Rigidity Properties in Dynamics and Geometry

动力学和几何中的刚性特性

基本信息

批准号：
2003712
负责人：
Ralf Spatzier
金额：
$ 57.1万
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：
2020
资助国家：
美国
起止时间：
2020-06-01 至 2024-05-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2003712&HistoricalAwards=false
关键词：
Rigidity Properties Dynamics Geometry

项目摘要

Dynamical systems and ergodic theory investigate the evolution of physical, biological or mathematical systems in time, such as turbulence in a fluid flow, changing planetary systems or the evolution of diseases. Fundamental ideas and concepts such as information, entropy, chaos and fractals have had a profound impact on our understanding of the world. Dynamical systems and ergodic theory have developed superb 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, ancient yet superbly active field in mathematics. It studies curves, surfaces and their higher dimensional analogues, their shapes, shortest paths, and maps between such spaces. Surveying the land for his principality, Gauss also developed the fundamental notions of geodesics and curvature, laying the groundwork for modern differential geometry. It has close links with physics and applied areas like computer vision. Geometry and dynamics are closely connected. Indeed, important dynamical systems such as the geodesic flow come from geometry, and vice versa one can use geometric tools to study dynamics. One main goal of this project studies symmetries of dynamical systems, especially when one system is unaffected by the changes brought on by the other. The quest is to study these systems via unexpected symmetries. Important examples arise from geometry when the space contains many flat subspaces. Finally group theory gets introduced in both dynamics and geometry via the group of symmetries of a geometry or dynamical situation. The principal investigator will continue training a new generation of researchers and mathematicians, and students at all levels in their mathematical endeavor. This project includes support for research training opportunities for graduate students and summer research experiences for undergraduates.This project will investigate rigidity phenomena in geometry and dynamics, especially actions of higher rank abelian and semi-simple Lie groups and their lattices. The latter is part of the Zimmer program. Particular emphasis will be put on hyperbolic actions of such groups. As higher rank semisimple Lie groups and their lattices contain higher rank abelian groups, the classification and rigidity problems for the abelian and semi-simple cases are closely related, with abundant cross fertilization. The goal is the classification of such actions. Closely related are the study of automorphism groups of geometric structures. Investigations in geometry will address higher rank Riemannian manifolds and their classification, introducing novel methods. The dynamics of geodesic and frame flows will also be studied, with investigations of discrete subgroups of Lie groups for rank rigidity and measure properties. Besides establishing new results, the principal investigator also strives to find and introduce novel methods for investigating these problems which will lend themselves to applications in other areas.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

动力学系统和厄贡理论会及时研究物理，生物学或数学系统的演变，例如流体流动中的湍流，变化的行星系统或疾病的演变。信息，熵，混乱和分形等基本思想和概念对我们对世界的理解产生了深远的影响。动力学系统和偏僻的理论开发了出色的工具，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。几何形状是一个高度发达，古老而又极具活跃的数学领域。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。高斯对土地的公国进行了调查，还制定了大地测量和曲率的基本概念，为现代差异几何形状奠定了基础。它与物理和应用领域（如计算机视觉）有着密切的联系。几何和动力学紧密连接。实际上，重要的动力系统（例如测量流）来自几何形状，反之亦然，可以使用几何工具来研究动力学。该项目的一个主要目标研究动态系统的对称性，尤其是当一个系统不受另一个系统的变化影响时。追求是通过意外的对称性研究这些系统。当空间包含许多平面子空间时，重要的例子是由几何形状产生的。最终，小组理论通过几何或动态情况的对称性组在动力学和几何形状中引入。首席研究人员将继续培训新一代的研究人员和数学家，并在其数学努力中各个级别的学生。该项目包括支持研究生的研究培训机会以及本科生的夏季研究经验。该项目将研究几何和动态的僵化现象，尤其是高级阿贝利安和半简单的谎言组及其晶格的较高行动。后者是Zimmer计划的一部分。特别重点将放在此类群体的双曲动作上。由于较高排名的半圣母谎言组及其晶格包含较高排名的阿贝利亚人群，因此阿贝尔和半简单病例的分类和僵化问题密切相关，并具有丰富的交叉施肥。目标是对此类行动的分类。密切相关的是对几何结构的自动形态群体的研究。几何学的研究将解决较高的riemannian歧管及其分类，并引入新方法。还将研究测量和框架流动的动力学，并研究了谎言组的离散亚组，以进行等级刚度和测量属性。除了建立新的结果外，首席研究人员还努力寻找并介绍新的方法来调查这些问题，这些方法将使自己适用于其他领域的应用。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响标准的评估来获得支持的。