Pseudoholomorphic Curves and Dynamics

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AbstractAward: DMS-0606588Principal Investigator: Krzysztof WysockiThe project has two themes: (1) the development of a generalapproach for studying non-linear elliptic equations arising insymplectic geometry and (2) applications of global methods ofsymplectic geometry to dynamical systems. Part 1 of the project,joint with Hofer and Zehnder, develops an analytical frameworkfor Symplectic Field Theory. It is devoted to a general nonlinearFredholm theory, which takes place on new spaces of locallyvarying dimensions called polyfolds. The second theme of theproject is the outgrowth of research of Wysocki and collaboratorson finite energy foliations. In one of the subprojects he willuse the theory of finite energy foliations to prove thatstar-shaped energy surfaces in four-dimensional space carry atleast two Hamiltonian periodic orbits. A long-term behavior ofarea preserving disk maps will be investigated in a jointsubproject with Hofer. To understand this behavior, the Floertheory will be combined with the theory of finite energyfoliations. In another part of the project, Wysocki will usefinite energy foliations to study the uniqueness of symplecticcapacities of convex domains in four-dimensional spaces. The problems studied in symplectic geometry were motivated bycelestial mechanics. For example, the motion of the planetarysystem can be described by a system of nonlinear differentialequations called Hamiltonian systems. The flow lines ofHamiltonian systems follow very complex patterns. This projectwill provide new tools for studying complexities of this behaviorand will lead to a better understanding of the structural aspectsof Hamiltonian flows on star-shaped energy surfaces. The newgeneral Fredholm theory aims at providing the rigorous analyticalfoundations of Symplectic Field Theory. These ideas should alsobe applicable to nonlinear partial differential equations arisingin mathematical physics.