Meeting Time: Fridays, 3:00 PM  5:00 PM  Location: Room 2135
Coordinators: Pavel Etingof, Victor Kac, and Andrei Negut
Date and Time  Speaker  

September 10, 3:00 PM  5:00 PM  Dennis Gaitsgory (Harvard) 
Factorization of the minimal model CFT and Langlands duality We will suggest a conjectural picture, which follows from the quantum geometric Langlands theory, according to which the minimal model CFT factors into the tensor product of two copies of WZW CFT (for a pair of Langlands dual groups). 
September 17, 3:00 PM  5:00 PM  Milen Yakimov (Northeastern University) 
Root of unity quantum cluster algebras: discriminants, CayleyHamilton algebras, and Poisson orders We will describe a theory of root of unity quantum cluster algebras, which includes various families of algebras from Lie theory and topology. All such algebras will be shown to be maximal orders in central simple algebras. Inside each of them, we will construct a canonical central subalgebra which is isomorphic to the underlying cluster algebra. It is a farreaching generalization of the De ConciniKacProcesi central subalgebras that play a fundamental role in the representation theory of quantum groups at roots of unity. An explicit formula for the corresponding discriminants will be presented. We will also show that all root of unity quantum cluster algebras have canonical structures of CayleyHamilton algebras (in the sense of Procesi) and Poisson orders (in the sense of De ConciniKacProcesi and BrownGordon). Their fully Azumaya loci will be shown to contain the underlying cluster A varieties. This is a joint work with Shengnan Huang, Thang Le, Bach Nguyen and Kurt Trampel. 
September 24, 3:00 PM  5:00 PM  Alexander Goncharov (Yale) 
Spectral description of noncommutative local systems on surfaces I will explain a cluster description of moduli spaces of Rvector bundles with flat connections over topological surfaces, where R is a noncommutative field. Examples include moduli spaces of Stokes data, which appear in the study of differential equations with irregular singularities on Riemann surfaces. This is a joint work with Maxim Kontsevich. 
October 1, 10:00 AM  12:00 PM (special time) 
Michael Finkelberg (HSE) 
KazhdanLusztig conjecture via zastava spaces This is a joint work with A.Braverman and H.Nakajima. We give yet another proof (a geometric one) of the famous KazhdanLusztig conjecture on the characters of irreducible modules in the category O over a complex semisimple Lie algebra (in the Koszuldual formulation). The proof proceeds by analysis of fixed points in the zastava spaces. 
October 8, 3:00 PM  5:00 PM  Eugene Gorsky (UC Davis) 
Tautological classes and symmetry in KhovanovRozansky homology We define a new family of commuting operators F_k in KhovanovRozansky link homology, similar to the action of tautological classes in cohomology of character varieties. We prove that F_2 satisfies ``hard Lefshetz property" and hence exhibits the symmetry in KhovanovRozansky homology conjectured by Dunfield, Gukov and Rasmussen in 2005. This is a joint work with Matt Hogancamp and Anton Mellit. 
October 15, 3:00 PM  5:00 PM (Virtual Seminar) 
Reimundo Heluani (IMPA) 
Finite dimensionality of conformal blocks on the torus We will review conditions on a vertex algebra V so that its space of conformal blocks on the torus is finite dimensional. This leads to conditions of V related to C_2 cofiniteness: the zeroth Poisson homology of Zhu's C_2 algebra R_V is finite dimensional. We analyze analogous conditions so that the higher chiral homology of V on the torus is finite dimensional, this leads to the obvious condition on the Poisson homology of Zhu's C_2 algebra, as well as some extra conditions on the full classical limit of V. This is joint work with J. V. Ekeren. 
October 22, 10:00 AM  12:00 PM (SPECIAL TIME) 
Anne Moreau (Université de Lille) 
Nilpotent orbits arising from admissible affine vertex algebras In this talk, I will give a simple description of the closure of the nilpotent orbits appearing as associated varieties of admissible affine vertex algebras in terms of primitive ideals. I will also connect these varieties with the cohomology of the small quantum groups associated with an lth root of unity. 
