Heegaard Floer homology and Low Dimensional Topology

本计画的主题是应用Heegaard Floer同调与规范理论于低维拓扑与纽结理论。Heegaard Floer同调中的新手术公式允许PI研究三流形中的手术问题的应用，以及结的手术特征。在不同的方向上，Floer同调用于研究3-球面中结的解结数。该项目包括结弗洛尔同源性的基础问题和组合技术的研究。在不同的方向，规范理论techniquesare用于搜索小奇异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。主要工具之一是一个相对较新的理论，称为Heegaard Floer同源性。这一理论与源于数学与数学物理相互作用的规范理论不变量有着密切的联系。Heegaard Floer同调理论的进展也将有助于我们更好地理解这些规范理论不变量。该项目的另一个组成部分是研究结的解结数。虽然这是结理论中的经典不变量，但解结过程也与结DNA的研究有关。测量复杂性的一种方法是研究链必须被酶切割多少次，然后重新组合才能解开。

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.