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Abstract: The metaplectic geometric Langlands program can be understood in two ways: as a twisted analogue of the geometric Langlands program, and as a geometric analogue of the Langlands program for covering groups, namely the metaplectic Langlands program. One of the key structures in the geometric Langlands program is the spectral action, which roughly says that the spectral side acts on the automorphic side, allowing one to spectrally decompose the automorphic category over the moduli of \check{G}-local systems. This can be viewed as a geometric incarnation of the automorphic-to-Galois direction.

One may hope that the spectral action carries over to the metaplectic setting, and this expectation was formulated by Gaitsgory and Lysenko under the name “metaplectic vanishing conjecture.” In this talk, I will discuss my recent work on constructing the spectral action in the Betti metaplectic context. I will focus on what changes in the metaplectic setting and on the categorical framework for studying factorizable actions of local systems of symmetric monoidal categories.

Abstract: Classical Harish-Chandra bimodules are related to many objects in representation theory such as infinite dimensional representations of complex groups viewed as real groups, category O and character sheaves. The notion of Harish-Chandra bimodules for quantum groups was introduced in the works about topological quantum field theory of surfaces. I will mention some results about quantum Harish-Chandra bimodules at odd order roots of unity, connect them to affine Soergel bimodules and coherent sheaves on Steinberg variety. 

Abstract: In joint work with Gurbir Dhillon, we prove a universal monodromic extension of Bezrukavnikov's two realizations of the affine Hecke category over the integers. I will discuss some ingredients that go into to the proof of the Arkhipov-Bezrukavnikov equivalence in this setting. These include analysis of multi-graded rings, deformation between characteristic zero and positive characteristic, and the tilting property of Whittaker averaged central sheaves. The proof of the latter is based on ideas of Faergeman-Raskin and Bezrukavnikov-Morton-Ferguson. 

Abstract: I will report on an incomplete project, whose goal is to geometrically realize representations of critical level affine Lie algebras at all central characters. To do so, we study these categories of representations from the point of view of the local geometric Langlands program. As an accidental consequence, we will prove some new results in the local Langlands program, both in geometry and in arithmetic. No knowledge of affine Lie algebras or of Langlands will be assumed. Some parts are joint work with a number of other mathematicians, including Dhillon, Eteve, Gaitsgory, Raskin and Varshavsky 

Abstract: I'll explain a tweak to Clausen and Scholze's theory of solid modules that is better-adapted to applications in geometry, as opposed to number theory. This theory replaces point-set topology when dealing with objects like the field of formal Laurent series. For instance, it can be used to streamline deformation theory in infinite type. 

Abstract: Fundamental Local Equivalence of Gaitsgory and Lurie is an instance of quantum local geometric Langlands correspondence. It asserts an equivalence between the affine category O and the category of twisted Whittaker D-modules on the affine flag variety for the dual group. A version of this conjecture was proved by Campbell, Dhillon, Raskin. In this talk I will explain our work in progress with Dhillon where we study the quasi-classical version of this equivalence: instead of modules over quantizations the categories we care about are those of equivariant modules over suitable commutative algebras. Our main result is a construction of a functor from the category O side to the Whittaker side that has several properties expected of the equivalence -- but we do not know how to prove that our functor is an equivalence. I will try to keep the exposition reasonably elementary. The talk is based on work in progress with Gurbir Dhillon. 

Abstract: Given a symplectic variety X, we construct a canonical category, which should be thought of as a category of modules over a quantization of the loop space of X. For example, if X = T^*M is a cotangent bundle, the category in question is a variant of the category of D-modules on the loop space of M. We discuss the potential application to the relative Langlands program, sheaves of vertex algebras (chiral quantizations), and line defects in quantum field theory. Based on work in progress with D. Kubrak. 

Abstract: In a talk I will discuss how interesting cases of Coulomb branches arise as Hamiltonian reductions of cluster varieties. The Coulomb branches are spaces of vacua for supersymmetric gauge theories with 8 supercharges. It's been long known that for 4d N=2 theories "of class S", compactified on a circle, those Coulomb branches are Fock-Goncharov cluster varieties. Recently a set of examples of Coulomb branches which have cluster structure on them was extended by both gauge 4d N=2 theories described through Braverman-Finkelberg-Nakajima construction, and by Coulomb branches of 5d N=1 theories compactified on a torus being identified with Goncharov-Kenyon cluster integrable systems.

In some cases, when known cluster varieties do not provide a satisfying description, it appears to be natural to use Hamiltonian reductions of known cluster varieties and show that the result is a cluster variety itself. In my talk I will discuss in detail the simplest example of 4d N=2 theory - abelian gauge theory - being almost a simplest possible cluster variety. And show that it is a multiplicative hypertoric variety using cluster Hamiltonian reduction. Then I will present more complicated examples, coming from 5d N=1 theories Coulomb branches being cluster Hamiltonian reductions of Goncharov-Kenyon integrable systems.

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