Heegaard Floer homology is family of invariants of objects studied in low-dimensional topology, including closed 3-manifolds, 4-dimensional cobordisms, knots and links in 3-manifolds, and contact structures on 3-manifolds. Bordered Heegaard Floer homology is an extension of Heegaard Floer homology to 3-manifolds with boundary, with good gluing properties. (Roughly, Heegaard Floer homology forms a (3+1)-dimensional topological field theory, and bordered Heegaard Floer homology is a (2+1+1)-dimensional extension of this field theory.) This project seeks to further develop bordered Heegaard Floer homology. The ultimate goals are to find practical ways of computing Heegaard Floer homology (and the Seiberg-Witten invariant); an axiomatic characterization of Heegaard Floer theory; and variants on Heegaard Floer theory capable of answering other topological questions.
As is taught in high-school geometry (and was known to the ancient Greeks), three-dimensional space -- the space we live in -- can be sliced into a family of non-intersecting (parallel) planes. It can also be sliced into a family of disks with boundary on the standard, unkotted circle (except that one disk will have a point missing): think of the disks as soap bubbles on a bubble-wand, and the family as coming from blowing on the soap harder or less hard. If you start with a knotted circle K, it may or may not be possible to slice space into a family of surfaces with boundary on K. So, it is natural to ask: for which knots K is there such a family of surfaces? Knots for which there is such a family are called "fibered knots", and finding ways to tell if a knot is fibered turns out to be both interesting and hard. Surprisingly, some of the most effective tools for answering this kind of question are closely related to ideas from modern mathematical physics, quantum field theory, and string theory, and have applications not just to 3-dimensional questions but also to 4-dimensional ones. A family of invariants called Heegaard Floer homology is one such tool. This project seeks to further understand the structure of Heegaard Floer homology, with the goals of developing ways to compute it more efficiently and finding other related tools adapted to different problems.
