Introduction

The seminar will deal with fusion and modular categories. These are abelian tensor
categories which gained an increasing amount of interest in the last few years.
Fusion categories may be viewed as a categorification of the notion of an algebra.
Indeed, one fundamental example, that of a fusion category with underlying abelian
category of G-graded vector spaces (G a finite group), is a categorification of the group
algbra k[G]. Fusion and modular categories arise in areas of mathematics such as:
Hopf algebra theory, topological quantum field theory, operator algebras and
representations of quantum groups, and in physics in the context of topological phases of
matter. Historically speaking, modular categories appeared first, even though they
are in fact a specific type of fusion categories.

We will dedicate the first talk of the seminar to give a general exposition to the subject,
as well as basic definitions. We would like to stress the fact that no knowledge in category
theory (beyond a few simple definitions, which we will review anyhow) is needed; the
seminar is intended for general mathematical audience, and everyone is invited.