Stanislav Smirnov
The 2001 Clay Research Award was made to Stanislav Smirnov for establishing the existence of the scaling limit of two-dimensional percolation, and for verifying John Cardy’s conjectured relation.
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The 2001 Clay Research Award was made to Stanislav Smirnov for establishing the existence of the scaling limit of two-dimensional percolation, and for verifying John Cardy’s conjectured relation.
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The 2000 Clay Research Award was made to Laurent Lafforgue for his work on the Langlands program.
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The 2000 Clay Research Award was made to Alain Connes for revolutionizing the field of operator alebras, for inventing modern non-commutative geometry, and for discovering that these ideas appear everywhere, including the foundations of theoretical physics.
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The 2002 Clay Research Award was made to Oded Schramm for his work in combining analytic power with geometric insights in the field of random walks, pecolation, and probability theory in general, especially for formulating stochastic Loewner evolution. His work opens new doors and reinvigorates research in these fields.
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The 2002 Clay Research Award was made to Manindra Agrawal for his work on primality testing; for finding, jointly with two undergraduate students, an algorithm that solves a modern version of a problem going back to the ancient Chinese and Greeks about how one can determine whether a number is prime in a time that increases polynomially with the size of the number; presented on October 30, 2002 in Cambridge, MA.
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The 2005 Clay Research Award was made to Nils Dencker for his complete resolution of a conjecture made by F. Trèves and L. Nirenberg in 1970. This conjecture posits an essentially geometric necessary and sufficient condition, Psi, for a pseudo-differential operator of principal type to be locally solvable, i.e., for the equation Pu = ƒ to have local solutions given a finite number of conditions on ƒ.
Dencker’s work provides a full mathematical understanding of the surprising discovery by Hans Lewy in 1957 that there exists a linear partial differential operator — a one-term, third-order perturbation of the Cauchy-Riemann operator — which is not locally solvable in this sense. The necessity of condition “Psi” was shown for operators in dimension 2 by R. Moyer in 1978 and in general by L. Hormander in 1981. The sufficiency of the condition has resisted many previous attacks.
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The 2005 Clay Research Award was made to Manjul Bhargava for his discovery of new composition laws for quadratic forms and for his work on the average size of ideal class groups.
The field of composition laws had lain dormant for 200 years since the pioneering work of C.F Gauss. The laws discovered by Bhargava were a complete surprise, and led him to another major breakthrough, namely, counting the number of quartic and quintic number fields with given discriminant. The ideal class group is an object of fundamental importance in number theory. Nonetheless, despite some conjectures of Cohen and Lenstra about this problem, there was not a single proven case before Bhargava’s work. Bhargava solved the problem for the 2-part of the class groups of cubic fields, in which case, curiously, the numerical evidence had led people to doubt the Cohen-Lenstra heuristics.
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The 2003 Clay Research Award was made to Terry Tao for his groundbreaking work in analysis, notably his optimal restriction theorems in Fourier analysis, his work on the wave map equations, his global existence theorems for KdV type equations, as well as for significant work in quite distant areas of mathematics, such as his solution with Allen Knutson of Horn’s conjecture, a fundamental problem about hermitian matrices that goes back to questions posed by Hermann Weyl in 1912.
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The 2004 Clay Research Award was made to Gérard Laumon and Ngô Bao Châu for their proof of the Fundamental Lemma for unitary groups.
The lemma is a conjectured identity between orbital integrals for two groups, e.g., the unitary groups U(n) and U(p)xU(q), where p+q = n. Combined with the Arthur-Selberg trace formula, it enables one to prove relations between automorphic forms on different groups and is a key step towards proving links between certain automorphic forms and Galois representations. This is one of the aims of the Langlands program, which seeks a far-reaching unification of ideas in number theory and representation theory. The result of Laumon and Ngô uses the equivariant cohomology approach introduced by Goresky, Kottwitz, and MacPherson, who proved the lemma in the split and equal valuation case. The proof for the unitary case, which is significant for applications, requires many new ideas, including Laumon’s deformation strategy and Ngô’s purity result which is based on a geometric interpretation of the endoscopy theory of Langlands and Kottwitz in terms of the Hitchin fibration.
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The 2004 Clay Research Award was made to Gérard Laumon and Ngô Bao Châu for their proof of the Fundamental Lemma for unitary groups.
The lemma is a conjectured identity between orbital integrals for two groups, e.g., the unitary groups U(n) and U(p)xU(q), where p+q = n. Combined with the Arthur-Selberg trace formula, it enables one to prove relations between automorphic forms on different groups and is a key step towards proving links between certain automorphic forms and Galois representations. This is one of the aims of the Langlands program, which seeks a far-reaching unification of ideas in number theory and representation theory. The result of Laumon and Ngô uses the equivariant cohomology approach introduced by Goresky, Kottwitz, and MacPherson, who proved the lemma in the split and equal valuation case. The proof for the unitary case, which is significant for applications, requires many new ideas, including Laumon’s deformation strategy and Ngô’s purity result which is based on a geometric interpretation of the endoscopy theory of Langlands and Kottwitz in terms of the Hitchin fibration.
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The 2004 Clay Research Award was made to Ben Green for his joint work with Terry Tao on arithmetic progressions of prime numbers.
These are equally spaced sequences of primes such as 31, 37, 43 or 13, 43, 73, 103. Results in the area go back to the work of Lagrange and Waring in the 1770’s. A major breakthrough came in 1939 when the Dutch mathematician Johannes van der Corput showed that there are an infinite number of three-term arithmetic progressions of primes. Green and Tao showed that for any n, there are infinitely many n-term progressions of primes. Their proof, which relies on results of Szemerédi (1975) and Goldston and Yildirim (2003), uses ideas from combinatorics, ergodic theory, and the theory of pseudorandom numbers. The Green-Tao result is a major advance in our understanding of the primes.
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The 2003 Clay Research Award was made to Richard Hamilton for his discovery of the Ricci Flow Equation and its development into one of the most powerful tools of geometric analysis. Hamilton conceived of his work as a way to approach both the Poincaré Conjecture and the Thurston Geometrization Conjecture.