Heegaard Diagrams and Holomorphic Disks

这个拓扑学项目研究了“Heegaard flower同调”，这是三维和四维空间的不变量。这种结构的一种变体，称为“结花同源”，为研究三维空间中的结曲线和回答有关此类物体的经典问题提供了新的方法。Heegaard flower同调和相关的结不变量都有两个版本：一个简化版本和一个更丰富的版本。简化的版本提供了有趣的三维信息，但更丰富的版本也揭示了理论的四维方面。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。eggaard flower同源性继续为解决老问题提供新方法。该奖项为研究生从事相关研究提供支持。在与Robert Lipshitz和Dylan Thurston的合作中，PI定义了“边界花同调”，这是一种从分解成简单组件的三维空间重建Heegaard花同调的简化版本的技术。在与萨博的合作中，PI使用了类似的方法来描述结花同源性的简化版本。该项目涉及进一步开发三维空间和结的边界技术，将它们扩展到更丰富的版本。本项目旨在对这些不变量提供更好的概念理解，并提供有效的计算技术来计算它们。本项目旨在研究边界花同源性，作为研究各种版本的Heegaard花同源性和knot花同源性的工具。项目的一部分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。在建立了该理论的大部分代数基础之后，PI现在将转向分析方面，着眼于构造具有环面边界的三流形的模块，以及计算沿环面分解的三流形的（非专门化）Heegaard flower同调的配对定理，根据与块相关的模块。在不同的方向上，边界花同源性被扩展到一个研究和计算（非专业）结花同源性的工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project in topology studies "Heegaard Floer homology," which is an invariant for three-and four-dimensional spaces. A variant of this construction, called "knot Floer homology," provides new methods for studying knotted curves in three-dimensional space and answering classical questions about such objects. Both Heegaard Floer homology and the associated knot invariant come in two versions: a simplified one, and a richer one. The simplified version provides interesting three-dimensional information, but the richer version also sheds light on four-dimensional aspects of the theory. 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. Heegaard Floer homology continues to provide novel methods for attacking old problems. The award provides support for graduate students to be engaged in related research.In collaboration with Robert Lipshitz and Dylan Thurston, the PI defined "bordered Floer homology," a technique for reconstructing the simplified version of Heegaard Floer homology from a three-dimensional space that is decomposed into simple component pieces. In collaboration with Szabo, the PI used similar methods to describe the simplified version of knot Floer homology. This project deals with further developing bordered techniques for three-dimensional spaces and also for knots, extending them to the richer versions. This project aims to both give a better conceptual understanding of these invariants, and to give effective computational techniques for calculating them. The project aims to study bordered Floer homology as a tool for studying various versions of Heegaard Floer homology and knot Floer homology. Part of the project will start by extending the bordered theory to include the full (unspecialized) Heegaard Floer homology for three-manifolds with torus boundary. Having established most of the algebraic foundations of this theory, the PI will now turn to the analytical aspects, with a view towards constructing modules for three-manifolds with torus boundary, and a pairing theorem for computing the (unspecialized) Heegaard Floer homology of a three-manifold decomposed along a torus, in terms of the modules assocaited to the pieces. In a different direction, bordered Floer homology is extended to a tool for studying and computing (unspecialized) knot Floer homology.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.