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2003 Birkhoff Prize 2003 Birkhoff Prize The 2003 George David Birkhoff Prize in Applied this is very difficult to check directly, Mather proved Mathematics was awarded at the 109th Annual that infinitesimal stability, a condition that can Meeting of the AMS in Baltimore in January 2003. often be verified constructively, implies stability, The Birkhoff Prize recognizes outstanding con- and he developed an algorithm for describing tributions to applied mathematics in the highest the local forms of these stable mappings. These and broadest sense and is awarded every three astonishing generalizations of the earlier work of years (until 2001 it was awarded usually every five Hassler Whitney have provided approaches to years). Established in 1967, the prize was endowed understand a variety of applied issues ranging by the family of George David Birkhoff (1884–1944), from the structure of the Pareto set of the utility who served as AMS president during 1925–26. The mapping in economics to phase transitions in prize is given jointly by the AMS and the Society physics. for Industrial and Applied Mathematics (SIAM). Switching to the theory of dynamical systems, The recipient must be a member of one of these Mather has made several major contributions. An societies and a resident of the United States, early highlight was his result with Richard McGehee Canada, or Mexico. The prize carries a cash award proving that binary collisions in the Newtonian of $5,000. 4-body problem could accumulate in a manner that The recipients of the Birkhoff Prize are chosen would force the system to expand to infinity in by a joint AMS-SIAM selection committee. For the finite time. He was a co-founder of Aubry-Mather 2003 prize the members of the selection commit- theory where, in particular, he proved that twist tee were: Douglas N. Arnold, Paul H. Rabinowitz, maps of an annulus possess so-called Aubry-Mather and Donald G. Saari (chair). invariant sets for any irrational rotation number. Previous recipients of the Birkhoff Prize are These sets are Cantor sets, and the diffeomorphism Jürgen K. Moser (1968), Fritz John (1973), James B. on them is equivalent to a rigid rotation of a circle. Serrin (1973), Garrett Birkhoff (1978), Mark Kac Since KAM theory, which extends research going (1978), Clifford A. Truesdell (1978), Paul R. back to the work of Birkhoff, provides informa- Garabedian (1983), Elliott H. Lieb (1988), Ivo Babusˇka tion about such situations when the rotation (1994), S. R. S. Varadhan (1994), and Paul H. number is Diophantine, Mather found the missing Rabinowitz (1998). circles in KAM theory. The 2003 Birkhoff Prize was awarded to JOHN Mather extended this work to multidimensional MATHER and to CHARLES S. PESKIN. The text that follows positive definite Lagrangian systems. He proved the presents the selection committee’s citation, a brief invariant sets he found here—called Mather sets— biographical sketch, and the awardee’s response are Lipschitz graphs over configuration space. He upon receiving the prize. also developed a variational method for con- structing shadowing trajectories first for twist John Mather maps and then for positive definite Lagrangian Citation systems. In the twist map setting, he established John Mather is a mathematician of exceptional the existence of heteroclinic orbits joining Aubry- depth, power, and originality. Mather sets in the same Birkhoff instability region. His earliest work included contributions to Currently he is doing seminal work on Arnold dif- foliation theory in topology and to the theory of fusion. In particular Mather proved the existence singularities for smooth and analytic maps on Rn of Arnold diffusion for a generic perturbation of an where he provided the rigorous foundations of a priori unstable integrable Hamiltonian system, this theory. Among his main contributions is a sta- solving the problem left standing from Arnold’s bility result. Here stability of a map means that any famous 1964 paper. nearby map is equivalent to it up to diffeomor- Mather is a member of the U.S. and Brazilian phisms of the domain and target manifolds. While National Academies of Sciences, a Guggenheim and 470 NOTICES OF THE AMS VOLUME 50, NUMBER 4 Sloan Fellow, and the winner of the 1978 John J. Charles S. Peskin Carty Medal from the U.S. Academy. Citation Biographical Sketch Charles Samuel Peskin has de- John N. Mather was born in Los Angeles, California, voted much of his career to on June 9, 1942. He received a B.A. from Harvard understanding the dynamics University in 1964 and a Ph.D. from Princeton of the human heart. Blurring University in 1967. From 1967 to 1969 he was pro- disciplinary boundaries, he fesseur associé (visiting professor) at the Institut des has brought an extraordinarily Hautes Études Scientifiques (IHÉS) in France. In broad range of expertise to 1969 he joined the faculty of Harvard University as bear on this