Heegaard Floer homology and Low Dimensional Topology

本课题的主题是Heegaard flower同调和规范理论在低维拓扑和结理论问题中的应用。heegard flower同调中的新手术公式允许PI研究三流形手术问题的应用，以及结的手术表征。在另一个方向上，花同源性用于研究3球中结的解结数。该项目包括结花同源性的基础问题和组合技术的研究。在另一个方向上，规范理论技术被用于寻找小的奇异4流形。本课题研究三维和四维空间以及结理论和低维拓扑中的一些核心问题。一个主要的工具是一个相对较新的理论，叫做Heegaard flower同源性。这一理论与测量理论不变量有着密切的联系，这些不变量起源于数学和数学物理之间的相互作用。预计Heegaard flower同调的进展也将导致对这些规范理论不变量的更好理解。该项目的另一个组成部分是对结的解结数的研究。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。测量其复杂性的一种方法是研究链需要被酶切割多少次并重新组合才能解结。

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.