Low Dimensional Topology and holomorphic disks

该奖项支持首席研究员在拓扑学方面的研究，拓扑学是一种对在连续变形（如拉伸和弯曲）下保持的空间属性的数学研究。本研究的中心主题是更好地理解一种高效的数学工具，称为结弗洛尔同源性，用于研究三维和四维空间和打结循环。PI和他的合作者在这方面做了基础性的工作。拟议的项目建立在他们最近的成功接近结弗洛尔同源性和相关的三维空间的不变量通过代数技术，从而使他们更容易处理。除了在拓扑学的抽象领域的应用外，这些方法还与物理学所激发的其他数学领域密切相关，例如规范理论和辛几何。该项目的完成将在不同的研究领域和方法之间架起新的桥梁。该项目还包括研究结的各种不变量之间的关系，以及结理论中的计算问题。Heegaard-Floer同调为研究低维拓扑中的问题提供了各种工具。该结构首先由Peter Ozsvath和主要研究者开发，通过研究Heegaard图和Heegaard曲面对称产品中的全纯圆盘，使用了拓扑学和辛几何的方法。 有特殊的三流形，包括纤维三流形和Dehn手术沿着结，在纽结弗洛尔同调和Heegaard-Floer同调的进展，预计将导致新的拓扑应用。相关的研究方向包括协调不变量、光滑切片纽结和光滑四维流形。该项目还旨在进一步开发代数结构来研究纽结理论中的问题，并研究不同纽结同源不变量之间的关系。

The award supports the principal investigator's research in topology, a mathematical study of properties of spaces that are preserved under continuous deformations, such as stretching and bending. The central theme in this research is to develop a better understanding of a highly effective mathematical tool known as the knot Floer homology that is used in the study of three- and four-dimensional spaces and knotted loops. The PI and his collaborators have done foundational work in this area. The proposed project builds on their recent success in approaching knot Floer homology and related invariants of three-dimensional spaces by means of algebraic techniques, thereby making them much more tractable. In addition to applications within the abstract field of topology, the methods are closely related to other areas of mathematics that are motivated by physics, such as gauge theory and symplectic geometry. The completion of this project would lead to new bridges between different research areas and methods. The project also includes investigating the relationship between various invariants for knots, and computational problems in knot theory.Heegaard-Floer homology provides various tools to study problems in low-dimensional topology. The construction, first developed by Peter Ozsvath and the principal investigator, uses methods both from topology and symplectic geometry through the study of Heegaard diagrams, and holomorphic disks in symmetric products of Heegaard surfaces. There are special families of three-manifolds, including fibered three-manifolds and Dehn surgeries along knots, where advances in knot Floer homology and Heegaard-Floer homology are expected to lead to new topological applications. Related research directions involve concordance invariants, smoothly slice knots and smooth four-manifolds. The project also aims to further develop algebraic constructions to study problems in knot theory, and investigate relationships between different knot homology invariants.