I study representation theory, usually with a categorical or combinatorial flavor to it. I am particularly interested in the interaction of representation theory with other areas of math. A recurring thread in my research is using tools and ideas from model theory to give new constructions in representation theory, and I tend to very excited about representation theoretic questions coming from geometry or topology, which has motivated a lot of my work. If you have a representation theory question coming from some other area of math, chances are I will be excited to hear about it!
Pre-Tannakian Categories, Deligne Categories, and Oligomorphic Groups
The definition of a pre-Tannakian category is somewhat technical, but it is in essence an axiomatization of categories with all the same internal structure as the finite dimensional representation theory of a group G. If a pre-Tannakian category admits a "fiber functor" to vector spaces (a concrete realization of it as a collection of vector spaces) then it is said to be Tannakian, and Tannakian reconstruction says that in this case it is actually equivalent to representations of some group. However not every pre-Tannakian category is Tannakian, and Deligne constructed explicit examples Rep(St), Rep(GLt), and Rep(Ot) of categories which are pre-Tannakian but not Tannakian, which interpolate the representation theories of the symmetric, general linear, and orthogonal groups.
I have been fascinated by these Deligne categories for years, and a lot of my work has been trying to understand and generalize them. My first project in graduate school was to apply the Deligne category machinery to prove a conjecture about representations of Weyl groups of type B. Then I defined a new integral form of Rep(St) with better reduction to positive characteristic, which motivated my definition with Etingof and Ostrik of p-adic dimensions in symmetric tensor categories. I also gave a new interpretation of Deligne's categories using model theory. This perspective has proved useful for understanding the internal structure of these categories - Daniil Kalinov and I used it to characterize algebra objects in Rep(St), Flake, Laugwitz, and I used it to describe the Drinfeld center of Rep(St), and it has been utilized further in the work of Kalinov, Utiralova, Etingof, Rains, and others.
Recently, Andrew Snowden and I found a new construction of pre-Tannakian categories using oligomorphic groups, a class of groups that normally arise in model theory. This gives another new interpretation of Rep(St), but also shows that the world of pre-Tannakian categories is much bigger than previously thought. We are still actively developing this theory. Since that first paper we have: extended the oligomorphic group set-up to linear groups, joined with Noah Snyder to analyze the Delannoy Category (a particularly interesting example), and extended that analysis to a more complicated circular version, then we introduced the oligomorphic analogs of Grothendieck's Galois categories, and used them to give a universality statement about our construction. There's been a flurry of activity in this area, and more results are on the way.
It has long been observed that many aspects of the representation theory of certain natural families of groups, such as the symmetric groups Sn or the general linear groups GLn, stabilize as n tends to infinity. For a long time this was regarded as something of a representation theoretic curiosity, however in 2012 Church, Ellenberg, and Farb observed that this sort of stability manifests itself in many other areas of mathematics and dubbed the phenomenon "representation stability". This area has seen a flurry of activity in recent years, with broad applications to a variety of different mathematical fields.
I have been heavily involved in developing the representation theoretic aspects of this theory, particularly the positive characteristic aspects of the story. In particular I proved a categorical periodicity theorem and a virtual stability theorem, which together describe the behavior of FI-modules in positive characteristic. I also applied representation stability to a problem in additive combinatorics, and have been developing the theory of representation stability for arithmetic groups.
Sam Hopkins and I defined a q-analog of the ring of integer valued polynomials.
Benkart, Halverson, and I computed dimensions of irreducible representations of the partition algebra.
I investigated representations of the group of monomial matrices try to understand restriction from GLn to Sn.
I wrote an appendix on a certain class of modular representations of Sn for a paper by Farb, Kisin, and Wolfson.
Chris Ryba and I proved a tensor-cube version of the Saxl conjecture.