Heegaard diagrams and holomorphic disks

在最近与Zoltan Szabo的合作中，主要研究人员构造了三维和四维空间的不变量，称为Heegaard Floer同调。在与Robert Lipshitz和Dylan瑟斯顿的进一步合作中，这些不变量被向下扩展，以定义有界Floer同调。带边Floer同调给出了二维曲面和具有二维边界的三维空间的不变量。这些有边界的不变量揭示了Heegaard Floer同调在其原始背景下的情况。特别地，它们可用于给出闭流形的某些Heegaard Floer不变量的概念计算。该项目旨在更好地理解不变量，将“边界理论”扩展到更广泛的背景下。在过去的25年里，起源于数学物理的方程的引入使我们对三维和四维空间的拓扑性质的理解有了很大的进步。研究人员与Zoltan Szabo合作开发了另一种对这些方程数据的更几何理解，称为“Heegaard Floer同调”，这促进了这一领域的进一步进展。这个三维和四维的故事也可以扩展到二维物体，这是一种新的理论，由研究人员与罗伯特·利普希茨和迪伦·瑟斯顿合作开发的“有边界的弗洛尔同源”。该提案旨在进一步开发这两种工具，并将其应用于研究拓扑问题。

In recent work with Zoltan Szabo, the principal investigator constructed invariants for three- and four-dimensional spaces, called Heegaard Floer homology. In further joint work with Robert Lipshitz and Dylan Thurston, these invariants are extended down, to define bordered Floer homology. Bordered Floer homology gives invariants for two-dimensional surfaces and three-dimensional spaces with two-dimensional boundary. These bordered invariants shed light on Heegaard Floer homology in its original setting. In particular, they can be used to give conceptual calculations of certain Heegaard Floer invariants for closed manifolds. The project aims to understands the invariants better, extending the "bordered theory" to a broader context.The introduction of equations with origins in mathematical physics has lead to great advances in our understanding of the topological properties of three and four-dimensional spaces over the past twenty-five years. Further progress in this area is facilitated by an alternative, more geometric understanding of the data derived from these equations, known as "Heegaard Floer homology", developed by the investigator in collaboration with Zoltan Szabo. This three and four-dimensional story can be extended to cover two-dimensional objects, as well, in a a new theory, "bordered Floer homology", developed by the investigator in collaboration with Robert Lipshitz and Dylan Thurston. The proposal aims to further develop both of these tools and apply them to study topological questions.