School of Mathematical Sciences
Monday, November 22, 2010
Schreiber 006, 12:15
A review of categorical Geometric Langlands
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