Representation theory without vector spaces

Classically, representation theory studies symmetries encoded as actions of group-like objects on vector spaces. A celebrated theorem of Deligne from 2002 explained that abstract categorical symmetries in characteristic zero are still governed by classical ``supersymmetries” and have super vector spaces lurking in the background. In positive characteristic the situation is much wilder: we don’t yet have a good understanding of what it is that is lurking. It is definite, though, that there is a lot there beyond vector and super vector spaces. I will try to describe where the new theory meets the old and what we do (not much) and don’t (a lot) know.