In this talk, I will introduce fusion categories as categorical versions of finite rings. We will discuss some examples which may already be familiar, like the category of representations of a finite group and the category of vector spaces graded over a finite group. Then, we will define Tambara-Yamagami categories, which are a certain type of fusion categories which have one simple object which is non-invertible. I will provide the classification results of Tambara and Yamagami on these categories and give some small examples. Time permitting, I will discuss current work in progress on how to generalize Tambara-Yamagami fusion categories to locally compact groups. 

This talk will not assume familiarity with category theory further than the definition of a category and a functor.

The Gelfand--Kirillov dimension is a classical invariant that measures the size of smooth representations of p-adic groups. It acquired particular relevance in the mod p Langlands program because of the work of  Breuil--Herzig--Hu--Morra--Schraen, who computed it for the mod p cohomology of GL_2 over totally real fields, and used it to prove several structural properties of the cohomology. In this talk, we will present a simplified proof of this result, which has the added benefit of working unchanged for nonsplit inner forms of GL_2. This is joint work with Bao V. Le Hung.