Meeting Time: Friday, 3:005:00 p.m.  Location: 2135
Contact: Pavel Etingof and Victor Kac
Date and Time  Speaker  

February 8, 35 p.m.  Dennis Gaitsgory (Harvard University) 
Metaplectic Whittaker sheaves and quantum groups In this talk we will explain the construction of (several versions of the) FLE, and equivalence between metaplectic Whittaker sheaves and quantum groups for a group G and modules for the Langlandsdual quantum group. The equivalence goes through relating both categories to that of Factorizable Sheaves. 
February 15, 35 p.m.  Brian Williams (Northeastern University)  Chiral Lie algebroids at the boundary of the Courant sigmamodel ChernSimons theory is a gauge theory describing the moduli space of flat connections on 3manifolds. When the 3manifold has a boundary endowed with a complex structure, the theory admits natural boundary conditions which are related to some familiar vertex algebras. A wellstudied example of such a correspondence is between the operators of the chiral WessZuminoWitten (WZW) model, described by the affine KacMoody vertex algebra, and ChernSimons theory. In this talk, we propose a general correspondence between a class of 3d topological field theories and vertex algebras using the language of factorization algebras and factorization categories. In addition to the standard CS/WZW example, we will see how the sheaf of chiral differential operators exists as the boundary chiral algebra of a certain 3d TFT labeled by a Courant algebroid. This is joint work with Pavel Safronov. 
February 22, 35 p.m.  Roman Bezrukavnikov (MIT) 
On dimensions of modular representations Let g be the Lie algebra of a semisimple algebraic group over an algebraically closed field of positive characteristic. In a joint work with Ivan Mirkovic we have established a relation between representations of g and canonical bases in homology of a Springer fiber predicted by Lusztig. In contrast with classical KazhdanLusztig theory, the definition of a canonical basis here does not give a concrete way to compute it. I will report on a current project with Ivan Losev where computing the basis is reduced to knowing affine KazhdanLusztig polynomials. This yields (reasonably) explicit formulas for dimensions of irreducible representations of g. 
March 1, 35 p.m.  Xiaomeng Xu (MIT) 
WKB analysis and integrable systems In this talk, I will first recall the Stokes matrices of differential equations with irregular singularities. I will then compute the WKB approximation of the Stokes matrices in the simplest case, and explore its relation with GelfandZeitlin systems, total positivity of matrices, and De ConciniKacProcesi braid group actions. This is the first step towards the understanding of the relation between WKB analysis and integrable systems proposed by Anton Alekseev, there are many problems remianed to be understood. 
March 8, 35 p.m.  Ivan Loseu (University of Toronto) 
HarishChandra bimodules over quantized symplectic
singularities We study a certain category of bimodules over a filtered algebra quantizing the algebra of functions on a conical symplectic singularity. The bimodules we care about are so called HarishChandra bimodules. This notion first appeared in the case of universal enveloping algebras of semisimple Lie algebras in the work of HarishChandra on representations of the corresponding complex Lie groups. Since then it was generalized to filtered quantizations of algebras of functions on affine Poisson varieties. The goal of this talk is to explain a classification of the simple HarishChandra bimodules with full support over quantizations of conical symplectic singularities (that have no slices of type E_8). We will see that these irreducible bimodules are in onetoone correspondence with the irreducible representations of a suitable finite group. The talk is based on arXiv:1810.07625. I will not assume any preliminary knowledge of conical symplectic singularities, their quantizations etc. 
March 15, 35 p.m.  Valerio Toledano Laredo (Northeastern University) 
PROPs, enveloping algebras without primitive elements, and quantisation functors. Let g be an infinitedimensional KacMoody algebra, and $\hat{U}$ the completion of its universal enveloping algebra with respect to highest weight representations. $\hat{U}$ is a topological Hopf algebra, which appears to have a rather unwieldy Hochschild (coalgebra) cohomology. When g is symmetrisable, Enriquez introduced a subalgebra $U$ of $\hat{U}$ which is universal among the the class of Manin triples, is large enough to define quantisation functors, and yet has a manageable Hochschild cohomology. The latter is in fact rather small since, for practical purposes, $U$ behaves like a universal enveloping algebra without primitive elements. I will give an alternative definition of $U$ using PROPs, which makes its properties far more manifest. I will then explain how this definition can be refined in various ways to accommodate in particular the 2categorical extension of EtingofKazhdan quantisation which Andrea Appel and I recently obtained. This is joint work with Andrea Appel. 
March 22, 35 p.m.  Bojko Bakalov (North Carolina State University) 
On the cohomology of vertex algebras and Poisson vertex algebras Following Beilinson and Drinfeld, we describe vertex algebras as Lie algebras for a certain operad of $n$ary chiral operations. This allows us to introduce the cohomology of a vertex algebra $V$ as a Lie algebra cohomology. When $V$ is equipped with a good filtration, its associated graded is a Poisson vertex algebra. We relate the cohomology of $V$ to the variational Poisson cohomology studied previously by De Sole and Kac. This talk is based on joint work with Alberto De Sole, Reimundo Heluani, Victor Kac, and Veronica Vignoli. 
March 29, 35 p.m.  Spring break 

April 5, 35 p.m.  Zhiwei Yun (MIT) 
Endoscopy for Hecke category and character sheaves Hecke categories are geometric incarnations of Hecke algebras, and they play an important role in the classification of irreducible representations of finite groups of Lie type. We consider a version of the Hecke category for the reductive group G with prescribed monodromy under the left and right actions of the maximal torus. We show that this monoidal category can essentially be identified with the usual Hecke category (with trivial torus monodromy) for an endoscopic group H of G, which is a reductive group of smaller dimension sharing a maximal torus with G but is not necessarily a subgroup of G. As a consequence, we show that character sheaves on G and on H are closely related. Joint work with G.Lusztig. 
April 12, 35 p.m.  Alexei Borodin (MIT) 
TBA TBA 
April 19, 35 p.m.  Andrei Okounkov (Columbia University) 
TBA TBA 
April 26, 35 p.m.  Simons lectures 

May 3, 35 p.m.  Pavel Safronov (University of Zurich) 
TBA TBA 
May 10, 35 p.m.  Pavel Galashin (MIT) 
TBA TBA 
S2008  F2008  S2009  F2009  S2010  F2010  S2011  F2011  S2012  F2012  S2013  F2013  S2014  F2014  S2015 F2015  S2016  F2016  S2017  F2017 S2018 F2018