Contact homology, dynamics, and embeddings

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This NSF award provides funding for a project to study certain questions in dynamical systems in symplectic geometry. These dynamical systems are mathematical models for physical processes such as the motion of the planets in the solar system. One basic goal is to understand periodic orbits; these correspond to repeating behavior such as a single planet revolving around a star. In particular, for a given dynamical system, it is important to understand how many periodic orbits exist, and whether or not a random configuration is close to a periodic one. The principal investigator will also study related questions about the existence of symplectic embeddings; these are mathematical transformations that can be used to relate different dynamical systems to each other.In order to study these kinds of dynamical questions, mathematical tools will be developed in the theory of embedded contact homology (ECH) of three-manifolds, and other kinds of contact homology in higher dimensions. The foundations of ECH will be extended to the case of Morse-Bott contact forms. ECH of unit cotangent bundles will be studied and related to string topology. Spectral invariants in periodic Floer homology (a variant of ECH) will be developed and used to study whether or not generic area-preserving maps of surfaces have dense periodic orbits. Cobordism maps on ECH will be used to study Lagrangian embeddings in four-manifolds, and to investigate generalizations of the Arnold chord conjecture on the existence of Reeb trajectories with ends on a given Legendrian knot. Nonequivariant and equivariant contact homology will be studied in three and higher dimensions. Symplectic capacities arising from these different kinds of contact homologies will be compared. Combinatorial Reeb dynamics on polytopes will be developed and used to perform computer experiments to test Viterbo's conjecture and other related conjectures about Reeb dynamics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.