Schedule, Notes/Handouts & Videos

σ-quiver varieties are fixed point loci of involution on quiver varieties. If we consider quiver varieties with 0-stability (i.e., categorical quotients), σ-quiver varieties are symplectic reduction of vector spaces. Thus we can consider the corresponding Coulomb branches. We take the case when σ-quiver varieties are moduli of SO-bundles over type A surfaces. We describe the Coulomb branches, as type D bow varieties.

The symplectic duality program studies varieties with conical symplectic singularities, as well as their partial symplectic resolutions. An ambitious program of Bellamy-Craw-Schedler proposes to classify all such resolutions via their so-called Hamiltonian-Cox space. In this talk, I'll describe progress I've made in this program. In particular, I will explain how, in joint work with Gwyn Bellamy, we carry out this program for one of the cornerstone varieties in the theory: the nilpotent cone of a complex semisimple Lie algebra.

From 3d mirror symmetry, any Nakajima quiver variety comes with a dual variety known as the Coulomb branch. A conjecture proposed by Bullimore–Dimofte–Gaiotto–Hilburn–Kim and, independently, by Okounkov asserts that the cohomology of the moduli space of quasimaps to a quiver variety admits a canonical action by the quantized coordinate ring of the dual Coulomb branch. In this talk, I will propose a refinement of this conjecture and discuss its proof. The construction relies on a -1 shifted symplectic structure on the moduli space of quasimaps and a compatible critical description of the BFN Coulomb branch. Time permitting, I will discuss the relation to critical Hall algebras and shifted Yangians.

Let $X=H\backslash G$ be a homogeneous spherical variety for a reductive group $G$, such that the dual group of $X$ is equal to the dual group of $G$. We prove a structure theorem for the cotangent space $T^*X$ ``up to codimension two,'' which strengthens some results of Knop. The theorem confirms observations of V.\ Lafforgue and Gaiotto, and a conjecture of Hameister, Luo, and Morrissey, and by their work implies a ``classical'' or ``Dolbeault'' limit of the global geometric relative Langlands conjecture. Conditionally on the local relative Langlands conjecture of [BZSV], the theorem also amounts to a description of the Braverman--Finkelberg--Nakajima ``Coulomb branch'' of hyperspherical symplectic vector spaces. Finally, we generalize the classical Chevalley--Richardson restriction theorem to this setting, giving a description of the invariant-theoretic double quotient $H\backslash G/H$ in terms of root data and invariants of the spherical variety.

A chiral quantization of a Poisson variety is a vertex algebra, whose classical limit is isomorphic to the algebra of functions on the arc space of the same variety. I will explain a way of thinking about chiral quantizations that makes manifest the relation to the more traditional story of deformation quantization. In the case where the variety is symplectic I will relate the space of chiral quantizations to geometry.

We will describe some recent progress, joint with Yasha Varshavsky and David Yang, on the local geometric Langlands conjecture, including its version with restricted variation and a quantum analogue thereof. .  

In joint work with Gurbir Dhillon, we prove that the universal monodromic affine Hecke category is equivalent to ind-coherent sheaves on the Steinberg stack, an extension of Bezrukavnikov's equivalence. I will describe the proof, assuming as input a coherent description of the Iwahori-Whittaker category. The main ingredients are categorical representation theory and analysis of t-structures. Working over the integers presents some new complications because then perfect complexes on a stack are typically not compact.

Coxeter groups are defined by gluing (as a colimit) from finite Coxeter subgroups. We propose a global analogue of this fact in Betti Langlands: for a degeneration family of algebraic curves over the complex numbers, we define a Coxeter gluing functor from the automorphic category of smooth curves to the limit of that of singular curves. We show that this functor is an equivalence if and only if the automorphic gluing functor of Nadler–Yun is an equivalence. In genus 1, we show that the Coxeter gluing functor is an equivalence. This is joint work in progress with David Nadler and Zhiwei Yun.