduction; Preliminaries; Reducing the proof of the main results to the construction Jussieu-CNRS, Paris, France of n-regular and n-semiregular N -covers: n A note to readers: Half of this book is in English and half is in French. Approximation within an Nn-cover; Constructing closed Nn-covers; Carrier and nerve theorems; Anticanonical maps and semiregularity; About 20 years ago Gross and Prasad formulated a conjecture Extending homeomorphisms by the use of a “brick partitionings” determining the restriction of an irreducible admissible technique; Proof of the main results; Constructing n-semiregular and representation of the group G = SO(n) over a local field to a subgroup 0 n-regular Nn-covers: Basic constructions in Nn-spaces; Core of a of the form G = SO(n − 1). The conjecture stated that for a given cover; Proof of Theorem 6.7; Bibliography; Index. pair of generic L-packets of G and G0, there is a unique non-trivial pairing, up to scalars, between precisely one member of each packet, Memoirs of the American Mathematical Society, Volume 223, where G and G0 are allowed to vary among inner forms; moreover, the Number 1048 relevant members of the L-packets are determined by an explicit April 2013, 92 pages, Softcover, ISBN: 978-0-8218-5366-5, 2010 formula involving local root numbers. For non-archimedean local Mathematics Subject Classification: 54-XX, Individual member fields this conjecture has now been proved by Waldspurger and US$41.40, List US$69, Institutional member US$55.20, Order code Mœglin, using a variety of methods of local representation theory; the Plancherel formula plays an important role in the proof. There is also MEMO/223/1048 a global conjecture for automorphic representations, which involves the central critical value of L-functions. This volume is the first of two volumes devoted to the conjecture and its proof for non-archimedean local fields. It contains two long articles by Gan, Gross, and Prasad, formulating extensions of the original Gross-Prasad conjecture to more general pairs of classical groups including metaplectic groups, and providing examples for low rank unitary groups and for representations with restricted ramification. It also includes two articles by Waldspurger: a short article deriving the local multiplicity one conjecture for special orthogonal groups from the results of Aizenbud-Gourevitch-Rallis-Schiffmann on orthogonal groups and a long article (which appeared in Compositio Mathematica in 2010) completing the first part of the proof of the Gross-Prasad conjecture by extending an integral formula relating multiplicities in the restriction problem to harmonic analysis from supercuspidal representations to general tempered representations here. This item will also be of interest to those working in number theory. A publication of the Société Mathématique 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.

518 Notices of the AMS Volume 60, Number 4 New AMS-Distributed Publications

Contents: W. T. Gan, B. H. Gross, and D. Prasad, Symplectic local Astérisque, Number 347 root numbers, central critical L-values, and restriction problems April 2013, 216 pages, Softcover, ISBN: 978-2-85629-350-8, 2010 Math- in the representation theory of classical groups; W. T. Gan, B. H. Gross, and D. Prasad, Restrictions of representations of classical ematics Subject Classification: 22E50, 22E55, Individual member groups: Examples; J.-L. Waldspurger, Une formule intégrale reliée US$72.90, List US$81, Order code AST/347 à la conjecture locale de Gross-Prasad, 2e partie : Extension aux représentations tempérées; J.-L. Waldspurger, Une variante d’un résultat de Aizenbud, Gourevitch, Rallis et Schiffmann. Differential Equations Astérisque, Number 346

November 2012, 318 pages, Softcover, ISBN: 978-2-85629-348-5, 2010 Mathematics Subject Classification: 22E50, 22E55, 11F70, 11R39, Analytic Aspects of 11S37, Individual member US$94.50, List US$105, Order code Problems in Riemannian AST/346 Geometry: Elliptic PDEs, Solitons and Computer Sur Les Conjectures de Imaging Gross et Prasad II Paul Baird, Université de Brest, Colette Mœglin and Jean-Loup France, Ahmad El Soufi, Université Waldspurger, Institut de François-Rabelais, Tours, France, Mathématiques de Jussieu-CNRS, and Ali Fardoun and Rachid Paris, France Regbaoui, Université de Brest, France, Editors

The conference on Analytic Aspects of Problems in Riemannian Geometry was held at the Centre de la Mer de l’Aber Wrac’h in Brittany, A note to readers: This book is in French. France, on May 9–13 2005. Situated in the heart of the Pays des Abers, The conjecture of Gross and Prasad determines, under some the centre provided a superb setting for a scientific meeting on topics assumptions, the restriction of an irreducible admissible at the forefront of current research in this field of mathematics. representation of a group G = SO(n) over a local field to a subgroup This book is a collection of articles based on the contributions of some 0 of the form G = SO(n − 1). For two generic L-paquets (more precisely of the speakers at the conference. The editors preferred to publish a 0 two generic Vogan’s L-packets), the first for G, the second for G , the work based on a coherent theme, and so have limited the number of 0 conjecture states that there is a unique pair (π, π ) in the product of papers. Some of the papers provide an introduction, while others 0 the two packets such that π appears in the restriction of π. Moreover, present new results. the parametrization of π and π 0 (in the usual parametrization of This item will also be of interest to those working in analysis. L-packets) is given by an explicit formula involving some -factors. A publication of the Société Mathématique de France, Marseilles (SMF), In this second volume of Astérisque devoted to the conjecture, the distributed by the AMS in the U.S., Canada, and Mexico. Orders from authors give its proof when the base field is non-archimedean. In the other countries should be sent to the SMF. Members of the SMF receive first paper, they consider an irreducible admissible and self-dual a 30% discount from list. representation of a group GL(N). They prove that the value at the center of symmetry of its -factor is given by an integral formula in Contents: F. Robert, Extremals for Hardy-Sobolev type inequalities: which the character of an extension of the representation to the The influence of the curvature; C. Arezzo and F. Pacard, On the twisted GL(N) appears. The second paper proves the conjecture for Kähler classes of constant scalar curvature metrics on blow ups; M. J. tempered representations. It is a consequence of the stabilization, in Gursky, Fully nonlinear equations, ellipticity, and curvature pinching; the sense of endoscopy theory, of the two integral formulas proved in M. Jumonji and H. Urakawa, Visualization of the eigenvalue the first paper above and in volume 346. Here the authors use some problems of the Laplacian for embedded surfaces and its applications; properties of L-packets that are still conjectural, but were probably T. Rivière, The role of integrability by compensation in conformal proved by Arthur. In the last paper with Mœglin, they extend the result geometric analysis; N. Hungerbühler and B. Roost, Mean curvature to non-tempered generic L-packets. It follows from the following fact flow solitons; R. Mazzeo and M. Saez, Self-similar expanding that they prove that the elements in these L-packets are irreducible solutions for the planar network flow. induced representations from tempered representations. Séminaires et Congrès, Number 22 A publication of the Société Mathématique de France, Marseilles (SMF), November 2012, 174 pages, Softcover, ISBN: 978-2-85629-330-0, 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 2010 Mathematics Subject Classification: 35J60, 46E35, 51E10, 53A10, a 30% discount from list. 53A30, 53C21, 53C42, 53C44, 53C55, 58J32, 58J50, 58J53, 65N30, Individual member US$54, List US$60, Order code SECO/22 Contents: J.-L. Waldspurger, Calcul d’une valeur d’un facteur ε par une formule intégrale; J.-L. Waldspurger, La conjecture locale de Gross-Prasad pour les représentations tempérées des groupes spéciaux orthogonaux; C. Mœglin and J.-L. Waldspurger, La conjecture locale de Gross-Prasad pour les groupes spéciaux orthogonaux : le cas général; References.

