Low Dimensional Topology and Heegaard Floer homology

这个项目的主要主题是研究纽结、三维流形和光滑的4-流形。其中一个核心工具是Heegaard Floer Homology，它是由Peter Ozsvath和首席调查员开发的。PI研究了这些技术在三维流形、切片纽结和光滑4-流形上的奇异结构的各种外科问题上的新应用。该建议还涉及Heegaard Floer同调的拓扑和计算方面。这涉及到对特殊的多点Heegaard图的研究，以建立该理论的拓扑版。在不同的方向上，Heegaard Floer同调的新进展被用于研究具有简单Floer同调的三维流形。该项目研究三维和四维空间以及纽结理论和低维拓扑中的中心问题。Heegaard Floer同调为低维数学对象提供了各种新的工具，如拓扑不变量和运算公式。这一理论与规范理论不变量密切相关，规范理论不变量源于数学和数学物理之间的相互作用。预计Heegaard Floer同调的进展也将导致对这些规范理论不变量的更好理解。

The main theme of this project is the study of knots, three dimensional manifolds, and smooth 4-manifolds. One of the central tools is Heegaard Floer homology that was developed by Peter Ozsvath and the Principal Investigator. The PI studies new applications of these techniques for various surgery problems for three dimensional manifolds, slice knots, and exotic structures on smooth 4-manifolds.The proposal also deals with topological and computational aspects of Heegaard Floer homology. This involves the study of special multiply pointed Heegaard diagrams to build up a topological version of the theory. In a different direction new advances in Heegaard Floer homology are used to study three-manifolds with simple Floer homologies.The project studies three and four dimensional spaces and central problems in Knot Theory and Low Dimensional Topology. Heegaard Floer homology provides various new tools, such as topological invariants and surgery formulas, for low-dimensional mathematical objects. This theory is closely related to gauge theoretical invariants that originated from interactions between Mathematics and Mathematical Physics. It is expected that advances in Heegaard Floer homology will lead to a better understanding of these gauge theoretical invariants as well.