Principal Investigator: Krzysztof Wysocki
In general terms, the goal of the project is to study analytical aspects of symplectic geometry and Hamiltonian dynamical systems. The overall aim of the project is the development of a general framework for studying nonlinear elliptic equations appearing in symplectic geometry. Many of the problems in symplectic geometry, like the Gromov-Witten theory and the Symplectic Field Theory, are based on the study of the moduli spaces of the first order nonlinear elliptic equations. These muduli spaces exhibit lack of compactness, however, they have nontrivial compactification. In the project, Wysocki jointly with Hofer and Zehnder continues the development of a new general Fredholm theory, which takes place in new smooth spaces called polyfolds. This general Fredholm theory has all the properties of the classical Fredholm theory, but is more flexible and applicable to problems with lack of compactness. In the project, the general Fredholm theory on polyfolds is applied to the Gromow-Witten theory and to the Symplectic Field Theory. In another part of the proposal, we use the theory of pseudholomorphic curves and the theory of generating functions to study multiplicity of closed characteristics on weakly dynamically convex energy surfaces.
Symplectic geometry has its origin in classical mechanics. For example, the motion of the planetary system can be described by a system of nonlinear differential equations called Hamiltonian systems. The flow lines of Hamiltonian systems follow very complex patterns. Hence it is of importance to get better understanding of Hamiltonian flows on its energy surfaces. The development of the general Fredholm theory on polyfolds will put the Symplectic Field Theory on solid analytical foundations. The methods developed in this project should have applications to larger classes of nonlinear partial differential equations of relevance in differential geometry and physics.
