Low Dimensional Topology and Heegaard Floer homology

低维拓扑和 Heegaard Florer 同调

• 批准号: 1006006
• 负责人: Zoltan Szabo
• 金额: $ 48.7万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1006006&HistoricalAwards=false
• 关键词: Low; Dimensional; Topology; Heegaard; Floer

The main theme of this project is the study of knots, three dimensional manifolds, and smooth 4-manifolds. One of the central tools is Heegaard Floer homology that was developed by Peter Ozsvath and the Principal Investigator. The PI studies new applications of these techniques for various surgery problems for three dimensional manifolds, slice knots, and exotic structures on smooth 4-manifolds.The proposal also deals with topological and computational aspects of Heegaard Floer homology. This involves the study of special multiply pointed Heegaard diagrams to build up a topological version of the theory. In a different direction new advances in Heegaard Floer homology are used to study three-manifolds with simple Floer homologies.The project studies three and four dimensional spaces and central problems in Knot Theory and Low Dimensional Topology. Heegaard Floer homology provides various new tools, such as topological invariants and surgery formulas, for low-dimensional mathematical objects. This theory is closely related 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.

该项目的主题是对结、三维流形和光滑 4 流形的研究。 核心工具之一是由 Peter Ozsvath 和首席研究员开发的 Heegaard Floer 同源性。 PI 研究这些技术在三维流形、切片结和光滑 4 流形上的奇异结构的各种手术问题中的新应用。该提案还涉及 Heegaard Floer 同调的拓扑和计算方面。这涉及对特殊多点 Heegaard 图的研究，以建立该理论的拓扑版本。 在不同的方向上，Heegaard Floer 同调的新进展被用来研究具有简单 Floer 同调的三流形。该项目研究三维和四维空间以及结理论和低维拓扑中的中心问题。 Heegaard Floer 同调为低维数学对象提供了各种新工具，例如拓扑不变量和手术公式。该理论与源于数学和数学物理之间相互作用的规范理论不变量密切相关。 预计 Heegaard Floer 同调性的进展也将导致人们更好地理解这些规范理论不变量。