My research is in geometric topology. Below are my ongoing and past projects.
Totally Symmetric Sets
Totally Symmetric Sets are subsets of groups satisfying properties regarding commutativity and conjugation. They aid classification of homomorphisms. It’s of particular interest in braid groups and mapping class groups. Much 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, Involve, a Journal of Mathematics 14.5 (2022): 853-870.
Bounding Knot Volumes via Subdivision
We can associate to hyperbolic knots an invariant called hyperbolic volume. At the SMALL Knot Theory REU, a group worked under Prof. Colin Adams on bounding volume using symmetries of the knot complement, building upon the work of Agol-Storm-Thurston and others.
- 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, European Journal of Mathematics 9, 100 (2023).
- Lower Bounds on Volumes of Hyperbolic 3-Manifolds via Decomposition, Submitted for Review
Topological Polynomials and the Twisted Rabbit
Starting Summer 2021, I’m working on a project with Caleb Partin under 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. The project is ongoing.