Langlands Program, Trace Formulas, and their Geometrization

TL;DR

This paper surveys the Langlands Program, focusing on trace formulas and their geometrization, highlighting recent work on proving the Functoriality Conjecture and categorification of the Langlands correspondence.

Contribution

It introduces a new approach to proving the Functoriality Conjecture using trace formulas and explores the geometrization and categorification aspects of the Langlands Program.

Findings

New approach to Functoriality Conjecture using trace formulas

Geometrization of trace formulas and their connection to categorification

Survey of recent joint work with Langlands and Ngo Bao Chau

Abstract

The Langlands Program relates Galois representations and automorphic representations of reductive algebraic groups. The trace formula is a powerful tool in the study of this connection and the Langlands Functoriality Conjecture. After giving an introduction to the Langlands Program and its geometric version, which applies to curves over finite fields and over the complex field, I give a survey of my recent joint work with Robert Langlands and Ngo Bao Chau (arXiv:1003.4578 and arXiv:1004.5323) on a new approach to proving the Functoriality Conjecture using the trace formulas, and on the geometrization of the trace formulas. In particular, I discuss the connection of the latter to the categorification of the Langlands correspondence.