Mathematics Colloquia and Seminars

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I will talk about my joint work with P.Etingof and D.Nikshych. A categorification is a procedure in which one replaces integer numbers by vector spaces, vector spaces by categories, maps between vector spaces by functors etc. Surprisingly enough such an abstract procedure is related to physics (here is a typical slogan: a categorification of $d-$dimensional topological field theory is $(d+1)-$dimensional topological field theory). In this talk I will explain the simplest way to categorify ring theory. In this theory rings are replaced by fusion categories (= semisimple rigid monoidal categories with finitely many simple objects) and modules over rings are replaced by module categories. Our main result is the following Theorem: fusion categories and module categories over them admit no deformations. Thus it is reasonable to try to classify these objects. The problem of classification of fusion categories and module categories over them appears to be closely related to Operator Algebras and to Conformal Field Theory. I will review some results in this direction.

3:45 Refreshments will be served before the talk in 551 Kerr Hall