Principal Investigator: Michael L. Hutchings
The main part of the project is to develop "embedded contact homology" (ECH), a new invariant of a contact three-manifold, which is defined in terms of periodic orbits of the Reeb flow and embedded pseudoholomorphic curves in the symplectization. ECH is conjecturally isomorphic to versions of the Seiberg-Witten and Ozsvath-Szabo Floer homologies. Some specific goals of the project are to further develop the analytical foundations of ECH; to create tools for computing ECH, particularly in terms of open book decompositions; to use ECH to obtain lower bounds on numbers of Reeb orbits; and to work towards extending ECH to a more general theory which would unify it with the Ozsvath-Szabo Floer homology. Some broader goals are to use ECH machinery to help compute symplectic field theory in three dimensions, and to explore Floer-theoretic invariants of families.
The embedded contact homology developed in this project lies on the interface between dynamics and low-dimensional topology. Dynamics is concerned with the behavior of physical systems over time, while low-dimensional topology studies the possible shapes of curved spaces in three and four dimensions. Embedded contact homology allows one to obtain deep topological information from an understanding of dynamics; and conversely to obtain important dynamical information, such as the existence of stable configurations, from topological conditions.
