A brief mathematical biography

I graduated from UC Berkeley in May 2008, where my advisor was Nicolai Reshetikhin. I then spent a year working with Arun Ram in Melbourne Australia and three years as an NSF postdoctoral fellow and CLE Moore Instructor at MIT. As of August 2012, I am an assistant professor at Loyola University in Chicago.

My research is mostly about quantum groups and related combinatorics, particularly quantum affine algebras. Recently I have also been working with quiver varieties and with Khovanov-Lauda-Rouquier algebras. Below you can find brief descriptions of some of my main projects. This is not an exhaustive list of my work; some of my papers do not fit nicely into these holes, and so are not mentioned here. You can also find more information in my CV and related documents.

I am most interested in crystals, which in this context are combinatorial objects associated to (usually highest weight) representations of a complex simple Lie algebra, or more generally a symmetrizable Kac-Moody algebra. They record certain leading order data of the representation theory, and can be used to give combinatorial answers to various representation theoretic questions, such as calculating tensor product multiplicities. They also let you draw nice pictures:

Affine MV polytopes

Mirkovic-Vilonen (MV) polytopes were developed by Anderson and by Kamnitzer as moment map images of certain cycles in an affine grassmannian. They can be used to realize the crystal of any finite dimensional representation of a complex simple Lie algebra. The polytopes encode a great deal of information, and show up in several natural contexts. For instance, as well as with the affine grassmannian, they have natural relationships with Lusztig's PBW bases, with quiver varieties, and (very recently) with Khovanov-Lauda-Rouquier (KLR) algebras. This project develops an analogue of this theory in affine types.

The crystal commutor and the half twist

One important property of the category of representations of a quantized universal enveloping algebra is that it is braided: there is a family of isomorphisms \( V \otimes W \rightarrow W \otimes V \) such that the action on any n-fold tensor product factors through the n-strand braid group. One might hope this descends to a braiding on the category of crystals, but it cannot. In fact it is easy to see that there is no braiding on the category of crystals. However, there is a related structure: the crystal commutor of Henriques-Kamnitzer. There the braid group is replaced by the fundamental group of the moduli space of marked real genus zero stable curves (sometimes called the cactus group).

Constructing and relating various realization of affine sl(n) crystals

The goal of this project (which formed the bulk of my thesis) was to have a concrete understanding of affine sl(n) crystals, and to understand how seemingly different realizations of those crystals were related. The early work was combinatorial, but the project has more recently used quite a bit of geometry, in particular the quiver varieties of Lusztig and Nakajima.