April 2013 Notices of the AMS 519 New AMS-Distributed Publications AMERICAN MATHEMATICAL SOCIETY Number Theory BESTSELLING TITLES FROM Séminaire Bourbaki Volume 2010/2011 Exposés 1027-1042

Analysis I A note to readers: This book is in French. Second Edition As in the preceding volumes of this seminar, Terence Tao, University of at which more than one thousand talks California, Los Angeles, CA have been presented, this volume includes sixteen survey lectures on topics of current Hindustan Book Agency; 2009; interest: three lectures on analysis and 347 pages; Hardcover; ISBN: 978-81- geometry, three about algebraic geometry, 85931-94-4; List US$40; AMS members three on diophantine geometry, two related to Langlands’ program, US$32; Order code HIN/28.R two about group theory, one on algebraic topology, one related to Ising’s model, and two about mathematical physics. A publication of the Société Mathématique de France, Marseilles (SMF), Analysis II 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 Second Edition a 30% discount from list. Terence Tao, University of California, Los Angeles, CA Contents: Novembre 2010: O. Glass, La méthode du retour en contrôlabilité et ses applications en mécanique des fluides; Hindustan Book Agency; 2009; E. Kowalski, Crible en expansion; H. Miller, Kervaire invariant one; 218 pages; Hardcover; ISBN: 978-81- W. Werner, Analyticité discrète du modèle d’Ising; Janvier 2011: 85931-95-1; List US$40; AMS members C. Breuil, Correspondance de Langlands p-adique, compatibilité US$32; Order code HIN/29.R local-global et applications; A. Chambert-Loir, Relations de dépendance et intersections exceptionnelles; A. Naor, Sparse quadratic forms and their geometric applications; F. Santambrogio, Inégalités isopérimétriques quantitatives via le transport optimal; Collected Papers of Avril 2011: T. C. Hales, The fundamental lemma and the Hitchin C.S. Seshadri fibration; A. Oancea, Invariants de Welschinger; T. Scanlon, A proof of the André-Oort conjecture via mathematical logic; C. Voisin, Volumes 1 and 2 Sections rationnelles de fibrations sur les surfaces et conjecture de Vikraman Balaji, Chennai Serre; Juin 2011: C. Houdayer, Invariant percolation and measured Mathematical Institute, India, theory of nonamenable groups; D. Huybrechts, A global Torelli V. Lakshmibai, Northeastern theorem for hyperkähler manifolds; P. Mironescu, Le déterminant University, Boston, MA, and jacobien; F. Planchon, Existence globale et scattering pour les M. Pavaman Murthy and solutions de masse finie de l’équation de Schrödinger cubique en Madhav V. Nori, University of dimension deux; Table par noms d’auteurs. Chicago, Illinois, Editors Astérisque, Number 348 Hindustan Book Agency; 2011; 1632 October 2012, 496 pages, Softcover, ISBN: 978-2-85629-351-5, 2010 pages; Hardcover; ISBN: 978-93-80250-17-5; Mathematics Subject Classification: 11N05, 93C20, 35Q30, 35Q31, List US$195; AMS members US$156; Order 11N35, 11N36, 20F69, 05C25, 55Q45, 60K35, 82B20, 52C26, 60J67, code HIN/50 81T40, 11S37, 11F80, 22E55, 11G10, 11G50, 14K12, 14K15, 65F50, 15A63, Individual member US$126.90, List US$141, Order code AST/348 Publications of Hindustan Book Agency. Distributed on an exclusive basis by the AMS in North America.

To view more Hindustan titles, go to ams.org/bookstore/hinseries

Contact the AMS: 1-800-321-4AMS (4267), in the U.S. and Canada, or 1-401-455-4000 (worldwide)

520 Notices of the AMS Volume 60, Number 4 Classified Advertisements Positions available, items for sale, services available, and more

California prints IMA. Salary and length of term as Director (7) Names of three references of the IMA are negotiable. UNIVERSITY OF CALIFORNIA, IRVINE Instructions for the electronic application The Director will be offered a tenured Department of Mathematics process can be found at: http://www. professorship in an academic department mathjobs.org. at the University of Minnesota. Applicants The Department of Mathematics at the Applications are welcome at any time. should have an outstanding track record University of California, Irvine, invites The review process starts March 1, 2013, of research and teaching accomplish- applications to fill an endowed chair and will continue until the position is ments. The candidate’s expertise and position, the UCI Excellence in Teaching filled. documented research activities must Endowed Chair in Mathematics. This chair UCI is an Equal Opportunity Employer demonstrate a strong potential toward position is made possible through a gift committed to excellence through diversity enhancing both the department’s research from the California Community Founda- and strongly encourages applications and the undergraduate and graduate tion and provides an annual discretionary from all qualified applicants, including teaching missions. However, while acting research fund to the successful candidate. women and minorities. as IMA director, teaching responsibilities 000018 are waived and other departmental duties The successful candidate, who is ex- will be determined at the discretion of the pected to be at the full-professor level, will department head. be an excellent research mathematician, The IMA was established in 1982 with fi- with national and international recogni- Minnesota nancial support from the National Science tion for scholarship, demonstrated excel- Foundation. The Institute connects scien- lence in teaching at the undergraduate UNIVERSITY OF MINNESOTA tists, engineers, and mathematicians in and graduate levels, and strong commit- Institute for Mathematics order to address scientific and technologi- and its Applications ment to service. Applicants in all areas of cal challenges in a collaborative, engaging Director mathematics will be considered. environment, developing transformative, Completed applications must be sub- The Board of Governors of the Institute for new mathematics and exploring its appli- mitted electronically through Mathjobs Mathematics and its Applications (IMA) cations, while training the next generation (http://www.mathjobs.org) and must and the University of Minnesota seek a of researchers and educators. contain: new Director of the IMA for an appoint- Applications, consisting of a cover letter (1) AMS cover sheet ment beginning July 2015. Distinguished and curriculum vitae, must be submit- (2) Curriculum vitae academic credentials, including a Ph.D. or ted through the MathJobs.org website, (3) Cover letter equivalent and a record of scientific lead- https://www.mathjobs.org/jobs/ (4) Research statement ership, are required. Candidates should jobs/4603. Consideration of applications (5) Teaching statement have the qualifications to provide scien- will begin March 15, 2013. Nominations (6) Selected reprints and/or pre- tific and administrative leadership to the and questions regarding the position

Suggested uses for classified advertising are positions available, books or 2013; September 2013 issue–July 1, 2013, October 2013 issue–July 26, 2013; lecture notes for sale, books being sought, exchange or rental of houses, November 2013 issue–August 29, 2013. and typing services. U.S. laws prohibit discrimination in employment on the basis of color, age, The 2013 rate is $3.50 per word with a minimum two-line headline. No sex, race, religion, or national origin. “Positions Available” advertisements from institutions outside the U.S. cannot be published unless they are discounts for multiple ads or the same ad in consecutive issues. For an accompanied by a statement that the institution does not discriminate on additional $10 charge, announcements can be placed anonymously. Cor- these grounds whether or not it is subject to U.S. laws. Details and specific respondence will be forwarded. wording may be found on page 1373 (vol. 44). Advertisements in the “Positions Available” classified section will be set Situations wanted advertisements from involuntarily unemployed math- with a minimum one-line headline, consisting of the institution name above ematicians are accepted under certain conditions for free publication. Call body copy, unless additional headline copy is specified by the advertiser. toll-free 800-321-4AMS (321-4267) in the U.S. and Canada or 401-455-4084 Headlines will be centered in boldface at no extra charge. Ads will appear worldwide for further information. in the language in which they are submitted. Submission: Promotions Department, AMS, P.O. Box 6248, Providence, There are no member discounts for classified ads. Dictation over the Rhode Island 02940; or via fax: 401-331-3842; or send email to [email protected]. AMS location for express delivery packages is telephone will not be accepted for classified ads. 201 Charles Street, Providence, Rhode Island 20904. Advertisers will be Upcoming deadlines for classified advertising are as follows: May 2013–Febru- billed upon publication. ary 28, 2013; June/July 2013 issue–April 26, 2013; August 2013 issue–May 29,

April 2013 Notices of the AMS 521 Classified Advertisements

should be addressed to the Search Com- detailed curriculum vitae, and three let- mittee Chair via e-mail at: search@ima. ters of recommendation to: umn.edu. Monica Musso The University of Minnesota is commit- Director ted to the policy that all persons shall have Departamento de Matemáticas equal access to its programs, facilities, Pontificia Universidad Católica de and employment without regard to race, Chile color, creed, religion, national origin, sex, Av. Vicuña Mackenna 4860 age, marital status, disability, public as- Santiago-CHILE; Moving? sistance status, veteran status, or sexual fax: (56-2) 552-5916; orientation. email: [email protected]. 000019 For full consideration, complete applica- tion materials must arrive by June 30, 2013. Virginia 000017

