Combinatorial structures arising from categorical representations

Abstract:

Linear representations of Lie algebras have a beautiful and well studied theory.  In the last decade we've discovered an equally rich and rigid theory of categorical representations of Lie algebras, which has had some important applications.  In this talk I'll survey some of these developments, with an emphasis on its combinatorial consequences, i.e. to crystal combinatorics and cactus group actions.  This talk will assume no prior knowledge about any of these topics.