Floer homology and low dimensional contact and symplectic geometry

该项目的主要目标是发展嵌入接触同调（ECH）及其在三维接触几何和四维辛几何中的应用。 ECH最近已被用来获得改进的温斯坦猜想在三维空间，证明阿诺德弦猜想在三维空间，并定义“ECH能力”，这给新的，有时尖锐的障碍，辛嵌入在四维空间。 该项目的一个目的是获得温斯坦猜想和和弦猜想的进一步改进，给出更好的Reeb轨道和Reeb和弦数量的下界以及关于它们的更多定性和定量信息。 该项目的另一部分是研究最近引入的ECH容量，以便更好地理解辛嵌入何时可能。 该项目的第三部分是通过继续发展ECH的基础，为未来的应用做准备，特别是开发cobordism映射和TQFT结构，缝合版本，并扩展到稳定的Hamilton结构。在这个项目中开发的嵌入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，而动力学研究物理系统随时间的发展。 嵌入接触同调编码了深层拓扑信息，这些信息可以用来获得关于动力学的新的具体信息，例如某些稳定构型的存在。

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.