VIRGINIA COMMONWEALTH Please make sure UNIVERSITY Mathematics and Applied Mathematics that the AMS Notices Virginia Commonwealth University invites applications for the position of Chair of the Department of Mathematics and Ap- and Bulletin find their plied Mathematics. We are looking for an accomplished mathematician who wants new home. to make a difference. The chair must have the vision and skill necessary to lead the department as it expands and strength- ens its Ph.D. program, contributes to the university’s goal of becoming a leading research university, and continues in its strong commitment to excellence in its undergraduate and graduate programs. The chair will be required to have a well- developed scholarly/research portfolio with evidence of multi-disciplinary appli- cations and external funding appropriate to complement and expand existing exper- tise in the department. For complete de- scription see http://www.pubinfo.vcu. edu/facjobs/facjob.asp?Item=4760. Submit applications to https://www. mathjobs.org/jobs/jobs/4286. Vir- • Email your new address to us: ginia Commonwealth University is an [email protected] Equal Opportunity/Affirmative Action Employer. Women, minorities and persons • or make the change yourself with disabilities are encouraged to apply. online at: 000015 www.ams.org/cml-update • or send the information to: Chile Member and Customer Services: PONTIFICIA UNIVERSIDAD CATOLICA DE CHILE American Mathematical Society Departamento de Matemáticas 201 Charles Street The Department of Mathematics invites Providence, RI 02904-2294 USA applications for one tenure-track position at the Assistant Professor level begin- Phone: (800) 321-4267 ning either March or August 2014. Appli- (US & Canada) cants should have a Ph.D. in mathematics, proven research potential either in pure (401) 455-4000 (Worldwide) or applied mathematics, and a strong commitment to teaching and research. The regular teaching load for assistant professors consists of three one-semester courses per year, reduced to two during the first two years. The annual salary will be approximately US$48,000. Please send a letter indicating your main research interests, potential collaborators in our department (www.mat.puc.cl),

522 Notices of the AMS Volume 60, Number 4 AMERICAN MATHEMATICAL SOCIETY

Mathematical Reviews/MathSciNet Associate Editor

Applications are invited for a full-time position as an Associate Editor of Mathematical Reviews/MathSciNet, to commence as soon as possible after July 1, 2013, preferably before August 31, 2013. The Mathematical Reviews (MR) division of the American Mathematical Society (AMS) is located in Ann Arbor, Michigan, in a beautiful historic building close to the campus of the University of Michigan. The editors are employees of the AMS; they also enjoy many privileges at the University. At present, the AMS employs over seventy people at Mathematical Reviews, including sixteen associate editors. MR’s mission is to develop and maintain the MR Database, from which MathSciNet is produced. An Associate Editor is responsible for a broad area of the mathematical sciences. Editors select articles and books for coverage, classify these items, determine the type of coverage, assign selected items for review to reviewers, and edit the reviews on their return. The successful applicant will have mathematical breadth with an interest in current developments, and will be willing to learn new topics in pure and applied mathematics. The ability to write well in English is essential. Preference will be given to applicants with expertise in one or more of the following areas: mathematical physics, dynamical systems, algebra, geometry, number theory. The applicant should normally have several years of relevant academic (or equivalent) experience beyond the Ph.D. Evidence of written scholarship in mathematics is expected. The twelve- month salary will be commensurate with the experience that the applicant brings to the position. Applications (including a curriculum vitae; bibliography; and the names, addresses, phone numbers, and email addresses of at least three references) should be sent to:

Dr. Graeme Fairweather email: [email protected] Executive Editor Tel: (734) 996-5257 Mathematical Reviews Fax: (734) 996-2916 P. O. Box 8604 URL: www.ams.org/mr-database Ann Arbor, MI 48107-8604 Applications received by April 15, 2013, will receive full consideration. The American Mathematical Society is an Affirmative Action/Equal Opportunity Employer. Meetings & Conferences of the AMS

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

Marston Conder, University of Auckland, Discrete ob- Chestnut Hill, jects with maximum possible symmetry. Alice Guionnet, École Normale Supérieure de Lyon, Massachusetts Title to be announced. Yanir Rubinstein, University of Maryland, Geometry: Boston College (very) local meets global.

April 6–7, 2013 Special Sessions Saturday – Sunday Algebraic and Geometric Structures of 3-manifolds, Ian Biringer, Tao Li, and Robert Meyerhoff, Boston College. Meeting #1088 Algorithmic Problems of Group Theory and Applications Eastern Section to Information Security, Delaram Kahrobaei, City Univer- Associate secretary: Steven H. Weintraub sity of New York Graduate Center and New York College of Announcement issue of Notices: January 2013 Technology, and Vladimir Shpilrain, City College of New Program first available on AMS website: February 21, 2013 York and City University of New York Graduate Center. Program issue of electronic Notices: April 2013 Arithmetic Dynamics and Galois Theory, John Cullinan, Issue of Abstracts: Volume 34, Issue 2 Bard College, and Farshid Hajir and Siman Wong, Univer- Deadlines sity of Massachusetts, Amherst. Combinatorics and Classical Integrability, Amanda For organizers: Expired For consideration of contributed papers in Special Ses- Redlich and Shabnam Beheshti, Rutgers University. sions: Expired Commuting Matrices and the Hilbert Scheme, Anthony For abstracts: Expired Iarrobino, Northeastern University, and Leila Khatami, Union College. The scientific information listed below may be dated. Complex Geometry and Microlocal Analysis, Victor W. For the latest information, see www.ams.org/amsmtgs/ Guillemin and Richard B. Melrose, Massachusetts Insti- sectional.html. tute of Technology, and Yanir A. Rubinstein, Stanford University. Invited Addresses Counting and Equidistribution on Symmetric Spaces, Roman Bezrukavnikov, Massachusetts Institute of Dubi Kelmer, Boston College, and Alex Kontorovich, Technology, Canonical bases and geometry. Yale University.

524 Notices of the AMS Volume 60, Number 4 Meetings & Conferences

Discrete Geometry of Polytopes, Barry Monson, Univer- Invited Addresses sity of New Brunswick, and Egon Schulte, Northeastern Gunnar Carlsson, Stanford University, Title to be an- University. nounced. Financial Mathematics, Hasanjan Sayit and Stephan Jesus A. De Loera, University of California, Davis, Title Sturm, Worcester Polytechnic Institute. to be announced. History and Philosophy of Mathematics, James J. Tatter- Brendan Hassett, Rice University, Title to be announced. sall, Providence College, and V. Frederick Rickey, United Raphael Rouquier, University of California Los Angeles, States Military Academy. Title to be announced. Homological Invariants in Low-dimensional Topology, John Baldwin, Joshua Greene, and Eli Grigsby, Boston Special Sessions College. Advances in Mathematical Biology, Liming Wang, Homology and Cohomology of Arithmetic Groups, California State University, Los Angeles, and Jiangguo Liu, Avner Ash, Boston College, Darrin Doud, Brigham Young Colorado State University. David Pollack University, and , Wesleyan University. Algebraic Geometry, Sebastian Casalaina-Martin, Uni- Hopf Algebras and their Applications, Timothy Kohl, versity of Colorado, Renzo Cavalieri, Colorado State Uni- Boston University, and Robert Underwood, Auburn Uni- versity, Brendan Hassett, Rice University, and Jonathan versity Montgomery. Wise, University of Colorado. Moduli Spaces in Algebraic Geometry, Dawei Chen and Algebras, Lattices and Varieties, Keith A. Kearnes and Maksym Fedorchuk, Boston College, and Joe Harris and Ágnes Szendrei, University of Colorado, Boulder. Yu-Jong Tzeng, Harvard University. Analysis of Dynamics of the Incompressible Fluids, Mimi Real and Complex Dynamics of Difference Equations Dai and Congming Li, University of Colorado, Boulder. with Applications, Ann Brett, Johnson and Wales Univer- Arithmetic Statistics and Big Monodromy, Jeff Achter, sity, and M. R. S. Kulenovic, University of Rhode Island. Colorado State University, and Chris Hall, University of Recursion and Definability, Rachel Epstein, Harvard Wyoming. University, Karen Lange, Wellesley College, and Russell Associative Rings and Their Modules, Greg Oman and Miller, Queens College and City University of New York Zak Mesyan, University of Colorado, Colorado Springs. Graduate Center. Cluster Algebras and Related Combinatorics, Gregg Research by Undergraduates and Students in Post- Musiker, University of Minnesota, Kyungyong Lee, Wayne Baccalaureate Programs, Chi-Keung Cheung, Boston College, David Damiano, College of the Holy Cross, Ste- State University, and Li Li, Oakland University. ven J. Miller, Williams College, and Suzanne L. Weekes, Combinatorial Avenues in Representation Theory, Rich- Worcester Polytechnic Institute. ard Green, University of Colorado Boulder, Anne Shepler, Topology and Generalized Cohomologies in Modern University of North Texas, and Nathaniel Thiem, Univer- Condensed Matter Physics, Claudio Chamon and Robert sity of Colorado Boulder. Kotiuga, Boston University. Combinatorial and Computational Commutative Alge- bra and Algebraic Geometry, Hirotachi Abo, University of Idaho, Zach Teitler, Boise State University, and Alexander Woo, University of Idaho. Boulder, Colorado Diophantine Approximation on Manifolds and Fractals: University of Colorado Boulder Dynamics, Measure Theory and Schmidt Games, Wolfgang Schmidt, University of Colorado at Boulder, and Lior Fish- April 13–14, 2013 man, University of North Texas. Saturday – Sunday Dynamical Systems: Thermodynamic Formalism and Connections with Geometry, Keith Burns, Northwestern Meeting #1089 University, and Dan Thompson, The Ohio State University. Western Section Dynamics and Arithmetic Geometry, Su-ion Ih, Uni- Associate secretary: Michel L. Lapidus versity of Colorado at Boulder, and Thomas J. Tucker, Announcement issue of Notices: January 2013 University of Rochester. Program first available on AMS website: February 28, 2013 Elliptic Systems and Their Applications, Wenxiong Chen, Program issue of electronic Notices: April 2013 Yeshiva University, and Congming Li, University of Colo- Issue of Abstracts: Volume 34, Issue 2 rado at Boulder. Extremal Graph Theory, Michael Ferrara, University of Deadlines Colorado Denver, Stephen Hartke, University of Nebraska- For organizers: Expired Lincoln, and Michael Jacobson, University of Colorado For consideration of contributed papers in Special Ses- Denver. sions: Expired Foundations of Computational Mathematics, Susan For abstracts: Expired Margulies, Pennsylvania State University, and Jesus De The scientific information listed below may be dated. Loera, University of California, Davis. For the latest information, see www.ams.org/amsmtgs/ Geometric Methods in the Representation Theory of Re- sectional.html. ductive Groups, J. Matthew Douglass, University of North

