Recent Advances in the Langlands Program

TL;DR

This paper reviews recent progress in the Langlands program, covering proofs of the Langlands correspondence in various settings, including function fields, geometric cases, and quantization of the Hitchin system, highlighting key developments and proofs.

Contribution

It summarizes recent significant advances and proofs in the Langlands program across different mathematical contexts and groups.

Findings

Proof of Langlands correspondence for GL(n) over function fields by Drinfeld and Lafforgue

Development of geometric Langlands correspondence for GL(n) by Gaitsgory, Vilonen, and the author

Work on quantization of the Hitchin system and Langlands correspondence for semisimple groups

Abstract

These are the notes for the lecture given by the author at the "Current Events" Special Session of the AMS meeting in Baltimore on January 17, 2003. Topics reviewed include the Langlands correspondence for GL(n) in the function field case and its proof by V.Drinfeld and L.Lafforgue, the geometric Langlands correspondence for GL(n) and its proof by D.Gaitsgory, K.Vilonen and the author, and the work of A.Beilinson and V.Drinfled on the quantization of the Hitchin system and the Langlands correspondence for an arbitrary semisimple algebraic group.