MIT Infinite Dimensional Algebra Seminar (Spring 2023)

Explicit approaches to GL(N) link homology (which categorifies GL(N) quantum invariant of links) are based of foams, which are cobordisms in 3D between planar graphs. I'll explain how to construct a topological theory which assigns state spaces to planar graphs and maps to cobordisms between them. It then allows to build explicit complexes for link homology groups. This approach is based on Robert-Wagner foam evaluation formula. We'll also explain unoriented version of this construction related to the 4-color theorem and Kronheimer-Mrowka 3-orbifold homology. If time allows, we'll review the universal construction in other contexts, where it relates to Deligne categories, automata and quasirepresentations.

Cluster algebras are a class of commutative algebras defined by quiver and simple exchange relations. Originating from the theory of total positivity in matrices, they appeared to have rich combinatorial properties and be natural language in Lie theory, algebraic geometry, and integrable systems. Recently it was observed by M.Bershtein, P.Gavrylenko, and A.Marshakov that deautonomizations of relativistic Toda chains, which have natural cluster structure on a phase space, can be solved by (dual) partition functions of instanton in 5d super Yang-Mills theory. They have also conjectured that any cluster integrable systems, which are classified by Newton polygons of their spectral curves, can be solved by properly modified partition functions of topological string on 3d local Calabi-Yau manifold determined by the same Newton polygon.

In my talk, I will review this conjecture and show a way for the self-consistent proof of it using the first non-trivial example of the relativistic Toda chain with two particles. The idea of the proof is based on the observation that there is a natural embedding of the partition function of topological string, defined as a statistical model of 1d-2d-3d Young diagrams, into the context of cluster algebras. The proposed solution can be checked in a "melting" limit, where the weights of boxes are going to 1, and a novel formula for Seiberg-Witten prepotential appears from the limit shape geometry. If time will permit, I will speculate on how the class of "toroidal" algebras might appear from the quantization of the setting.

Let R be a hensilian discrete valuation domain (e.g., R=K[[t]]) with fraction field F, and let X be a Severi-Brauer variety over F. We say that X is tame if it has a point over an unramified extension of F (this is always the case if the residue field of R is perfect).

We prove that X has a proper semistable model over R if and only if X is tame.

The geometric special fiber of the semistable model we construct is a certain quiver Grassmannian. The proof makes use of a description of hereditary R-orders (i.e. orders whose Jacobson radical is a projective module over the order) in the algebra of matrices over F obtained by Brumer.

The talk is based on a joint work in progress with Alexei Kubanov and Constantin Shramov.

Several sorts of categorical resolutions for singular algebraic varieties have been constructed by Kuznetsov, Lunts, van den Bergh and other authors. Mostly, these constructions are rathre complicated, as they use gluing of some categories, and it is not so easy to study their structure. We propose a more “concrete” construction for the case of singular curve, where the resulting category arises as a category of sheaves over a non-commutative curve.

A non-commutative curve is a pair (X,A), where X is an algebraic curve and A is a sheaf of OX-algebras coherent as a sheaf of modules. For such a curve we construct a categorical resolution for the derived category, calculate its global homological dimension, find a semi-orthogonal decomposition of the derived category and obtain an upper bound for the Rouquier dimension. If X is rational, we construct a tilting complex that establishes a derived equivalence of the curve (X,A) to a finite dimensional quasi-hereditary algebra.

Holomorphic Floer Theory (HFT) is the name of the project which we have been developing jointly with Maxim Kontsevich since 2014. This talk is based on several sections of the large file which we plan to publish in the future (probably in the form of several papers).

HFT studies questions of "Floer-theoretical nature" in the framework of complex symplectic manifolds. Similarly to the Homological Mirror Symmetry (HMS) there are "A-side" and "B-side" of HFT. But differently from the HMS in the HFT the equivalence of A and B sides relates categories associated to the SAME complex symplectic manifold. The equivalence has a form of the (generalized) Riemann-Hilbert correspondence.

In the first part of my talk I plan to review some basic ideas and examples of HFT. In the second part I am going to explain how these ideas lead to a conjectural approach to Chern-Simons theory. In particular I plan to discuss the corresponding Hodge structure of infinite rank as well as a conceptual reason for resurgence (Borel resummability) of perturbative expansions in the Chern-Simons theory.