April 2013 Notices of the AMS 525 Meetings & Conferences

Texas, Gerhard Roehrle, Ruhr-Universitaet Bochum, and Vladimir Markovic, California Institute of Technology, Rahbar Virk, University of Colorado, Boulder. Title to be announced. Harmonic Analysis of Frames, Wavelets, and Tilings, Ve- Endre Szemerédi, Siemens Corporate Technology, Title ronika Furst, Fort Lewis College, Keri Kornelson, Univer- to be announced (Erdo˝s Memorial Lecture). sity of Oklahoma, and Eric Weber, Iowa State University. Eitan Tadmor, University of Maryland, Title to be an- Noncommutative Geometry and Geometric Analysis, nounced. Carla Farsi and Alexander Gorokhovsky, University of Colorado, Boulder. Special Sessions Nonlinear Waves and Integrable Systems, Christo- Algebraic and Geometric Combinatorics, Sung Y. Song, pher W. Curtis, University of Colorado, Boulder, Anton Iowa State University, and Paul Terwilliger, University of Dzhamay, University of Northern Colorado, Willy Here- Wisconsin-Madison. man, Colorado School of Mines, and Barbara Prinari, Analysis, Dynamics and Geometry In and Around University of Colorado, Colorado Springs. Teichmuller Spaces, Alistair Fletcher, Northern Illinois Number Theory with a Focus on Diophantine Equations University, Vladimir Markovic, California Institute of and Recurrence Sequences, Patrick Ingram, Colorado Technology, and Dragomir Saric, Queens College CUNY. State University, and Katherine E. Stange, University of Commutative Algebra and its Environs, Olgur Celikbas Colorado, Boulder. and Greg Piepmeyer, University of Missouri, Columbia. Set Theory and Boolean Algebras, Natasha Dobrinen, Commutative Ring Theory, Michael Axtell, University University of Denver, and Don Monk, University of Colo- of St. Thomas, and Joe Stickles, Millikin University. rado, Boulder. Computability and Complexity in Discrete and Con- Singular Spaces in Geometry, Topology, and Algebra, tinuous Worlds, Jack Lutz and Tim McNicholl, Iowa State Greg Friedman, Texas Christian University, and Laurentiu University. Maxim, University of Wisconsin, Madison. Computational Advances on Special Functions and Themes in Applied Mathematics: From Data Analysis Tropical Geometry, Lubjana Beshaj, Oakland University, through Fluid Flows and Biology to Topology, Hanna and Emma Previato, Boston University. Makaruk, Los Alamos National Laboratory, and Robert Control Theory and Qualitative Analysis of Partial Owczarek, University of New Mexico, and Enfitek, Inc. Differential Equations, George Avalos, University of Nebraska-Lincoln, and Scott Hansen, Iowa State University. Discrete Methods and Models in Mathematical Biology, Ames, Iowa Dora Matache, University of Nebraska-Omaha, and Ste- Iowa State University phen J. Willson, Iowa State University. Extremal Combinatorics, Steve Butler and Ryan Martin, April 27–28, 2013 Iowa State University. Generalizations of Nonnegative Matrices and Their Sign Saturday – Sunday Patterns, Minerva Catral, Xavier University, Shaun Fallat, Meeting #1090 University of Regina, and Pauline van den Driessche, Central Section University of Victoria. Associate secretary: Georgia Benkart Geometric Elliptic and Parabolic Partial Differential Announcement issue of Notices: February 2013 Equations, Brett Kotschwar, Arizona State University, and Program first available on AMS website: March 14, 2013 Xuan Hien Nguyen, Iowa State University. Program issue of electronic Notices: April 2013 Graphs, Hypergraphs and Counting, Eva Czabarka and Issue of Abstracts: Volume 34, Issue 2 Laszlo Szekely, University of South Carolina. Kinetic and Hydrodynamic PDE-based Descriptions of Deadlines Multi-scale Phenomena, James Evans and Hailiang Liu, For organizers: Expired Iowa State University, and Eitan Tadmor, University of For consideration of contributed papers in Special Ses- Maryland. sions: Expired Logic and Algebraic Logic, Jeremy Alm, Illinois College, For abstracts: Expired and Andrew Ylvisaker, Iowa State University. Multi-Dimensional Dynamical Systems, Jayadev The scientific information listed below may be dated. Athreya, University of Illinois, Urbana-Champaign, Jona- For the latest information, see www.ams.org/amsmtgs/ than Chaika, University of Chicago, and Joseph Rosen- sectional.html. blatt, University of Illinois at Urbana-Champaign. Numerical Analysis and Scientific Computing, Hailiang Invited Addresses Liu, Songting Luo, James Rossmanith, and Jue Yan, Iowa Kevin Costello, Northwestern University, Title to be State University. announced. Numerical Methods for Geometric Partial Differential Marianne Csornyei, University of Chicago, Title to be Equations, Gerard Awanou, University of Illinois at Chi- announced. cago, and Nicolae Tarfulea, Purdue University.

