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

February 4, 3:00 PM  5:00 PM (Virtual Seminar) 
Catharina Stroppel (Mathematical Institute of the University of Bonn) 
Motivic Springer theory Many interesting algebras in (geometric) representation theory arise as convolution algebras. Based on these examples we develop a general framework using Chow rings and Chow motives. Chow motives are objects in a weights structure of the triangulated derived category of motives. I will explain weight structure and weight complex functors and try to explain why it might be interesting for representation theorists. We finally indicate formality results using motives instead of perverse sheaves. 
February 11, 3:00 PM  5:00 PM (Virtual Seminar) 
Jethro van Ekeren (IMPA) 
Chiral homology, the Zhu algebra and identities of RogersRamanujan type
The notion of chiral homology of a chiral algebra was introduced by Beilinson and Drinfeld, generalising conformal blocks. The construction of a chiral algebra from a conformal vertex algebra and a smooth complex curve provides a large supply of interesting examples, but in general the chiral homology of these examples seems not to be well understood. Motivated by questions in the representation theory of vertex algebras, we study the behaviour of the chiral homology of families of elliptic curves degenerating to a nodal curve. After introducing chiral homology in general, I will explain how to develop explicit complexes to compute it in the case of interest, relate it to the Hochschild homology of the corresponding Zhu algebra, and establish links with identities of RogersRamanujan type and their generalisations. (Joint work with R. Heluani) 
February 18, 3:00 PM  5:00 PM  Andrei Negut (MIT Mathematics) 
Generators and relations for quantum loop groups
I will describe a program that uses shuffle algebras to yield generatorsandrelations presentations for quantum loop groups. The main idea is that the necessary relations are dual to the socalled wheel conditions that describe the shuffle algebras in question, and we will use this to get a complete presentation of two interesting algebras that arise in geometric representation theory: Ktheoretic Hall algebras of quivers, and Hall algebras of coherent sheaves on curves over finite fields (the latter project joint work with Francesco Sala and Olivier Schiffmann). 
February 25, 4:00 PM  6:00 PM (Virtual Seminar Special Time) 
Steven Sam (UC San Diego) 
Curried Lie Algebras
A representation of $gl(V)$ is a map $V \otimes V^* \otimes M \to M$ satisfying some conditions, or via currying, it is a map $V \otimes M \to V \otimes M$ satisfying different conditions. The latter formulation can be used in more general symmetric tensor categories where duals may not exist, such as the category of sequences of symmetric group representations under the induction product. Several other families of Lie algebras have such "curried" descriptions and their categories of representations have nice compact descriptions as representations of diagram categories, such as the (walled) Brauer category, partition category, and variants. I will explain a few examples in detail and how we came to this definition. This is joint work with Andrew Snowden. 
March 4, 9:00 AM  11:00 AM (Virtual Seminar Special Time) 
Sylvain Carpentier (Columbia University) 
Quantization of integrable differential difference equations
We present a new approach to the problem of quantising integrable systems of differentialdifference equations.
The main idea is to lift these systems to systems defined on free associative algebras and look for the ideals there that are stabilized by the new dynamics.
In a reasonable class of candidate ideals, there are typically very few that are invariant for the first equation in the hierarchy.
Once these ideals are picked the challenge is to prove that the whole hierarchy of equations stabilizes them.
We will discuss these ideas using as a key example the hierarchy of the Bogoyavlensky equation.

