Brauer Groups in Algebraic Topology I

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.