526 Notices of the AMS Volume 60, Number 4 Meetings & Conferences

Operator Algebras and Topological Dynamics, Benton Sergiu Klainerman, Princeton University, Title to be L. Duncan, North Dakota State University, and Justin R. announced. Peters, Iowa State University. George Lusztig, Massachusetts Institute of Technology, Partial Differential Equations, Gary Lieberman and Title to be announced. Paul Sacks, Iowa State University, and Mahamadi Warma, Stefan Papadima, Institute of Mathematics of the Ro- University of Puerto Rico at Rio Piedras. manian Academy, Title to be announced. Probabilistic and Multiscale Modeling Approaches in Dan Timotin, Institute of Mathematics of the Romanian Cell and Systems Biology, Jasmine Foo, University of Min- Academy, Title to be announced. nesota, and Anastasios Matzavinos, Iowa State University. Srinivasa Varadhan, New York University, Title to be Quasigroups, Loops, and Nonassociative Division Alge- announced. bras, C. E. Ealy Jr. and Annegret Paul, Western Michigan University, Benjamin Phillips, University of Michigan Special Sessions Dearborn, J. D. Phillips, Northern Michigan University, Algebraic Geometry, Marian Aprodu, Institute of and Petr Vojtechovsky, University of Denver. Mathematics of the Romanian Academy, Mircea Mustata, Ring Theory and Noncommutative Algebra, Victor University of Michigan, Ann Arbor, and Mihnea Popa, Camillo, University of Iowa, and Miodrag C. Iovanov, University of Illinois, Chicago. University of Bucharest and University of Iowa. Articulated Systems: Combinatorics, Geometry and Ki- Stochastic Processes with Applications to Physics and nematics, Ciprian S. Borcea, Rider University, and Ileana Control, Jim Evans and Arka Ghosh, Iowa State University, Streinu, Smith College. Jon Peterson, Purdue University, and Alexander Roiter- Calculus of Variations and Partial Differential Equations, shtein, Iowa State University. Marian Bocea, Loyola University, Chicago, Liviu Ignat, Topology of 3-Manifolds, Marion Campisi and Alexan- Institute of Mathematics of the Romanian Academy, Mihai der Zupan, University of Texas at Austin. Mihailescu, University of Craiova, and Daniel Onofrei, Zero Forcing, Maximum Nullity/Minimum Rank, and University of Houston. Colin de Verdiere Graph Parameters, Leslie Hogben, Iowa Commutative Algebra, Florian Enescu, Georgia State State University and American Institute of Mathematics, University, and Cristodor Ionescu, Institute of Mathemat- and Bryan Shader, University of Wyoming. ics of the Romanian Academy. Discrete Mathematics and Theoretical Computer Science, Sebastian Cioaba, University of Delaware, Gabriel Istrate, Alba Iulia, Romania Universitatea de Vest, Timisoara, Ioan Tomescu, Univer- sity of Bucharest, and Marius Zimand, Towson University. University of Alba Iulia Domain Decomposition Methods and their Applications in Mechanics and Engineering, Lori Badea, Institute of June 27–30, 2013 Mathematics of the Romanian Academy, and Marcus Sar- Thursday – Sunday kis, Worcester Polytechnic Institute. Geometry and Topology of Arrangements of Hyper- Meeting #1091 surfaces, Daniel Matei, Institute of Mathematics of the First Joint International Meeting of the AMS and the Ro- Romanian Academy, and Alexandru I. Suciu, Northeastern manian Mathematical Society, in partnership with the University. “Simion Stoilow” Institute of Mathematics of the Romanian Harmonic Analysis and Applications, Ciprian Demeter, Academy. Indiana University, Bloomington, and Camil Muscalu, Associate secretary: Steven H. Weintraub Cornell University. Announcement issue of Notices: January 2013 Hopf Algebras, Coalgebras, and their Categories of Program first available on AMS website: Not applicable Representations, Miodrag C. Iovanov, University of Program issue of electronic Notices: Not applicable Bucharest and University of Iowa, Susan Montgomery, Issue of Abstracts: Not applicable University of Southern California, and Siu-Hung Ng, Iowa State University. Deadlines Local and Nonlocal Models in Wave Propagation and For organizers: To be announced Diffusion, Anca V. Ion, Institute of Mathematical Statis- For consideration of contributed papers in Special Ses- tics and Applied Mathematics of the Romanian Academy, sions: To be announced Petronela Radu, University of Nebraska, Lincoln, and For abstracts: To be announced Lorena Bociu, North Carolina State University. Mathematical Finance, Stochastic Analysis, and Partial The scientific information listed below may be dated. Differential Equations, Lucian Beznea, Institute of Math- For the latest information, see www.ams.org/amsmtgs/ ematics of the Romanian Academy, Paul Feehan, Rutgers internmtgs.html. University, Victor Nistor, Pennsylvania State University, Camelia Pop, University of Pennsylvania, and Mihai Sirbu, Invited Addresses University of Texas, Austin. Viorel Barbu, Universitatea Al. I. Cuza, Title to be an- Mathematical Models in Life and Environment, Gabri- nounced. ela Marinoschi, Institute of Mathematical Statistics and

April 2013 Notices of the AMS 527 Meetings & Conferences

Applied Mathematics of the Romanian Academy, and Invited Addresses Fabio Augusto Milner, Arizona State University. Michael Hill, University of Virginia, Title to be an- Mathematical Models in Materials Science and Engineer- nounced. ing, Marian Bocea, Loyola University, Chicago, and Bogdan Suzane Lenhart, University of Tennessee and NIMBioS, Vernescu, Worcester Polytechnic Institute. Using optimal control of PDEs to investigate population Noncommutative Ring Theory and Applications, Toma questions. Albu, Institute of Mathematics of the Romanian Academy, Ralph McKenzie, Vanderbilt University, Title to be an- and Lance W. Small, University of California, San Diego. Nonlinear Evolution Equations, Daniel Tataru, Univer- nounced. sity of California, Berkeley, and Monica Visan, University Victor Moll, Tulane University, Title to be announced. of California, Los Angeles. Special Sessions Operator Algebra and Noncommutative Geometry, Marius Dadarlat, Purdue University, and Florin Radulescu, Algebraic Coding Theory (Code: SS 5A), Steve Szabo, Institute of Mathematics of the Romanian Academy and Eastern Kentucky University, and Heide Gluesing- University of Rome Tor Vergata. Luerssen, University of Kentucky. Operator Theory and Function Spaces, Aurelian Gheon- Applied Analysis and Inverse Problems (Code: SS 9A), dea, Institute of Mathematics of the Romanian Academy Peijun Li, Purdue University, Jiguang Sun, Michigan Tech- and Bilkent University, Mihai Putinar, University of nological University, and Yongzhi Steve Xu, University of California, Santa Barbara, and Dan Timotin, Institute of Louisville. Mathematics of the Romanian Academy. Combinatorial Commutative Algebra (Code: SS 4A), Probability and its Relation to Other Fields of Mathemat- Juan Migliore, University of Notre Dame, and Uwe Nagel, ics, Krzysztof Burdzy, University of Washington, and University of Kentucky. Mihai N. Pascu, Transilvania University of Bras¸ov. Commutative Rings, Ideals, and Modules (Code: SS 3A), Random Matrices and Free Probability, Ioana Dumitriu, Ela Celikbas and Olgur Celikbas, University of Missouri- University of Washington, and Ionel Popescu, Georgia Columbia. Institute of Technology and Institute of Mathematics of Extremal Graph Theory (Code: SS 2A), Jozsef Balogh, the Romanian Academy. Several Complex Variables, Complex Geometry and University of Illinois at Urbana-Champaign, and Louis Dynamics, Dan Coman, Syracuse University, and Cezar DeBiasio and Tao Jiang, Miami University, Oxford, OH. Joita, Institute of Mathematics of the Romanian Academy. Finite Universal Algebra (Code: SS 6A), Ralph McKenzie, Topics in Geometric and Algebraic Topology, Stefan Vanderbilt University, and Matthew Valeriote, McMaster Papadima, Institute of Mathematics of the Romanian Acad- University. emy, and Alexandru I. Suciu, Northeastern University. Fixed Point Theorems and Applications to Integral, Dif- ference, and Differential Equations (Code: SS 8A), Jeffrey W. Lyons, Nova Southeastern University, and Jeffrey T. Louisville, Kentucky Neugebauer, Eastern Kentucky University. Harmonic Analysis and Partial Differential Equations University of Louisville (Code: SS 10A), Russell Brown and Katharine Ott, Uni- versity of Kentucky. October 5–6, 2013 Recent Advances on Commutative Algebra and Its Ap- Saturday – Sunday plications (Code: SS 11A), Hamid Kubsman and Jinjia Meeting #1092 Li, University of Louisville, and Hamid Rahmati, Miami University. Southeastern Section Associate secretary: Brian D. Boe Set Theory and Its Applications (Code: SS 1A), Paul Lar- Announcement issue of Notices: June 2013 son, Miami University, Justin Moore, Cornell University, Program first available on AMS website: August 22, 2013 and Grigor Sargsyan, Rutgers University. Program issue of electronic Notices: October 2013 Weak Dependence in Probability and Statistics (Code: SS Issue of Abstracts: Volume 34, Issue 3 7A), Cristina Tone, Ryan Gill, and Kiseop Lee, University of Louisville. Deadlines For organizers: Expired For consideration of contributed papers in Special Ses- sions: June 18, 2013 For abstracts: August 13, 2013

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

528 Notices of the AMS Volume 60, Number 4 Meetings & Conferences

Recent Advances in Harmonic Analysis and Partial Dif- Philadelphia, ferential Equations (Code: SS 1A), Cristian Gutiérrez and Irina Mitrea, Temple University. Recent Developments in Noncommutative Algebra Pennsylvania (Code: SS 6A), Edward Letzter and Martin Lorenz, Temple Temple University University. Several Complex Variables and CR Geometry (Code: October 12–13, 2013 SS 7A), Andrew Raich, University of Arkansas, and Yuan Saturday – Sunday Zhang, Indiana University-Purdue University Fort Wayne.

