Support varieties for finite tensor categories

Support varieties are important tools in representation
theory, starting with representations of finite groups
and finite group schemes. Much of the theory has been
successfully adapted to many other settings, for example
finite tensor categories, while much is still unknown.
This talk will include a survey of some known results
and some recent developments which are joint work with
Petter Andreas Bergh and Julia Yael Plavnik. We will
focus on a tensor product property for support varieties,
explaining that it can be reduced to checking it for the
indecomposable periodic objects, and as a consequence
it holds for all symmetric finite tensor categories
over algebraically closed fields of characteristic 0.