SPEAKER: Ben Webster
TITLE: Categorification and Lie algebras
ABSTRACT: It's a long established principle that an interesting way to think about numbers as the sizes of sets or dimensions of vector spaces, or better yet, the Euler characteristic of complexes. You can't have a map between numbers, but you can have one between sets or vector spaces. For example, Euler characteristic of topological spaces is not functorial, but homology is. One can try to extend this idea to a bigger stage, by, say, taking a vector space, and trying to make a category by defining morphisms between its vectors. This approach (interpreted suitably) has been a remarkable success with the representation theory of semi-simple Lie algebras (and their associated quantum groups). I'll try to explain why this approach is interesting on its own terms, and its connection to more classical representation theory.