The Langlands Correspondence

Introduction

R. Langlands conjectured the existence of a correspondence between automorphic spectrums of Hecke algebras and representations of Galois groups of global fields. The existence of such correspondence is one of the main conjectures in mathematics. Even if not known in full generality, it leads to proofs of Fermat and Sato-Tate conjectures.

Project Overview

This project is on three aspects of the Langlands correspondence.

First Aspect

The first part of this project is a description of the spectrum of Hecke algebras on the space generated by pseudo Eisenstein series of cuspidal automorphic forms of Levi subgroups. In the simplest non-trivial case, the precise description is a conjecture of Langlands.

This conjecture is proven in my work with A. Okounkov, by an unexpected topological interpretation. I expect this approach to work in a number of other cases.

Second Aspect

The second part of this project is an extension of the Langlands correspondence to a completely new area of fields of rational functions on curves over local fields.

This extension of the Langlands correspondence to a new area could lead to new interplays between Representation Theory and Number Theory.

Third Aspect

The third part of the project is on a categorification of the Langlands correspondence necessary for establishing the strong form of this correspondence.