Fall 2022 MATH 274 001 LEC

Topics in Algebra
Schedule: 
SectionDays/TimesLocationInstructorClass
001 LECTuTh 05:00PM - 06:29PMEvans 740Edward Frenkel31406
UnitsEnrollment StatusSession
4Open2022 Fall, August 24 - December 09
Additional Information: 

Prerequisites Math 261A, 261B. Consent of instructor for undergrads.

Description The goal of this course is to give an overview of the Langlands Program for complex curves, with the focus on its "analytic" version recently developed by P. Etingof, E. Frenkel, and D. Kazhdan.

The Langlands correspondence for complex curves has been traditionally formulated in terms of sheaves rather than functions. Etingof, Frenkel, and Kazhdan have recently formulated an analytic (or function-theoretic) version as a spectral problem for an algebra of commuting operators acting on half-densities on the moduli space of G-bundles over a complex algebraic curve. Moreover, they have conjectured (and proved in some cases) a description of the joint spectrum of this algebra in terms of the Langlands dual group of G. The course will start with an introduction to various forms of the Langlands correspondence. Then the analytic Langlands correspondence and related topics will be discussed. In particular, its interpretation in terms of a quantum integrable system obtained by "doubling" the celebrated quantum Hitchin system (also known as the Gaudin system in genus 0), i.e. combining the holomorphic and anti-holomorphic degrees of freedom.

recent talk by Frenkel provides a brief summary of what to expect in this course:

Recommended Reading  E. Frenkel, Survey of the Langlands Program

E. Frenkel, Is there an analytic theory of automorphic functions for complex algebraic curves?

P. Etingof, E. Frenkel, and D. Kazhdan, An analytic version of the Langlands correspondence for complex curves

P. Etingof, E. Frenkel, and D. Kazhdan, Hecke operators and analytic Langlands correspondence for curves over local fields

P. Etingof, E. Frenkel, and D. Kazhdan, Analytic Langlands correspondence for PGL(2) on P^1 with parabolic structures over local fields

Course Webpage will be on bCourses