CAREER: Floer-theoretic approaches to low-dimensional topology

Heegaard Floer同调是在低维拓扑中研究的对象的不变量族，包括三维流形、四维余边线、三维流形中的纽结和链环以及三维流形上的接触结构。带边Heegaard Floer同调是Heegaard Floer同调的推广，具有很好的粘性。(粗略地说，Heegaard Floer同调形成了(3+1)维拓扑场论，而有界Heegaard Floer同调是该场论的(2+1+1)维推广。)这个项目寻求进一步发展边界Heegaard Floer同调。最终目标是找到计算Heegaard Floer同调(和Seiberg-Witten不变量)的实用方法；Heegaard Floer理论的公理刻画；以及能够回答其他拓扑问题的Heegaard Floer理论的变体。正如高中几何中所教授的(古希腊人知道的)，三维空间--我们生活的空间--可以分割成一族不相交(平行)的平面。它也可以被切成一系列圆盘，其边界在标准的未打结的圆圈上(除了一个圆盘将缺少一个点)：将这些圆盘想象成泡泡棒上的肥皂泡，而这个家族是通过将肥皂吹得更硬或更小而产生的。如果你从一个纽结的圆K开始，那么可能也可能不可能将空间分割成边界在K上的曲面族。因此，很自然地会问：对于哪个纽结K，有这样的曲面族？有这样一个家庭的结被称为“纤维结”，而找到方法来辨别一个结是否是纤维结被证明是既有趣又困难的。令人惊讶的是，回答这类问题的一些最有效的工具与现代数学物理、量子场论和弦理论的思想密切相关，不仅适用于3维问题，也适用于4维问题。一个叫做Heegaard Floer同调的不变量族就是这样一个工具。这个项目寻求进一步了解Heegaard Floer同调的结构，目的是开发更有效的计算方法，并找到适合不同问题的其他相关工具。

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.