Informal Categorification Seminar
Spring 2023

I will discuss joint work with Peter Johnson and Slava Krushkal which introduces an invariant of a particular class of 3-manifolds that unifies and extends two theories with quite different origins and structures. The first is lattice cohomology, due to Némethi, whose degree zero part is described by a certain graph and is isomorphic to Heegaard Floer homology for a large subclass of such 3-manifolds. The second theory is the Z-hat q-series of Gukov-Pei-Putrov-Vafa, a power series which conjecturally recovers SU(2) quantum invariants at roots of unity. I will explain lattice cohomology, Z-hat, and our unification of these theories. I will also point out some key features of the new invariant. Time permitting, I will discuss work in progress with Peter Johnson and Sunghyuk Park on extending these constructions to knot complements. 

One of the most classical form of Howe duality relates the commuting action of two general linear Lie algebra on a polynomial ring.

We will first discuss this duality in the specific case of gl(2) and gl(n). Specifically, we will explain how the polynomial ring decomposes as a direct sum of finite dimensional bimodules over gl(2) x gl(n).

With D. Tubbenhauer and P. Vaz, we construct a generalization of this duality in a context where Verma modules replace the finite dimensional representations of gl(2). Surprisingly the dual picture leaves the realm of lowest and highest weight modules.

As an application, we use a quantized version of this duality to study the irreducibility of the Lawrence--Krammer--Bigelow representations of braid groups.

Let L be a link in a thickened annulus.  Grigsby-Licata-Wehrli showed that the annular Khovanov homology of L is equipped with an action of sl_2(\wedge), the exterior current algebra of the Lie algebra sl_2.

We will discuss how this structure can be understood in the setting of L_\infty-algebras and modules. We show that sl_2(\wedge) is an L_\infty-algebra and that the annular Khovanov homology of L is an L_\infty-module over sl_2(\wedge). Up to L_\infty-quasi-isomorphism, this structure is invariant under Reidemeister moves, and the higher L_\infty-operations can be computed using explicit formulas. 

Extended Khovanov arc algebras are finite-dimensional graded algebras which naturally appear in a variety of contexts such as link homology, representation theory and symplectic geometry. We will explain how to describe these algebras and their Koszul duals via quivers with relations which allows us to compute (part of) their Hochschild cohomology groups, settling a conjecture by C. Stroppel. As a consequence we obtain explicit A∞ deformations of extended Khovanov arc algebras which can also be viewed as A∞ deformations of Fukaya-Seidel categories associated to Hilbert schemes of points on type A Milnor fibres. These talks are based on https://arxiv.org/abs/2211.03354.
Recording. 

I will survey the general homological theory of hopfological algebra through the example of p-differential graded dual zigzag algebras (Koszul dual to the zigzag algebra defined by Khovanov-Seidel-Thomas). This is based on an earlier joint work with Joshua Sussan and a work in progress with Ben Cooper and Joshua Sussan. 

Extended Khovanov arc algebras are finite-dimensional graded algebras which naturally appear in a variety of contexts such as link homology, representation theory and symplectic geometry. We will explain how to describe these algebras and their Koszul duals via quivers with relations which allows us to compute (part of) their Hochschild cohomology groups, settling a conjecture by C. Stroppel. As a consequence we obtain explicit A∞ deformations of extended Khovanov arc algebras which can also be viewed as A∞ deformations of Fukaya-Seidel categories associated to Hilbert schemes of points on type A Milnor fibres. These talks are based on https://arxiv.org/abs/2211.03354 

There is a well-known map from the affine Hecke algebras in type A to those in finite type A called the evaluation map.

Pulling back representations along this map gives the so-called evaluation modules for Hecke algebras in affine type A.

In this talk, I will explain how the evaluation map lifts to a monoidal functor from affine type A Soergel bimodules to the bounded homotopy category of finite type A Soergel bimodules, and give some applications to 2-representation theory.

This is joint work with Marco Mackaay and Vanessa Miemietz. 

I will give an overview of Kronheimer and Mrowka's construction for a knot K in S^3 of a spectral sequence having as E_2 page the knot homology associated to the universal rank 2 Frobenius system F_5 and converging to a version of instanton Floer homology. In particular, I will focus on describing certain interesting local systems on the configuration space, and their role in the construction of the invariant. 

Bott-Samelson spaces are geometric counterparts of (Bott-Samelson) Soergel bimodules and their Hochschild homology via equivariant cohomology. This was initially explored by Webster and Williamson in an algebro-geometric setting. From an algebraic topology perspective, these spaces were assembled into filtered equivariant stable link homotopy types by Kitchloo. In this talk we explore some of the known and speculated relations between these spaces and structures in link homologies, with an eye towards annular webs and foams. 

Khovanov homology for links in RP3 was described by Gabrovsek.  We will discuss how to deform this construction to arrive at a Lee homology theory and Rasmussen invariant for such links, as well as describe some applications for freely 2-periodic knots in S3.