I’m a doctoral candidate studying mathematical physics at the University of Texas at Dallas. I’m interested in fusion categories and their application to topological phases of matter.

Fusion categories are ubiquitous. They show up in the representation theory for things like quantum groups and affine Lie algebras, vertex algebras, and Hopf algebras. They show up in places like operator algebras as the even halves of (finite depth) subfactors. Throw in some extra stuff and you get modular categories which are related to things like conformal field theory and topological quantum field theory. These last two are important because of their applications to physics. Specifically, topological matter (things like fractional quantum Hall effect liquids) is something whose ground state is described by TQFTs. These are cool because they give rise to quasi-particles called “anyons” that can be used to store quantum information as part of the topological quantum computing paradigm.

One of the reasons fusion categories are nice is that there are certain finiteness conditions assumed. This mean they can be constructed and classified by solving families of cubic equations called “pentagon,” “hexagon,” and “pivotal.” Unfortunately, the number and complexity of these present serious hurdles. Moreover, given a solution there are group actions which take you to other ones. Related to this I’ve developed an interest in applying geometric invariant theory to studying fusion categories. Solutions to pentagon, etc. equations lead to algebraic schemes on which act algebraic groups. These can be used to construct quotient schemes whose points provide information about gauge and monoidal classes. More generally, I’m interested in how the problem of determining monoidal classes of Grothendieck equivalent fusion categories can be cast in geometric terms and how algebro-geometric tools can be used to gain information about fusion categories.

I am currently on the market. You can find my CV here, my research statement here, and my teaching statement here. These are all current as of March 2016.