Rigidity Phenomena in Geometry and Dynamics

几何和动力学中的刚性现象

• 批准号: 0203735
• 负责人: Ralf Spatzier
• 金额: $ 22.2万
• 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: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0203735&HistoricalAwards=false
• 关键词: Rigidity; Phenomena; Geometry; Dynamics

ABSTRACT DMS - 0203735.The 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 principal investigator will study rigidity properties of actions of higher rank abelian and semisimple Lie groups and their lattices with the ultimate goal of classifying such systems under suitable geometric or dynamical hypotheses. In particular, he will study higherrank hyperbolic abelian actions, and actions by semisimple groups and their lattices preserving affine and geometric structures. He will also investigate invariant measures of algebraic higher rank abelian actions. The investigator will also 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 its close ties with physics and other sciences and applied areas (computer vision e.g.)as well as internal aesthetic reasons. Geometry and dynamics are closely related as some important dynamical systems originate from geometry, and geometry also provides tools to study dynamical systems. One 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.

摘要DMS -0203735。提出的研究在于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，主要研究者将研究高级阿贝利安和半圣母谎言组及其晶格的僵化特性，其最终目标是在适当的几何或动态假设下对此类系统进行分类。 特别是，他将研究高光双曲线阿贝尔的作用，以及半圣像群体及其晶格的作用，以保留仿射和几何结构。他还将研究代数高级阿贝尔行动的不变措施。 研究人员还将调查里曼尼亚歧管（尤其是较高排名）及其测量流。 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，现在研究其与物理，其他科学和应用领域（计算机视觉）以及内部美学原因的紧密联系。 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，该项目研究的一个主要目标。这种系统的重要例子来自几何形状。