Heegaard Floer homology, knots, and three-manifolds

Heegaard Floer 同调、结和三流形

• 批准号: 1309152
• 负责人: Zoltan Szabo
• 金额: $ 45.24万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2017-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1309152&HistoricalAwards=false
• 关键词: Heegaard; Floer; homology; knots; three

The central themes of this project are Low Dimensional Topology and Knot Theory. One of the main techniques involved is Heegaard Floer homology, developed by Peter Ozsvath and the principal investigator. The investigator will study knots, three-manifolds and smooth 4-manifolds by using recent advances and developing new techniques. In a different direction the PI investigates the relationship between different homological invariants for knots and links. The PI also studies foundational problems in Heegaard Floer homology and knot Floer homology.The project studies various three-dimensional spaces (three-manifolds), and also embedded loops inside three-manifolds. Heegaard Floer homology, developed by the principal investigator and Peter Ozsvath, plays a central role in the project. This theory has close ties to gauge theoretical invariants that originated from interactions between Mathematics and Mathematical Physics. Some of the recent results in Knot and Heegaard Floer homologies are also related to different mathematical fields, such as combinatorial constructions and algebraic techniques. A better understanding of the central problems in the project are expected to lead to further discoveries and new bridges between different mathematical fields. The project is also aimed to enhance graduate student research and introduce graduate students to the areas of Low Dimensional Topology and Heegaard Floer invariants.

这个项目的中心主题是低维拓扑和纽结理论。其中一项主要技术是Heegaard Floer同源性，由Peter Ozsvath和首席研究员开发。研究者将利用最新的进展和开发新的技术来研究结，三流形和光滑4流形。在不同的方向上，PI研究结和链接的不同同调不变量之间的关系。PI还研究Heegaard Floer同调和knot Floer同调的基础问题。该项目研究各种三维空间（三维流形），以及三维流形内的嵌入环。Heegaard Floer同源性由首席研究员和Peter Ozsvath开发，在该项目中发挥了核心作用。这一理论与源于数学与数学物理相互作用的规范理论不变量有着密切的联系。Knot和Heegaard Floer同调的一些最新结果也与不同的数学领域有关，例如组合构造和代数技巧。更好地理解该项目中的中心问题有望导致进一步的发现和不同数学领域之间的新桥梁。该项目还旨在加强研究生的研究，并向研究生介绍低维拓扑和Heegaard Floer不变量。