problem: mathe- associate professor and was promoted to profes- matical modeling, differential sor in 1971. He was a visiting professor at Princeton equations, numerical analysis, high performance computing, University in 1974–75 and joined the faculty of fluid dynamics, physiology, Princeton University as professor in 1975. He was neuroscience, physics, and en- a visiting professor at IHÉS in 1982–83 and at the gineering. His primary tool for Eidgenössische Technische Hochschule in Zurich in understanding the heart is 1989–90. computer simulation. In work John Mather Mather was an editor of the Annals of Mathe- spanning more than two matics from 1990 to 2001 and has been an editor decades, much of it with David of the Annals of Math. Studies from 1990 to the McQueen, Peskin has devel- present. oped a computer model that Mather was a Sloan Fellow in 1970–72 and a simulates blood circulation Guggenheim Fellow in 1989–90. He was elected a through the four chambers of member of the National Academy of Sciences in the heart and in and out of the 1988 and a member of the Brazilian Academy of surrounding circulatory sys- Sciences in 2000. He received the John J. Carty tem along with the deforma- Medal of the National Academy of Sciences in 1978 tion of the cardiac muscle and and the Ordem Nacional do Mérito Científico from the valves. This virtual heart enables experimentation in sil- the Brazilian Academy of Sciences in 2000. ico that would be impossible Mather’s current research is in the area of in vivo and is of tremendous Hamiltonian dynamics. In the past he has worked value to the study of normal in the theories of singularities of mappings and heart function and a variety foliations. of pathologies, to plan inter- Response ventions, and to design pros- It is a pleasure to accept the Birkhoff Prize for my thetic devices. work in singularities of mappings, the theory of fo- Peskin’s computer simula- liations, and Hamiltonian dynamics. I greatly ap- tions are based on the im- Charles S. Peskin preciate the generous citation of my achievements, mersed boundary method, a as well as the honor of the prize. While I have not unique numerical method he (yet) worked on applications of mathematics as developed for the solution of dynamic fluid-struc- such, I have always been fascinated by theoretical ture interactions. This method, which is built on a mathematical questions that originated in appli- novel approach to couple a fluid description in Eulerian coordinates to a solid description in La- cations, for example, the n-body problem in New- grangian coordinates, was originally designed to de- tonian mechanics. Poincaré showed long ago that scribe the flow of blood around cardiac valve sur- the study of the dynamics of area-preserving map- faces. But it has found much wider use, allowing pings of surfaces provides important insights into simulation of a variety of complex systems, such this problem. G. D. Birkhoff greatly extended Poin- as the inner ear, swimming fish, locomoting mi- caré’s work on area-preserving mappings, and his crobes, flowing suspensions, and filaments flapping work was one of the inspirations for my contribu- in soap films. The development and analysis of the tion to Aubry-Mather theory. immersed boundary method is an ongoing and ac- I am grateful to my teachers at Harvard University tive field of study. and Princeton University, as well as colleagues and While the heart is a large biological motor, much friends, from whom I have learned so much. I also of Peskin’s recent research concerns biological wish to express my appreciation for the system of motors at the smallest scales. Here too he brings higher education, which makes a career of mathe- innovative mathematical modeling and computa- matical research possible. tional simulation to bear, exploring and explaining APRIL 2003 NOTICES OF THE AMS 471 the microscopic machinery inside cells which Peskin’s other honors are the MacArthur Fellow- harness Brownian motion for transport and motility. ship (1983–88), SIAM Prize in Numerical Analysis A former MacArthur fellow, Charles Peskin is a and Scientific Computing (1986), Gibbs Lecturer member of the American Academy of Arts and (1993), Cray Research Information Technology Sciences, the National Academy of Sciences, and the Leadership Award (joint with David M. McQueen, Institute of Medicine. 1994), Sidney Fernbach Award (1994), Mayor’s Award Biographical Sketch for Excellence in Science and Technology (1994), and Charles S. Peskin was born in New York City in 1946. von Neumann Lecturer (1999). He is a fellow of His mathematical education began at the Ethical Cul- the American Institute for Medical and Biological ture School, where arithmetic was done with sticks Engineering (since 1992), fellow of the American tied together, when possible, in bundles of ten to Academy of Arts and Sciences (since 1994), mem- explain the decimal system.
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