Invariants in Low-Dimensional Topology

这个项目的主要课题是低维拓扑中的Floer同调和相关不变量。Floer同调是莫尔斯理论的一个无限维版本，它已被用来构造各种不变量的结，3流形，4流形等反过来，这些不变量可以回答微妙的问题，有关各自的拓扑对象。一个来源的不变量与众多的拓扑应用是Heegaard Floer理论。例如，纽结的Heegaard Floer不变量（称为纽结Floer同源性）能够检测纽结的属。最初，所有的Heegaard Floer不变量都是用对称乘积中的伪全纯曲线定义的。最近，纽结、3-流形和4-流形的Heegaard Floer不变量都得到了基于网格图的组合描述。这个项目的一个重点是提高我们对这些组合描述的理解。PI还将致力于寻找结Floer同调和其他结不变量之间的联系，例如Khovanov-Rozansky同调;将Heegaard Floer理论扩展到带角的流形;以及利用平坦联络的模空间构造新的三流形的Floer理论不变量。拟议的研究是拓扑学，这是一个研究几何对象（如弯曲空间）的数学领域（流形）和空间中的节点。研究流形和纽结的一种有用方法是通过拓扑量子场论（TQFT），这是数学物理学中用于探索宇宙量子理论的某些玩具模型。TQFT是拓扑学家感兴趣的，因为它们包含了关于空间在不同维度上的可能形状的信息。一个重要的问题是这些形状的分类，这在四维空间中特别困难。 因为我们的宏观时空是四维的，所以四维形状的性质是量子物理学家和宇宙学家寻找宇宙几何模型的重要输入。这个项目的目标是研究的TQFT的持有最有希望为我们的理解四维形状。

The main topic of this project is Floer homology and related invariants in low-dimensional topology. Floer homology is an infinite dimensional version of Morse theory which has been used to construct various invariants of knots, 3-manifolds, 4-manifolds, etc. In turn, these invariants can answer subtle questions about the respective topological objects. One source of invariants with numerous topological applications is Heegaard Floer theory. For example, the Heegaard Floer invariant for knots (called knot Floer homology) is able to detect the genus of a knot. Originally, all the Heegaard Floer invariants were defined in terms of pseudo-holomorphic curves in symmetric products. Recently, the Heegaard Floer invariants of knots, 3-manifolds and 4-manifolds have all been given combinatorial descriptions, based on grid diagrams. One focus of this project is to improve our understanding of these combinatorial descriptions. The PI will also work on finding connections between knot Floer homology and other knot invariants, such as the Khovanov-Rozansky homologies; on extending Heegaard Floer theory to manifolds with corners; and on constructing new Floer-theoretic invariants of three-manifolds using moduli spaces of flat connections.The proposed research is on topology, an area of mathematics that studies geometric objects such as curved spaces (manifolds), and knots in space. One useful method of studying manifolds and knots is through topological quantum field theories (TQFT's), which are certain toy models used in Mathematical Physics to explore quantum theories about the universe. TQFT's are of interest to topologists because they contain information about the possible shapes of space in various dimensions. An important problem is the classification of these shapes, and this is particularly difficult in four dimensions. Because our macroscopic space-time has four dimensions, the properties of four-dimensional shapes are an essential input for quantum physicists and cosmologists looking for geometric models for the universe. The goal of this project is to study the TQFT's which hold the most promise for our understanding of four-dimensional shapes.