Heegaard diagrams and holomorphic disks

在最近与首席研究员佐尔坦·萨博（Zoltan Szabo）的合作中，他构造了三维和四维空间的不变量，称为Heegaard flower同调。在与Robert Lipshitz和Dylan Thurston进一步的联合工作中，这些不变量被向下扩展，以定义有边界的flower同调。边界花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阐明了Heegaard花在其原始设置中的同源性。特别是，它们可以用来给出闭流形的某些Heegaard花不变量的概念计算。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，数学物理中起源方程的引入使我们对三维和四维空间的拓扑特性的理解取得了巨大的进步。研究者与Zoltan Szabo合作开发的“Heegaard Floer同源性”（Heegaard Floer homology），促进了对这些方程数据的另一种更几何的理解，从而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，在一个新的理论中，“边界花同调”，由研究者与罗伯特·利普希茨和迪伦·瑟斯顿合作开发。该提案旨在进一步发展这两种工具，并将它们应用于拓扑问题的研究。

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.