Roughly, 2-Segal sets are simplicial sets such that higher-dimensional simplices can be uniquely described by triangulated polygons formed out of 2-simplices. In a sense that I will make precise, 2-Segal sets can be viewed as categorified associative algebras. As a TQFT Club member, you might ask, “Are there 2-Segal sets that correspond to (commutative) Frobenius algebras?” The answer is yes, commutativity and Frobenius structures come from asking the simplicial set to possess additional compatible structure maps. I’ll give an overview of these correspondences as well as some background as to how I arrived at this topic from the world of Poisson geometry. This is based on joint works with Ivan Contreras, Walker Stern, and Sophia Marx.

I will discuss joint work with Agustina Czenky. We introduce a $(3-ε)$-dimensional TQFTs which is generated, in some sense, by the derived category of quantum group representations. This TQFT is valued in the $∞$-category of dg vector spaces, and the value on a genus $g$ surface is a $g$-th iterate of the Hochschild cohomology for the aforementioned category. I will explain how this TQFT arises as a derived variant of the usual Reshetikhin–Turaev theory and, if time allows, I will discuss the possibility of introducing local systems into the theory. Our interest in local systems comes from proposed relationships with 4-dimensional non-topological QFT.

We construct an action of sl(2) on equivariant Khovanov–Rozansky link homology. We will discuss some topological applications and show how the construction simplifies in characteristic p. This is joint with You Qi, Louis-Hadrien Robert, and Emmanuel Wagner.