The stable trace formula, automorphic forms, and Galois representations (08w5040)


(University of Toronto)

Michael Harris (Universite de Paris 7)

Eric Urban (Institut Mathématique de Jussieu)

Vinayak Vatsal (University of British Columbia)


The Langlands program, initiated by Robert Langlands in the 1960s, is a grand unification of two of the oldest branches of mathematics: number theory, the study of solutions of equations in whole numbers, enjoyed since antiquity for its purely aesthetic qualities but with surprising applications in the treatment and storage of information; and function theory on groups, which originated in the study of symmetry in classical physics. Langlands duality, as it is sometimes called, provides a way to interpret results in number theory in terms of function theory, and vice versa, providing a new perspective in each case that was otherwise unattainable. As a vision guiding research it has been one of the great success stories of mathematics in the past half century, and
versions of Langlands' duality principle have been influential in numerous other branches of mathematics, as well as in mathematical physics. In mathematics itself the most striking success of the Langlands program has been its use by Andrew Wiles in his proof of Fermat's Last Theorem, completed in joint work with Richard Taylor.

The workshop to be held at the Banff International Research Station on August 17 - 22, 2008 aims to present recent progress, largely due to James Arthur, in Canada, and to Gerard Laumon, Ngo Bao-Chau, Jean-Loup Waldspurger, and Jean-Pierre Labesse, in France, toward the development of a stable trace formula, a crucial tool for confirming the predictions of the Langlands program. We especially hope to report on applications to number theory of the sort developed by Wiles in the course of his work on Fermat's Last Theorem.

The Banff International Research Station for Mathematical Innovation and Discovery (BIRS) is a collaborative Canada-US-Mexico venture that provides an environment for creative interaction as well as the exchange of ideas, knowledge, and methods within the Mathematical Sciences, with related disciplines and with industry. The research station is located at The Banff Centre in Alberta and is supported by Canada's Natural Science and Engineering Research Council (NSERC), the US National Science Foundation (NSF), Alberta's Advanced Education and Technology, and Mexico's Consejo Nacional de Ciencia y Tecnologí­a (CONACYT).