Overview
My research is in low-dimensional topology, the study of the shapes of 3- and 4-dimensional spaces (manifolds) and of curves and surfaces contained therein. Classifying smooth 4-dimensional manifolds, in particular, has been a deep challenge for topologists for many decades; unlike in higher dimensions, there is not enough "wiggle room" to turn topological problems into purely algebraic ones. Many of my projects reveal new complications in the topology of 4-manifolds, particularly related to embedded surfaces. My main tools come from Heegaard Floer homology, a powerful package of invariants derived from symplectic geometry. I am also interested in the interrelations between different invariants of knots in 3-space, particularly the connections between knot invariants arising from gauge theory and symplectic geometry and those coming from representation theory.
Current Appointments & Affiliations
Associate Professor of Mathematics
·
2020 - Present
Mathematics,
Trinity College of Arts & Sciences
Recent Publications
A surgery formula for knot Floer homology
Journal Article Quantum Topology · January 1, 2024 Let K be a rationally null-homologous knot in a 3-manifold Y, equipped with a non-zero framing λ, and let Yλ(K) denote the result of λ-framed surgery on Y. Ozsváth and Szabó gave a formula for the Heegaard Floer homology groups of Yλ ... Full text CiteNew constructions and invariants of closed exotic 4-manifolds
Preprint · July 16, 2023 Link to item CiteA note on rationally slice knots
Journal Article New York Journal of Mathematics · January 1, 2023 Kawauchi proved that every strongly negative amphichiral knot (Formula Presented) bounds a smoothly embedded disk in some rational homology ball VK, whose construction a priori depends on K. We show that VK is inde-pendent of K up to ... CiteRecent Grants
Four-Manifolds and Categorification
ResearchPrincipal Investigator · Awarded by National Science Foundation · 2022 - 2026Low-Dimensional topology, Floer Homology, and Categorification
ResearchPrincipal Investigator · Awarded by National Science Foundation · 2017 - 2022View All Grants
Education, Training & Certifications
Columbia University ·
2010
Ph.D.
Harvard University ·
2005
A.B.