Mathematical Sciences: Differential Structures

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Five investigators will study related problems involving various aspects of differential geometry, algebraic varieties, and dynamical systems. The first will study variational problems arising from curvature considerations. The second will study isospectral deformations, optimal structures on Riemannian manifolds, and fibrations of spheres by great spheres. The third will investigate Einstein metrics, isotropic irreducible Riemannian manifolds, and the isotropy representation of symmetric spaces. A topologist will study stratified spaces, particularly algebraic varieties. And the fifth will study hyperbolic geometry in higher dimensions, representations of braid groups, link groups, and invariant links. A central goal of the differential geometers working on this project is to understand spacial curvature. In particular they will study the relationship between curvature and the overall shape of the space, or what sorts of functions may exist on the space. The topologist and dynamical systems theorist will use their understanding of the shape of the space and allowable dynamical properties to restrict curvature in various ways.