Heegaard Diagrams and Holomorphic Disks

这个拓扑学项目研究的是“Heegaard Floer同调”，它是三维和四维空间的不变量。这种结构的一个变种，称为“纽结Floer同调”，为研究三维空间中的纽结曲线和回答关于这类物体的经典问题提供了新的方法。Heegaard Floer同调和相关的纽结不变量都有两个版本：简化的版本和更丰富的版本。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该理论的四个维度。Heegaard Floer同调汇集了不同数学学科的工具，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，其方式部分受到现代物理学的启发。因此，它位于一个肥沃的知识十字路口，为邻近的学科带来了新的视角。Heegaard Floer同源理论继续为解决老问题提供新的方法。该奖项为研究生从事相关研究提供支持。在与罗伯特·利普希茨和迪伦·瑟斯顿的合作下，PI定义了“边界Floer同调”，这是一种从被分解成简单组件的三维空间重建Heegaard Floer同调的简化版本的技术。在与Szabo的合作中，PI使用了类似的方法来描述纽结Floer同调的简化版本。这个项目致力于进一步开发用于三维空间和结的边界技术，将它们扩展到更丰富的版本。这个项目的目的既是为了更好地从概念上理解这些不变量，也是为了给出计算它们的有效计算技术。该项目旨在研究边界Floer同调，作为研究Heegaard Floer同调和纽结Floer同调的不同版本的工具。该项目的一部分将通过扩展边界理论来包括具有环面边界的三流形的完全(非特殊的)Heegaard Floer同调。在建立了这个理论的大部分代数基础之后，PI现在将转向分析方面，以期构造具有环面边界的三维流形的模，以及一个配对定理，用于计算沿环面分解的三流形的(非特殊的)Heegaard Floer同调，其依据是与片断相关联的模。在不同的方向上，边界Floer同源被扩展到一个用于研究和计算结Floer同源的工具。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.