Superrigidity, Actions on Manifolds and CAT(0) Geometry

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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.