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The Fields Medalists

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The Fields Medalists August 19, 2010 THE FIELDS MEDALISTS Elon Lindenstrauss Ngô Bảo Châu Stanislav Smirnov Cédric Villani Citation: “For the proof of con- Citation: “For his proofs of Citation: “For his results on Citation: “For his proof of the formal invari- n o n l i n e a r m e a s u r e Fundamental ance of L a n d a u rigidity in er- Lemma in percolation damping and godic theory, the theory of and the pla- convergence and their ap- automorphic nar Ising to equilibrium plications to f o r m s model in sta- for the Boltz- number the- through the t i s t i c a l mann equa- ory.” introduction physics.” tion.” Lindenstrauss has made far- of new algebro-geometric reaching advances in ergodic methods.” theory, the study of measure It was predicted in the 1990s, One of the fundamental and and used in many studies, that initially very controversial the- preserving transformations. His In the 1960's and 70's Robert the scaling limit of various two ories of classical physics is work on a conjecture of Langlands formulated various dimensional models in statisti- Boltzmann's kinetic theory of Furstenberg and Margulis con- basic unifying principles and cal physics has an unexpected gases. Instead of tracking the cerning the measure rigidity of conjectures relating automor- symmetry, namely it is confor- individual motion of billions of higher rank diagonal actions in phic forms on different groups, mally invariant. Smirnov was individual atoms it studies the homogeneous spaces has led Galois representations and L- the first to prove this rigorously evolution of the probability that to striking applications. Specifi- functions. These led to what for two important cases: perco- a particle occupies a certain cally, jointly with Einsiedler and today is referred to as the lation on the triangular lattice position and has a certain ve- Katok, he established the con- Langlands programme. The and the planar Ising model. The locity. The equilibrium proba- jecture under a further hypoth- main tool in establishing some proof is elegant and it is based bility distributions are well esis of positive entropy. It has cases of these conjectures is on extremely insightful combi- known for more than a hun- impressive applications to the the trace formula and in apply- natorial arguments. Smirnov's dred years, but to understand classical Littlewood Conjecture ing it for the above purposes a work gave the solid foundation whether and how fast conver- in the theory of diophantine ap- central difficulty intervenes: to for important methods in statis- gence to equilibrium occurs proximation. Developing these establish some natural identi- tical physics like Cardy's For- has been very difficult. Villani as well other powerful ergodic ties in harmonic analysis on mula, and provided an (in collaboration with Desvil- theoretic and arithmetical local groups as well as ones all-important missing step in the lettes) obtained the first result ideas, Lindenstrauss resolved connected to arithmetic geo- theory of Schramm-Loewner on the convergence rate for the arithmetic quantum unique metric objects. This problem Evolution in the scaling limit of initial data not close to equilib- ergodicity conjecture of Rud- became known as the Funda- various processes. rium. Later in joint work with nick and Sarnak in the theory of mental Lemma. After many his student Mouhut he rigor- modular forms. His work is ex- advances by a number of re- Professor, University of ously established the so-called ceptionally deep and its impact searchers in 2004, Laumon Geneva. Born in 1970 in St. Pe- non-linear Landau damping for goes far beyond ergodic theory. and Ngô established the Fun- tersburg, Russia. He studied the kinetic equations of damental Lemma for a special mathematical analysis with Vik- plasma physics, settling a Professor, Hebrew University, family of groups, and recently tor Havin at St. Petersburg long-standing debate. He has 2008- , Professor, Princeton Ngô established the Lemma in University, USA (2004-2010); State University. After graduat- been one of the pioneers in general. Born in Jerusalem, 1970. B. Sc ing in 1992 he moved to the the applications of optimal (Mathematics and Physics), Caltech where he received his transport theory to geometric Ngô's brilliant proof of this im- The Hebrew University, Ph. D. in 1996 under Nikolai and functional inequalities. He portant long standing conjec- Jerusalem, 1991; M. Sc. Math- Makarov. After short stints at wrote a very timely and accu- ematics, The Hebrew Univer- ture is based in part on the the Institute of Advanced Study, rate book on mass transport. sity, 1995; Ph.D. in introduction of novel geometric Mathematics, The Hebrew Uni- objects and techniques into Princeton and the MPIM, Bonn, versity, 1999. this sophisticated analysis. His Smirnov spent an important Born in 1973 in France. After achievement, which lies at the part of his career in Stockholm, studying mathematics at École Awards: The