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  

September 9, 3:00 PM  5:00 PM (Room 2135) 
Victor Kac (MIT) 
Unitary representations of vertex algebras I will explain the definition of unitarity of a representation of a conformal vertex algebra and describe the classification of unitary representations of minimal quantum affine Walgebras. I will also explain how to compute the characters of these representations. 
September 16, 3:00 PM  5:00 PM (Room 2135) 
Leonid Rybnikov (Harvard University) 
sl_2opers and arc diagrams We define a natural, purely geometrical bijection between the set of monodromyfree sl_2opers (aka projective structures) on the projective line with the prescribed type of regular singularities at prescribed real marked points and the set of arc diagrams of FrenkelKirillovVarchenko. The former set indexes the solutions of Bethe ansatz equations for the Gaudin model (according to Feigin and Frenkel), while the latter one indexes the canonical base in a tensor product of U_q(sl_2)modules (via the SchechtmanVarchenko isomorphism). Both sets carry a natural action of the cactus group, i.e., the fundamental group of the real DeligneMumford space of stable rational curves with marked points. We prove that our bijection is compatible with this cactus group action. Moreover, this cactus group action coincides with that on an appropriate tensor product of sl_2crystals coming from the coboundary category formalism (joint work with Nikita Markarian). 
September 23, 3:00 PM  5:00 PM (Room 2135) 
no seminar (Vogan conference) 

September 30, 3:00 PM  5:00 PM (Room 2135) 
no seminar (retreat weekend) 

October 7, 3:00 PM  5:00 PM (Room 2135)  Vasily Krylov (MIT Mathematics) 
Subregular nilpotent orbits and explicit character formulas for modules over affine Lie algebras.
The talk is based on the joint work with Roman Bezrukavnikov and Victor Kac (arXiv:2209.08865). Let g be a simple Lie algebra and let $\hat{g}$ be the corresponding affine Lie algebra. It is known that characters of irreducible (highest weight) representations of $\hat{g}$ can be computed in terms of values at q=1 of affine (inverse) KazhdanLusztig polynomials. These values can be computed recursively but there are no explicit formulas for them in general. The goal of this talk is to describe certain cases when we can compute the values above explicitly resulting in explicit formulas for characters of certain irreducible $\hat{g}$modules (partly generalizing results of Kac and Wakimoto). The calculation relies on the description of the corresponding module over the affine Hecke algebra in terms of the equivariant $K$theory of the Springer resolution. Time permitting we will discuss possible generalizations. 
October 14, 3:00 PM  5:00 PM (Room 2135) 
Maria Gorelik (UC Berkeley) Vladimir Hinich (UC Berkeley) 
Root groupoid and related Lie superalgebras.
This talk in based on a joint work with V. Serganova and V. Hinich, arXiv:2209.06253 [1].

October 21, 3:00 PM  5:00 PM (Room 2135) 
Ben Davison (University of Edinburgh) 
BPS algebras for totally negative 2CalabiYau categories
To certain 2CalabiYau categories A one may associate a BPS Lie algebra, which in certain cases can be shown to generate the BorelMoore homology of the stack of objects in A under a PBWtype theorem. An example of such a case is the category of representations of a preprojective algebra associated to a quiver Q, where this Lie algebra recovers the Kac polynomials of Q as the cohomologically graded dimensions of its (dimension vector)graded pieces. In this talk I will concentrate on "totally negative" 2CY categories. Examples are provided by categories of representations of the preprojective algebra of Q in the case in which Q has enough arrows, or the category of semistable Higgs bundles on a curve C of genus of at least 2, or the category of representations of the fundamental group of the same C. For totally negative 2CY categories, in joint work with Hennecart and SchlegelMejia, we prove that the BPS Lie algebra is a free Lie algebra, with generators identified with the intersection cohomology of coarse moduli spaces. As applications, we prove

October 28, 3:00 PM  5:00 PM (Room 2135) 
Arun Kannan (MIT Mathematics) 
Symmetric Tensor Categories and New Constructions of Exceptional Simple Lie
I will present new constructions of several of the exceptional simple Lie super algebras
with integer Cartan matrix in characteristic p = 3 and p = 5, which were classified in [1].
These include the Elduque and Cunha Lie superalgebras. Specifi cally, let αp denote the kernel
of the Frobenius endomorphism on the additive group scheme Ga over an algebraically closed field
of characteristic p. The Verlinde cate gory Verp is the semisimplification of the representation
category Rep αp, and Verp contains the category of super vector spaces as a full subcategory.
Each exceptional Lie superalgebra we construct is realized as the image of an exceptional
Lie algebra equipped with a nilpotent derivation of order at most p under the semisimplification
functor from Rep αp to Verp. The content of this talk can primarily be found in [2] and [3].

November 4, 3:00 PM  5:00 PM (Room 2135) 
Brian Williams (Boston University) 
Dolbeault AGT and infinite dimensional exceptional super Lie algebras
Nakajima and Grojnowski initiated the relationship between infinitedimensional Lie algebras and the Hilbert scheme of points on a complex surface. In physics, a similar correspondence between a certain conformal field theory and the cohomology of the moduli space of instantons was laid out by Alday, Gaiotto, and Tachikawa. One way to contextualize these relationships is through the topological twist of a mysterious sixdimensional superconformal field theory. We will use a new mathematical understanding of this theory to propose, and provide evidence for, enhancements of our understanding of the cohomology of instanton moduli spaces which will involve appearances of certain exceptional super Lie algebras. 
November 11, 3:00 PM  5:00 PM (Room 2135) 
no seminar (holiday) 

November 18, 3:00 PM  5:00 PM (Room 2135) 
Iva Halacheva (Northeastern University) 
Cotangent Schubert calculus and Lagrangian correspondences
In recent work, the study of partial flag varieties and the Schubert bases of their equivariant cohomology has been extended to cotangent bundles and SegreSchwartzMacPherson classes. I will discuss the behavior of these bases in the restriction in cohomology from type A to type C Grassmannians. When considering their cotangent bundles, this behavior has a further geometric interpretation in terms of MaulikOkounkov stable envelopes and Lagrangian correspondences. Both settings can be considered from the combinatorial perspective of puzzle rules, which in turn are interpreted as quantum integrable systems via Rmatrices for the sl(3) Yangian. This is joint work with Allen Knutson and Paul ZinnJustin. 
November 25, 3:00 PM  5:00 PM (Room 2135) 
no seminar (holiday) 

December 9, 3:00 PM  5:00 PM (Room 2135) 
Ilya Dumanski (MIT Mathematics) 
A geometric approach to the FeiginLoktev fusion product
FeiginLoktev's fusion product of cyclic graded modules over the current algebra was introduced in 1998. Although its definition is elementary, very little is known about its properties in general. We introduce a way to study it geometrically. This approach also gives BorelWeiltype theorems for the BeilinsonDrinfeld Grassmannian and the global convolution diagram. We will also discuss a relation to quantum loop group representations and the category of coherent perverse sheaves on the affine Grassmannian. 