Mathematical Sciences: Rigidity Phenomena in Differential Geometry and Dynamical Systems

小行星9626173 拟议的研究位于差分的接口 几何学和动力系统 一个主要目标是调查 “高阶”系统的动力学和几何结构。 特别是，研究人员打算将双曲型 行动的高阶阿贝尔和半单李群或行动，保持几何结构，并建立其他 刚性性能 研究人员还计划研究 黎曼流形（特别是高阶流形）及其 测地线流 几何的，动力学的和群论的 工具在这项研究中使用。 几何学是数学中最古老的领域之一，通常 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地图。 微分几何起源于制图学，现在 由于美学原因及其与物理学的密切联系， 其他科学和应用领域（例如计算机视觉）。 动力学 系统是一个相对较新的领域，研究进化 物理或数学系统随时间的变化（例如， 流体流）。 来自动力学的新思想和概念（例如混沌， 分形）从根本上改变了我们对世界的看法。 动力系统对科学产生了重大影响， 工程. 例如，符号动力学 为计算机科学开发高效安全的代码。 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。

9626173 Spatzier The proposed research lies at the interface of differential geometry and dynamical systems. A principal goal is to investigate the dynamical and geometric structures of "higher rank" systems. In particular, the investigator intends to classify hyperbolic actions of higher rank abelian and semisimple Lie groups or actions which preserve geometric structures, and also establish other rigidity properties. The investigator also plans to study Riemannian manifolds (especially higher rank ones) and their geodesic flows. Both geometric, dynamical and group theoretic tools are used in this research. Geometry is one of the oldest fields in mathematics, and generally studies curves, surfaces and their higher dimensional analogues, their shapes, shortest paths, and maps between such spaces. Differential geometry had its roots in cartography, and is now studied for aesthetic reasons and its close ties with physics and other sciences and applied areas (computer vision e.g.). Dynamical systems is a relatively new field that investigates the evolution of a physical or mathematical system over time (e.g. turbulence in a fluid flow). New ideas and concepts from dynamics (e.g. chaos, fractals) have changed our perception of the world fundamentally. Dynamical systems have had a major impact on the sciences and engineering. Symbolic dynamics for instance has been instrumental for developing efficient and safe codes for computer science. Tools and ideas from smooth dynamics are used as far afield as cell biology and meteorology.