MIT Infinite-Dimensional Algebra Seminar (Spring 2024)

Meeting Time: Friday, 3:00-5:00 p.m. | Location: 2-135

Contact: Pavel Etingof and Victor Kac

Zoom Link: https://mit.zoom.us/j/94469771032

Meeting ID
944 6977 1032

For the Passcode, please contact Pavel Etingof at etingof@math.mit.edu.

Schedule of Talks

Date Speaker
February 9 Elijah Bodish
(MIT)

Spin link homology and webs in type B

In their study of GL(N)-GL(m) Howe duality, Cautis-Kamnitzer-Morrison observed that the GL(N) Reshetikhin-Turaev link invariant can be computed in terms of quantum gl(m). This idea inspired Cautis and Lauda-Queffelec-Rose to give a construction of GL(N) link homology in terms of Khovanov-Lauda's categorified quantum gl(m). There is a Spin(2n+1)-Spin(m) Howe duality, and a quantum analogue which was first studied by Wenzl. In the first half of the talk I will explain how to use this duality to compute the Spin(2n+1) link polynomial, and present calculations which suggest that the Spin(2n+1) link invariant is obtained from the GL(2n) link invariant by folding. In the second part of the talk, I will introduce the parallel categorified constructions and explain how to use them to define Spin(2n+1) link homology.

This is based on joint work in progress with Ben Elias and David Rose.

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February 16 Yasuyuki Kawahigashi
(University of Tokyo)

Quantum 6j-symbols and braiding

I will explain certain 4-tensors appearing in studies of two-dimensional topological order from a viewpoint of subfactor theory of Jones and alpha-induction there, which is a tensor functor arising from a modular tensor category and a Frobenius algebra in it. They are understood with quantum 6j-symbols and braiding.

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February 23 Kenta Suzuki
(MIT)

Affine Kazhdan-Lusztig polynomials on the subregular cell: with an application to character formulae

(joint with Vasily Krylov) I will explain the computation of special values of parabolic affine inverse Kazhdan-Lusztig polynomials, which give explicit formulas for certain irreducible representations of affine Lie algebras that generalize Kac and Wakimoto's results. By Bezrukavnikov's equivalence, the canonical basis in the subregular part of the anti-spherical module can be identified with irreducible objects in the exotic t-structure on the equivariant derived category of the subregular Springer fiber. We describe the irreducible objects explicitly using an equivariant derived McKay correspondence. In doing so, we identify the module with a module Lusztig defines, which compatibly extends to the regular cell.

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March 1 Vadim Vologodsky
(IAS)

On the de Rham cohomology of the local system P^{1/h}

Let X -->S be a smooth family of algebraic varieties, P an invertible function on X. Consider the relative de Rham cohomology of the asymptotic D-module P^{1/h}: = (O_X, hd + dP/P). In the limit, when h goes to 0, the cohomology depends only on the neighborhood of the critical locus of function P. I will explain how, when working with algebraic varieties over Z/nZ, the above "asymptotic" formula for the cohomology becomes exact. I will also explain some applications to the study of the KZ equations over Z/nZ. The talk is based on a joint work with Alexander Varchenko.

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March 8 Zhiwei Yun
(MIT)

Counting indecomposable G-bundles over a curve

In an influential 1980 paper of Victor Kac, he proved (among many other things) that the number of absolutely indecomposable representations of a quiver over a finite field behaves like point-counting on a variety over F_q. This variety has been made precise by Crawley-Boevey and Van den Bergh using deformed preprojective algebras.

A parallel problem is to count absolutely indecomposable vector bundles on a curve over a finite field. About 10 years ago, Schiffmann proved that the number of such (with degree coprime to the rank) is equal to the number of stable Higgs bundles of the same rank and degree (up to a power of q). Dobrovolska, Ginzburg and Travkin gave a slightly different formulation of this result and a very different proof. Neither argument obviously generalizes to G-bundles for other reductive groups G.

In joint work with Konstantin Jakob, we generalize the above-mentioned results to G-bundles. Namely, we show that the number of absolutely indecomposable G-bundles (suitably defined) on a curve over a finite field can be expressed using the number of stable (parabolic) G-Higgs bundles on the same curve.

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March 15 Ivan Cherednik
(University of North Carolina at Chapel Hill)

From DAHA superpolynomials for algebraic links to motivic ones

The focus will be on a recent construction of the motivic superpolynomials for arbitrary singularities (multi-branch and non-square-free). They will be introduced from scratch, which includes the definition of varieties of torsion-free sheaves of any rank over curve singularities. Our motivic superpolynomials are q,t,a-generalizations of orbital integrals associated with Affine Springer Fibers of type A in the case of the most general characteristic polynomials. I will not use the theory of ASF. The key conjecture is their coincidence with the DAHA superpolynomials of the corresponding (colored) algebraic links. The latter (due to Cherednik-Danilenko) will be defined. This coincidence can be seen as a high-level Shuffle Conjecture. As an application, the DAHA vertex will be considered and its relation to the q-theory of Riemann’s zeta. Also, q,t,a-deformations of the modified rho-invariants of algebraic knots will be discussed; classically, rho is defined via the Atiyah-Patodi-Singer eta invariant, but I will need only some formulas in this talk. See https://arxiv.org/abs/2304.02200.

March 22 Do Kien Hoang
(Yale University)

Geometry of the fixed points loci and discretization of Springer fibers in classical types

Consider a simple algebraic group $G$ of classical type and its Lie algebra $\mathfrak{g}$. Let $(e,h,f) \subset \mathfrak{g}$ be an $\mathfrak{sl}_2$-triple and $Q_e= C_G(e,h,f)$. The torus $T_e$ that comes from the $\mathfrak{sl}_2$-triple acts on the Springer fiber $\mathcal{B}_e$. Let $\mathcal{B}_e^{gr}$ denote the fixed point loci of $\mathcal{B}_e$ under this torus action. Our main geometric result is that when the partition of $e$ has up to $4$ rows, the derived category $D^b(\mathcal{B}_e^{gr})$ admits a complete exceptional collection that is compatible with the $Q_e$-action. The objects in this collection give us a finite set $Y_e$ that is naturally equipped with a $Q_e$-centrally extended structure. We prove that the set $Y_e$ constructed in this way coincides with a finite set that has appeared in various contexts in representation theory. For example, a direct summand $J_c$ of the asymptotic Hecke algebra is isomorphic to $K_0(Sh^{Q_e}(Y_e\times Y_e)$. The left cells in the two-sided cell $c$ corresponding to the orbit of $e$ are in bijection with the $Q_e$-orbits in $Y_e$. Our main numerical result is an algorithm to compute the multiplicities of the $Q_e$-centrally extended orbits that appear in $Y_e$.

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March 29 Spring Break
April 5 Dan Freed
(Harvard University)

Finiteness and fusion categories

Fusion categories satisfy stringent finiteness conditions: they are finite semisimple, abelian, rigid, etc. General finiteness conditions occur in higher category theory in the form of dualizability in symmetric monoidal categories. Together with Constantin Teleman, we apply topological quantum field theory--in particular, boundary theories therein--to characterize fusion categories among all tensor categories: fusion categories are tensor categories that are (1) dualizable and (2) the regular module category is also dualizable. The talk will include an exposition of relevant parts of topological field theory.

April 12 Andrei Neguț
(MIT)
April 19 Alexandre Goncharov
April 26 Milen Ykimov
May 3 Roman Bezrukavnikov
(MIT)
May 10 Ivan Loseu

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