Said Business School
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).
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|
|5:00||Andrew Wiles||Solving Equations|
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.
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.