March 11, 9:00 AM  11:00 AM (Virtual Seminar Special Time) 
Quoc Ho (Hong Kong University of Science and Technology) 
Revisiting mixed geometry
I will present joint work with Penghui Li on our theory of graded sheaves on Artin stacks. Our sheaf theory comes with a sixfunctor formalism, a perverse tstructure in the sense of BeilinsonBernsteinDeligneGabber, and a weight (or cot)structure in the sense of Bondarko and Pauksztello, all compatible, in a precise sense, with the sixfunctor formalism, perverse tstructures, and Frobenius weights on elladic sheaves. The theory of graded sheaves has a natural interpretation in terms of mixed geometry à la BeilinsonGinzburgSoergel and provides a uniform construction thereof. In particular, it provides a general construction of graded lifts of many categories arising in geometric representation theory and categorified knot invariants. Historically, constructions of graded lifts were done on a casebycase basis and were technically subtle, due to Frobenius' nonsemisimplicity. Our construction sidesteps this issue by semisimplifying the Frobenius action itself. As an application, I will conclude the talk by showing that the category of constructible Bequivariant graded sheaves on the flag variety G/B is a geometrization of (and is equivalent to) the DGcategory of bounded chain complexes of Soergel bimodules. 
March 18, 3:00 PM  5:00 PM (Virtual Seminar) 
Alexander Tsymbaliuk (Purdue) 
BGGtype relations for transfer matrices of rational spin chains and the shifted Yangians
In this talk, I will discuss: (1) the new BGGtype resolutions of finite dimensional representations of simple Lie algebras that lead to BGGrelations expressing finitedimensional transfer matrices via infinitedimensional ones, (2) the factorization of infinitedimensional ones into the product of two Qoperators, (3) the construction of a large family of rational Lax matrices from antidominantly shifted Yangians. This talk is based on the joint works with R.Frassek, I. Karpov, and V.Pestun. 
April 1, 3:00 PM  5:00 PM (Virtual Seminar) 
Ralf Köhl (University of Kiel) 
Topological KacMoody groups  Discussing the topology proposed by Kac and Peterson
This topology induces the Lie topology on the Levi factors of parabolics of spherical type, in the indefinite cases it provides new examples for Kramer's theory
of topological twin buildings (which he developed in 2002 for loop groups), and in the Archimedian case it is possible to determine their fundamental groups,
actually providing a structural explanation for the (known by classification) fundamental groups of semisimple split real Lie groups.

April 8, 3:00 PM  5:00 PM  Dennis Gaitsgory (Harvard) 
Screening operators revisited
We’ll revisit the theorem of Feigin and Frenkel that says that screening operators that act between Wakimoto modules satisfy quantum Serre relations. We’ll use this to giving an alternative construction of (the Iwahori) variant of KazhdanLusztig equivalence. 
April 15, 3:00 PM  5:00 PM  Zhiwei Yun (MIT) 
Conjugacy classes in the Weyl group and nilpotent orbits.
The Weyl group and the nilpotent orbits are two basic
objects attached to a semisimple Lie group. In this talk, I will
describe two very different geometric constructions relating these two objects,
due to KazhdanLusztig, Lusztig, and myself. 
April 22, 3:00 PM  5:00 PM  Aleksandra Utiralova (MIT Mathematics) 
HarishChandra bimodules in complex rank
The Deligne tensor categories are defined as an interpolation of
the categories of representations of groups GL_n, O_n, Sp_{2n} or S_n
to the complex values of the parameter n. One can extend many
classical representationtheoretic notions and constructions to this
context. These complex rank analogs of classical objects provide
insights into their stable behavior patterns as n goes to infinity.

April 29, 3:00 PM  4:00 PM  Ben Webster (University of Waterloo + Perimeter Institute) 
Noncommutative resolutions and Coulomb branches
Coulomb branches are a new construction of symplectic singularities based in 3dimensional N=4 supersymmetric quantum field theory. Even in the case where these recover wellknown singularities such as nilcones and symmetric powers of C^2, they still shed new light on these varieties. In particular, they are a presentation which is much better adapted to the construction of tilting generators using the approach of Bezrukavnikov and Kaledin. I'll discuss how this leads to interesting noncommutative symplectic resolutions of Coulomb branches, and a description of the wallcrossing functors for the categories of coherent sheaves on resolutions. 
May 6, 3:00 PM  5:00 PM  Roman Bezrukavnikov (MIT Mathematics) 
New geometric approaches to the small quantum group
I will talk about relations between modules over the small quantum group $u_q$ at a root of unity and geometry of a specific affine Springer fiber F. Cohomology of F is related to the center of $u_q$, while (Koszul dual of) the category of $u_q$ modules is related to microlocal sheaves on F.Based on joint project with Pablo Boixeda Alvarez, Peng Shan and Eric Vasserot, and with Boixeda Alvarez, Michael McBreen and Zhiwei Yun. 
May 13, 3:00 PM  5:00 PM  Nate Harman (Institute of Advanced Study) 
Oligomorphic Groups and PreTannakian Categories
I will discuss a new construction of preTannakian categories associated to oligomorphic groups  a class of groups arising in model theory. This gives a new concrete realization of Deligne's interpolation categories Rep $(S_t)$, as well as new examples of preTannakian categories in characteristic zero which are not interpolations or ultraproducts of Tannakian categories. 