Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain this perspective and illustrate its applications to representation theory following joint work with Nadler as well as Brochier, Gunningham, Jordan and Preygel.[-]

Kapustin and Witten introduced a powerful perspective on the geometric Langlands correspondence as an aspect of electric-magnetic duality in four dimensional gauge theory. While the familiar (de Rham) correspondence is best seen as a statement in conformal field theory, much of the structure can be seen in the simpler (Betti) setting of topological field theory using Lurie's proof of the Cobordism Hypothesis. In these lectures I will explain this perspective and illustrate its applications to representation theory following joint work with Nadler as well as Brochier, Gunningham, Jordan and Preygel.[-]

I will discuss the transfer of Harish-Chandra characters under the local theta correspondence, in particular in the (almost) equal rank case. More precisely, if $G X H$ is a dual pair in the equal rank setting, it is known that discrete series (resp. tempered) representations lifts to discrete series (resp. tempered) representations. If two such representations correspond under theta lifting, one can ask how their Harish-Chandra characters are related. I will define a space of test functions on each group and a correspondence of their orbital integrals induced by the Weil representation, and show that the resulting transfer of invariant distribution carries the character of a tempered representation to that of its theta lift. I will also explain how the transfer of test functions can be understood geometrically, by relating it to the moment map arising in theta correspondence.[-]

We will report on an ongoing project to understand geometric Langlands in genus one, in particular a version that depends only on the topology of the curve (as appears in physical descriptions of the subject). The emphasis will be on the realization of the automorphic and spectral categories as the center/cocenter of the affine Hecke category. We will mention work with D. Ben-Zvi and A. Preygel that accomplishes this on the spectral side, then focus on ongoing work with D. Ben-Zvi, building on work with P. Li, that we expect will lead to a parallel automorphic result.[-]