Rigidity Phenomena in Geometry and Dynamics

提出的研究介于动力系统、群论和几何之间。主要的，研究者计划在这些地区研究“高阶”系统的动力学和几何结构。它们自然而然地出现在看似完全不同的领域，例如在数论或研究拉普拉斯谱时。研究人员将致力于研究高阶阿贝尔和半单李群及其格子作用的刚性性质，努力在适当的几何或动力学假设下对这类系统进行分类。特别是，他将研究环面和齐次空间上的高阶双曲阿贝尔作用量及其上循环，以及秩为2的一般Cartan作用量。这些特例为更一般的猜想提供了检验。研究人员还将研究保持射影、仿射和其他几何结构的半单群及其格的作用。此外，研究者还将分析黎曼流形(特别是高球秩黎曼流形)和更一般的奇异空间及其测地线流。这项研究将使用几何、动力学和群论工具。动力学系统和遍历理论研究物理或数学系统随时间的演变，如流体流动中的湍流。混沌、分形学等新思想、新概念改变了我们对世界的认识。动力学和遍历理论提供了极好的数学工具，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和古老的领域，具有惊人的生命力。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。微分几何起源于地图学，始于19世纪的高斯。它与物理学和其他科学以及计算机视觉等应用领域密切相关。几何学和动力学是密切相关的。事实上，重要的动力系统来自几何，反之亦然，几何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两个动力系统何时通勤，即一个系统不受另一个系统带来的变化的影响。当空间包含许多平坦的子空间时，这种系统的重要例子来自几何学。群论通过研究几何或动力学情形的对称群，或通过研究作用于空间的对称群的动力学和几何行为，最终进入动力学和几何学。

The research proposed lies between dynamical systems, group theory and geometry. Principally, the investigator plans to study dynamical and geometric structures of "higher rank" systems in these areas. These appear naturally in seemingly quite separate areas, for example in number theory or in studying the spectrum of the Laplacian. The investigator will work on rigidity properties of actions of higher rank abelian and semi-simple Lie groups and their lattices striving to classify such systems under suitable geometric or dynamical hypotheses. In particular, he will study higher rank hyperbolic abelian actions and their cocycles on tori and homogeneous spaces as well as general Cartan actions of rank 2. These special cases provide tests for more general conjectures. The investigator will also study actions by semi-simple groups and their lattices preserving projective, affine and other geometric structures. In addition, the investigator will analyze Riemannian manifolds (especially those of higher spherical rank) and more general singular spaces and their geodesic flows. Geometric, dynamical and group theoretic tools will be used in this research.Dynamical systems and ergodic theory investigate the evolution of a physical or mathematical system over time, such as turbulence in a fluid flow. New ideas and concepts such as chaos and fractals have changed our understanding of the world. Dynamics and ergodic theory provide excellent mathematical tools, and have a strong 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 a highly developed and ancient field in mathematics of amazing vigor. It studies curves, surfaces and their higher dimensional analogues, their shapes, shortest paths, and maps between such spaces. Differential geometry had its roots in cartography, starting with Gauss in the nineteenth century. It is closely linked with physics and other sciences and applied areas such as computer vision. Geometry and dynamics are closely related. Indeed, important dynamical systems come from geometry, and vice versa geometry 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 when the space contains many flat subspaces. Group theory finally enters both dynamics and geometry by studying the group of symmetries of a geometry or dynamical situation, or by investigating the dynamical and geometric behavior of the group of symmetries acting on a space.