Applications and Refinements of Floer Homology

本文的研究内容是Floer同调及其在低维拓扑中的应用。Floer同调是Morse理论的无限维版本，已被用来构造纽结、3-流形、4-流形等的各种不变量。反过来，这些不变量可以回答关于各自拓扑对象的微妙问题。具有大量拓扑应用的不变量的一个来源是Heegaard Floer理论。例如，纽结的Heegaard Floer不变量(称为纽结Floer同调)能够检测纽结的亏格。最初，所有Heegaard Floer不变量都是用对称乘积中的伪全纯曲线定义的。最近，纽结-Floer同调给出了几种组合刻画。这个项目的一个重点是找到Heegaard Floer三流形和四流形不变量的组合描述。在其他方面，PI将致力于寻找纽结Floer同调与其他纽结不变量之间的联系，如Khovanov-Rozansky同调；从几何上解释Khovanov-Rozansky同调；发展Floer同伦理论；以及利用平坦连通的模空间构造新的三维流形的Floer理论不变量。Floer同调在拓扑量子场论的构建中起着核心作用。这些是数学物理中用来发展关于宇宙的量子理论的玩具模型。它们也引起了拓扑学家的兴趣，他们研究空间在不同维度上的可能形状。一个重要的问题是这些形状的分类，这在四个维度上尤其困难。Floer同调及其相关的不变量是检测四维形状性质的一些最有用的工具。因为我们的宏观时空有四个维度，这是量子物理学家和宇宙学家寻找宇宙几何模型的必要输入。此外，最近弗洛尔同源在生物学中发现了令人惊讶的应用，更准确地说，在DNA打结的分析中。

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.