Abstract: Consider a simple algebraic group G of classical type and its Lie algebra g. Let F be an algebraically closed field of characteristic p>> 0, we write g_F for the F-form of g. Let U^{\chi}_{F, \lambda} be the central reduction of U(g_F) with respect to a pair \lambda\in \mathfrak{h}^*/W and \chi\in \mathfrak{g}^{*}_\mathbb{F}^{(1)}. We focus on the case where \lambda is integral regular and \chi is nilpotent. In [BM12], there exists an algebra A^{0}_e (the fiber of the noncommutative Springer resolution over a nilpotent element e\in g) equipped with an isomorphism K_0(U^{\chi}_{F, \lambda}- mod)\cong K_0(A^{0}_e-mod)$. This isomorphism sends classes of simple modules to classes of simple modules. Let Y_e be the set of simple modules of A^{0}_e. Let Q_e be the reductive part of the centralizer of e in G. As Q_e acts on A^{0}_e by algebra automorphisms, the finite set Y_e has the structure of a Q_e-centrally extended set. In this work, we study this centrally extended structure of Y_e when the partition of e has few rows. In particular, we first prove that Y_e can be determined by certain numerical invariants of the Springer fiber B_e. Next, we introduce a variety B_e^{gr} \subset B_e that shares the same set of numerical invariants as B_e. We then reconstruct Y_e from B_e^{gr} using categorical tools. The main result is that the derived category D^b(B_e^{gr}) admits a complete exceptional collection that is compatible with the Q_e-action. The objects in this collection can be regarded as the points of Y_e in a suitable sense. As applications, we obtain an algorithm to compute the multiplicities of orbits in Y_e, which provides some numerical information on cells in affine Weyl group.

Abstract: We will define a stratification on the moduli space of local systems on the punctured disc and prove some properties of it. This stratification has a counterpart for categories with an action of the loop group, and we will describe a conjectural application of this formalism to the representation theory of W-algebras. 

Abstract:  I'll describe a perturbative classical field theory defined on 11-manifolds with a rank 6 transversely holomorphic foliation and a transverse Calabi–Yau structure. The theory has an infinite-dimensional algebra of gauge symmetries preserving the trivial background, which is $L_\infty$ equivalent to a Lie 2-extension of the infinite-dimensional exceptional simple super Lie algebra $E(5|10)$. Conjecturally, this theory describes the minimal twist of eleven-dimensional supergravity. After describing this conjecture, and evidence for it, I'll describe twisted avatars of the $AdS_7\times S^4$ and $AdS_4\times S^7$ backgrounds, and how two other infinite-dimensional exceptional simple super Lie algebras $E(3|6)$ and $E(1|6)$ appear as asymptotic symmetries. Enumerating gravitons on such backgrounds naturally leads to refinements of generating functions of representation-theoretic significance, such as the MacMahon function. Time permitting, I'll explain how our results combined with holographic techniques can be used to produce enhancements of familiar vertex algebras such as the Heisenberg and Virasoro algebras, to holomorphic factorization algebras in three complex dimensions, and furnish geometric constructions of representations thereof. This talk is based on joint work with Ingmar Saberi and Brian Williams. 

Abstract: The main theorem of this talk will be that the affine closure of the cotangent bundle of the basic affine space (also known as the universal hyperkahler implosion) has symplectic singularities for any reductive group, where essentially all of these terms will be defined in the course of the talk. After discussing some motivation for the theory of symplectic singularities, we will survey some of the basic facts that are known about the universal hyperkahler implosion and discuss how they are used to prove the main theorem. Time permitting, we will also discuss a recent result, joint with Harold Williams, which identifies the universal hyperkahler implosion in type A with a Coulomb branch in the sense of Braverman, Finkelberg, and Nakajima, confirming a conjectural description of Dancer, Hanany, and Kirwan. 

Abstract: A 2-group is a categorical generalization of a group: it's a category with a multiplication operation which satisfies the usual group axioms only up to coherent isomorphisms. The isomorphism classes of its objects form an ordinary group, G. Given a 2-group G with underlying group G, we can similarly define a categorical generalization of the notion of principal bundles over a manifold (or stack) X, and obtain a bicategory Bun_G(X), living over the category Bun_G(X) of ordinary G-bundles on X. For G finite and X a Riemann surface, we prove that this gives a categorification of the Freed--Quinn line bundle, a mapping-class group equivariant line bundle on Bun_G(X) which plays an important role in Dijkgraaf--Witten theory (i.e. Chern--Simons theory for the finite group G). I will not assume background knowledge on 2-groups, 2-group bundles, or the Freed--Quinn line bundle; I will provide the necessary definitions and an outline of the proof of the categorification result. Time permitting, I will discuss work-in-progress regarding applications of this result, and generalizations to Chern--Simons theory in the case that G is a compact group. This talk is based on joint work with Daniel Berwick-Evans, Laura Murray, Apurva Nakade, and Emma Phillips.

