My research interests lie in the intersection of combinatorics and algebra. I am interested in the combinatorics of Coxeter groups, especially the combinatorics of reduced
expressions (sometimes called reduced words or decompositions) for elements of a Coxeter group. One popular way to study Coxeter groups is to study the collection \(\mathcal{R}(w)\) of reduced expressions
for a fixed group element \(w\). A classic theorem (usually attributed to Matsumoto or Tits) tells us that this set has a surprising structure: every two reduced expressions for the same group element
are related by a sequence of so-called braid and commutation moves. In other words, we can encode the structure of \(\mathcal{R}(w)\) with a connected graph (the Matsumoto graph), where reduced expressions are related by an edge if they differ by
either a braid relation or a commutation relation. This graph reveals surprising connections to other areas of mathematics. For instance, in the case that \(w\) is a permution, the Matsumoto graph is the Hasse diagram for a lattice.
The focus of my current research is to classify the maximal components of the graph that only involve braid moves.