Mathematical Sciences: Topology, Geometry and Analysis of Group Actions and Foliations

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The problems addressed in this research project are in the area of differential topology. Their common theme is to develop relations between the geometry and topology of group actions and foliations, their cohomology invariants, and the analytic invariants of differential operators naturally associated with their geometry. Steven Hurder will work on rigidity of Anosov actions of "large" groups; on the "topology-at-infinity" of open manifolds, based on K-theory invariants of differential operators; on the classifications of low-dimensional Anosov diffeomorphisms, using secondary classes and moduli spaces for quasi-symmetric maps; and on the spectrum of elliptic operators on leaves of foliations, using the dynamics of the geodesic flow of the foliation. This collection of problems concerns natural objects, manifolds, i.e. locally Euclidean geometric spaces. The properties dealt with are such things as symmetry of these objects and flows defined on them, all to be investigated by a variety of techniques from algebra and analysis.