On fusion categories
A fusion category is a semisimple tensor category with duality. Such categories arise in several areas of mathematics and physics - conformal field theory, operator algebras, representation theory of quantum groups, and others. In this talk we present the results of our joint work with Pavel Etingof and Victor Ostrik on the structure and properties of such categories. We show that the global dimension of a fusion category is always positive and that fusion categories and functors between them are undeformable (in particular, the number of categories realizing a given fusion rule is finite). We also develop the theory of Frobenius-Perron dimensions in a fusion category.