The classification of fusion categories via operator algebras

Abstract: 

Fusion categories may be considered as generalization of the representation categories of finite groups, and they appear in various areas in mathematics and mathematical physics, such as representation theory of quantum groups, conformal field theory, and operator algebras.

There have been several attempts to classify fusion categories in recent years, and in this talk I will focus on

the classification of specific classes, called near-group categories, and more generally,  quadratic categories.

While this is a purely algebraic finite dimensional problem, our approach goes through infinite dimensional objects, called operator algebras, consisting of bounded operators acting on Hilbert spaces.