Rigidity Phenomena in Geometry and Dynamics

几何和动力学中的刚性现象

基本信息

批准号：
1307164
负责人：
Ralf Spatzier
金额：
$ 29.52万
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：
2013
资助国家：
美国
起止时间：
2013-07-01 至 2017-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1307164&HistoricalAwards=false
关键词：
Rigidity Phenomena Geometry Dynamics

项目摘要

The research proposed lies between dynamical systems, group theory and geometry. Principally, the investigator plans to study dynamical and geometric structures of "higher rank" systems in these areas. These appear naturally in seemingly quite separate areas, for example in number theory or in studying the spectrum of the Laplacian. The investigator will work on rigidity properties of actions of higher rank abelian and semi-simple Lie groups and their lattices striving to classify such systems under suitable geometric or dynamical hypotheses. In particular, he will study higher rank hyperbolic abelian actions and their cocycles on tori and homogeneous spaces as well as general Cartan actions of rank 2. These special cases provide tests for more general conjectures. The investigator will also study actions by semi-simple groups and their lattices preserving projective, affine and other geometric structures. In addition, the investigator will analyze Riemannian manifolds (especially those of higher spherical rank) and more general singular spaces and their geodesic flows. Geometric, dynamical and group theoretic tools will be used in this research.Dynamical systems and ergodic theory investigate the evolution of a physical or mathematical system over time, such as turbulence in a fluid flow. New ideas and concepts such as chaos and fractals have changed our understanding of the world. Dynamics and ergodic theory provide excellent mathematical tools, and have a strong 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 a highly developed and ancient field in mathematics of amazing vigor. It studies curves, surfaces and their higher dimensional analogues, their shapes, shortest paths, and maps between such spaces. Differential geometry had its roots in cartography, starting with Gauss in the nineteenth century. It is closely linked with physics and other sciences and applied areas such as computer vision. Geometry and dynamics are closely related. Indeed, important dynamical systems come from geometry, and vice versa geometry 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 when the space contains many flat subspaces. Group theory finally enters both dynamics and geometry by studying the group of symmetries of a geometry or dynamical situation, or by investigating the dynamical and geometric behavior of the group of symmetries acting on a space.

该研究提出的位于动力学系统，群体理论和几何学之间。主要是，研究人员计划研究这些领域“较高等级”系统的动态和几何结构。这些自然是在看似很独立的领域，例如在数字理论或研究拉普拉斯的频谱中。研究人员将致力于高级阿贝尔和半简单谎言组的僵化特性及其在适当的几何或动力学假设下努力对此类系统进行分类的晶格。特别是，他将研究较高的柔双曲线阿贝尔（Abelian）的行动及其对托里和同质空间的共生以及等级2的一般cartan行动。这些特殊案例为更一般的猜想提供了测试。研究人员还将研究半简单组及其晶格的行动，以保存投影，仿射和其他几何结构。此外，研究者将分析Riemannian歧管（尤其是球形级别的歧管）和更一般的奇异空间及其地球流量。这项研究将使用几何，动力学和组理论工具。跨性别的系统和厄贡理论研究了物理或数学系统随时间的演变，例如流体流动中的湍流。混乱和分形等新思想和概念改变了我们对世界的理解。动力学和千古理论提供了出色的数学工具，并对科学和工程产生了强烈的影响。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。几何是令人惊叹的活力数学中的一个高度发达和古老的领域。它研究了此类空间之间的曲线曲线，表面及其更高的类似物，它们的形状，最短路径和地图。差异几何形状源于制图，从19世纪的高斯开始。它与物理和其他科学以及计算机视觉等应用领域紧密相关。几何和动态密切相关。实际上，重要的动力系统来自几何形状，反之亦然的几何形状为研究动态系统提供了工具。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。当空间包含许多平面子空间时，这种系统的重要示例来自几何形状。群体理论最终通过研究几何或动态情况的对称性组，或研究作用在空间上的对称性组的动力学和几何行为，从而进入动力学和几何形状。