Floer homology and low dimensional contact and symplectic geometry

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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.