Meeting #1093 Eastern Section St. Louis, Missouri Associate secretary: Steven H. Weintraub Announcement issue of Notices: June 2013 Washington University Program first available on AMS website: To be announced October 18–20, 2013 Program issue of electronic Notices: October 2013 Friday – Sunday Issue of Abstracts: Volume 34, Issue 3 Meeting #1094 Deadlines Central Section For organizers: Expired Associate secretary: Georgia Benkart For consideration of contributed papers in Special Ses- Announcement issue of Notices: August 2013 sions: June 25, 2013 Program first available on AMS website: September 5, 2013 For abstracts: August 20, 2013 Program issue of electronic Notices: October 2013 Issue of Abstracts: Volume 34, Issue 4 The scientific information listed below may be dated. For the latest information, see www.ams.org/amsmtgs/ Deadlines sectional.html. For organizers: March 20, 2013 For consideration of contributed papers in Special Ses- Invited Addresses sions: July 2, 2013 Patrick Brosnan, University of Maryland, Title to be For abstracts: August 27, 2013 announced. Xiaojung Huang, Rutgers University, Title to be an- The scientific information listed below may be dated. nounced. For the latest information, see www.ams.org/amsmtgs/ Barry Mazur, Harvard University, Title to be announced sectional.html. (Erdo˝s Memorial Lecture). Robert Strain, University of Pennsylvania, Title to be Invited Addresses announced. Ronny Hadani, University of Texas at Austin, Title to be announced. Special Sessions Effie Kalfagianni, Michigan State University, Title to be Contact and Symplectic Topology (Code: SS 5A), Joshua announced. M. Sabloff, Haverford College, and Lisa Traynor, Bryn Jon Kleinberg, Cornell University, Title to be announced. Mawr College. Vladimir Sverak, University of Minnesota, Title to be announced. Difference Equations and Applications (Code: SS 9A), Michael Radin, Rochester Polytechnic Institute. Special Sessions Geometric and Spectral Analysis (Code: SS 3A), Thomas Algebraic and Combinatorial Invariants of Knots (Code: Krainer, Pennsylvania State Altoona, and Gerardo A. Men- SS 1A), Heather Dye, McKendree University, Allison Hen- doza, Temple University. rich, Seattle University, and Louis Kauffman, University Higher Structures in Algebra, Geometry and Physics of Illinois. (Code: SS 2A), Jonathan Block, University of Pennsylvania, Computability Across Mathematics (Code: SS 2A), Wes- Vasily Dolgushev, Temple University, and Tony Pantev, ley Calvert, Southern Illinois University, and Johanna University of Pennsylvania. Franklin, University of Connecticut. History of Mathematics in America (Code: SS 4A), in Low Dimensions (Code: SS 4A), Thomas L. Bartlow, Villanova University, Paul R. Wolfson, William H. Kazez, University of Georgia, and Rachel Rob- West Chester University, and David E. Zitarelli, Temple erts, Washington University in St. Louis. University. Groupoids in Analysis and Geometry (Code: SS 6A), Alex Mathematical Biology (Code: SS 8A), Isaac Klapper, Kumjian, University of Nevada at Reno, Markus Pflaum, Temple University, and Kathleen Hoffman, University of University of Colorado, and Xiang Tang, Washington Maryland, Baltimore County. University in St. Louis.

April 2013 Notices of the AMS 529 Meetings & Conferences

Interactions between Geometric and Harmonic Analysis (Code: SS 3A), Leonid Kovalev, Syracuse University, and Baltimore, Maryland Jeremy Tyson, University of Illinois, Urbana-Champaign. Baltimore Convention Center, Hilton Bal- Noncommutative Rings and Modules (Code: SS 5A), Greg Marks and Ashish Srivastava, St. Louis University. timore, and Baltimore Marriott Inner Har- bor Hotel Riverside, California January 15–18, 2014 Wednesday – Saturday University of California Riverside Meeting #1096 November 2–3, 2013 Joint Mathematics Meetings, including the 120th Annual Saturday – Sunday Meeting of the AMS, 97th Annual Meeting of the Math- Meeting #1095 ematical Association of America (MAA), annual meetings of the Association for Women in Mathematics (AWM) and Western Section the National Association of Mathematicians (NAM), and the Associate secretary: Michel L. Lapidus winter meeting of the Association for Symbolic Logic (ASL), Announcement issue of Notices: August 2013 with sessions contributed by the Society for Industrial and Program first available on AMS website: September 19, Applied Mathematics (SIAM). 2013 Associate secretary: Georgia Benkart Program issue of electronic Notices: November 2013 Announcement issue of Notices: October 2013 Issue of Abstracts: Volume 34, Issue 4 Program first available on AMS website: November 1, 2013 Deadlines Program issue of electronic Notices: January 2013 For organizers: April 2, 2013 Issue of Abstracts: Volume 35, Issue 1 For consideration of contributed papers in Special Ses- Deadlines sions: July 15, 2013 For organizers: April 1, 2013 For abstracts: September 10, 2013 For consideration of contributed papers in Special Ses- sions: To be announced The scientific information listed below may be dated. For abstracts: To be announced For the latest information, see www.ams.org/amsmtgs/ sectional.html.

Invited Addresses Knoxville, Tennessee Michael Christ, University of California, Berkeley, Title University of Tennessee, Knoxville to be announced. Mark Gross, University of California, San Diego, Title March 21–23, 2014 to be announced. Friday – Sunday Matilde Marcolli, California Institute of Technology, Title to be announced. Meeting #1097 Paul Vojta, University of California, Berkeley, Title to Southeastern Section be announced. Associate secretary: Brian D. Boe Announcement issue of Notices: To be announced Special Sessions Program first available on AMS website: To be announced Computer, Mathematics, Imaging, Technology, Network, Program issue of electronic Notices: To be announced Health, Big Data, and Statistics (Code: SS 3A), Subir Ghosh, Issue of Abstracts: To be announced University of California, Riverside. Developments in Markov Chain Theory and Methodol- Deadlines ogy (Code: SS 2A), Jason Fulman, University of California, For organizers: August 21, 2013 Riverside, and Mark Huber, Claremont McKenna College. For consideration of contributed papers in Special Ses- The Mathematics of Planet Earth (Code: SS 1A), John sions: To be announced Baez, University of California, Riverside. For abstracts: To be announced

530 Notices of the AMS Volume 60, Number 4 Meetings & Conferences Baltimore, Maryland Lubbock, Texas University of Maryland, Baltimore County Texas Tech University

March 29–30, 2014 April 11–13, 2014 Saturday – Sunday Friday – Sunday

Meeting #1098 Meeting #2000 Eastern Section Central Section Associate secretary: Steven H. Weintraub Associate secretary: Georgia Benkart Announcement issue of Notices: January 2014 Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Program issue of electronic Notices: March 2014 Issue of Abstracts: To be announced Issue of Abstracts: To be announced Deadlines Deadlines For organizers: September 18, 2013 For organizers: August 29, 2013 For consideration of contributed papers in Special Ses- For consideration of contributed papers in Special Ses- sions: To be announced sions: To be announced For abstracts: To be announced For abstracts: To be announced The scientific information listed below may be dated. For the latest information, see www.ams.org/amsmtgs/ Albuquerque, New sectional.html. Mexico Special Sessions Analysis and Applications of Dynamic Equations on Time University of New Mexico Scales (Code: SS 2A), Heidi Berger, Simpson College, and Raegan Higgins, Texas Tech University. April 5–6, 2014 Topology and Physics (Code: SS 1A), Razvan Gelca and Saturday – Sunday Alastair Hamilton, Texas Tech University.

Meeting #1099 Western Section Tel Aviv, Israel Associate secretary: Michel L. Lapidus Announcement issue of Notices: To be announced Bar-Ilan University, Ramat-Gan and Program first available on AMS website: To be announced Tel-Aviv University, Ramat-Aviv Program issue of electronic Notices: April 2014 June 16–19, 2014 Issue of Abstracts: To be announced Monday – Thursday Deadlines The 2nd Joint International Meeting between the AMS and For organizers: September 5, 2013 the Israel Mathematical Union. Associate secretary: Michel L. Lapidus For consideration of contributed papers in Special Ses- Announcement issue of Notices: To be announced sions: To be announced Program first available on AMS website: To be announced For abstracts: February 11, 2014 Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced The scientific information listed below may be dated. For the latest information, see www.ams.org/amsmtgs/ Deadlines sectional.html. For organizers: To be announced For consideration of contributed papers in Special Ses- Special Sessions sions: To be announced The Inverse Problem and Other Mathematical Methods For abstracts: To be announced Applied in Physics and Related Sciences (Code: SS 1A), Hanna Makaruk, Los Alamos National Laboratory, and The scientific information listed below may be dated. Robert Owczarek, University of New Mexico and Enfitek, For the latest information, see www.ams.org/amsmtgs/ Inc. internmtgs.html.

