Research

My interests are primarily in algebraic combinatorics, but I also have work in extremal combinatorics, graph theory, and discrete geometry. My research in algebraic combinatorics connects with topology, polyhedral geometry, representation theory, billiards, free probability, classical probability, and integrable probability. Here is my research statement.  

I wrote my dissertation about a specific combinatorial operator on the symmetric group called the stack-sorting map, which I found to have striking and deep connections with free probability. 

Here is a biref summary of some of my recent interests: 

• I have initia a topic that I call combinatorial billiards, which draws inspiration from dynamical algebraic combinatorics and mathematical billiards. This topic concerns billiard systems that are rigid and discretized. These systems can usually be modeled combinatorially or algebraically, and they are often defined using the Coxeter arrangement of a Coxeter group as the set of reflective hypersurfaces. 

• I have been exploring random subwords in Coxeter groups and Hecke algebras. This leads to fascinating new stochastic processes that are analogues and deformations of more classical processes. 

• I am interested in the connections among quiver representation theory, polyhedral geometry, and lattice theory. I like to prove that certain interesting posets are lattices.  

• I study triangulations of Coxeter permutohedra that come from quiver representation theory, along with their algebraic and topological consequences. I am interested in using these triangulations to obtain elegant, type-uniform, explicit presentations of pure braid groups. 

My curriculum vitae contains a (usually) up-to-date list of my publications and preprints, all of which are available on arXiv.