Integrable cross-ratio maps are solutions to one of the discrete integrable equations on quad-graphs. These maps may be of interest to many mathematicians; just to name a few uses, discrete holomorphic functions, orthogonal circle packings, and polygon recutting are all special cases of integrable cross-ratio maps. The goal of my research is to find an…

A fundamental problem in knot theory is determining when two distinct knot diagrams represent the same knot. This is traditionally addressed through the use of invariants such as the knot group, Alexander polynomial, and Jones polynomial, among others. While the knot group distinguishes prime knots, it is known to be incomplete as a knot invariant.…

Generalizing unknotting number, n-adjacent knots have n crossings such that changing any non-empty subset of them results in the unknot. We determine the 2-adjacent knots through 12-crossings, with one exception. Using Heegaard Floer d-invariants and the Alexander polynomial, we develop a new technique to obstruct 2-adjacency, and we prove conjectures of Ito and Kato regarding…

This talk will begin by constructing a new algebra based on a generalization of the set of relations for Ozsváth-Szabó's bordered Heegaard Floer algebra $B(n,k)$. Then, I will strengthen an existing result by establishing an isomorphism between our new algebra and the algebra $\widetilde{B}(\mathcal{V})$ associated with an arrangement of real affine hyperplanes. If time permits, I’ll…