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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：
1607260
负责人：
Ralf Spatzier
金额：
$ 32.2万
依托单位：
Regents of the University of Michigan - Ann Arbor
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2016
资助国家：
美国
起止时间：
2016-09-01 至 2020-08-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1607260&HistoricalAwards=false
关键词：
Rigidity Phenomena Geometry Dynamics

项目摘要

Dynamical systems and ergodic theory investigate the evolution of a physical or mathematical system over time, such as turbulence in a fluid flow or changing planetary systems. New ideas and concepts such as information, entropy, 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 example 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. Alternatively, these are systems with unexpected symmetries 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.This project centers on problems between dynamical systems, group theory and geometry. There are two main goals: First, establish exponential mixing properties for several different systems in dynamics, in particular frame flows from Riemannian geometry and solenoids coming from noninvertible systems. The principal investigator (PI) will draw tools from dynamics, geometry and number theory to accomplish these goals. Second, prove rigidity properties in geometry and in dynamical systems, in particular when the system and spaces in question are "higher rank", e.g. when spaces have flat subspaces or the dynamics has nontrivially commuting elements. Such systems 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. The PI will employ tools from geometry, dynamics, Lie groups, and specifically exponential mixing properties. The PI will also investigate discrete faithful representations of hyperbolic groups in p-adic Lie groups, equilibrium states for partially hyperbolic dynamical systems and spherical higher rank in Riemannian geometry.

动力系统和遍历理论研究物理或数学系统随时间的演化，例如流体流动中的湍流或变化的行星系统。信息、熵、混沌和分形等新的思想和概念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。几何学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。微分几何起源于制图学，始于世纪的高斯。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。几何学和动力学密切相关。事实上，重要的动力系统来自几何，反之亦然，几何提供了研究动力系统的工具。该项目的一个主要目标是研究两个动力系统何时交换，即当一个系统不受另一个系统所带来的变化的影响时。或者，这些系统具有意想不到的对称性，当空间包含许多平坦子空间时，这种系统的重要例子来自几何学。群论通过研究几何或动力学情形的对称群，或通过研究作用于空间的对称群的动力学和几何行为，最终进入动力学和几何学。这个项目集中在动力系统，群论和几何之间的问题。有两个主要目标：首先，建立指数混合性质的几个不同的系统的动力学，特别是框架流从黎曼几何和不可逆系统的。主要研究者（PI）将从动力学，几何学和数论中提取工具来实现这些目标。其次，证明几何和动力系统中的刚性性质，特别是当系统和空间的问题是“高秩”，例如，当空间有平坦的子空间或动力学有非平凡的交换元素。这样的系统自然地出现在看似完全独立的领域，例如数论或拉普拉斯算子的频谱研究。研究人员将研究高阶阿贝尔和半单李群及其晶格的刚性性质，努力在合适的几何或动力学假设下对此类系统进行分类。 PI将采用几何学，动力学，李群，特别是指数混合性质的工具。 PI还将研究双曲群在p-adic李群中的离散忠实表示，部分双曲动力系统的平衡态和黎曼几何中的球形高阶。