Group Actions, rigidity and geometry

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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).