Rigidity Properties in Dynamics and Geometry

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变，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、混沌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。动力系统和遍历理论已经发展出了高超的工具，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、古老而又极为活跃的领域。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。在为他的公国测量土地的同时，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。几何和动力学是紧密相连的。事实上，像测地线流这样重要的动力系统来自于几何学，反之亦然，人们可以使用几何学工具来研究动力学。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所带来的变化的影响时。我们的目标是通过意想不到的对称性来研究这些系统。重要的例子出现在几何中，当空间包含许多平坦子空间时。最后，将群论引入到动力学和几何中，通过几何或动力情况下的对称群。首席研究员将继续培养新一代的研究人员和数学家，以及各级学生在数学方面的努力。该项目包括为研究生提供研究培训机会和为本科生提供暑期研究经验。本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简单李群及其格的作用。后者是齐默项目的一部分。将特别强调这类群体的夸张行为。由于高阶半单李群及其格中含有高阶阿贝尔群，因此阿贝尔群与半单李群的分类和刚性问题密切相关，具有丰富的交叉效应。目标是对这些行为进行分类。与此密切相关的是几何结构的自同构群的研究。几何学的研究将涉及高阶黎曼流形及其分类，引入新的方法。也将研究测地线和框架流的动力学，并研究李群的秩刚性和测量性质的离散子群。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研究这些问题，这些方法将有助于在其他领域的应用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.