Heegaard Diagrams and Holomorphic Disks

该提案涉及研究“Heegaard flower同源性”，这是研究者与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，它们还可以用于解决老的拓扑问题，包括关于经典结的Dehn手术的具体问题。这个项目的目的是更好地理解这些不变量，考虑到最近在理论计算方面的进展，并进一步应用于结理论和三维和四维流形的拓扑。在过去的二十年里，在数学物理中引入有起源的方程，使我们对三维和四维空间的拓扑性质的理解取得了巨大的进步。这一领域的进一步进展是由另一种方法促进的，对这些方程得出的数据有更多的几何理解，被称为“Heegaard Floer同源”，这是研究者与Zoltan Szabo合作开发的一种新理论。Heegaard flower同源性已经为三维和四维空间的结构提供了新的思路。这些新方法一直是研究者研究的对象，目前的建议涉及使用这些技术的进一步发展。

The proposal deals with studying ``Heegaard Floer homology,'' the new invariants for three-and four-dimensional spaces constructed by the investigator in collaboration with Zoltan Szabo. These invariants give new insight into earlier invariants for spaces constructed using ``gauge theoretic'' techniques: the Donaldson and Seiberg-Witten invariants. In addition, they can be used to address older topological questions, including specifically questions about Dehn surgery on classical knots. This project aims to understand these invariants better, in veiw of recent computational advances in the theory, and to give further applications to knot theory and the topology of three- and four-dimensional manifolds.The introduction of equations with origins in mathematical physics has has lead to great advances in our understanding of the topoligical properties of three and four-dimensional spaces over the past twenty 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'', a new theory developed by the investigator in collaboration with Zoltan Szabo. Heegaard Floer homology has already shed new light on the structure of three- and four-dimensional spaces. These new methods have been the object of study of the investigator, and the present proposal deals further developments using these techniques.