Superrigidity, Actions on Manifolds and CAT(0) Geometry

提出的研究是研究动力系统和几何之间的相互作用。Furstenberg， Mostow和margulis的工作表明，理解某些动力系统是研究对称和局部对称空间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。研究者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对称空间之间的映射。这提高了纯动态应用的可能性。紧流形的微分同态群作用于平方可积黎曼度量空间，该空间自然是非正曲率的无限维空间。研究者将试图利用这些关系来证明齐默长期存在的猜想。动力系统是一个重要且相对年轻的数学领域，它研究物理或数学系统随时间的演变（例如流体流动）。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对具有“多对称性”的空间的几何性质的研究。提出的研究将扩展我们对几何和动力学之间这种深层联系的理解。

The proposed research studies the interaction between dynamicalsystems and geometry. Work of Furstenberg, Mostow and Margulisshows that understanding certain dynamical systems is animportant tool in the study geometric properties of symmetric andlocally symmetric spaces. . One important property of symmetricspaces 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 symmetric spaces. This raises thepossibility of purely dynamical applications. The group ofdiffeomorphisms of a compact manifold acts on the space ofsquare-integrable Riemannian metrics, which is naturally aninfinite-dimensional space of non-positive curvature. Theinvestigator will attempt to exploit these relationships to provelong-standing conjectures of Zimmer.Dynamical systems is an important and relatively young field ofmathematics that investigates the evolution of physical ormathematical systems over time (e.g. fluid flow). 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 the study of geometric properties ofspaces with "many symmetries". The proposed research will extendour understanding of this deep connection between geometry anddynamics.