Applications and Refinements of Floer Homology

Floer同调性的应用和改进

• 批准号: 0803465
• 负责人: Ciprian Manolescu
• 金额: $ 14.1万
• 依托单位: Columbia University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-06-01 至 2008-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0803465&HistoricalAwards=false
• 关键词: Applications; Refinements; Floer; Homology

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.

本文主要研究Floer同调及其在低维拓扑中的应用。Floer同调是莫尔斯理论的一个无限维版本，它已被用来构造各种不变量的结，3流形，4流形等反过来，这些不变量可以回答微妙的问题，有关各自的拓扑对象。一个来源的不变量与众多的拓扑应用是Heegaard Floer理论。例如，纽结的Heegaard Floer不变量（称为纽结Floer同源性）能够检测纽结的属。最初，所有的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-弗洛尔同调被给出了几种组合描述。这个项目的一个重点是找到组合描述的Heegaard Floer三，四流形不变量。在其他方面，PI将致力于寻找结Floer同调和其他结不变量之间的联系，如Khovanov-Rozansky同调;几何解释Khovanov-Rozansky同调;发展Floer同伦理论;并使用平坦连接的模空间构造新的三流形的Floer理论不变量。Floer同调在拓扑量子场论的构造中起着核心作用。这些是数学物理学中用来发展宇宙量子理论的玩具模型。它们也是拓扑学家感兴趣的，拓扑学家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，这在四维空间中特别困难。Floer同调和相关的不变量是检测四维形状性质的最有用的工具。因为我们的宏观时空有四维，这是量子物理学家和宇宙学家寻找宇宙几何模型的重要输入。此外，最近Floer同源性在生物学中发现了令人惊讶的应用，更确切地说，在DNA打结的分析中。