Heegaard Diagrams and Holomorphic Disks

该提案涉及研究 Heegaard Floer 同源性，这是研究人员与 Zoltan Szabo 合作构建的三维和四维空间的新不变量。这些不变量为使用规范理论技术构建的空间的早期不变量提供了新的见解：Donaldson 和 Seiberg-Witten 不变量。 此外，它们还可用于解决较旧的拓扑问题，包括有关经典结的 Dehnsurgery 的具体问题。该项目旨在更好地理解这些不变量，并进一步应用结理论和三维和四维流形拓扑。起源于数学物理的方程的引入使我们在过去二十年中对三维和四维空间的拓扑性质的理解取得了巨大进步。对从这些方程导出的数据进行另一种更几何的理解，促进了该领域的进一步进展。这些另一种更几何的方法一直是研究者的研究对象，目前的提案涉及这些方法的进一步发展。

The proposal deals with studying Heegaard Floer homology, the newinvariants for three-and four-dimensional spaces constructed by theinvestigator in collaboration with Zoltan Szabo. These invariants givenew insight into earlier invariants for spaces constructed usinggauge theoretic techniques: the Donaldson and Seiberg-Witteninvariants. In addition, they can be used to address oldertopological questions, including specifically questions about Dehnsurgery on classical knots. This project aims to understand theseinvariants better, and give further applications to knot theory andthe topology of three- and four-dimensional manifolds.The introduction of equations with origins in mathematical physics has lead to great advances in our understanding of the topologicalproperties of three and four-dimensional spaces in the past twentyyears. Further progress in this area is facilitated by an alternative,more geometric understanding of the data derived from these equations.These alternative more geometric methods have been the object of studyof the investigator, and the present proposal deals with furtherdevelopments of these methods.