Tensor products and trivalent vertex bimodules in Heegaard Floer homology
Location
I will start by summarizing a higher tensor product operation recently defined in joint work with Raphael Rouquier, along with its relationship to Heegaard Floer homology. Then I will discuss some bimodules for trivalent vertices, as well as singular and nonsingular crossings, that should be part of an approach to HFK similar to Ozsvath-Szabo's recent approach but that are better adapted to the higher tensor product. I will also try to describe how categorified skew Howe duality suggests trivalent vertex bimodules as a basic starting point for categorified link invariants in type A, how it specifies certain higher maps that should act on these bimodules and relations among them, and how one can use these ideas to define maps for braid cobordisms. If time permits I will discuss these higher maps algebraically for the trivalent vertex bimodules under consideration (finding a topological or diagrammatic explanation for these higher maps is an interesting open problem).