Lecture

Modular fusion categories are mathematical structures appearing in mathematical physics (high energy and now condensed matter physics) and low dimensional topology (3d topological field theories).
The course will give an (elementary and example based) introduction to them and their very rough structure theory, governed by
the so-called Witt group of modular categories.

The course is meant to provide an informal and elementary introduction to
some of the notions of vertex operator algebras and conformal field theory,
to the modularity properties of the corresponding characters, and to the
“modular linear differential equations” satisfied by these modular forms

Let $k$ be a field. The collection of (isomorphism classes of) central division algebras over $k$ can be organized into an abelian group $\mathrm{Br}(k)$, called the Brauer group of $k$. In this series of talks, I'll describe some joint work with Mike Hopkins on a variant of the Brauer group which arises in algebraic topology, controlling the classification of certain cohomology theories known as Morava $K$-theories.

An axiom system for special quantum field theories was introduced over 25 years ago by Segal and Atiyah.
It has been much elaborated and developed, particularly in the topological case.  In these three lectures I will
discuss aspects of this mathematical theory and applications to problems in physics.