This talk will give an overview of what is currently known about the structure of the Hochschild cohomology ring and describe some of the applications of this theory.

We begin with an introduction to Hochschild cohomology, an important invariant in the representation theory of algebras, and discuss the fact that the low-dimensional Hochschild cohomology groups have interpretations relating to derivations and to deformation theory.

I will then discuss joint work with Solberg, where we used the Hochschild cohomology ring to construct a support variety for any module over a finite-dimensional algebra. This was motivated by the work of Carlson, who introduced the support variety of a module over a group algebra; this is now a powerful invariant in the modular representation theory of finite groups. I will present an overview of our generalization using Hochschild cohomology and show that, under certain finite generation conditions, we have analogues of many of the properties of the group ring situation