Four-manifolds, Dehn surgery, and Floer homology

Geometric topology is a field which studies the intrinsic structure of shapes. The primary investigator (PI) will focus on properties of knots, three- and four-dimensional shapes, which appear in aspects of biology, physics, and computation. The PI will develop tools which help to measure the complexity of these objects, as well as constructing exciting new topological structures in four dimensions. Broader impacts will be through mentoring at the undergraduate and graduate levels, organizing seminars, and disseminating mathematical resources. The PI will produce new exotic smooth four-manifolds using hands-on constructions and invariants from Heegaard Floer homology. He will also study surfaces in four-manifolds, including finding exotic embeddings of surfaces in four-manifolds and constructing new non-orientable surfaces in the four-sphere. The PI will apply instanton gauge theory to a number of topological problems. For example, he will make progress on the Cosmetic Surgery Conjecture using the Chern-Simons filtration; show that nullhomotopic knots are determined by their complements, answering a question of Boileau from the famous Kirby Problem list; and constrain the Heegaard genus of three-manifolds through homology cobordism invariants. The PI will establish foundational results about Heegaard Floer homology, including giving a topological characterization of Heegaard Floer homology solid tori and establishing structural results about the module structure on Heegaard Floer homology. The PI will also continue mentorship of PhD students; organizing seminars and working groups with the goal of enhancing the mathematical community in the Research Triangle region; and writing Mathematical Reviews. 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.