October 29, 3:00 PM  5:00 PM (Virtual Seminar  Time To Be Confirmed) 
Anton Mellit (University of Vienna) 
Affine Springer fibers, open Hessenberg varieties, and nabla positivity. I will talk about the positive part of a certain affine Springer fiber studied by Goresky, Kottwitz, and MacPherson, and a certain interesting open subvariety. The Hilbert series of their BorelMoore homology turns out to be related to reproducing kernels of the BergeronGarsia nabla operator. This operator is easy to define in the basis of modified Macdonald polynomials, but producing explicit combinatorial evaluations of this operator is usually difficult and (conjecturally) relates to interesting Hilbert series associated to various moduli spaces. Our work is motivated by the nabla positivity conjecture of Bergeron, Garsia, Haiman, and Tesler that predicts that nabla evaluated on a Schur function is sometimes positive, sometimes negative. We categorify this conjecture and reduce it to a vanishing conjecture for the interesting open variety. It turns out, each irreducible S_n representation mysteriously prefers to live in certain degrees and weights in the cohomology. This is a joint work with Erik Carlsson. 
November 5, 3:00 PM  5:00 PM  YiZhi Huang (Rutgers) 
Vertex operator algebras and tensor categories In 1988, based on the fundamental conjectures on operator product expansion and modular invariance, Moore and Seiberg observed that there should be tensor categories with additional structures associated to rational conformal field theories. Since then, tensor category structures from conformal field theories have been constructed, studied and applied to solve mathematical problems. Mathematically, conformal field theories can be constructed and studied using the representation theory of vertex operator algebras. In this talk, I will give a survey on the constructions and studies of various tensor category structures on module categories for vertex operator algebras. 
November 12, 3:00 PM  5:00 PM  Pavel Etingof (MIT) 
Introduction to the analytic Langlands correspondence I will review an analytic approach to the geometric Langlands correspondence, following my work with E. Frenkel and D. Kazhdan,
arXiv:1908.09677, arXiv:2103.01509, arXiv:2106.05243. This approach was developed by us in the last couple of years and involves ideas from previous and ongoing works of a number of mathematicians and mathematical physicists  Kontsevich, Langlands, Nekrasov, Teschner, GaiottoWitten and others. One of the goals of this approach is to understand singlevalued real analytic eigenfunctions of the quantum Hitchin integrable system. The main method of studying these functions is realizing them as the eigenbasis for certain compact normal commuting integral operators the Hilbert space of L2 halfdensities on the (complex points of) the moduli space Bun(G,X) of principal Gbundles on a smooth projective curve X, possibly with parabolic points. These operators actually make sense over any local field, and over nonarchimedian fields are a replacement for the quantum Hitchin system. We conjecture them to be compact and prove this conjecture in the genus zero case (with parabolic points) for G=PGL(2).

November 19, 3:00 PM  5:00 PM  Kent Vashaw (MIT) 
On the spectrum and support of a finite tensor category The tools of support varieties (initiated by Carlson in 1983) and tensor triangular geometry (initiated by Balmer in 2005) have played an important role in the study of monoidal triangulated categories, with stable categories of finite tensor categories forming one of the principal classes of examples. The relationship between support varieties and tensor triangular geometry has been used in many cases to classify the thick ideals of the category in question, a fundamental problem. We will discuss work of BuanKrauseSolberg and NakanoV.Yakimov, which defined and developed noncommutative versions of Balmer's theory, and will proceed to describe new methods for determining the Balmer spectrum and thick ideals of a monoidal triangulated category. This is joint work with Daniel Nakano and Milen Yakimov. 
December 3, 10:00 AM  12:00 PM (special time) 
Michela Varagnolo (Université de CergyPontoise) 
K theoretic Hall algebras and coherent categorification of quantum groups I will explain an isomorphism between the positive half o a quantum toroidal group and the Ktheoretic Hall algebra of a preprojective algebra of affine type. There is an analogue result in the finite type case. For type A_1 this allows to propose a coherent categorification of the quantum affine sl(2). Surprisingly it may be computed using KLR algebras. The talk is based on two joint works, one with E. Vasserot, the other with P. Shan and E. Vasserot. 