2005 Meeting

October 11, 2005

Said Business School
Oxford University

The 2005 Annual Meeting of the Clay Mathematics Institute featured presentation of the Clay Research Awards and a special public lecture, "Solving Equations," by Professor Andrew Wiles. This year's meeting was held at Oxford University following "Euclid and His Heritage," a public conference celebrating the first digital edition of the oldest extant manuscript of the Elements (888 AD).

Program

All lectures are 50 minutes in duration, followed by a 10 minute question and answer period.

Tuesday, October 11
2:00 Jim Carlson Opening Remarks
2:10 Jim Carlson Presentation of the Clay Research Awards
2:30 Manjul Bhargava Higher composition laws and applications
3:30 Nicolas Lerner Presentation of Nils Dencker's Work on the Resolution of the Nirenberg-Treves Conjecture
4:20 Tea break
5:00 Andrew Wiles Solving Equations
6:00 Reception

Presentation of Nils Dencker's Work on the Resolution of the Nirenberg-Treves Conjecture

We start by describing the content of the 1970 Nirenberg-Treves conjecture, and then we review part of the history and developments of local solvability questions. Next we describe the main ideas used by Nils Dencker, stressing the new geometric vision and ideas that he brought forward for this problem. This talk is devised for a general mathematical audience, with no particular expertise on solvability problems.

Higher composition laws and applications

In 1801, Gauss laid down the remarkable law of composition of binary quadratic forms which would play such a critical role in number theory in the decades to follow. This law of composition still remains today one of the primary tools for understanding and computing with the class groups of quadratic fields. It is thus only natural to ask whether higher analogues of this composition law might exist that could shed light on the structure of other algebraic number rings and fields. In this talk, we show that Gauss's law of composition is only one of at least twenty composition laws of its kind that can be used to yield information on number rings and their class groups. We also discuss various applications of these new composition laws to Lie theory, to the theory of prehomogeneous vector spaces, and to analytic number theory.

"Solving Equations," a public lecture by Andrew Wiles

Professor Wiles will survey some of the recent progress in solving classical equations, notably the Fermat equation and the equations representing elliptic curves. He will consider the problem of finding rational solutions as well as the problem of finding solutions by repeated extraction of roots.

poster

Poster images: 128 kb, 4.5 mb. You are welcome to download and print the poster.

Manjul Bhargava

Manjul Bhargava was born in Hamilton, Ontario, Canada but spent most of his early years in Long Island, New York. He received his A.B. in Mathematics summa cum laude from Harvard University in 1996 and his Ph.D. from Princeton University under the advisorship of Andrew Wiles in 2001. After brief visiting positions at the Institute for Advanced Study and Harvard University, he joined the faculty of Princeton University as Professor of Mathematics in 2003. An accomplished tabla player whose primary research interests lie in number theory, representation theory, and algebraic geometry, Bhargava has received numerous awards and honors, including three Derek Bok Awards for Excellence in Teaching (1993-95), the Hoopes Prize for Excellence in Scholarly Work and Research from Harvard University (1996), the AMS-MAA-SIAM Morgan Prize for Outstanding Undergraduate Research in Mathematics (1997), the MAA Merten M. Hasse Prize for Exposition (2003), a Packard Foundation Fellowship in Science and Engineering (2004), and the AMS Blumenthal Award for the Advancement of Pure Mathematics (2005). Bhargava was also the Clay Mathematics Institute's first five-year Research Fellow (2000-05). He was named one of Popular Science magazine's "Brilliant 10" in 2002.

Nils Dencker

Nils Dencker was born 1953 in Lund, Sweden. He received his Ph.D. from Lund University in 1981 under the direction of Lars Hormander. After spending 1981-1983 as a C.L.E. Moore instructor at MIT, he returned to Lund University. He has been the Director of Studies at the Department of Mathematics 2001-2003. Dencker received in 2003 the Gaarding prize from the Royal Physiographic Society in Lund. He is currently the vice chairman of the Swedish Mathematical Society.

Dencker's research interests lie in the microlocal analysis of partial differential equations and the calculus of pseudodifferential operators. He has studied the propagation of polarization (vector valued singularities) for systems of partial differential equations, for example in double refraction. Dencker has also studied the pseudospectra (spectral instability) of semi-classical partial differential equations and the solvability of partial differential equations.

Andrew Wiles

Princeton University Professor Andrew Wiles is one of the world's preeminent number theorists celebrated worldwide for his proof of Fermat's Last Theorem. He has received numerous awards and honors including the Schock Prize from the Swedish Academy in 1995, and the Wolf Prize from the Israeli Wolf Foundation in 1996. He was elected a Fellow of the Royal Society in 1989, and knighted by the Queen of England in 2000.