Meeting Time: Friday, 3:005:00 p.m.  Location: 2135
Contact: Pavel Etingof and Victor Kac
Zoom Link: https://mit.zoom.us/j/94469771032
Meeting ID
944 6977 1032
For the Passcode, please contact Pavel Etingof at etingof@math.mit.edu.
Date  Speaker  

February 9  Elijah Bodish (MIT) 
Spin link homology and webs in type B In their study of GL(N)GL(m) Howe duality, CautisKamnitzerMorrison observed that the GL(N) ReshetikhinTuraev link invariant can be computed in terms of quantum gl(m). This idea inspired Cautis and LaudaQueffelecRose to give a construction of GL(N) link homology in terms of KhovanovLauda's categorified quantum gl(m). There is a Spin(2n+1)Spin(m) Howe duality, and a quantum analogue which was first studied by Wenzl. In the first half of the talk I will explain how to use this duality to compute the Spin(2n+1) link polynomial, and present calculations which suggest that the Spin(2n+1) link invariant is obtained from the GL(2n) link invariant by folding. In the second part of the talk, I will introduce the parallel categorified constructions and explain how to use them to define Spin(2n+1) link homology. This is based on joint work in progress with Ben Elias and David Rose. Audio Transcript (Requires MIT Login) 
February 16  Yasuyuki Kawahigashi (University of Tokyo) 
Quantum 6jsymbols and braiding I will explain certain 4tensors appearing in studies of twodimensional topological order from a viewpoint of subfactor theory of Jones and alphainduction there, which is a tensor functor arising from a modular tensor category and a Frobenius algebra in it. They are understood with quantum 6jsymbols and braiding. Audio Transcript (Requires MIT Login) 
February 23  Kenta Suzuki (MIT) 
Affine KazhdanLusztig polynomials on the subregular cell: with an application to character formulae (joint with Vasily Krylov) I will explain the computation of special values of parabolic affine inverse KazhdanLusztig polynomials, which give explicit formulas for certain irreducible representations of affine Lie algebras that generalize Kac and Wakimoto's results. By Bezrukavnikov's equivalence, the canonical basis in the subregular part of the antispherical module can be identified with irreducible objects in the exotic tstructure on the equivariant derived category of the subregular Springer fiber. We describe the irreducible objects explicitly using an equivariant derived McKay correspondence. In doing so, we identify the module with a module Lusztig defines, which compatibly extends to the regular cell. Audio Transcript (Requires MIT Login) 
March 1  Vadim Vologodsky (IAS) 
On the de Rham cohomology of the local system P^{1/h} Let X >S be a smooth family of algebraic varieties, P an invertible function on X. Consider the relative de Rham cohomology of the asymptotic Dmodule P^{1/h}: = (O_X, hd + dP/P). In the limit, when h goes to 0, the cohomology depends only on the neighborhood of the critical locus of function P. I will explain how, when working with algebraic varieties over Z/nZ, the above "asymptotic" formula for the cohomology becomes exact. I will also explain some applications to the study of the KZ equations over Z/nZ. The talk is based on a joint work with Alexander Varchenko. Audio Transcript (Requires MIT Login) 
March 8  Zhiwei Yun (MIT) 
Counting indecomposable Gbundles over a curve In an influential 1980 paper of Victor Kac, he proved (among many other things) that the number of absolutely indecomposable representations of a quiver over a finite field behaves like pointcounting on a variety over F_q. This variety has been made precise by CrawleyBoevey and Van den Bergh using deformed preprojective algebras. A parallel problem is to count absolutely indecomposable vector bundles on a curve over a finite field. About 10 years ago, Schiffmann proved that the number of such (with degree coprime to the rank) is equal to the number of stable Higgs bundles of the same rank and degree (up to a power of q). Dobrovolska, Ginzburg and Travkin gave a slightly different formulation of this result and a very different proof. Neither argument obviously generalizes to Gbundles for other reductive groups G. In joint work with Konstantin Jakob, we generalize the abovementioned results to Gbundles. Namely, we show that the number of absolutely indecomposable Gbundles (suitably defined) on a curve over a finite field can be expressed using the number of stable (parabolic) GHiggs bundles on the same curve. Audio Transcript (Requires MIT Login) 
March 15  Ivan Cherednik (University of North Carolina at Chapel Hill) 
From DAHA superpolynomials for algebraic links to motivic ones The focus will be on a recent construction of the motivic superpolynomials for arbitrary singularities (multibranch and nonsquarefree). They will be introduced from scratch, which includes the definition of varieties of torsionfree sheaves of any rank over curve singularities. Our motivic superpolynomials are q,t,ageneralizations of orbital integrals associated with Affine Springer Fibers of type A in the case of the most general characteristic polynomials. I will not use the theory of ASF. The key conjecture is their coincidence with the DAHA superpolynomials of the corresponding (colored) algebraic links. The latter (due to CherednikDanilenko) will be defined. This coincidence can be seen as a highlevel Shuffle Conjecture. As an application, the DAHA vertex will be considered and its relation to the qtheory of Riemann’s zeta. Also, q,t,adeformations of the modified rhoinvariants of algebraic knots will be discussed; classically, rho is defined via the AtiyahPatodiSinger eta invariant, but I will need only some formulas in this talk. See https://arxiv.org/abs/2304.02200. 
March 22  Do Kien Hoang (Yale University) 
Geometry of the fixed points loci and discretization of Springer fibers in classical types Consider a simple algebraic group $G$ of classical type and its Lie algebra $\mathfrak{g}$. Let $(e,h,f) \subset \mathfrak{g}$ be an $\mathfrak{sl}_2$triple and $Q_e= C_G(e,h,f)$. The torus $T_e$ that comes from the $\mathfrak{sl}_2$triple acts on the Springer fiber $\mathcal{B}_e$. Let $\mathcal{B}_e^{gr}$ denote the fixed point loci of $\mathcal{B}_e$ under this torus action. Our main geometric result is that when the partition of $e$ has up to $4$ rows, the derived category $D^b(\mathcal{B}_e^{gr})$ admits a complete exceptional collection that is compatible with the $Q_e$action. The objects in this collection give us a finite set $Y_e$ that is naturally equipped with a $Q_e$centrally extended structure. We prove that the set $Y_e$ constructed in this way coincides with a finite set that has appeared in various contexts in representation theory. For example, a direct summand $J_c$ of the asymptotic Hecke algebra is isomorphic to $K_0(Sh^{Q_e}(Y_e\times Y_e)$. The left cells in the twosided cell $c$ corresponding to the orbit of $e$ are in bijection with the $Q_e$orbits in $Y_e$. Our main numerical result is an algorithm to compute the multiplicities of the $Q_e$centrally extended orbits that appear in $Y_e$. Audio Transcript (Requires MIT Login) 
March 29  Spring Break 

