Applications and Refinements of Floer Homology

Floer同调性的应用和改进

• 批准号: 0852439
• 负责人: Ciprian Manolescu
• 金额: --
• 依托单位: University of California-Los Angeles
• 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 2011-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0852439&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.

拟议的研究是关于浮动同源性及其在低维拓扑中的应用。 Floer同源性是Morse理论的无限尺寸版本，用于构建各种不变的结，3个manifolds，4-manifolds等。反过来，这些不变的人可以回答有关相应拓扑对象的细微问题。 Heegaard Floer理论是具有众多拓扑应用的不变性来源。例如，针对结的Heegaard漂浮物不变（称为结式同源性）能够检测结的属。最初，所有Heegaard浮子不变性均根据对称产品中的伪形曲线定义。最近，打结浮子同源性已获得了几种组合描述。该项目的一个重点是为Heegaard Floer 3和四个manifold不变式找到组合描述。在其他方向上，PI将致力于寻找结式浮子同源性与其他结中的连接，例如Khovanov-Rozansky同源性；几何地插图Khovanov-Rozansky同源物； 开发浮动同质理论；并使用平坦连接的模量空间来构建三个manifolds的新浮动理论不变。这些是在数学物理学中用于开发有关宇宙的量子理论的玩具模型。拓扑师也很感兴趣，他们研究了各个维度的空间形状。一个重要的问题是对这些形状的分类，这在四个维度上特别困难。浮点同源性和相关的不变性是检测四维形状属性的一些最有用的工具。由于我们的宏观时空具有四个维度，因此对于寻找宇宙几何模型的量子物理学家和宇宙学家来说，这是必不可少的输入。此外，最近的浮子同源性发现了生物学中令人惊讶的应用，更确切地说是在DNA结的分析中。