CAREER: Heegaard Floer homology and low-dimensional topology

Low-dimensional topology is the study of spaces of dimensions three and four. A fundamental goal is to understand what properties remain unchanged under stretching or bending; for example, the number of holes is an invariant property of a space. This project aims to make progress on topological questions using an algebraic invariant called Heegaard Floer homology. In parallel with its research goals, the project includes mentoring and outreach activities. In addition, the PI plans to organize workshops which will provide speaking and collaboration opportunities for junior researchers. Heegaard Floer homology plays a prominent role in modern low-dimensional topology. In particular, the theory has been influential in the study of knot concordance, leading to the construction of fruitful invariants and structural results on the concordance group. Building on recent developments, the PI plans to construct new knot concordance invariants and four-ball genus bounds. In a different direction, the PI will study spectral sequences involving Heegaard Floer homology and Khovanov homology, a combinatorially-defined link invariant. Finally, the PI aims to give new bounds on difficult-to-compute knot invariants such as unknotting number. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.