Floer homology in low-dimensional topology

This project focuses on an area of mathematics called low-dimensional topology, which concerns the study of shapes and spaces in three and four dimensions. The physical universe is modeled on such a space, and indeed many of the tools mathematicians use to study these spaces come from physics. One of these tools, called instanton Floer homology, comes from an area called gauge theory and is related to Maxwell's equations for electricity and magnetism. Another, Heegaard Floer homology, comes from an area called symplectic geometry and is related to classical mechanics. Although these tools come from disparate areas of mathematics and physics, they share striking similarities. One of the PI's central aims is to unify these and other Floer-theoretic tools in ways that explain these similarities, and to use this unification to discover new features of spaces in dimensions three and four. This project will also support the mathematical community through the PI's mentorship of graduate students at Boston College, his organization of the Boston Graduate Topology Seminar, and his work on new problem lists in low-dimensional topology. Several of the most prominent Floer-theoretic invariants of 3-manifolds seem to encode the same information, indicating deep connections between fields like symplectic geometry and gauge theory. The PI aims to unify these invariants in the spirit of Eilenberg and Steenrod’s axiomatization of ordinary homology in the 1950’s. His proposal contains two distinct lines of attack. One of these, the so-called monopole category, will also yield the first cut-and-paste approach to the Seiberg--Witten invariants of closed 4-manifolds. In a more purely topological direction, the PI will use Floer theory in a novel way to study the rich theories of Heegaard splittings and smooth 4-manifold trisections. Finally, the PI will use recent breakthroughs relating the Floer homology of knots with the fixed point dynamics of surface diffeomorphisms to address important open problems in Dehn surgery, like the cabling conjecture and the classification of knots with elliptic surgeries, and to study the question of which knots are detected by their 4-dimensional traces. 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.