Research
My research focuses on the intersection of geometric topology and group theory. In particular I study hyperbolic 3-manifolds, mapping class groups, and non-positively curved groups. Recently I have become interested in applications of geometry and topology to machine learning and data analysis.
Abstract: We prove that for hyperbolic fibered knots in any closed, connected, oriented 3-manifold the volume and genus are unrelated. As an application we answer a question of Hirose, Kalfagianni, and Kin about volumes of mapping tori that are double branched covers.
Automorphisms of the k-curve graph. With S. Agrawal, T. Aougab, Y. Chandran, M. Loving, R. Shapiro, and Y. Xiao. Michigan Math Journal, 1(1): 1-39, 2021. [pdf ], [arxiv]
Abstract: Given a natural number k and an orientable surface S of finite type, define the k-curve graph to be the graph with vertices corresponding to isotopy classes of essential simple closed curves on S and with edges corresponding to pairs of such curves admitting representatives that intersect at most k times. We prove that the automorphism group of the k-curve graph of a surface S is isomorphic to the extended mapping class group for all k sufficiently small with respect to the Euler characteristic of S. We prove the same result for the so-called systolic complex, a variant of the curve graph whose complete subgraphs encode the intersection patterns for any collection of systoles with respect to a hyperbolic metric. This resolves a conjecture of Schmutz Schaller.