Or, the part of the story I left out for 10 years

The last time I updated this page I was a doctoral candidate. Well, I’m happy to say, I got my doctorate in 2016 from University of Texas at Dallas in the Physics department studying “Arithmetic Properties of Anyonic Systems and Topological Phases.” So technically this makes me a physicist. However, my advisor was a mathematician and the contents of my dissertation are all about geometric invariants on affine schemes and algorithms for computing those invariants. I’d say that’s enough to qualify as interdisciplinary. Here’s a quick run down of what that’s about, because I still get asked about it to this day. If you want to read the whole thing, my dissertation is here.

A brief sojourn on the classification of fusion categories.

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 – in computer science terms, there are these things called complexity classes that define how “hard” a problem is. Solving systems of polynomial equations fit into one of the hardest classes called “EXPSPACE” and they are “complete” for this class, meaning that all of the hardest problems in this class are equivalent.

One of the things we want to do with fusion categories is classify them – we want to know what properties they have so that we can make general statements about them. A simple way that we want to classify them is by rank – or how many finite generating objects they have. One way to do this would be to solve the pentagon equations (notedly hard above) and then count the solutions, but there’s a wrinkle. In general, there are infinitely many solutions to the pentagon equations based on what are called automorphisms. There are infinitely many choices of automorphism, and so we can get infinitely many solutions. Thus, we restrict ourselves to finding solutions that are representatives of an automorphism equivalence class.

This leads us to ‘geometric invariants.’ The equations defining fusion categories lead to algebraic schemes on which act algebraic groups (our automorphisms). These can be used to construct quotient schemes whose points provide information about equivalence classes. This implies that there are invariants in the polynomial variables that can always be constructed to tell when two different solutions are different. The central result of my dissertation was that these invariants always exist and for a specific class of fusion categories (called multiplicity-free) I provided an algorithm for constructing these invariants.

This worked for a new class of fusion categories related to the quantum double of the group S_3, including the discovery of a new fusion category that doesn’t admit braiding (so can’t create an anyonic system). All of this is covered in one of my grad school papers here. Additionally, I also worked on the classification of fusion categories related to an infinite family related to the representation categories of the B series of affine Lie algebras. Here, we had to use a new approach because the algorithm described above only works for single categories, but we were able to discover and prove a pattern for a set of invariants that always classify the fusion categories. We were able to prove their classification and came up with an integer sequence that describes their numbers. Those results can be found here.

The Adjunct Hustle

So all of that got me my doctorate. I am neither physicist, nor mathematician, and yet both. I also have some sprinkling of computer scientist in there, but that’s informal. Once I got my doctorate though, I had to get a job and here is where the story is not so happy.

Academic jobs are hard to find especially if you work in a niche area. While trying to find a permanent position, I set myself up on what I call “The Adjunct Hustle” I would teach physics at 2, 3, up to 4 different institutions a semester to make bill money while trying to continue to do research and apply for jobs. This is not something I recommend anyone ever try to do. To keep this short I’ll just say that I’m grateful to University of Texas at Dallas, Collin College, Cedar Valley Community College, and University of North Texas for helping to keep me afloat during those years. I learned a lot about communication and how to talk about ideas from different perspectives. I am absolutely the better for my time spent as a teacher and should I retire from research I would be perfectly happy to have a second career teaching at a community college level.