Invariants in Low-Dimensional Topology

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The main topic of this project is Floer homology and related invariants in low-dimensional topology. Floer homology is an infinite dimensional version of Morse theory which has been used to construct various invariants of knots, 3-manifolds, 4-manifolds, etc. In turn, these invariants can answer subtle questions about the respective topological objects. One source of invariants with numerous topological applications is Heegaard Floer theory. For example, the Heegaard Floer invariant for knots (called knot Floer homology) is able to detect the genus of a knot. Originally, all the Heegaard Floer invariants were defined in terms of pseudo-holomorphic curves in symmetric products. Recently, the Heegaard Floer invariants of knots, 3-manifolds and 4-manifolds have all been given combinatorial descriptions, based on grid diagrams. One focus of this project is to improve our understanding of these combinatorial descriptions. The PI will also work on finding connections between knot Floer homology and other knot invariants, such as the Khovanov-Rozansky homologies; on extending Heegaard Floer theory to manifolds with corners; and on constructing new Floer-theoretic invariants of three-manifolds using moduli spaces of flat connections.The proposed research is on topology, an area of mathematics that studies geometric objects such as curved spaces (manifolds), and knots in space. One useful method of studying manifolds and knots is through topological quantum field theories (TQFT's), which are certain toy models used in Mathematical Physics to explore quantum theories about the universe. TQFT's are of interest to topologists because they contain information about the possible shapes of space in various dimensions. An important problem is the classification of these shapes, and this is particularly difficult in four dimensions. Because our macroscopic space-time has four dimensions, the properties of four-dimensional shapes are an essential input for quantum physicists and cosmologists looking for geometric models for the universe. The goal of this project is to study the TQFT's which hold the most promise for our understanding of four-dimensional shapes.