I’ll discuss recent work on the structure of Heisenberg categories and their generalizations, especially the higher-level Heisenberg categorifications defined by Brundan.  This includes the long sought-after identification of their Grothendieck groups with Heisenberg algebras, and a construction of categorical Kac-Moody actions from Heisenberg actions which unifies essentially all examples of categorical Kac-Moody actions of interest to representation theorists.  This is joint work with Jonathan Brundan and Alistair Savage. Read more about Heisenberg v. Kac-Moody

The Hilbert scheme of the plane (or more generally, the resolution of the type A Kleinian singularity) has a surprising description as a Coulombbranch in the sense of Braverman, Finkelberg and Nakajima; this gives a rather odd looking presentation of the projective coordinate ring of this variety.  It turns out that this presentation is very well adapted to describing the tilting generators constructed using Bezrukavnikov and Kaledin's perspective; of course, there are many other interesting aspects of the combinatorics of coherent sheaves on these varieties, which hopefully the... Read more about Coherent sheaves on Hilbert schemes through the Coulomb lens