Rigidity Phenomena in Geometry and Dynamics

摘要：项目负责人：Ralf spatzier提出的研究方向是动力系统与微分几何的结合。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、几何甚至其他看似完全不同的领域，如数论。研究者将研究高阶阿贝尔和半单李群及其格的作用的刚性性质，最终目的是在适当的几何或动力学假设下对这类系统进行分类。特别地，他将研究高阶双曲作用，半单群的作用及其格保持仿射和几何结构。研究者还将研究1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相对年轻的领域，研究物理或数学系统随时间的演变（例如流体流动中的湍流）。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。动力学安德古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产生了重大影响。例如，符号动力学在为计算机科学开发高效和安全的代码方面发挥了重要作用。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等领域。几何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。微分几何起源于地图学，现在因其与物理学、其他科学和应用领域（如计算机视觉）以及内在美学原因的密切联系而受到研究。几何与动力学密切相关，因为一些重要的动力系统起源于几何，而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态系统交换，即当一个系统不受另一个系统带来的变化的影响。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.