Heegaard Floer homology and Low Dimensional Topology

Heegaard Florer 同调和低维拓扑

• 批准号: 0704053
• 负责人: Zoltan Szabo
• 金额: $ 62.7万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0704053&HistoricalAwards=false
• 关键词: Heegaard; Floer; homology; Low; Dimensional

The main theme of this project is the application of Heegaard Floer homologyand gauge theory for problems in low-dimensional topology and knot theory. New surgery formulas in Heegaard Floer homology allow the PI to study applications for surgery problems in three-manifolds, and surgery characterization for knots.In a different direction Floer homology is used to investigate unknotting numbers for knots in the 3-sphere. The project includes foundational problems for knot Floer homology and the study of combinatorial techniques as well. In a different direction, gauge theoretical techniquesare used in the search for small exotic 4-manifolds. The project studies three and four dimensional spaces and some central problemsin knot theory and low dimensional Topology. One of the main tools is a relatively new theory called Heegaard Floer homology. This theory has close ties to gauge theoretical invariants that originated from interactions between Mathematics and Mathematical Physics. It is expected that advances in Heegaard Floer homology will lead to a better understanding of these gauge theoretical invariants as well.Another integral part of the project is the study of unknotting numbers for knots.While this is a classical invariant in knot theory, the unknotting process is also relevant in the study of knotted DNA. A way to measure the complexity is to study how many times the strand has to be cut by an enzyme and recombine itself to be unknotted.

该项目的主题是 Heegaard Floer 同调和规范理论在低维拓扑和纽结理论问题中的应用。 Heegaard Floer 同源性中的新手术公式允许 PI 研究三流形中手术问题的应用以及结的手术表征。在不同的方向上，Floer 同源性用于研究 3 球体中结的解结数。该项目包括结弗洛尔同源性的基本问题以及组合技术的研究。在不同的方向上，规范理论技术被用来寻找小型奇异的 4 流形。该项目研究三维和四维空间以及纽结理论和低维拓扑中的一些核心问题。主要工具之一是一种相对较新的理论，称为 Heegaard Floer 同源性。该理论与衡量源自数学和数学物理之间相互作用的理论不变量有着密切的联系。预计 Heegaard Floer 同调性的进展也将导致对这些规范理论不变量的更好理解。该项目的另一个组成部分是对结的解结数的研究。虽然这是结理论中的经典不变量，但解结数该过程也与打结 DNA 的研究相关。衡量复杂性的一种方法是研究链必须被酶切割多少次并重新组合才能解开。