Abstract: We introduce the notions of (G,q)-opers and Miura (G,q)-opers, where G is a simply-connected complex simple Lie group, and prove some general results about their structure. We then establish a one-to-one correspondence between the set of (G,q)-opers of a certain kind and the set of nondegenerate solutions of a system of Bethe Ansatz equations. Additionally we associate to a (G,q)-oper a class of meromorphic sections of a G-bundle, satisfying certain difference equations, which we refer to as generalized q-Wronskians. We show that the QQ-systems and their extensions emerge as the relations between generalized minors, thereby putting the Bethe Ansatz equations in the framework of cluster mutations known in the theory of double Bruhat cells.

Abstract: I will introduce the basics of toroidal algebras and certain extensions of them, together with their representations using vertex operator algebraic techniques. I will also introduce certain extensions of toroidal algebras including a generalization to lie super algebras. The purpose of this generalization will be related to the other main topic of the lecture, which will be the equivariant cohomology of moduli spaces of sheaves on elliptic surfaces which serve as weight spaces for representations of the algebras.  The action of autoequivalences of the derived category of the elliptic surfaces will be identified with automorphisms of toroidal algebras. Time permitting, we will discuss other representations of toroidal lie algebras related to flags of sheaves and their relations with superconformal algebras.

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Abstract: Among the nilpotent orbits in a simple Lie algebra are the special nilpotent orbits, which play an important role in representation theory.  Some of the geometry of the closure of a nilpotent orbit can be understood by taking a transverse slice to a smaller orbit in the closure.  This talk concerns a classification of two types of such transverse slices:  (1) those between adjacent special nilpotent orbits; and (2) those between a special nilpotent orbit and a certain non-special nilpotent orbit in its closure.  The slices in part (1) exhibit a duality, which extends an observation of Kraft and Procesi for type A.  The slices in part (2) are related to a conjecture of Lusztig on special pieces.  This talk is based on two preprints with Baohua Fu, Daniel Juteau, and Paul Levy.

Abstract: I will discuss families of multilinear k-ary operations ("brackets") that naturally arise in QFT. The brackets physically describe BRST anomalies generated by interactions/deformations of QFTs in perturbation theory, and are analogous to the beta-functions that describe quantum violations of scale symmetry due to interactions. Besides being formally interesting, I will show that the brackets are highly computable (requiring only a first course in QFT to compute), and contain familiar information like anomalies and OPEs. Time permitting, I will discuss how these brackets are very strongly constrained in Holomorphic-Topological scenarios, and a higher-dimensional analogue of Kontsevich's formality theorem which implies the absence of perturbative corrections to HT theories with more than 1 topological direction. Based on arXiv:2403.13049.

Abstract: The geometric Satake equivalence is a celebrated construction (with contributions by Lusztig, Ginzburg, Drinfeld and Mirkovic-Vilonen) that realizes the category of representations of a connected reductive group as a category of perverse sheaves on the affine Grassmannian of the Langlands dual group. In the setting of l-adic coefficients, Zhu and Richarz have studied a variant of this construction in a "ramified" situation, where the group of which one takes the affine Grassmannian can be a non constant group scheme over formal loops. In this talk I'll explain a version of this equivalence for general coefficients; the Tannakian group on the dual side is then a certain group of fixed points for automorphisms of a reductive group, which is not necessarily smooth. This is joint work with P. Achar, J. Lourenço and T. Richarz. 

Abstract: I will review the highlights of Strominger’s program of celestial holography, focusing on emerging connections to twisted holography, topological strings and twistor theory. A running example of the talk will be holography for certain self-dual theories placed on an asymptotically flat, scalar-flat Kahler geometry known as Burns space. This is based on work done in collaboration with Kevin Costello and Natalie M. Paquette.

Abstract: This talk is based on arXiv:2305.09451, arXiv:2403.18011 and work in progress. I will discuss several deformations of gravitational celestial chiral algebras which are closely related to $w_{1+\infty}$ and give bulk interpretations of the respective deformations. Some of these deformations arise naturally from a backreaction in self-dual Einstein gravity analogous to part of the recent top-down construction of Costello, Paquette and Sharma and I will highlight similarities and differences.

 Abstract: The Bethe subalgebras of the Yangian Y(gl(n)) form a family of maximal commutative subalgebras indexed by points of the Deligne-Mumford compactification of the moduli space M(0,n+2). When considering a point C in the real locus of this parameter space, the corresponding Bethe subalgebra B(C) acts with simple spectrum on a given tame representation of Y(gl(n)). This results in an unramified covering, whose fiber over C is the set of eigenlines for the action of B(C). I will discuss the identification of each fiber with a collection of Gelfand-Tsetlin keystone patterns, which carry a gl(n)-crystal structure, as well as the monodromy action realized by a type of cactus group. This is joint work with Anfisa Gurenkova and Leonid Rybnikov.

Abstract: I will propose two classes of algebras which are generalizations/enhancements of the Virasoro algebra in conformal field theory. The first class of examples exists in any dimension, and like the Virasoro Lie algebra, are built from central extensions of vector fields. The second example exists only in dimension three and appears as an enhancement of conformal symmetry in the famous AGT correspondence.