An analytic version of the Langlands correspondence for complex curves
Edward Frenkel (Berkeley)
Title: An analytic version of the Langlands correspondence for complex curves
Abstract: The Langlands correspondence for complex curves was traditionally formulated in terms of sheaves rather than functions. In 2018, Robert Langlands asked whether it is possible to construct a function-theoretic version. Together with Pavel Etingof and David Kazhdan, we have formulated a function-theoretic version as a spectral problem for an algebra of commuting operators acting on (a dense subspace of) the Hilbert space of half-densities on the moduli space of G-bundles over a complex algebraic curve. These operators include (self-adjoint extensions of) differential operators (both holomorphic and anti-holomorphic) as well as integral operators, which are analytic analogues of the Hecke operators. I will start with a brief introduction to both the sheaf-theoretic and function-theoretic versions and explain in what sense they complement each other. I will then present some of the results and conjectures from my joint work with Etingof and Kazhdan.
The talk will be independent from the talks given by the speaker in the past two weeks.
Recording taken from the Arithmetic Geometry and Quantum Field Theory seminar, organised by Jeff Harvey and Minhyong Kim