Heegaard Floer homology and Low Dimensional Topology

这个项目的主要主题是Heegaard Floer同调和规范理论在低维拓扑和纽结理论中的应用。Heegaard Floer同调中的新外科公式允许PI研究三维流形中外科问题的应用，以及纽结的外科特征.在不同的方向上，Floer同调被用于研究三维球面中纽结的解结数.该项目包括纽结Floer同调的基本问题和组合技术的研究。在不同的方向上，规范理论技术被用来寻找小的奇异的4-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一些中心问题。其中一个主要工具是一个相对较新的理论，称为Heegaard Floer同调。这一理论与测量源于数学和数学物理之间的相互作用的理论不变量有着密切的联系。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的理解。该项目的另一个组成部分是研究纽结的解结数。虽然这是纽结理论中的经典不变量，但解结过程在打结DNA的研究中也是相关的。衡量复杂性的一种方法是研究一条链需要被酶切割多少次，然后自我重组才能解开。

The main theme of this project is the application of Heegaard Floer homologyand gauge theory for problems in low-dimensional topology and knot theory. New surgery formulas in Heegaard Floer homology allow the PI to study applications for surgery problems in three-manifolds, and surgery characterization for knots.In a different direction Floer homology is used to investigate unknotting numbers for knots in the 3-sphere. The project includes foundational problems for knot Floer homology and the study of combinatorial techniques as well. In a different direction, gauge theoretical techniquesare used in the search for small exotic 4-manifolds. The project studies three and four dimensional spaces and some central problemsin knot theory and low dimensional Topology. One of the main tools is a relatively new theory called Heegaard Floer homology. This theory has close ties 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.Another integral part of the project is the study of unknotting numbers for knots.While this is a classical invariant in knot theory, the unknotting process is also relevant in the study of knotted DNA. A way to measure the complexity is to study how many times the strand has to be cut by an enzyme and recombine itself to be unknotted.