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，拓扑学是对连续变形（如拉伸和弯曲）下保留的空间特性的数学研究。本研究的中心主题是更好地理解一种高效的数学工具，称为结花同源性，用于研究三维和四维空间和结环。PI和他的合作者在这个领域做了基础工作。该计划建立在他们最近成功地利用代数技术接近三维空间的结花同调和相关不变量的基础上，从而使它们更加易于处理。除了在拓扑抽象领域的应用之外，这些方法还与其他由物理驱动的数学领域密切相关，例如规范理论和辛几何。这个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 - flower同调为研究低维拓扑问题提供了多种工具。该构造首先由Peter Ozsvath和首席研究员提出，通过研究Heegaard图和Heegaard曲面对称积中的全纯圆盘，使用了拓扑和辛几何的方法。三流形有特殊的家族，包括纤维三流形和沿结的Dehn手术，其中结花同源性和heegaard - flower同源性的进展有望导致新的拓扑应用。相关研究方向包括协调不变量、光滑切片结和光滑四流形。该项目还旨在进一步发展代数结构来研究结理论中的问题，并研究不同结同调不变量之间的关系。

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.