Rigidity Phenomena in Geometry and Dynamics

几何和动力学中的刚性现象

• 批准号: 0604857
• 负责人: Ralf Spatzier
• 金额: $ 23.88万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0604857&HistoricalAwards=false
• 关键词: Rigidity; Phenomena; Geometry; Dynamics

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.

Abstractaward：DMS-0604857原理研究者：Ralf Spatzierthe研究提出的研究位于动态系统和差异几何形状的界面。 它的主要目标是对“高级”系统的动态和几何结构进行研究。 这样的系统似乎是自然的印象系统，几何形状，甚至是其他看似退出的领域，例如数字理论。 研究者将研究高级Abelian和Emisimple Lie offer组及其晶格的作用的最终目标的僵化特性，其晶格将在适当的几何或动力学下对此类系统进行分类。 特别是，他将研究较高的级别双波拉贝利亚的行动，以及半圣像群体及其保持仿射和几何结构的作用。 评估者还将研究排名组中一个半圣母谎言组的离散组动作的刚性属性。 他将分析Riemannian歧管（尤其是Higherrank的流动）及其地球流量。 在这项研究中将使用几何，动力学和组理论工具。跨性别的系统和千古理论是相对较年轻的领域，这是随着时间的流逝研究物理或数学系统的演变（例如流体流中的湍流）。 来自混乱和分形等动力的新思想和概念从根本上讲我们对世界的看法。 动力学和er虫理论提供了数学工具和分析。 动力学系统对科学和工程产生了重大影响。 实例的符号动态在开发有效且安全的代码中有用。 流畅动态的工具和想法被远场用作细胞生物学和气象。 几何形状是数学上最古老的领域，通常是研究表面，表面及其较高的尺寸类似物，它们的形状，最短的路径，最短的路径和地图。差异几何形状扎根于制图中，并且与物理学和其他科学和其他科学和计算机的理由相结合，并且是既关键的，又是计算机和计算机（计算机远方的）。 几何和动力学是密切相关的可加工重要动力系统源自几何形状，几何也提供了研究动态系统的工具。 当两个动态系统访问时，即当一个系统不受另一个系统的变化影响时，该项目的目标是研究该项目的目标。 从几何形状进行此类系统化的重要例子。