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

September 15  Valerio ToledanoLaredo (North Eastern U) 
Pure Braid Group Actions on Category O Let g be a symmetrisable KacMoody algebra. The corresponding quantum Weyl group operators give an action of the braid group of g on integrablemodules over the quantum group U_{h}g. Answering a question of Pavel Etingof, we prove that the restriction of this action to the pure braid group extends to all (not necessarily integrable) category O modules for Uhg. We also show that this action describes the monodromy of the Casimir connection of g on category O modules. This is joint work with Andrea Appel, and is based on https://arxiv.org/abs/2208.05331 [1] 
September 22  Student Holiday 

September 29  Leonid Rybnikov (MIT) 
Cactus flower curves, integrable systems, and crystals. The moduli space of rational curves with n+1 marked points and a tangent vector at the first of them (called framing) has a natural compactification F_n analogous to the DeligneMumford one for the space of nonframed curves, called moduli space of cactus flower curves. Contrary to the usual DeligneMumford compactification, it is singular  but still has many nice properties. I will explain how the real locus of this moduli space controls coboundary monoidal categories that are finite and concrete (i.e. endowed with a faithful monoidal functor to finite sets). The main example of such a category is the category of Kashiwara crystals for a semisimple Lie algebra g. Using the moduli space of cactus flower curves, we can determine the category of Kashiwara gcrystals from some quantum integrable spin chains related to g (namely, XXX Heisenberg chain and its Gaudin degeneration), without referring to canonical bases or to Stembridge axioms. This also describes the monodromy of solutions to Bethe ansatz in the corresponding integrable systems in terms of the Schutzenberger involutions on Kashiwara crystals. This is a joint work in progress with Aleksei Ilin, Joel Kamnitzer, Yu Li, and Piotr Przytycki. Video Link (Requires MIT Login) 
October 6  Dennis Gaitsgory (Max Planck Institute for Mathematics, Bonn) 
The geometric Langlands functor and parabolic induction. It turns out that the most involved step in the recently obtained proof of the geometric Langlands conjecture is concerned with the compatibility of the Langlands functor with Constant Terms. We will explain how to prove it via a representationtheoretic result about representations of affine algebras at the critical level. 
October 13  Sam Raskin (Yale U) 
The geometric Langlands functor Following Gaitsgory's talk on Eisenstein series from the week before, we will explain some tricks for how to conclude that the geometric Langlands functor is an equivalence. This work is all joint with Gaitsgory, and some parts are joint with Arinkin, Beraldo, Chen, Faergeman, and Lin. Video Link (Requires MIT Login) 
October 20  Ivan Danilenko (UC Berkeley) 
Stable envelopes and homological mirror symmetry Homological mirror symmetry predicts an equivalence between the derived category of equivariant coherent sheaves on the additive Coulomb branch X and a version of the wrapped Fukaya category on the multiplicative Coulomb Y branch with a superpotential. Under decategorification, we get an interesting identification between the equivariant Ktheory of X and homology of Y. In this talk, we will show how special bases in the equivariant Ktheory (fixed points, stable envelopes) are mirrored by cycles on Y. This is work in progress with Andrey Smirnov, with insights from a joint project with Mina Aganagic, Yixuan Li, Vivek Shende, and Peng Zhou. 
October 27  Daniil Klyuev (MIT) 
Analytic Langlands correspondence for $G=PGL(2,\mathbb{C})$ Analytic Langlands correspondence was proposed by Etingof, Frenkel and Kazhdan based on ideas and results of Langlands, Teschner, BravermanKazhdan and Kontsevich. Let $X$ be a smooth irreducible projective curve over $\mathbb{C}$, $G$ be a semisimple group. On one side of this conjectural correspondence there are $G^{\vee}$opers on $X$ satisfying a certain condition ({\it real} opers), where $G^{\vee}$ is Langlands dual group. On the other side there are certain operators on $L^2(Bun_G)$, called Hecke operators, where $Bun_G$ is the variety associated to the moduli stack of stable $G$bundles on $X$ and $L^2(Bun_G)$ is a Hilbert space of squareintegrable halfdensities. I will describe the main picture and present new results in this direction. Partially based on joint projects with A. Wang and S. Raman. Video Link (Requires MIT Login) 
November 3  Alexander Braverman (U of Toronto) 
Hecke operators over local fields, CalogeroMoser varieties and lifting Let $G$ be a split reductive group over a finite field $k$. A wellknown result of Prasad says that the space of (complexvalued) functions with finite support on the moduli space of $G$bundles on $P^1$ over $k$ endowed with Borel structures at $0$ and infinity is naturally isomorphic to the regular bimodule over the IwahoriHecke algebra of $G$ over the field $k((t))$. We shall present a generalization of this result in two directions: first, we replace Borel structures at $0$ and infinity with a full trivialization at those points, and also we replace the finite field $k$ with a local nonarchimedian field $F$. We shall also discuss two types of applications of this generalization:
This is a joint work in progress with D. Kazhdan. Video Link (Requires MIT Login) 
November 10  Veterans Day 

November 17  Lauren Williams (Harvard U) 
Cluster algebras and tilings for the $m=4$ amplituhedron The amplituhedron $A_{n,k,m}(Z)$ is the image of the positive Grassmannian under a map induced by a positive linear map. It was originally introduced in physics by ArkaniHamed and Trnka in order to give a geometric interpretation of scattering amplitudes and in particular the BrittoCachazoFengWitten recurrence. The amplituhedron has since been linked to cluster algebras and interesting objects from combinatorics. I will give an introduction to the amplituhedron and discuss our recent proof of the cluster adjacency conjecture for BCFW tiles, as well as the BCFW tiling conjecture. Based on joint work with EvenZohar, Lakrec, Parisi, ShermanBennett, and Tessler. 
November 24  Thanksgiving 

December 1  Alex Postnikov (MIT) 
Polytopes, tilings, total positivity, clusters, and amplitudes. This talk is complementary to Lauren Williams' recent talk. We'll
discuss several motivating examples from algebra/geometry/physics and
related combinatorial objects (plabic graphs, tilings, and
positroids). We'll emphasize the connections between these objects
and polyhedral geometry. Video Link (Requires MIT Login) 
December 8  Jae Hee Lee (MIT) 
Quantum Steenrod operations of symplectic resolutions Quantum connections of symplectic manifolds X are flat connections defined from Gromovâ€”Witten theory of X, acting on singular cohomology of X. When the target manifold X is a symplectic resolution, the quantum connection of X is known to recover and generalize wellknown flat connections from representation theory. Video Link (Requires MIT Login) 