Heegaard Diagrams and Holomorphic Disks

这个项目在拓扑学研究“Heegaard Floer同调”，这是一个不变量的三维和四维空间。这种构造的一个变体，称为“结弗洛尔同源”，提供了新的方法来研究三维空间中的打结曲线，并回答有关此类对象的经典问题。Heegaard Floer同调和相关的纽结不变量都有两个版本：一个简化的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。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，为邻近的学科带来新的视角。Heegaard Floer同源性继续为解决老问题提供新的方法。PI与Robert Lipshitz和Dylan Thurston合作，定义了“边界Floer同调”（bordered Floer homology），这是一种将Heegaard Floer同调从三维空间分解为简单的组成部分，重建简化版本的技术。在与Szabo的合作中，PI使用类似的方法来描述结Floer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，并将其扩展到更丰富的版本。 这个项目的目的是提供一个更好的概念理解这些不变量，并提供有效的计算技术来计算它们。该项目旨在研究边界Floer同源性，作为研究Heegaard Floer同源性和knot Floer同源性的各种版本的工具。 该项目的一部分将开始扩展边界理论，包括完整的（非专门化的）Heegaard Floer同调与环面边界的三流形。建立了大部分的代数基础，这一理论，PI现在将转向分析方面，以期建设模块的三个流形与环面边界，和配对定理计算（非专门化）Heegaard Floer同源性的三个流形分解沿着一个环面，在模块assocaited件。在不同的方向，边界Floer同源性扩展到一个工具，用于研究和计算（非专业化）结Floer homology.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

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.