I am interested in knot theory and low-dimensional geometric topology, and in particular the relationship between knot invariants and the geometry of knot complements.
Talks
- Invited talk — AMS Fall Eastern Sectional Meeting, Special Session on Knots and Braids — October 2026
- Lightning talk — Advances in Geometric and Quantum Topology — July 2026
Publications and Preprints
- Synthetic Geometry in Hyperbolic Simplices with Barry Minemyer. Involve, a Journal of Mathematics. 15 (2023), Issue 5, 885-904. arXiv.
Abstract
Let $\tau$ be an $n$-simplex and let $g$ be a metric on $\tau$ with constant curvature $\kappa$. The lengths that $g$ assigns to the edges of $\tau$, along with the value of $\kappa$, uniquely determine all of the geometry of $(\tau, g)$. In this paper we focus on hyperbolic simplices ($\kappa = -1$) and develop geometric formulas which rely only on the edge lengths of $\tau$. Our main results are distance and projection formulas in hyperbolic simplices, as well as a projection formula in Euclidean simplices. We also provide analogous formulas in simplices with arbitrary constant curvature $\kappa$.
