Heegaard Diagrams and Holomorphic Disks

该提案涉及研究Heegaard Floer同调，这是由研究人员与Zoltan Szabo合作构建的三维和四维空间的新不变量。这些不变量为用规范理论技巧构造的空间的早期不变量提供了新的见解：Donaldson和Seiberg-Wittenant不变量。此外，它们还可以用来解决较老的拓扑问题，特别是关于经典纽结上的DehnSurgery的问题。这个项目旨在更好地理解这些不变量，并进一步应用于纽结理论和三维和四维流形的拓扑。起源于数学物理的方程的引入，在过去的二十年里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。这一领域的进一步进展是通过对从这些方程得到的数据的另一种更几何的理解来促进的。这些替代的更几何的方法一直是研究人员的研究对象，本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.