Pseudoholomorphic Curves and Dynamics

Pennsylvania State Univ University Park, University Park PA

Investigators

Abstract

Abstract Award: DMS-0606588 Principal Investigator: Krzysztof Wysocki The project has two themes: (1) the development of a general approach for studying non-linear elliptic equations arising in symplectic geometry and (2) applications of global methods of symplectic geometry to dynamical systems. Part 1 of the project, joint with Hofer and Zehnder, develops an analytical framework for Symplectic Field Theory. It is devoted to a general nonlinear Fredholm theory, which takes place on new spaces of locally varying dimensions called polyfolds. The second theme of the project is the outgrowth of research of Wysocki and collaborators on finite energy foliations. In one of the subprojects he will use the theory of finite energy foliations to prove that star-shaped energy surfaces in four-dimensional space carry at least two Hamiltonian periodic orbits. A long-term behavior of area preserving disk maps will be investigated in a joint subproject with Hofer. To understand this behavior, the Floer theory will be combined with the theory of finite energy foliations. In another part of the project, Wysocki will use finite energy foliations to study the uniqueness of symplectic capacities of convex domains in four-dimensional spaces. The problems studied in symplectic geometry were motivated by celestial 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. This project will provide new tools for studying complexities of this behavior and will lead to a better understanding of the structural aspects of Hamiltonian flows on star-shaped energy surfaces. The new general Fredholm theory aims at providing the rigorous analytical foundations of Symplectic Field Theory. These ideas should also be applicable to nonlinear partial differential equations arising in mathematical physics.