Rigidity Phenomena in Geometry and Dynamics

AbstractAward：DMS-0604857首席研究员：Ralf Spatzier提出的研究位于动力系统和微分几何的界面。 它的主要目标是研究“高阶”系统的动力学和几何结构。 这样的系统自然地出现在动力系统、几何学甚至其他看似完全不同的领域，如数论中。 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，并采取行动的半简单群体和theirlattices保持仿射和几何结构。 研究者还将研究一阶半单李群中离散群作用的刚性性质。 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是研究物理或热力学系统随时间演变（例如流体流动中的湍流）的相对年轻的领域。 混沌和分形等动力学的新思想和概念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有重大影响。 例如，符号动力学在开发计算机科学的高效和安全代码方面起到了重要作用。 平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以及应用领域（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. 这个项目的一个主要目标是研究当两个动力系统交换时，即当一个系统不受另一个系统所带来的变化的影响时。 这种系统的重要例子来自几何学。

AbstractAward: DMS-0604857Principal Investigator: Ralf SpatzierThe research proposed lies at the interface of dynamical systemsand differential geometry. Its principal goal is theinvestigation of the dynamical and geometric structures of"higher rank" systems. Such systems appear naturally indynamical systems, geometry and even other seemingly quitediffernt areas such as number theory. The investigator willstudy rigidity properties of actions of higher rank abelian andsemisimple 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, and actions by semisimple groups and theirlattices preserving affine and geometric structures. Theinvestigator will also investigate rigidity properties of actionsof discrete groups in rank one semisimple Lie groups. Inaddition, he will analyze Riemannian manifolds (especially higherrank ones) and their geodesic flows. Geometric, dynamical andgroup theoretic tools will be used in this research.Dynamical systems and ergodic theory are relatively young fieldsthat investigate the evolution of a physical or mathematicalsystem over time (e.g. turbulence in a fluid flow). New ideasand concepts from dynamics such as chaos and fractals havechanged our perception of the world fundamentally. Dynamics andergodic theory provide the mathematical tools and analysis forthese investigations. Dynamical systems have had a major impacton the sciences and engineering. Symbolic dynamics for instancehas been instrumental in developing efficient and safe codes forcomputer science. Tools and ideas from smooth dynamics are usedas far afield as cell biology and meteorology. Geometry is oneof 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. Onemain goal of this project studies when two dynamical systemscommute, i.e. when one system is unaffected by the changesbrought on by the other. Important examples of such systemsarise from geometry.