Heegaard diagrams and holomorphic disks

该提案研究了“Heegaard flower同源性”，这是研究者与Zoltan Szabo合作构建的三维和四维空间的固有特征。这种结构的一种变体，称为“结花同源”，可用于研究三维空间中的结曲线。一个三维空间可以由沿着边界组合在一起的小块组成。该建议的另一个方面涉及如何根据与其组成部分相关的数据重建三维空间的Heegaard flower同调。这种被称为“边界花同源性”的重建过程是与Robert Lipshitz和Dylan Thurston合作研究的。本文旨在进一步强化边界理论，探索边界理论的应用。更好地理解这些结构将导致这些新结构进一步应用于结理论和三维和四维空间的拓扑结构。heeggaard flower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，以一种部分受现代物理学启发的方式研究结理论和低维拓扑中的问题。正因为如此，它处于一个肥沃的智力十字路口，为邻近的学科带来了新的视角，并为解决老问题提供了新的方法。本研究旨在研究边界花同源性，作为研究各种版本Heegaard花同源性和结花同源性的工具。部分建议将从扩展边界理论开始，以包括具有环面边界的三流形的完整（非专门化）Heegaard flower同调。在不同的方向上，边界花同源性将扩展为一个研究和计算结花同源性的工具。这些结构的应用包括从结花同态中研究新的和谐同态。

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.