The subject of symplectic topology was developed in order to answer qualitative questions concerning classical mechanics, such as the existence and number of periodic orbits in conservative dynamical systems. Since the inception of the subject, results on rigidity, asserting various constraints on the dynamics, have coexisted with results on flexibility, yielding constructions that at first glance were counter-intuitive. While the most important developments of the last three decades concern rigidity, several new instances of symplectic flexibility have been discovered more recently. The goal of the current research project is to further develop both flexible and rigid methods in a search of precise description of the boundary between the two parts of symplectic topology.
The main objectives of the project are: -- development of effective methods of construction of symplectic structures; -- development of symplectic and contact pseudoisotopy theories, as necessary steps in understanding topology of groups of symplectic and contact transformations; -- systematic development of a piecewise linear version of symplectic topology as a necessary step in the development of topological Hamiltonian dynamics; and -- further development of symplectic field theory. The work on the project is expected to provide new methods for construction of symplectic structures on closed manifolds and to develop new effective techniques for computing symplectic and contact invariants of symplectic field theory.