We introduce a monoidal analogue of Jantzen filtrations in the framework of monoidal categories with generic braidings. It leads to a deformation of the multiplication of the Grothendieck ring. We conjecture, and we prove in many remarkable situations, that this deformation is associative so that our construction yields a quantization of the Grothendieck ring as well as analogs of Kazhdan-Lusztig polynomials. As a first main example, for finite-dimensional representations of simply-laced quantum affine Kac-Moody algebras, we prove the associativity and that the resulting quantization coincides with the quantum Grothendieck ring constructed in a geometric manner. Hence, it yields a unified representation-theoretic interpretation of the quantum Grothendieck ring. As a second main example, we establish an analogous result for representations of quiver Hecke algebras associated with adapted reduced words. This is a joint work with Ryo Fujita.

I will review the construction of polynomial solutions of KZ equations modulo an integer and the properties of the solutions, in particular, their p-adic limit.

We develop a general approach to reduction along strong Dirac maps, which are a broad generalization of Poisson moment maps. These methods recover a number of familiar constructions in Poisson and quasi-Poisson geometry. We use them to give several new reduction procedures in the quasi-Poisson setting, including a quasi-Poisson analogue of Whittaker reduction. This is joint work with Maxence Mayrand.

In a fundamental sequence of works from the 1990s, De Concini, Kac and Procesi constructed a Poisson geometric framework for the study of the irreducible representations of big quantum groups at roots of unity. We will describe an extension of this framework to all contragredient quantum supergroups at root of unity. The approach of De Concini, Kac, and Procesi relied on a reduction to rank two cases, which is not possible in the super case since there are 13 kinds of additional Serre relations on up to 4 generators. We use a new approach that relies on Nichols algebras and perfect pairings between restricted and non-restricted integral forms. In particular, this gives new proofs of the classical results, independent on Serre relations. The methods apply to a larger, axiomatically defined class of algebras (consisting of the Drinfeld doubles of diagonal pre-Nichols algebras that have 1-parameter deformations). This class includes the quantizations in characteristic 0 of the 34-dimensional Kac-Weisfeler Lie algebra in characteristic 2 and the 10-dimensional Brown Lie algebra in characteristic 3. This is a joint work with Nicolas Andruskiewitsch and Ivan Angiono (University of Cordoba).

I will explain how ideas from the (geometric) Langlands program help solve the following purely algebraic problem: describe the ring of conjugation-invariant functions on the scheme of commuting pairs in a complex reductive group. The answer was known up to nilpotents, and we show that this ring is indeed reduced. We also describe the ring of invariant functions on the derived version of the commuting scheme. The proof brings in seemingly unrelated objects such as the affine Hecke category and character sheaves (of the Langlands dual group).

This is joint work with Penghui Li and David Nadler.

The goal of this talk is to describe some t-structures on the derived category of CohG (St) of relevance to representation theory in zero and positive characteristic. In more detail, in positive characteristic, we describe t-structures whose hearts are equivalent to the modular analogs of the category of Harish-Chandra bimodules and the category O. This is based on arXiv:2302.05782. The characteristic 0 is a work in progress/ in preparation. I establish a t-structure whose heart is equivalent to the category O for a quantum group at a root of unity. Time permitting, I will discuss some techniques going into proving the latter equivalence including an approach to the double affine representation theory.

Let Z be the germ of a complex hypersurface isolated singularity of equation f. We consider the family of analytic D-modules generated by the powers of 1/f and relate it to the pole order filtration on the de Rham cohomology of the complement of {f=0}. This work builds on Vilonen’s characterization of the intersection homology D-module.

I will describe a theory which generalizes Heegard-Floer theory from gl(1|1) to arbitrary Lie (super) algebras. The theory gives rise to homological invariants of links and categorifies quantum group link invariants. The corresponding category of A-branes has many special features which render it solvable explicitly. In this talk, I will describe how the theory is solved, and how homological link invariants arise from it. I will focus on the two simplest cases, the gl(1|1) theory itself, and the su(2) theory, categorifying respectively the Alexander and the Jones polynomials

Given a semisimple element in the loop Lie algebra of a reductive group, we construct a quasi-coherent sheaf on a partial resolution of the trigonometric commuting variety of the Langlands dual group. The construction uses affine Springer theory and Coulomb branch algebras of Braverman-Finkelberg-Nakajima. We also consider a quantization of this construction for homogeneous elements. This is a joint work with Oscar Kivinen and Alexei Oblomkov.