Hopf algebras arise naturally in many different areas of mathematics, such as combinatorics, topology, and mathematical physics. Many commonly studied algebraic structures are either Hopf algebras themselves or are directly related to them: for example, groups, Lie algebras, and quantum groups. Hopf algebras act on rings, generalizing the notion of a group of automorphisms. Modules for a Hopf algebra can be added (direct sum) and multiplied (tensor product), giving their categories of modules the structure of tensor categories. In this first talk, we will define Hopf algebras and their actions on rings, give examples, and explain how their modules fit into this larger picture of tensor categories.

This series of talks will be devoted to the theory of fusion categories which is one way to give a mathematical description of quantum symmetries. We will discuss basic definitions, examples and constructions of such categories, as well as some structure theory and classification results.

Modular tensor categories encode the topological symmetries of 2-dimensional bosonic topological phases of matter, as the algebraic underpinnings of (2+1)D TQFTs. I will describe the main constructions of modular tensor categories, a few classification results and some open questions.

Reference List for Introduction to Modular Categories (compiled by E. Rowell)

Current research on Hopf algebras includes programs to classify some types and to understand their categories of modules. In this second talk, we will consider this second problem for nonsemisimple Hopf algebras, and homological techniques for approaching it. We will define the cohomology ring and state a finite generation conjecture for finite dimensional Hopf algebras and more generally for finite tensor categories. We will survey what is known and touch on some recent research.

I will speak on finite tensor categories and their appearances in mathematical physics. I will describe the "boundary" between finite-dimensional Hopf algebras and tensor categories.

I'll introduce Graeme Segal's functorial/geometric framework for conformal field theories, with emphasis on connections to other areas of mathematics such as modular tensor categories and vertex operator algebras.