Rigidity Phenomena in Geometry and Dynamics

摘要奖：DMS-0906085首席研究员：Ralf Spatzier提出的研究介于动力系统和几何学之间。 它的主要目标是研究这些领域中“高阶”系统的动力学和几何结构。 这些系统也自然地出现在完全独立的领域，如数论。研究者将研究高阶阿贝尔和半单李群及其格作用的刚性性质，最终目标是在适当的几何或动力学假设下对此类系统进行分类。特别是，他将研究高阶双曲阿贝尔行动的环面和嘉当行动的秩2。 这些特殊情况本身就很重要，它们将成为更一般性论述的试验场。此外，在这些情况下，还可以使用其他分类工具。 研究者还将研究半单群及其保持仿射和其他几何结构的格的作用。此外，他将分析黎曼流形（特别是高阶流形）及其测地线流。本研究将使用几何、动力学和群论工具。动力系统和遍历理论研究物理或数学系统随时间的演化（例如，湍流）。 新的思想和概念，如混沌和分形，从根本上改变了我们对世界的看法。动力学和遍历理论为这些研究提供了数学工具和分析。动力系统对科学和工程学产生了重大影响。例如，符号动力学在为计算机科学开发有效和安全的代码方面起到了重要作用。 光滑动力学的工具和思想被广泛应用于细胞生物学和气象学。 几何学是数学中最古老的领域之一，通常研究曲线，曲面及其高维类似物，它们的形状，最短路径以及这些空间之间的映射。微分几何起源于制图学，现在因其与物理学和其他科学和应用领域（例如计算机视觉）的密切联系而受到研究。以及内在的审美原因。几何与动力学有着密切的联系，一些重要的动力系统起源于几何，几何也为研究动力系统提供了工具.该项目的主要目的之一是研究两个动力系统何时发生交换，即当一个系统不受另一个系统变化的影响时。这种系统的重要例子来自几何学。

AbstractAward: DMS-0906085Principal Investigator: Ralf SpatzierThe research proposed lies between dynamical systems andgeometry. Its principal goal is the investigation of thedynamical and geometric structures of "higher rank" systems inthese areas. These systems also appear naturally in seeminglyquite separate areas such as number theory. The investigator willstudy rigidity properties of actions of higher rank abelian andsemi-simple Lie groups and their lattices with the ultimate goalof classifying such systems under suitable geometric or dynamicalhypotheses. In particular, he will study higher rank hyperbolicabelian actions on tori and Cartan actions of rank 2. Thesespecial cases, important in their own right, will serve astesting ground for more general conjectures. Also, additionaltools for classification are available in these cases. Theinvestigator will also study actions by semi-simple groups andtheir lattices preserving affine and other geometric structures.In addition, he will analyze Riemannian manifolds (especiallyhigher rank ones) and their geodesic flows. Geometric, dynamicaland group theoretic tools will be used in this research.Dynamical systems and ergodic theory investigate the evolution ofa physical or mathematical system over time (e.g. turbulence in afluid flow). New ideas and concepts such as chaos and fractalshave changed our perception of the world fundamentally. Dynamicsand ergodic theory provide the mathematical tools and analysisfor these investigations. Dynamical systems have had a majorimpact on the sciences and engineering. Symbolic dynamics forinstance has been instrumental in developing efficient and safecodes for computer science. Tools and ideas from smooth dynamicsare used as far afield as cell biology and meteorology. Geometryis one of the oldest fields in mathematics, and generally studiescurves, surfaces and their higher dimensional analogues, theirshapes, shortest paths, and maps between such spaces.Differential geometry had its roots in cartography, and is nowstudied for its close ties with physics and other sciences andapplied areas (computer vision e.g.) as well as internalaesthetic reasons. Geometry and dynamics are closely related assome important dynamical systems originate from geometry, andgeometry also provides tools to study dynamical systems. One maingoal of this project studies when two dynamical systems commute,i.e. when one system is unaffected by the changes brought on bythe other. Important examples of such systems arise fromgeometry.