Research

I’m involved in math research in topology. Below are my ongoing and past projects.

Totally Symmetric Sets

Totally Symmetric Sets are subsets of a group satisfying two defining properties regarding commutativity and conjugation. They are preserved under homomorphisms and are used to classify homomorphisms between groups. It’s of particular interest in braid groups and mapping class groups. The majority of the math is done at the Georgia Tech REU under the mentorship of Prof. Dan Margalit and Dr. Kevin Kordek.

Bounding Knot Volumes via Subdivision

We can associate a notion of volume to a hyperbolic knot. It’s a powerful invariant for knot complements. At the SMALL Knot Theory REU, a group of us worked under the guidance of Prof. Colin Adams on bounding volume using symmetries of the knot complement, building upon the work of Agol-Storm-Thurston and various other papers.

Topological Polynomials and the Twisted Rabbit

This summer, I’m working on a project with Caleb Partin under the mentorship of Prof. Dan Margalit on the n-eared twisted rabbit problem using techniques in topology and dynamical systems. The work builds on papers of Bartoldi-Nekrashevich and Belk-Lanier-Margalit-Winarski. There will probably be papers forthcoming about the Twisted n-eared Rabbit problem and/or structural theorems regarding the Liftable Mapping Class Groups associated to different topological polynomials.