Heegaard Diagrams and Holomorphic Disks

该提案涉及研究“Heegaard Floer同源性”，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。这些不变量提供了新的见解早期不变量的空间构造使用"规范理论“的技术：唐纳森和塞伯格-威滕不变量。 此外，它们还可以用来解决更古老的拓扑问题，特别是关于经典结的Dehn手术的问题。这个项目的目的是更好地理解这些不变量，在veiw最近的计算进展的理论，并给予进一步的应用纽结理论和三维和四维流形的拓扑结构。引入方程的数学物理起源已经导致了我们的理解的三维和四维空间的拓扑性质在过去的二十年中的巨大进步。这一领域的进一步进展是由另一种更几何的理解来自这些方程的数据，被称为“Heegaard Floer同源性”，一个新的理论，由研究人员与佐尔坦Szabo合作开发。 Heegaard Floer同调已经为三维和四维空间的结构提供了新的线索。这些新方法一直是研究人员的研究对象，本提案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.