There's a close connection between preprojective algebras associated to Dynkin diagrams and the convex geometry of Coxeter groups (as detailed in this paper of Hugh's). Specifically, brick modules of the preprojective algebra biject with shards of the Coxeter group. This paper proves a similar result for preprojective algebras of non-Dynkin diagrams, though we have to impose a couple additional conditions on bricks. My main contributions to this paper were showing that the extra conditions we impose on bricks actually are necessary (by using a computer search to find a brick which satisfied one but not the other) and fixing up the technical details of the framework we use for dealing with non-simply-laced diagrams.
One thing that almost all infinite families of finite and affine Coxeter groups have in common is that their Coxeter diagrams are obtained by inserting a long path into a fixed diagram. This paper looks at this "insert a long path" construction for any diagram, and shows how some related constructions (the root poset and the arrangement of shards) stabilize. The original inspiration for this came from wanting a nice uniform description of the bricks of type D preprojective algebras, though this paper is purely combinatorial.
This came out of the 2015 REU at the University of Washington. The basic idea is to build up a general foundation for talking about the sandpile group or critical group of a graph, allowing for coefficient rings other than Z, and to connect this to a bunch of other things. I wasn't directly involved with much of this paper, but I computed a lot of examples and worked on the material that became section 7.
My senior thesis, an exposition on dessins d'enfants. Written under the guidance of Professor Jim Morrow. The plan was to explain the arguments in Ernesto Girondo and Gabino González-Diez's Introduction to Compact Riemann Surfaces and Dessins d'Enfants in a way I could understand better.
The number of spanning trees of an undirected Cayley graph of the cyclic group of order n is always n times a perfect square. This note arose out of trying to find a nice algebraic explanation of this fact using sandpile groups.