Mathematical Sciences: Symplectic and Contact Geometry and Topology, and Their Applications

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9626430 Eliashberg This project will concentrate on several areas of symplectic and contact geometry and topology in interaction with other areas of mathematics. In particular, there will be continuing study of 3-dimensional contact structures and the relationship between contact geometry and the theory of codimension-1 foliations. The theory of confoliations initiated in joint work with W. P. Thurston should help to clarify this relationship. Another important part of the project will be devoted to symplectic geometry of Stein manifolds, where one of the main tools should be an analog of Morse-Smale theory for plurisubharmonic functions on Stein manifolds. Yet another focus (joint with H. Hofer) will be contact homology theory, which should deliver new invariants of contact manifolds in dimension 3 and higher. Finally, a significant part of the project (joint with M. Gromov) will involve the symplectization of Morse theory. The results here should help in understanding which part of stable Morse theory can be generalized to the problems of Lagrangian intersections. Symplectic geometry and topology have played an important role in such different areas of mathematics as low-dimensional topology, Hamiltonian dynamics, foliation theory, sub-Riemannian geometry, complex geometry and the theory of functions of several complex variables. Despite much recent progress, many basic questions remain unanswered. Moreover, the discovery of new interactions with these other fields creates new demands for the interior development of symplectic geometry and topology. The research under this project should answer some of the old problems of the field and discover further applications. ***