The proposed research is on Floer homology and its applications to 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, knot Floer homology has been given several combinatorial descriptions. One focus of this project is to find combinatorial descriptions for the Heegaard Floer three- and four-manifold invariants as well. In other directions, the PI will work on finding connections between knot Floer homology and other knot invariants, such as the Khovanov-Rozansky homologies; intepreting the Khovanov-Rozansky homologies geometrically; developing Floer homotopy theory; and constructing new Floer-theoretic invariants of three-manifolds using moduli spaces of flat connections.
Floer homology plays a central role in the construction of topological quantum field theories. These are toy models used in Mathematical Physics to develop quantum theories about the universe. They are also of interest to topologists, who study 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. Floer homology and the associated invariants are some of the most useful tools for detecting properties of four-dimensional shapes. Because our macroscopic space-time has four dimensions, this is an essential input for quantum physicists and cosmologists looking for geometric models for the universe. Furthermore, recently Floer homology has found surprising applications in biology, more precisely in the analysis of DNA knotting.
