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Recent advances in the Langlands Program


Speaker: Edward Frenkel, UC Berkeley
Location: 693 Kerr
Start time: Mon, Dec 9 2002, 4:10PM

What is the structure of the Galois group of the algebraic closure of the field of functions on an algebraic curve defined over a finite field? This question is at the heart of the Langlands Program, launched by Robert Langlands in the late 60's, which tied together seemingly unrelated objects in number theory, algebraic geometry, and the theory of automorphic functions. The Langlands conjecture (in the function field case) predicts that there is a correspondence between n-dimensional representations of the Galois group and the automorphic representations of the group GL(n) over the ring of adeles of the function field (for n=1 this correspondence was known from the abelian class field theory). This conjecture has been proved in the 80's by V. Drinfeld in the case when n=2 and recently by L. Lafforgue for an arbitrary n in a monumental effort for which both of them have been awarded the Fields Medals. In another development, A. Beilinson, V. Drinfeld, G. Laumon and others have reformulated the Langlands conjecture geometrically, for curves over an arbitrary ground field, such as the field of complex numbers. In the geometric Langlands conjecture automorphic representations are replaced by certain sheaves on the moduli space of rank n vector bundles on the curve.

There will be a dinner after the colloquium.