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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.

动力系统和遍历理论研究物理、生物或数学系统在时间上的演化，如流体流动中的湍流、变化的行星系统或疾病的演化。信息、熵、混沌和分形学等基本思想和概念深刻地影响了我们对世界的理解。动力系统和遍历理论发展了极好的工具，并对科学和工程产生了强大的影响。例如，符号动力学在为计算机科学开发高效而安全的代码方面发挥了重要作用。来自平滑动力学的工具和想法被应用到远至细胞生物学和气象学的领域。几何是数学中一个高度发达、古老而又异常活跃的领域。它研究曲线、曲面和它们的高维类似物、它们的形状、最短路径和这些空间之间的映射。为了他的公国而测量土地，高斯还发展了测地线和曲率的基本概念，为现代微分几何奠定了基础。它与物理学和计算机视觉等应用领域有着密切的联系。几何学和动力学是紧密相连的。事实上，重要的动力系统，如测地线流，来自几何，反之亦然，人们可以使用几何工具来研究动力学。这个项目的一个主要目标是研究动力系统的对称性，特别是当一个系统不受另一个系统带来的变化的影响时。我们的追求是通过意想不到的对称性来研究这些系统。当空间包含许多平坦子空间时，重要的例子来自几何学。最后，通过几何或动力学情形的对称群，在动力学和几何学中引入群论。首席研究员将继续培训新一代研究人员和数学家，以及各级学生在数学方面的努力。这个项目包括支持研究生的研究培训机会和本科生的暑期研究经验。这个项目将研究几何和动力学中的刚性现象，特别是高阶阿贝尔和半单李群及其格子的作用。后者是齐默尔计划的一部分。将特别强调这类团体的夸张行动。由于高阶半单李群及其格包含高阶阿贝尔群，交换半单李群与半单李群的分类和刚性问题是密切相关的，互为丰富。目标是对这类行为进行分类。与此密切相关的是几何结构的自同构群的研究。几何方面的研究将介绍高阶黎曼流形及其分类，并引入新的方法。还将研究测地线流和框架流的动力学，并研究李群的离散子群的秩刚性和测度性质。除了建立新的结果，首席研究员还努力寻找和引入新的方法来调查这些问题，这将有助于他们自己在其他领域的应用。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。