Anna and Lajos crossroads between algebraic where he came in 1998. Be- Normale Supérieure in Paris Erds Prize in Mathematics geometry, group theory and came professor at the Royal In- (1992-96), he became assis- 2009; Michael Bruno Memorial automorphic forms, is leading stitute of Technology and tant professor there. He re- Award (given by the Rothschild to many striking advances in researcher at the Swedish ceived his PhD in 1998. In “Yad Hanadiv” Foundation) Royal Academy of Sciences in 2000 he became a full profes- 2008; European Mathematical the Langlands programme as 2001. sor at École Normale Society Prize 2004; Salem well as subjects linked with it. Prize 2003; Clay Mathematical Awards: St. Petersburg Mathe- Supérieure de Lyon. In 2009 Institute Long Term Prize Fel- Professor,the Faculté des Sci- matical Society Prize (1997), he was appointed director of low 2003-2005; Leonard M. ences at Orsay. Born 1972 in Clay Research Award (2001), the Institut Henri Poincaré in and Eleanor B. Blumenthal Hanoi. After secondary school, Gran Gustafsson Research Paris, and part-time visitor of Award for the Advancement of he moved to France. He did Prize (2001), Rollo Davidson the Institut des Hautes Études Research in Pure Mathematics his PhD in Orsay under the su- Prize (2002), EMS Prize Scientifiques. 2001. pervision of Gérard Laumon. (2004). REFLEXIONS August 19, Thursday The Rolf Nevanlinna Prize The Gauss Prize The Chern Medal Award Citation: “For smoothed analysis of Linear Citation: “For fundamental contributions to Citation: “For his role in the formulation of Programming, algo- number theory, oper- the modern theory of rithms for graph- ator theory and har- non-linear elliptic based codes and monic analysis, and partial differential applications of graph his pivotal role in the equations and for theory to Numerical development of mentoring numerous Computing.” wavelets and mul- students and post- tiresolution analysis”. docs in this area”. Linear Programming is one of the most Meyer has made Nirenberg is one of useful tools in ap- fundamental contri- the outstanding ana- plied mathematics. The oldest algorithm bution to a number of mathematical areas. lysts and geometers of the 20th Century for Linear Programming, the Simplex Around 1970, he developed the theory of and his work has had a major influence in Method, works very well in practice, but model sets in number theory, which has the development of several areas of math- mathematicians have been perplexed become an important tool in the mathemat- ematics and their applications. He has about this efficacy and have tried for long ical study of quasicrystals -- space-filling made fundamental contributions to the un- to establish this as a mathematical theo- structures that are ordered but lack trans- derstanding of linear and non-linear partial rem. Spielman and his co-author Shenhua lational symmetry -- and aperiodic order in differential equations (PDEs) and related Teng developed a beautiful method and general. Together with Ronald Coifman aspects of complex analysis and geometry, proved that, while there may be patholog- and Alan MacIntosh he proved the continu- the basic mathematical tools of modern sci- ical examples where the method fails, ity of the Cauchy integral operator on all ence. He developed intricate connections slight modifications of any pathological ex- Lipschitz curves, a long-standing problem between analysis and differential geometry ample yields a “smooth” problem on which in analysis. and applied them to the theory of fluid flow the Simplex method works very well. and other physical phenomena. Meyer played a leading role in the modern A second major contribution of Spielman development of wavelet theory, which has Nirenberg’s name is associated with sev- is in the area of coding. Much of the pres- had a spectacular impact in information eral major developments in analysis in the ent-day communication uses coding, ei- sciences, statistics and technology. Fourier last 65 years. His theorem with August ther for preserving secrecy or for ensuring analysis is a universal tool in applied math- Newlander on the existence of almost com- error correction. An important technique to ematics, and due in a large measure to plex structures has become a classic. One make both coding and decoding efficient Meyer’s work, wavelet theory has become of the most widely quoted results in analy- is based on extremely well-connected the new name for Fourier analysis. He con- sis is that a priori estimates for general lin- graphs called expanders. Spielman and structed the first non-trivial wavelet bases ear elliptic systems, which he obtained with his co-authors have done foundational and wavepackets that dramatically ex- Shmuel Agmon and Avron Douglis. His fun- work on such codes and have designed tended the expressing power of wavelets. damental work with Fritz John on functions very efficient methods for coding and de- This led to many applications in practice – of bounded mean oscillation was crucial for coding.
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