Heegaard diagrams and holomorphic disks

Heegaard 图和全纯盘

• 批准号: 1405114
• 负责人: Peter Ozsvath
• 金额: $ 35.31万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-07-01 至 2019-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1405114&HistoricalAwards=false
• 关键词: Heegaard; diagrams; holomorphic; disks

The proposal studies "Heegaard Floer homology," which is an invaraint for three-and four-dimensional spaces constructed by the investigator in collaboration with Zoltan Szabo. A variant of this construction, called "knot Floer homology", can be used to study knotted curves in three-dimensional space. A three-dimensional space can be built up out of smaller pieces which fit together along their boundaries. A further aspect of the proposal deals with how to reconstruct the Heegaard Floer homology of a three-dimensional space in terms of data associated to its component pieces. This reconstruction procedure, entitled "bordered Floer homology", is studied in collaboration with Robert Lipshitz and Dylan Thurston. The proposal aims to further strengthen the bordered theory, and explore its applications. A better understanding of these constructions will lead to further applications of these new constructions to knot theory and the topology of three- and four-dimensional spaces. Heegaard Floer homology brings together tools from various mathematical disciplines, including symplectic geometry, analysis, and homological algebra to study problems in knot theory and low-dimensional topology, in a way which was partially inspired by modern physics. As such, it lies at a fertile intellectual crossroads, bringing new perspectives to neighboring subjects, and providing novel methods for attacking old problems.The proposal aims to study bordered Floer homology as a tool for studying various versions of Heegaard Floer homology and knot Floer homology. Part of the proposal will start by extending the bordered theory to include the full (unspecialized) Heegaard Floer homology for three-manifolds with torus boundary. In a different direction, bordered Floer homology will be extended to a tool for studying and computing knot Floer homology. Applications of these structures include a study of new concordance homomorphisms from knot Floer homology.

该提案研究了“Heegaard Floer同调”，这是一个由研究者与Zoltan Szabo合作构建的三维和四维空间。这种构造的一个变体，称为“knot Floer homology”，可以用来研究三维空间中的打结曲线。 一个三维空间可以由沿着沿着它们的边界组合在一起的小块组成。另一方面的建议涉及如何重建Heegaard弗洛尔同源的三维空间的数据相关的组成部分。 这个重建过程，题为“边界Floer同源性”，研究合作与罗伯特Lipshitz和迪伦瑟斯顿。该建议旨在进一步加强边界理论，并探索其应用。更好地理解这些结构将导致进一步应用这些新的结构，纽结理论和三维和四维空间的拓扑结构。Heegaard Floer同调汇集了各种数学学科的工具，包括辛几何，分析和同调代数，以研究纽结理论和低维拓扑中的问题，其部分灵感来自现代物理学。因此，它位于一个肥沃的智力十字路口，带来了新的视角，相邻的科目，并提供新的方法来攻击老problems.The建议的目的是研究边界Floer同调作为一种工具，研究各种版本的Heegaard Floer同调和结Floer同调。 部分提案将开始扩展边界理论，包括完整的（非专门化）Heegaard Floer同源的三个流形与环面边界。在另一个方向上，加边Floer同调将被扩展为研究和计算纽结Floer同调的工具。这些结构的应用包括研究新的协调同态从结弗洛尔同源。