Floer homology and surfaces in 3- and 4-manifolds

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This project investigates a variety of questions in low-dimensional topology, the study of the global shapes of 3- and 4-dimensional spaces, which has a wide variety of applications ranging from astrophysics (the shape of the universe) to biochemistry (the knotting of DNA molecules) to theoretical physics. Much of the interesting topological information about a 3- or 4-dimensional space is captured in the types of surfaces that are embedded in that space, and this project aims to obtain restrictions on the types of such surfaces that can be found in various classes of spaces. The PI's primary tools come from Heegaard Floer homology, a package of invariants that has become one of the dominant areas of research in low-dimensional topology since its introduction in the early 2000s by Ozsváth and Szabó. These techniques bring together several different fields of mathematics, including representation theory, homological algebra, analysis, and mathematical physics, and the PI hopes to elucidate the connections between these different areas and expand the discourse among researchers in these fields. The specific goals of the project are (1) to understand the relationship between the Heegaard Floer homology of a 3-manifold and topological properties such as the existence of incompressible surfaces, taut foliations, and left-orderings on the fundamental group; (2) to solve concrete topological problems regarding smoothly embedded surfaces in 4-manifolds, in part by using new computational techniques made possible by bordered Floer homology; and (3) to use the recently-developed spanning tree models for knot Floer homology and Khovanov homology to shed light on the long-conjectured relationship between the two theories.