The set of all possible configurations of a physical system generally has the structure of a symplectic manifold. When one restricts to configurations with a fixed energy, one typically obtains a contact manifold. Understanding the geometry of symplectic manifolds and contact manifolds is thus important to understanding the dynamics of physical systems. The following two geometric questions about such manifolds are of particular interest. First, one would like to understand Reeb orbits on contact manifolds, which correspond to physical behavior which repeats over time. Second, one would like to understand when one symplectic manifold can be symplectically embedded into another, in order to better understand the relations between different symplectic manifolds. Contact homology is a powerful tool, currently under development, which can be applied to both of these questions.
The project will develop the foundations, computation, and applications of various kinds of contact homology. In particular, embedded contact homology (ECH) will be extended to a functor on three-dimensional contact manifolds and four-dimensional strong symplectic cobordisms. The foundations of ECH and other kinds of contact homology will be constructed directly in terms of holomorphic curves when possible, in order to more closely relate them to geometry. ECH capacities (quantitative invariants which obstruct symplectic embeddings in four dimensions) will be computed in more examples and related to geodesic flows and Hamiltonian dynamics. Analogues of ECH capacities for other kinds of contact homology, such as cylindrical contact homology and rational symplectic field theory, will be constructed and studied. These new tools will be used to explore whether the Weinstein conjecture can be extended by increasing the lower bound on the number of Reeb orbits or proving the existence of short Reeb orbits.