Group Actions, rigidity and geometry

摘要奖：DMS-0541917首席研究员：大卫费舍尔拟议的研究介于动力系统和几何。Furstenberg，Mostow，Margulis等人的工作表明，某些动力系统是研究重要几何空间，特别是对称和局部对称空间的几何性质的重要工具。.对称空间的一个重要性质是它们具有非正曲率。调查员计划在几个相互关联的方向上扩展这项工作。对称空间的研究将被推广到包括更一般的非正曲率空间以及更一般的空间之间的映射，重点是无限维空间。 研究者将尝试利用这些关系来证明Zimmer长期存在的问题。紧致流形的复同态群作用在平方可积黎曼度量空间上，而度量空间自然是一个无限维的非正曲率空间。 在以前的工作中，PI广泛研究了某些群体作用于“平坦”无限维空间的动力学，目前的一些工作可以被视为将这些结果推广到具有曲率的空间。通常，如果一个人对一个已经理解的动力系统进行开孔，那么对邻近动力系统的研究就可以归结为无限维空间的性质。然而，如果一个人想对动力系统进行分类，而不把它假设为对那些已经很好理解的动力系统的扰动，那么他就不得不在具有曲率的空间中进行分类。动力系统是一个年轻而重要的数学领域，它研究物理或物理系统随时间的演化（例如流体流动），而微分几何是一个更经典的领域，它倾向于研究空间中曲线和形状的静态配置。来自动力系统理论的新思想，如混沌和分形，对我们对世界的感知产生了深远的影响。动力系统最深刻和最有影响力的数学应用之一是研究具有“许多对称性”的空间的几何性质。PI的研究可以被看作是现代数学总体发展的一部分，在现代数学中，动力学和微分几何的思想相互作用，导致了旧理论的证明和令人兴奋的新发现。一个反复出现的关键思想是，具有许多对称性的空间实际上必须是齐次的，即任何具有足够对称性的空间实际上在每一点上看起来都是一样的。PI在这些主题上的工作与不同的研究领域有关系，包括计算机科学（Kazhdan的扩展图和性质（T））和天体力学（KAM理论和太阳系的稳定性）。

AbstractAward: DMS-0541917Principal Investigator: David FisherThe proposed research lies between dynamical systems andgeometry. Work of Furstenberg, Mostow, Margulis and others showsthat certain dynamical systems are an important tool for studyinggeometric properties of important geometric spaces, particularlysymmetric and locally symmetric spaces. . One important propertyof symmetric spaces is that they have non-positive curvature. Theinvestigator plans to extend this work in several inter-relateddirections. The study of symmetric spaces will be generalized toinclude more general spaces of non-positive curvature as well asmore general maps between spaces with an emphasis on infinitedimensional spaces. The investigator will attempt to exploitthese relationships to prove long-standing conjectures ofZimmer. The group of diffeomorphisms of a compact manifold actson the space of square-integrable Riemannian metrics, which isnaturally an infinite-dimensional space of non-positivecurvature. In previous work the PI has extensively studieddynamics of certain groups acting on ``flat" infinite dimensionalspaces, and some present work can be viewed as generalizing thoseresult to spaces that have curvature. Typically if one is aperturbing a dynamical system one already understands, the studyof nearby dynamical systems can be reduced to properties of aflat infinite dimensional space. However, if one wants toclassify dynamical systems which one does not assume areperturbations of ones that are well-understood, one is forced ina space that has curvature.Dynamical systems is a young and important field of mathematicsthat investigates the evolution of physical or mathematicalsystems over time (e.g. fluid flow) while differential geometryis a more classical field that tends to study staticconfigurations of curves and shapes in space. New ideas fromdynamical systems theory such as chaos and fractals have had aprofound impact on our perception of the world. One of thedeepest and most influential mathematical applications ofdynamical systems has been to the study of geometric propertiesof spaces with "many symmetries". The PI's research can be viewedas part of a general development in modern mathematics in whichideas from dynamics and differential geometry interact to lead toboth proofs of old conjectures and exciting new discoveries. Akey idea that appears repeatedly is that spaces that possess manysymmetries must actually be homogeneous, i.e. any space withenough symmetry actually looks the same at every point. The PI'swork on these topics has relationships with diverse areas ofresearch including computer science (expander graphs and property(T) of Kazhdan) and celestial mechanics (KAM theory and stabilityof the solar system).