Recent advances in the Langlands Program

Abstract

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