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.

动力学系统和厄贡理论会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或变化的行星系统。信息，熵，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。另外，这些系统具有意外的对称性系统，当该空间包含许多平面子空间时，几何形状引起了此类系统的重要示例。群体理论最终通过研究几何或动态状况的对称性组，或研究作用于空间上的对称对称的动力学和几何行为，从而进入动力学和几何形状。该项目基于动态系统之间的问题，组理论集中在动态系统之间和几何。有两个主要目标：首先，为动力学中的几个不同系统建立指数混合属性，特别是来自riemannian几何形状和来自非不可逆转系统的螺线管的框架流。首席研究员（PI）将从动态，几何学和数理论中汲取工具，以实现这些目标。其次，证明几何和动态系统中的刚度属性，特别是当所讨论的系统和空间是“较高的等级”时，例如当空间具有平坦的子空间或动力学时，具有非平整通勤元素。这种系统自然而然地在看似非常独立的领域，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。 PI将采用几何，动力学，谎言组，特别是指数混合属性中的工具。 PI还将研究P-Adic Lie群体中双曲线基团的离散忠实表示形式，部分双曲动力学系统的平衡状态和Riemannian几何形状中的球形较高等级。