Title: Some new perspectives in the Langlands program

Speaker: Professor Matthew Emerton (University of Chicago)

Date, Time, Location: Wednesday, November 20th @4:30PM in Math Hall 520

Abstract:

The goal of this talk is to explain some recent (so-called “categorical”) perspectives on the Langlands program. I will gently lead up to these new developments, beginning with background and examples aimed at introducing some of the ideas of the Langlands program to non-experts.

The underlying theme of the talk will be a common one in representation theory: if we have a commuting action of two groups (or algebras) on a vector space, then we can use the resulting decomposition into irreducible representations to (attempt to) induce a correspondence between irreducible representations of one group (or algebra) and the other. The Langlands program for the group GL_2 over the field Q of rational numbers (the “first non-abelian case” of the Langlands program) implements this idea by taking the vector spaces to be the first cohomology groups of modular curves (certain finite volume quotients of the complex upper half-plane). These curves have an inordinate amount of symmetry, as a result of which the cohomology gets a commuting action of three different objects: p-adic Lie groups, so-called Hecke algebras, and the absolute Galois group of Q. The rich interaction of these actions gives rise to a vast amount of number theory; starting with some key examples, I will pursue one aspect of this story, building up to a statement of the categorical Langlands correspondence for the p-adic Lie group GL_2(Q_p).

Print this page