Rigidity Phenomena in Differential Geometry and Dynamical Systems

微分几何和动力系统中的刚性现象

• 批准号: 9971556
• 负责人: Ralf Spatzier
• 金额: $ 9.73万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1999
• 资助国家: 美国
• 起止时间: 1999-08-01 至 2003-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9971556&HistoricalAwards=false
• 关键词: Rigidity; Phenomena; Differential; Geometry; Dynamical

AbstractSpatzierThe research proposed lies at the interface of dynamical systems and differential geometry. Its principal goal is the investigation of the dynamical and geometric structures of "higher rank" systems. Such systems naturally appear both in dynamical systems and geometry. In particular, the investigator will study rigidity properties of hyperbolic actions of higher rank abelian and semisimple Lie groups and their lattices with the ultimate goal of classifying such systems. The investigator also plans to investigate Riemannian manifolds (especially higher rank ones) and their geodesic flows. Geometric, dynamical and group theoretic tools will be used in this research.Dynamical systems and ergodic theory are relatively new fields that investigate the evolution of a physical or mathematical system over time (e.g. turbulence in a fluid flow). New ideas and concepts from dynamics (e.g. chaos, fractals) have changed our perception of the world fundamentally. Dynamics and ergodic theory provide the mathematical tools and analysis for these investigations. Dynamical systems have had a major impact on the sciences and engineering. Symbolic dynamics for instance has been instrumental in developing efficient and safe codes for computer science. Tools and ideas from smooth dynamics are used as far afield as cell biology and meteorology. Geometry is one of the oldest fields in mathematics, and generally studies curves, surfaces and their higher dimensional analogues, their shapes, shortest paths, and maps between such spaces. Differential geometry had its roots in cartography, and is now studied for aesthetic reasons and its close ties with physics and other sciences and applied areas (computer vision e.g.). Geometry and dynamics are closely related as some important dynamical systems originate from geometry, and geometry also provides tools to study dynamical systems. The main goal of this project studies when two dynamical systems commute, i.e. when one system is unaffected by the changes brought on by the other. Important examples of such systems arise from geometry.

AbstractSpatzierThe研究提出的是动态系统和差异几何形状的界面。 它的主要目标是研究“高级”系统的动力学和几何结构。 这样的系统自然出现在动态系统和几何形状中。 特别是，研究者将研究高级阿贝利安和半圣母谎言组及其晶格的双曲线作用的刚性特性，其最终目标是对此类系统进行分类。 研究人员还计划调查里曼尼（Riemannian）流形（尤其是等级的流动）及其地球流量。 这项研究将使用几何，动力学和群体理论工具。跨性别的系统和厄贡理论是相对较新的领域，可以随着时间的推移研究物理或数学系统的演变（例如流体流中的湍流）。 动态（例如混乱，分形）的新思想和概念从根本上改变了我们对世界的看法。 动力学和千古理论为这些研究提供了数学工具和分析。动态系统对科学和工程产生了重大影响。 例如，符号动态在开发计算机科学的高效且安全的代码方面发挥了作用。 流畅动态的工具和想法与细胞生物学和气象学一样远。 几何是数学中最古老的领域之一，通常研究曲线，表面及其较高的尺寸类似物，它们的形状，最短路径和此类空间之间的地图。 差异几何形状起源于制图，现在出于美学原因及其与物理，其他科学和应用领域的紧密联系（例如计算机视觉）。 几何和动力学密切相关，因为一些重要的动态系统源自几何形状，几何也提供了研究动态系统的工具。 当两个动态系统通勤时，即当一个系统不受另一个系统带来的变化影响时，该项目的主要目标是研究。这种系统的重要例子来自几何形状。