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 Saharon Shelah 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 Soviet Union 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 Cornell University, 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, United Kingdom) 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: Yakov Sinai 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 George David Birkhoff, 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 Moscow State University. 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, Akshay Venkatesh, 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