Symplectic Geometry and Dynamics

摘要奖：DMS-1104470首席研究员：赫尔穆特·霍夫第一个子项目将与辛场理论(SFT)的完成有关。这是目前最普遍、最全面的辛不变量理论。这个项目有三个组成部分，即多重Fredholm理论、尺度光滑分析和将SFT表示为具有运算的多重Fredholm问题的解计数的代数表示。第二个子项目是多重Fredholm理论的副产品，它试图构建一个基于Fredholm问题的等价类的同调理论。由此产生的好处是，研究非线性偏微分方程时的横截性问题实际上不会显式出现。它们将隐藏在(一次性)证明中，证明空间的新同调理论自然同构于有理奇异同调。第三个子项目涉及有限能量叶层，或者在面积保持圆盘映射的背景下，或者在受限平面三体问题的背景下。有限能谱是由PI及其合作者提出的，是研究低维哈密顿系统动力学的重要工具。受限圆平面三体问题不仅从纯学术角度来说很有趣，而且在科学航天任务的轨道设计中也有应用。在雅可比能量的某些区域内，有限能量叶允许将受限三体问题的动力学简化为面积保持的圆盘映射的动力学。同样的理论也有助于理解迭代保面积圆盘映射的长期行为。该项目将研究有限能量分层理论、非辛结构和像熵这样的核心动力系统概念之间的关系。令人惊讶的是，许多物理系统在时间上演化，允许描述为哈密顿系统。这类系统描述了不可压缩理想流体的流动，卫星在天体引力作用下的运动，或带电粒子在磁场中的运动。哈密顿系统是一类非常特殊的动力系统，它不仅可以用动力系统理论的方法来研究，而且还可以用一种更奇特的被称为辛几何的几何来研究。这是一种基于面积概念的几何，与通常的以长度和距离为基本概念的几何不同。这一领域的最新进展已经发现了一些实际用途。例如，这一新观点被用于加速器物理学中控制和确保粒子束稳定性的算法，并被称为辛跟踪。这项研究的明确目的是整合来自这两个不同角度的哈密顿系统的方法。这应该会产生新的方法，从而有可能解决迄今为止似乎遥不可及的问题。例如，使用新方法开发算法似乎从长远来看是可行的，这将为科学空间任务找到燃料效率高的轨道。目前的技术可以极快地验证所提出的轨道的性质。然而，目前还没有好的方法来寻找这样的霍尔比特。这相当于大海捞针，使用了非常大量的计算时间。

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.