The PI proposes to conduct research on topological aspects of symplectic and contact manifolds using recent developments in the theory of J-holomorphic curves. In one direction, continuing previous work in collaboration with R. Hind, the PI will investigate symplectic embeddings of domains and related symplectic packing questions, with an eye for proving a general packing stability property. Related problems to study are several possible estimates of symplectic capacities. A second project, with D. Gay, will describe 3-dimensional contact analogues of 4-dimensional ellipsoid embeddings. An array of 3-dimensional contact techniques will be used with possible applications to the 4-dimensional symplectic embedding questions. In a third objective the PI continues previous work using J-holomorphic curves techniques to pursue topological aspects of symplectomorphism groups of manifolds such as ruled 4-dimensional surfaces.
In contrast with classical geometries, symplectic geometry is predominantly built around the notion of area (of two dimensional symplectic submanifolds). Embedding and packing questions are fundamental rigidity questions originally motivated by Hamiltonian dynamics, and the quest for recurrence properties of Hamiltonian automorphisms. Answers to such packing questions will guide our understanding of symplectic spaces. Due to their very computational nature, these problems have the possibility of involving students in guided computer experimentation.