April 2013 Notices of the AMS 531 Meetings & Conferences

Special Sessions Mirror Symmetry and Representation Theory, David San Francisco, Kazhdan, Hebrew University, and Roman Bezrukavnikov, Massachusetts Institute of Technology. California Nonlinear Analysis and Optimization, Boris Morduk- San Francisco State University hovich, Wayne State University, and Simeon Reich and October 25–26, 2014 Alexander Zaslavski, The Technion-Israel Institute of Saturday – Sunday Technology. Western Section Qualitative and Analytic Theory of ODE’s, Yosef Yom- Associate secretary: Michel L. Lapidus din, Weizmann Institute. Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Program issue of electronic Notices: October 2014 Eau Claire, Wisconsin Issue of Abstracts: To be announced Deadlines University of Wisconsin-Eau Claire For organizers: March 25, 2014 For consideration of contributed papers in Special Ses- September 20–21, 2014 sions: To be announced Saturday – Sunday For abstracts: September 3, 2014 Central Section Associate secretary: Georgia Benkart Announcement issue of Notices: To be announced Greensboro, North Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Carolina Issue of Abstracts: To be announced University of North Carolina, Greensboro

Deadlines November 8–9, 2014 Saturday – Sunday For organizers: February 20, 2014 Southeastern Section For consideration of contributed papers in Special Ses- Associate secretary: Brian D. Boe sions: To be announced Announcement issue of Notices: To be announced For abstracts: August 5, 2014 Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced

Halifax, Canada Deadlines Dalhousie University For organizers: April 8, 2014 For consideration of contributed papers in Special Ses- October 18–19, 2014 sions: To be announced For abstracts: To be announced Saturday – Sunday Eastern Section Associate secretary: Steven H. Weintraub San Antonio, Texas Announcement issue of Notices: To be announced Henry B. Gonzalez Convention Center and Program first available on AMS website: To be announced Grand Hyatt San Antonio Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced January 10–13, 2015 Saturday – Tuesday Deadlines Joint Mathematics Meetings, including the 121st Annual For organizers: March 18, 2014 Meeting of the AMS, 98th Annual Meeting of the Math- ematical Association of America (MAA), annual meetings For consideration of contributed papers in Special Ses- of the Association for Women in Mathematics (AWM) and sions: To be announced the National Association of Mathematicians (NAM), and the For abstracts: To be announced winter meeting of the Association of Symbolic Logic (ASL),

532 Notices of the AMS Volume 60, Number 4 Meetings & Conferences with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Chicago, Illinois Associate secretary: Steven H. Weintraub Announcement issue of Notices: October 2014 Loyola University Chicago Program first available on AMS website: To be announced October 3–4, 2015 Program issue of electronic Notices: January 2015 Saturday – Sunday Issue of Abstracts: Volume 36, Issue 1 Central Section Associate secretary: Georgia Benkart Deadlines Announcement issue of Notices: To be announced For organizers: April 1, 2014 Program first available on AMS website: To be announced For consideration of contributed papers in Special Ses- Program issue of electronic Notices: October 2015 sions: To be announced Issue of Abstracts: To be announced For abstracts: To be announced Deadlines For organizers: March 10, 2015 For consideration of contributed papers in Special Ses- Las Vegas, Nevada sions: To be announced University of Nevada, Las Vegas For abstracts: To be announced April 18–19, 2015 Fullerton, California Saturday – Sunday Western Section California State University, Fullerton Associate secretary: Michel L. Lapidus October 24–25, 2015 Announcement issue of Notices: To be announced Saturday – Sunday Program first available on AMS website: To be announced Western Section Program issue of electronic Notices: To be announced Associate secretary: Michel L. Lapidus Issue of Abstracts: To be announced Announcement issue of Notices: To be announced Program first available on AMS website: To be announced Deadlines Program issue of electronic Notices: October 2015 For organizers: September 18, 2014 Issue of Abstracts: To be announced For consideration of contributed papers in Special Ses- sions: To be announced Deadlines For abstracts: To be announced For organizers: March 27, 2015 For consideration of contributed papers in Special Ses- sions: To be announced Porto, Portugal For abstracts: To be announced University of Porto Seattle, Washington June 11–14, 2015 Washington State Convention Center and Thursday – Sunday the Sheraton Seattle Hotel First Joint International Meeting involving the American Mathematical Societry (AMS), the European Mathematical January 6–9, 2016 Society (EMS), and the Sociedade de Portuguesa Matematica Wednesday – Saturday (SPM). Joint Mathematics Meetings, including the 122nd Annual Associate secretary: Georgia Benkart Meeting of the AMS, 99th Annual Meeting of the Math- Announcement issue of Notices: To be announced ematical Association of America (MAA), annual meetings Program first available on AMS website: To be announced of the Association for Women in Mathematics (AWM) and Program issue of electronic Notices: To be announced the National Association of Mathematicians (NAM), and the winter meeting of the Association of Symbolic Logic (ASL), Issue of Abstracts: Not applicable with sessions contributed by the Society for Industrial and Deadlines Applied Mathematics (SIAM). Associate secretary: Michel L. Lapidus For organizers: To be announced Announcement issue of Notices: October 2015 For consideration of contributed papers in Special Ses- Program first available on AMS website: To be announced sions: To be announced Program issue of electronic Notices: January 2016 For abstracts: To be announced Issue of Abstracts: Volume 37, Issue 1

April 2013 Notices of the AMS 533 Meetings & Conferences

Deadlines For consideration of contributed papers in Special Ses- For organizers: April 1, 2015 sions: To be announced For consideration of contributed papers in Special Ses- For abstracts: To be announced sions: To be announced For abstracts: To be announced Baltimore, Maryland Baltimore Convention Center, Hilton Bal- Atlanta, Georgia timore, and Baltimore Marriott Inner Har- Hyatt Regency Atlanta and Marriott At- bor Hotel lanta Marquis January 16–19, 2019 January 4–7, 2017 Wednesday – Saturday Wednesday – Saturday Joint Mathematics Meetings, including the 125th Annual Joint Mathematics Meetings, including the 123rd Annual Meeting of the AMS, 102nd Annual Meeting of the Math- Meeting of the AMS, 100th Annual Meeting of the Math- ematical Association of America (MAA), annual meetings ematical Association of America, annual meetings of the of the Association for Women in Mathematics (AWM)and Association for Women in Mathematics (AWM) and the the National Association of Mathematicians (NAM), and the National Association of Mathematicians (NAM), and the winter meeting of the Association of Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and winter meeting of the Association of Symbolic Logic, with Applied Mathematics (SIAM). sessions contributed by the Society for Industrial and Ap- Associate secretary: Steven H. Weintraub plied Mathematics (SIAM). Announcement issue of Notices: October 2018 Associate secretary: Brian D. Boe Program first available on AMS website: To be announced Announcement issue of Notices: October 2016 Program issue of electronic Notices: To be announced Program first available on AMS website: To be announced Issue of Abstracts: To be announced Program issue of electronic Notices: January 2017 Issue of Abstracts: Volume 38, Issue 1 Deadlines Deadlines For organizers: April 2, 2018 For consideration of contributed papers in Special Ses- For organizers: April 1, 2016 sions: To be announced For consideration of contributed papers in Special Ses- For abstracts: To be announced sions: To be announced For abstracts: To be announced San Diego, California San Diego Convention Center and San Diego Marriott Hotel and Marina January 10–13, 2018 Wednesday – Saturday Joint Mathematics Meetings, including the 124th Annual Meeting of the AMS, 101st Annual Meeting of the Math- ematical Association of America (MAA), annual meetings of the Association for Women in Mathematics (AWM) and the National Association of Mathematicians (NAM), and the winter meeting of the Association of Symbolic Logic (ASL), with sessions contributed by the Society for Industrial and Applied Mathematics (SIAM). Associate secretary: Georgia Benkart Announcement issue of Notices: October 2017 Program first available on AMS website: To be announced Program issue of electronic Notices: To be announced Issue of Abstracts: To be announced

Deadlines For organizers: April 1, 2017

534 Notices of the AMS Volume 60, Number 4 Meetings and Conferences of the AMS

Associate Secretaries of the AMS Southeastern Section: Brian D. Boe, Department of Math- ematics, University of Georgia, 220 D W Brooks Drive, Athens, Central Section: Georgia Benkart, University of Wisconsin- GA 30602-7403, e-mail: [email protected]; telephone: Madison, Department of Mathematics, 480 Lincoln Drive, 706-542-2547. Madison, WI 53706-1388; e-mail: [email protected]; Western Section: Michel L. Lapidus, Department of Math- telephone: 608-263-4283. ematics, University of California, Surge Bldg., Riverside, CA Eastern Section: Steven H. Weintraub, Department of Math- 92521-0135; e-mail: [email protected]; telephone: ematics, Lehigh University, Bethlehem, PA 18105-3174; e-mail: 951-827-5910. [email protected]; telephone: 610-758-3717.

