## Bernd Sturmfels

Bernd Sturmfels was appointed as a Clay Senior Scholar during July 2004 to participate in *Geometric Combinatorics* at PCMI.

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Bernd Sturmfels was appointed as a Clay Senior Scholar during July 2004 to participate in *Geometric Combinatorics* at PCMI.

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Richard Stanley was appointed as a Clay Senior Scholar during July 2004 to participate in *Geometric Combinatorics* at PCMI.

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Jim Carlson was President of the Clay Mathematics Institute 2003-2012.

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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.

Home — 2004

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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Akshay Venkatesh completed his undergraduate degree at the University of Western Australia, Perth, and received his PhD from Princeton University in 2002 under the direction of Peter Sarnak. His mathematical interests center around number theory and automorphic forms. He is particularly interested inequidistribution questions on homogeneous spaces, and the interplay between ergodic and spectral techniques. Akshay was appointed as a Clay Research Fellow for a term of two years beginning 2004.