Rigidity Phenomena in Geometry and Dynamics

摘要DMS - 0203735。提出的研究是在动力系统和微分几何的界面。它的主要目标是研究“高阶”系统的动力学和几何结构。这样的系统自然地出现在动力系统和几何学中。特别是，首席研究员将研究高阶阿贝尔和半单李群及其格的作用的刚性性质，最终目标是在合适的几何或动力学假设下对这些系统进行分类。特别地，他将研究高阶双曲阿贝尔作用，半单群的作用及其格保持仿射和几何结构。他也将研究代数高阶阿贝尔动作的不变测度。研究者还将研究黎曼流形（特别是高阶流形）及其测地线流。本研究将使用几何、动力学和群论工具。动力系统和遍历理论是相对较新的领域，研究物理或数学系统随时间的演变（例如流体流动中的湍流）。来自动力学的新思想和概念（如混沌、分形）从根本上改变了我们对世界的看法。动力学和遍历理论为这些研究提供了数学工具和分析。动力系统对科学和工程产生了重大影响。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映射。微分几何起源于地图学，现在因其与物理学和其他科学和应用领域（例如计算机视觉）以及内部美学原因的密切联系而受到研究。几何与动力学密切相关，因为一些重要的动力系统起源于几何，而几何也为研究动力系统提供了工具。这个项目的一个主要目标是研究当两个动态系统交换时，即当一个系统不受另一个系统带来的变化的影响。这种系统的重要例子来自几何学。

ABSTRACT DMS - 0203735.The research proposed lies at the interface of dynamical systems and differential geometry. Its principal goal is the investigation of the dynamical and geometric structures of "higher rank" systems. Such systems naturally appear both in dynamical systems and geometry. In particular, the principal investigator will study rigidity properties of actions of higher rank abelian and semisimple Lie groups and their lattices with the ultimate goal of classifying such systems under suitable geometric or dynamical hypotheses. In particular, he will study higherrank hyperbolic abelian actions, and actions by semisimple groups and their lattices preserving affine and geometric structures. He will also investigate invariant measures of algebraic higher rank abelian actions. The investigator will also investigate Riemannian manifolds (especially higher rank ones)and their geodesic flows. Geometric, dynamical and group theoretic tools will be used in this research.Dynamical systems and ergodic theory are relatively new fields that investigate 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. Dynamics and ergodic theory provide the mathematical tools and analysis for these investigations. Dynamical systems have had a major impact on the sciences and engineering. Symbolic dynamics for instance 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 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 its close ties with physics and other sciences and applied areas (computer vision e.g.)as well as internal aesthetic reasons. Geometry and dynamics are closely related as some important dynamical systems originate from geometry, and geometry also 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. Important examples of such systems arise from geometry.