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.

本文所提出的研究是在动力系统和微分几何的界面上进行的。它的主要目标是研究“高阶”系统的动力学和几何结构。这样的系统自然出现在动力系统和几何学中。特别是，研究者将研究高阶交换李群和半单李群及其格的双曲作用的刚性性质，最终目的是对这类系统进行分类。这位研究者还计划研究黎曼流形(特别是高阶流形)及其测地线流。这项研究将使用几何、动力学和群论工具。动力学系统和遍历理论是相对较新的领域，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)。来自动力学(如混沌、分形学)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而安全的代码方面发挥了重要作用。来自平滑动力学的工具和想法被应用到远至细胞生物学和气象学的领域。几何是数学中最古老的领域之一，一般研究曲线、曲面和它们的高维类似物、它们的形状、最短路径和这些空间之间的映射。微分几何起源于制图学，现在被研究是出于美学原因，以及它与物理学、其他科学和应用领域(例如计算机视觉)的密切联系。几何与动力学有着密切的联系，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。这个项目的主要目标是研究两个动力系统何时通勤，即一个系统不受另一个系统带来的变化的影响。这类系统的重要例子来自几何学。