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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.

提出的研究介于动力系统、群论和几何之间。主要，研究者计划在这些地区研究“高阶”系统的动力学和几何结构。它们自然地出现在看似完全不同的领域，例如在数论或研究拉普拉斯谱。研究者将研究高阶阿贝尔和半简单李群及其格的作用的刚性性质，努力在适当的几何或动力学假设下对这些系统进行分类。特别是，他将研究环面和齐次空间上的高阶双曲阿贝尔作用及其环，以及一般的2阶Cartan作用。这些特殊情况为更一般的猜想提供了检验。研究者还将研究半单群及其格保持射影、仿射和其他几何结构的行为。此外，研究者将分析黎曼流形（特别是高球阶流形）和更一般的奇异空间及其测地线流。本研究将使用几何、动力学和群论工具。动力系统和遍历理论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。新的思想和概念，如混沌和分形已经改变了我们对世界的理解。动力学和遍历理论提供了优秀的数学工具，并对科学和工程产生了强烈的影响。例如，符号动力学在为计算机科学开发高效和安全的代码方面发挥了重要作用。光滑动力学的工具和思想被广泛应用于细胞生物学和气象学。几何是数学中一个高度发展的古老领域，具有惊人的活力。它研究曲线、曲面及其高维类似物、它们的形状、最短路径以及这些空间之间的映射。微分几何起源于19世纪由高斯提出的制图学。它与物理学和其他科学以及计算机视觉等应用领域密切相关。几何学和动力学是密切相关的。事实上，重要的动力系统来自几何学，反之亦然，几何学提供了研究动力系统的工具。这个项目的一个主要目标是研究当两个动态系统交换时，即当一个系统不受另一个系统带来的变化的影响。这种系统的重要例子出现在几何中，当空间包含许多平坦子空间时。群论最终通过研究几何或动力情况下的对称群，或通过研究作用于空间的对称群的动力和几何行为而进入动力学和几何。