Heegaard Diagrams and Holomorphic Disks

这项国家科学基金会奖支持研究开发新的工具来研究三维和四维空间以及打结曲线，将许多数学学科的技术结合在一起。由于这些物体与我们的物理世界和时空密切相关，这一研究路线部分受到了现代物理学的启发。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，并为解决旧问题提供了新的方法。PI和他的合作者开创了新的不变量，被称为“Heegaard-Floer同调”和“纽结Floer同调”，并开发了一种被称为“有界Floer同调”的技术，以有效地利用空间的简单组成部分。由该奖项资助的研究涉及进一步发展三维空间和纽结曲线的这些有边界技术，以更好地从概念上理解这些不变量，并为研究它们提供有效的计算技术。PI与Zoltan Szabo合作，构建了三维和四维空间的不变量，称为“Heegaard-Floer同调”。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，以部分受到现代物理学启发的方式研究纽结理论和低维拓扑中的问题。这种结构的一个变体，称为“纽结Floer同调”，被用来研究三维流形中的纽结。在与罗伯特·利普希茨和迪伦·瑟斯顿的合作中，PI定义了“边界Floer同调”，这是一种从被分解成简单组成部分的三流形重建Heegaard-Floer同调的一个变体的技术。在这项由该奖项资助的研究中，PI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。在不同的方向上，边界Floer同调被扩展为研究和计算纽结Floer同调的工具。

This National Science Foundation award supports research to develop new tools to study three- and four-dimensional spaces as well as knotted curves, bringing together techniques from many mathematical disciplines. As these objects closely relate to our physical world and the space-time, this line of research is 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 PI and his collaborators pioneered new invariants, known as "Heegaard-Floer homology" and "knot Floer homology," and developed a technique known as "bordered Floer homology" for effectively utilizing simple component pieces of a space. Research funded by this award deals with further developing these bordered techniques for three-dimensional spaces and for knotted curves, to get both a better conceptual understanding of these invariants, and for giving effective computational techniques for studying them.In collaboration with Zoltan Szabo, the PI constructed an invariant for three- and four-dimensional spaces known as the "Heegaard-Floer homology." 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 that was partially inspired by modern physics. A variant of this construction, called "knot Floer homology," is used to study knots in three-dimensional manifolds. In collaboration with Robert Lipshitz and Dylan Thurston, the PI defined "bordered Floer homology," a technique for reconstructing one variant of Heegaard-Floer homology from a three-manifold that is decomposed into simple component pieces. In the research funded by this award, the PI 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. In a different direction, bordered Floer homology is extended to a tool for studying and computing knot Floer homology.