CAREER: Floer-theoretic approaches to low-dimensional topology

gagaard flower同调是在低维拓扑中研究的对象的不变量族，包括闭合的3流形、4维协阵、3流形中的结点和连杆以及3流形上的接触结构。有边Heegaard花同调是Heegaard花同调对带边界的3流形的推广，具有良好的粘接性质。（大致来说，Heegaard flower同调形成了一个（3+1）维的拓扑场论，而有边Heegaard flower同调是这个场论的（2+1+1）维的扩展。）该项目寻求进一步发展边界Heegaard花同源性。最终目标是找到计算Heegaard flower同调（和Seiberg-Witten不变量）的实用方法；Heegaard flower理论的公理化表征以及heeggaard flower理论的变体，能够回答其他拓扑问题。正如高中几何（古希腊人所知）所教的那样，三维空间——我们生活的空间——可以被切成一系列不相交（平行）的平面。它也可以被切成一个圆盘族，其边界在标准的、不打结的圆上（除了一个圆盘会少一个点）：把圆盘想象成泡泡棒上的肥皂泡，而这个家族则是通过用力或不用力地吹肥皂而产生的。如果你从一个打结的圆K开始，它可能会也可能不会把空间切成一系列边界在K上的曲面，所以，很自然地要问：对于哪个结点K有这样的曲面族？有这样一个家族的结被称为“纤维结”，找到方法来判断一个结是否是纤维结是既有趣又困难的。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，并且不仅适用于三维问题，也适用于四维问题。称为Heegaard flower同源的不变量族就是这样一个工具。该项目旨在进一步了解Heegaard flower同源性的结构，目标是开发更有效地计算它的方法，并找到适用于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.