Symplectic Geometry and Dynamics

点击翻译按钮获取中文摘要

AbstractAward: DMS-1104470Principal Investigator: Helmut HoferThe first subproject will be concerned with the completion ofsymplectic field theory (SFT). This is currently the mostgeneral and most comprehensive theory of symplecticinvariants. There are three ingredients to this project, namelythe polyfold Fredholm theory, the scale-smooth analysis and theformulation of SFT as an algebraic presentation of the solutioncount of a polyfold Fredholm problem with operations. The secondsubproject, a by-product of the polyfold Fredholm theory,attempts to construct a homology theory based on equivalenceclasses of Fredholm problems. The resulting advantage would bethat transversality issues in studying nonlinear partialdifferential equations would actually not occur explicitly. Theywould be hidden in the (one-time) proof that the new homologytheory of a space would be naturally isomorphic to the rationalsingular homology. A third subproject is concerned with finiteenergy foliations, either in the context of area-preservingdisk-maps or the restricted planar three-body problem. Finiteenergy foliations were introduced by the PI and his collaboratorsand are an important tool in the study of the dynamics oflow-dimensional Hamiltonian systems. The restricted circularplanar three-body problem is not only interesting from a purelyacademic viewpoint, but is also used in the orbit design forscientific space missions. In certain regimes of the Jacobienergy, finite energy foliations allow to reduce the dynamics ofthe restricted three-body problem to that of the dynamics of anarea-preserving disk-map. Then again the same theory facilitatesthe understanding of the long-term behavior of iteratedarea-preserving disk maps. The project will study therelationship between the theory of finite energy foliations, asymplectic construction, and core dynamical systems notions likeentropy.It is a surprising fact that many physical systems evolving intime, allow a description as a Hamiltonian system. Systems ofthis kind describe the flow of an incompressible ideal fluid, themovement of a satellite under the gravitational forces ofcelestial bodies, or the movement of charged particles in amagnetic field. Hamiltonian systems are a very particular classof dynamical systems, which can be studied not only by themethods from dynamical systems theory, but also by a more exotickind of geometry called symplectic geometry. This is a geometrybased on the notion of area, in contrast to usual geometries,which have length and distance as their fundamentalnotions. Recent advances in this field already found somepractical uses. For example, this novel point of view has beenused in accelerator physics in algorithms controlling andinsuring stability of a beam of particles and called symplectictracking. It is the explicit purpose of this research tointegrate the approaches to Hamiltonian systems coming from thesetwo different perspectives. This should result in new methods,which should make it possible to attack problems, which so farseemed unreachable. For example it seems long term be feasible touse the new methods to develop algorithms, which would findfuel-efficient orbits for scientific space missions. Currenttechnology can verify the properties of proposed orbits extremelyfast. However, currently there is no good method for finding suchorbits. It rather compares to finding a needle in a haystack,using very large amounts of computing time.