April 5  Dan Freed (Harvard University) 
Finiteness and fusion categories Fusion categories satisfy stringent finiteness conditions: they are finite semisimple, abelian, rigid, etc. General finiteness conditions occur in higher category theory in the form of dualizability in symmetric monoidal categories. Together with Constantin Teleman, we apply topological quantum field theoryin particular, boundary theories thereinto characterize fusion categories among all tensor categories: fusion categories are tensor categories that are (1) dualizable and (2) the regular module category is also dualizable. The talk will include an exposition of relevant parts of topological field theory. Video Recording (Requires MIT Login) 
April 12  Andrei Neguț (MIT) 
Shrubby quivers on the torus and quantum loop groups Algebraic structures behind BPS counting on toric CalabiYau threefolds have recently been realized mathematically in terms of the quantum loop group associated to a certain quiver drawn on a torus. In this talk, we give a generatorsandrelations presentation of the reduced version of this quantum loop group, assuming the quiver satisfies a technical condition we call "shrubbiness". Video Recording (Requires MIT Login) 
April 19 Cancelled  Alexander Goncharov (Yale University) 
Exponential volumes in Geometry and Representation Theory Let S be a topological surface with holes. Let M(S,L) be the moduli space parametrising hyperbolic structures on S with geodesic boundary, and a given set L of lengths of the boundary circles. It carries the WeilPeterson volume form. The volumes of spaces M(S,L) are finite. M.Mirzakhani proved remarkable recursion formulas for them, related to several areas of Mathematics. However if S is a surface P with polygonal boundary, e.g. just a polygon, similar volumes are infinite. We consider a variant of these moduli spaces, and show that they carry a canonical exponential volume form. We prove that exponential volumes are finite, and satisfies unfolding formulas generalizing Mirzalkhani's recursions. This part of the talk is based on the joint work with Zhe Sun. There is a generalization of these moduli spaces for any split simple real Lie group G, with canonical exponential volume forms. When the modular group of the surface P is finite, our exponential volumes are finite for any G. When P are polygons, they provide a commutative algebra of positive Whittaker functions for the group G. The tropical limits of the positive Whittaker are the (zonal) spherical functions for the group G. 
April 26  Milen Yakimov (Northeastern University) 
Reflective centres of module categories and quantum Kmatrices Braided monoidal categories have applications in various situations, in particular their universal Rmatrices give solutions of the quantum YangBaxter equation and representations of braid groups of type A. There are powerful methods for constructing them: Drinfeld doubles of Hopf algebras and Drinfeld centres of monoidal categories. On the other hand, universal Kmatrices, leading to solutions of the reflection equation and representations of braid groups of type B are much less well understood. We will describe a construction of reflective centers of module categories. It gives rise to braided module categories and a quantum double construction (a la Drinfeld) for universal Kmatrices. From this perspective, quantum Rmatrices come from categorical versions of centers of algebras and quantum Kmatrices come from categorical versions of centers of bimondules. This is a joint work with Robert Laugwitz and Chelsea Walton. 
May 3  Roman Bezrukavnikov (MIT) 

May 10  Ivan Loseu 