Abstract: What is generally referred to as Geometric Langlands is a certain paradigm of problems in algebraic geometry that are motivated by the classical Langlands program in number theory.

Let X be an algebraic curve over complex numbers, i.e., a compact Riemann surface. The primary object of study is the moduli space of vector bundles (of a given rank n) on X, denoted Bun_n(X). In the talk I will explain in what sense this algebro-geometric object is an analog of the automorphic quotient considered by number theorists. Now, the geometric analog of the space of automorphic functions is the category D(Bun_n(X)), the derived category of D-modules on X (this notion will also be introduced).

The category D(Bun_n(X)) acquires an action of a large commuting family of operators (known as Hecke operators), and the basic questions that one asks is to find a spectral decomposition of this action. In the talk we'll argue that the eigenvalues for this action are parameterized by another natural geometric object--namely the moduli space of n-dimensional local systems on X.

We'll also explain in what way this is analogous to the Langlands parameterization of automorphic forms by Galois representations.

Coffee will be served at 12:00 before the lecture
at Schreiber building 006