Informal Categorification Seminar
Fall 2022

I will give an overview of Lipshitz-Sarkar's and Lawson-Lipshitz-Sarkar's construction of a stable homotopy type refining sl(2) link homology, outline its key features, and discuss some applications. I will then discuss work with Krushkal and Willis on defining an sl(2) action on the annular Khovanov spectrum, where new intricacies arise due to cancellations. Time permitting, I will explain Lawson-Lipshitz-Sarkar's spectral refinement of Khovanov's arc algebras and tangle bimodules and mention work in progress with Gerhardt and Willis.

We will go through and try to explain portions of the paper in the title. The main theorem of the paper is that many categories appearing in type A categorical representation theory admit an sl2 action on their morphism spaces. We will explain the structure of these morphism spaces as sl2 representations, our primary example being the Soergel category. We then explain how the sl2 action on the morphisms is conjecturally related to decompositions of objects into indecomposables.

We continue from last week and state the real main theorem of the paper and do some examples. We then define the core of a sl2 representation and explain how the core of morphism spaces in certain sl2-categories seems to contain information about splitting of objects into direct summands. Time permitting, we will then define the sl2-enriched category of a sl2-category and consequences.

In this talk I will give an overview of the program of categorification of Verma modules and explain some of its applications. An emphasis will be given to open problems and future challenges.

We introduce a theory of stated SL(n)-skein algebras of surfaces, which provides a geometric/combinatorial interpretation for the quantum groups Oq(sl(n)) and other related notions from quantum algebra. They also quantize the SL(n)-character varieties of surfaces, are examples of quantum cluster algebras, and are closely related to Reshetikhin-Turaev quantum invariants of links, factorization homology, and the lattice gauge theory.

We construct actions of sl(2) on foams which are compatible with the Robert-Wagner foam evaluation formula. As a consequence we obtain actions of sl(2) on link homology. This is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.

The Khovanov complex of a link L in a thickened annulus carries a filtration; the associated graded complex gives rise to the annular Khovanov homology of L. Grigsby-Licata-Wehrli show that this annular homology admits an action by the Lie algebra sl2. Using the techniques of Lipshitz-Sarkar, one can define a stable homotopy lift of the annular Khovanov homology of L. In this talk I will describe (in part) how to lift the sl2-action to the stable homotopy category as well, illustrating some features of how one might hope to lift signed maps with cancellations via framed flow categories. This is joint work with Ross Akhmechet and Slava Krushkal

We construct actions of sl(2) on foams which are compatible with the Robert-Wagner foam evaluation formula. As a consequence we obtain actions of sl(2) on link homology. This is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.

Plethystic substitutions are a useful operation on the ring of symmetric functions. We will discuss two recent papers where categorified instances of this notion have appeared, in connection to Schur functors and link homology. We will conclude with some questions involving annular foams, Soergel bimodules, Heisenberg categorification, and Macdonald polynomials.

Following a joint work with Hogancamp and Mellit, I will define an action of both a commutative family of operators and a Lie algebra sl(2) in Khovanov-Rozansky homology.

The action of sl(2) implies the symmetry of the homology conjectured by Dunfield, Gukov and Rasmussen. I will also discuss some examples and further properties of these operators, following a joint work with Chandler.