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.
This project studied the geometry of three-dimensional contact manifolds and four-dimensional symplectic manifolds. These kinds of geometries arise naturally in studying the set of all possible configurations of a physical system. The outcomes of the project all involved embedded contact homology (ECH). ECH is a structure associated to three-dimensional contact manifolds which has significant applications to three-dimensional contact geometry and four-dimensional symplectic geometry, some of which were developed in the project. One outcome of the project was to construct maps on ECH of contact three-dimensional manifolds induced by four-dimensional symplectic manifolds. These cobordism maps are an important technical tool for addressing various problems in three-dimensional contact geometry and four-dimensional symplectic geometry. For example, they were used to prove the three-dimensional case of the Arnold chord conjecture, a long-standing conjecture asserting the existence of certain trajectories in three-dimensional contact manifolds. ECH was used in the project in a different way to prove that every Reeb vector field on a closed three-dimensional contact manifold has at least two closed orbits. This extends the famous Weinstein conjecture, which asserts that there is always at least one closed orbit. These closed orbits describe ways that a physical system can repeat its behavior over time. Another outcome of the project was to compute the ECH capacities of concave toric domains. Concave toric domains are a certain kind of four-dimensional symplectic manifold. ECH capacities are a tool for deciding when one four-dimensional symplectic manifold can be embedded into another in a way which preserves the symplectic structure. The development of ECH capacities is helping answer basic questions about symplectic geometry, the fundamental geometry underlying classical mechanics, in four dimensions.
