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.
- Paper with A. Chudnovsky, K. Kordek, and C. Partin:
- Finite Quotients of Braid Groups, Geometriae Dedicata 207, 409-416 (2020)
- Paper with K. Kordek and C. Partin:
- Upper Bounds for Totally Symmetric Sets, Accepted for publication at Involve.
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.
- Paper with everyone (C. Adams, M. Capovilla-Searle, D. Li, J, McErlean, A. Simons, N, Stewart, and X. Wang):
- Generalized Augmented Cellular Alternating Links in Thickened Surfaces are Hyperbolic, preprint
- Lower Bounds on Volumes of Hyperbolic 3-Manifolds via Decomposition, in preparation
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.