The Meetings and Conferences section of the Notices 2015 gives information on all AMS meetings and conferences January 10–13 San Antonio, Texas p. 532 approved by press time for this issue. Please refer to the page Annual Meeting numbers cited in the table of contents on this page for more April 18–19 Las Vegas, Nevada p. 533 detailed information on each event. Invited Speakers and June 11–14 Porto, Portugal p. 533 Special Sessions are listed as soon as they are approved October 3–4 Chicago, Illinois p. 533 by the cognizant program committee; the codes listed October 24–25 Fullerton, California p. 533 are needed for electronic abstract submission. For some meetings the list may be incomplete. Information in this 2016 issue may be dated. Up-to-date meeting and conference January 6–9 Seattle, Washington p. 533 information can be found at www.ams.org/meetings/. Annual Meeting

Meetings: 2017 2013 January 4–17 Atlanta, Georgia p. 534 April 6–7 Chestnut Hill, Massachusetts p. 524 Annual Meeting April 13–14 Boulder, Colorado p. 525 April 27–28 Ames, Iowa p. 526 2018 June 27–30 Alba Iulia, Romania p. 527 January 10–13 San Diego, California p. 534 October 5–6 Louisville, Kentucky p. 528 Annual Meeting October 12–13 Philadelphia, Pennsylvania p. 529 2019 October 18–20 St. Louis, Missouri p. 529 January 16–19 Baltimore, Maryland p. 534 November 2–3 Riverside, California p. 530 Annual Meeting

2014 Important Information Regarding AMS Meetings January 15–18 Baltimore, Maryland p. 530 Potential organizers, speakers, and hosts should refer to Annual Meeting page 274 in the the February 2013 issue of the Notices for gen- March 21–23 Knoxville, Tennessee p. 530 eral information regarding participation in AMS meetings and March 29–30 Baltimore, Maryland p. 531 conferences. April 5–6 Albuquerque, New Mexico p. 531 Abstracts April 11–13 Lubbock, Texas p. 531 Speakers should submit abstracts on the easy-to-use interac- June 16–19 Tel Aviv, Israel p. 531 tive Web form. No knowledge of is necessary to submit September 20–21 Eau Claire, Wisconsin p. 532 an electronic form, although those who use may submit October 18–19 Halifax, Canada p. 532 abstracts with such coding, and all math displays and simi- October 25–26 San Francisco, California p. 532 larily coded material (such as accent marks in text) must November 8–9 Greensboro, North Carolina p. 532 be typeset in . Visit http://www.ams.org/cgi-bin/ abstracts/abstract.pl. Questions about abstracts may be sent to [email protected]. Close attention should be paid to specified deadlines in this issue. Unfortunately, late abstracts cannot be accommodated.

Conferences in Cooperation with the AMS: (see http://www.ams.org/meetings/ for the most up-to-date infor- mation on these conferences.) July 22–26, 2013: Samuel Eilenberg Centenary Conference (E100), Warsaw, Poland.

April 2013 Notices of the AMS 535 NEW TITLES IN MATHEMATICS from CAMBRIDGE UNIVERSITY PRESS!

Manifold Mirrors High Accuracy Singularities of The Crossing Paths of the Computing Methods the Minimal Model Arts and Mathematics Fluid Flows and Wave Phenomena Program Felipe Cucker Tapan Sengupta János Kollár $90.00: Hardback: 978-0-521-42963-4: 432 pp. $140.00: Hardback: 978-1-107-02363-5: 620 pp. In collaboration with $29.99: Paperback: 978-0-521-72876-8 Sándor Kovács Mathematical Modelling A Course in Cambridge Tracts in Mathematics in One Dimension $85.00: Hardback: 978-1-107-03534-8: 380 pp. Mathematical Analysis An Introduction via Difference and Volume 1: Foundations and Differential Equations Elementary Real Analysis Systematic Jacek Banasiak D. J. H. Garling Program Design AIMS Library of Mathematical Sciences $125.00: Hardback: 978-1-107-03202-6: 304 pp. From Clarity to Efficiency $25.99: Paperback: 978-1-107-65468-6: 120 pp. $50.00: Paperback: 978-1-107-61418-5 Yanhong Annie Liu

Third Edition! $99.00: Hardback: 978-1-107-03660-4: 272 pp. Coherence in Three-Dimensional $39.99: Paperback: 978-1-107-61079-8 Category Theory Modern Computer Algebra Nick Gurski Joachim von zur Gathen and Torsors, Étale Homotopy Jürgen Gerhard Cambridge Tracts in Mathematics and Applications to $120.00: Hardback: 978-1-107-03903-2: 815 pp. $70.00: Hardback: 978-1-107-03489-1: 280 pp. Rational Points Non-Hausdorff Topology Edited by Alexei Skorobogatov Combinatorics of and Domain Theory London Mathematical Society Lecture Note Minuscule Representations Selected Topics in Point-Set Topology Series R. M. Green $65.00: Paperback: 978-1-107-61612-7: 470 pp. Jean Goubault-Larrecq Cambridge Tracts in Mathematics New Mathematical Monographs $80.00: Hardback: 978-1-107-02624-7: 328 pp. $99.00: Hardback: 978-1-107-03413-6: 504 pp. Prices subject to change.

www.cambridge.org/us/mathematics 800.872.7423 @cambUP_maths Journal of Journal of Journal of Applied Mathematics Calculus of Variations Complex Analysis

International Journal of Journal of Stochastic Analysis Chaos Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013

Journal of

Journal of Function Discrete Mathematics Spaces and Applications

Journal of

Complex Journal of Journal of Systems Industrial Mathematics Mathematics

Hindawi Publishing Corporation Hindawi Publishing Corporation Volume 2013 Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 http://www.hindawi.com http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013

Journal of Journal of Optimization Probability and Statistics

Journal of Operators Mathematical Problems ISRN in Engineering Algebra Hindawi Publishing Corporation Volume 2013 Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013

ISRN ISRN ISRN Applied ISRN ISRN Computational Discrete Mathematics Biomathematics Combinatorics Mathematics Mathematics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013

ISRN ISRN ISRN ISRN ISRN Mathematical Mathematical Operations Probability and Geometry Analysis Physics Research Statistics Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation Hindawi Publishing Corporation http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 http://www.hindawi.com Volume 2013 American Mathematical Society Distribution Center

35 Monticello Place, Pawtucket, RI 02861 USA 0002-9920(201304)60:4;1-U Neil Trudinger , by Alberto, by Facchini

, Tokyo, Japan and Jeddah, Saudi Tokyo, lshii, Hitoshi

(Algebra, Pure Mathematics) Mathematics) Pure (Algebra, Efim Zelmanov , Bielefeld, Germany and Jeddah, Germany Claus Michael Ringel, Bielefeld, (Analysis, Applied Mathematics), Applied Mathematics), (Analysis, Ari Laptev , Ohio, USA and Jeddah, Saudi Arabia S. K. Jain, Ohio, , Ohio, USA and Jeddah, Saudi Arabia Neil Robertson, Ohio, , Jeddah, Saudi Arabia, Nevai Paul Finite rank Bargmann-Toeplitz operators with non-compactly operators supported symbols, Bargmann-Toeplitz rank Finite Grigori Rozenblum by Arabia, Arabia, Saudi Arabia, (Differential Equations, Applied Mathematics), Applied Mathematics), Equations, (Differential Editors Executive USA, , San Diego, Efim Zelmanov Editors Associate Italy and Jeddah, Saudi Arabia, , Pavia, Brezzi Franco articlesForthcoming include: Schwarz Fritz , by equations f Loewy of linear differential decomposition Sole Cherif Mihoubi und Patrick 1/2, by f Optimal and isodual ternary of rate cyclic codes und Oleg Safronov Molchanov Stanislav , by spectra of elliptic operators f Negative f Direct-sum of modules with semilocal endomorphism rings decompositions Aims and Scope publish original will journal, open access Sciences, a peer-reviewed Bulletin of Mathematical The sciences. of mathematical in all branches interest of highest quality and of broad work research giving value expository articles Bulletin will publish well-written The (40–50 pages) of exceptional of the art and short state on a specific topic, the latest articles significant 10 pages) containing (about Most of the expository articles of wider interest. will be invited. results Board Editorial Mathematics), Pure S. K. Jain (Algebra, f Bulletin of Mathematical Sciences Sciences Bulletin of Mathematical Jeddah, Saudi Arabia University, King Abdulaziz Launched by