Symplectic Geometry and Dynamics

Institute For Advanced Study, Princeton NJ

Investigators

Abstract

Abstract Award: DMS-1104470 Principal Investigator: Helmut Hofer The first subproject will be concerned with the completion of symplectic field theory (SFT). This is currently the most general and most comprehensive theory of symplectic invariants. There are three ingredients to this project, namely the polyfold Fredholm theory, the scale-smooth analysis and the formulation of SFT as an algebraic presentation of the solution count of a polyfold Fredholm problem with operations. The second subproject, a by-product of the polyfold Fredholm theory, attempts to construct a homology theory based on equivalence classes of Fredholm problems. The resulting advantage would be that transversality issues in studying nonlinear partial differential equations would actually not occur explicitly. They would be hidden in the (one-time) proof that the new homology theory of a space would be naturally isomorphic to the rational singular homology. A third subproject is concerned with finite energy foliations, either in the context of area-preserving disk-maps or the restricted planar three-body problem. Finite energy foliations were introduced by the PI and his collaborators and are an important tool in the study of the dynamics of low-dimensional Hamiltonian systems. The restricted circular planar three-body problem is not only interesting from a purely academic viewpoint, but is also used in the orbit design for scientific space missions. In certain regimes of the Jacobi energy, finite energy foliations allow to reduce the dynamics of the restricted three-body problem to that of the dynamics of an area-preserving disk-map. Then again the same theory facilitates the understanding of the long-term behavior of iterated area-preserving disk maps. The project will study the relationship between the theory of finite energy foliations, a symplectic construction, and core dynamical systems notions like entropy. It is a surprising fact that many physical systems evolving in time, allow a description as a Hamiltonian system. Systems of this kind describe the flow of an incompressible ideal fluid, the movement of a satellite under the gravitational forces of celestial bodies, or the movement of charged particles in a magnetic field. Hamiltonian systems are a very particular class of dynamical systems, which can be studied not only by the methods from dynamical systems theory, but also by a more exotic kind of geometry called symplectic geometry. This is a geometry based on the notion of area, in contrast to usual geometries, which have length and distance as their fundamental notions. Recent advances in this field already found some practical uses. For example, this novel point of view has been used in accelerator physics in algorithms controlling and insuring stability of a beam of particles and called symplectic tracking. It is the explicit purpose of this research to integrate the approaches to Hamiltonian systems coming from these two different perspectives. This should result in new methods, which should make it possible to attack problems, which so far seemed unreachable. For example it seems long term be feasible to use the new methods to develop algorithms, which would find fuel-efficient orbits for scientific space missions. Current technology can verify the properties of proposed orbits extremely fast. However, currently there is no good method for finding such orbits. It rather compares to finding a needle in a haystack, using very large amounts of computing time.