Floer homology and low dimensional contact and symplectic geometry

该项目的主要目标是开发嵌入式接触同调（ECH）及其在三维接触几何和四维辛几何中的应用。 ECH 最近被用来在三个维度上对韦恩斯坦猜想进行改进，在三个维度上证明阿诺德弦猜想，并定义“ECH 容量”，它为四个维度的辛嵌入提供了新的、有时是尖锐的障碍。 该项目的目标之一是进一步完善韦恩斯坦猜想和弦猜想，为里布轨道和里布弦的数量提供更好的下界，以及有关它们的附加定性和定量信息。 该项目的另一部分是研究最近引入的 ECH 能力，以便更好地了解辛嵌入何时可行。 该项目的第三部分是通过继续发展 ECH 的基础，特别是开发共配图和 TQFT 结构、缝合版本以及稳定哈密顿结构的扩展，为未来的应用做好准备。该项目中开发的嵌入式接触同调提供了低维拓扑和动力学之间的桥梁。 低维拓扑学关注三维和四维弯曲空间的整体结构，而动力学则研究物理系统随时间的发展。 嵌入式接触同源性编码深层拓扑信息，可用于获取有关动力学的新的具体信息，例如某些稳定配置的存在。

The main goal of the project is to develop embedded contact homology (ECH) and its applications to three-dimensional contact geometry and four-dimensional symplectic geometry. ECH has recently been used to obtain refinements of the Weinstein conjecture in three dimensions, to prove the Arnold chord conjecture in three dimensions, and to define "ECH capacities" which give new and sometimes sharp obstructions to symplectic embeddings in four dimensions. One aim of the project is to obtain further refinements of the Weinstein conjecture and chord conjecture, giving better lower bounds on the numbers of Reeb orbits and Reeb chords as well as additional qualitative and quantitative information about them. Another part of the project is to study the recently introduced ECH capacities in order to better understand when symplectic embeddings are possible. A third part of the project is to prepare for future applications by continuing to develop the foundations of ECH, in particular to develop cobordism maps and TQFT structure, the sutured version, and the extension to stable Hamiltonian structures.The embedded contact homology developed in this project provides a bridge between low-dimensional topology and dynamics. Low-dimensional topology is concerned with the global structure of curved spaces in three and four dimensions, while dynamics studies the development of physical systems over time. Embedded contact homology encodes deep topological information which can be used to obtain new concrete information about dynamics, such as the existence of certain stable configurations.