Mathematical Sciences: Geometrical Structures on Manifolds and Deformations of Discrete Groups

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William Goldman plans to continue his investigation into the deformation theory of discrete groups and geometric structures. This investigation is currently divided into five (closely related) topics: 1) the deformation theory of flat bundles over Kahler manifolds, and relations with symplectic geometry, 2) real projective structures on surfaces, 3) affinely flat manifolds, including flat Lorentz manifolds, 4) the deformation space of flat PSL(2,R)-bundles over surfaces, relations with (possibly singular) geometric structures, and 5) complex hyperbolic Kleinian groups. Goldman and his students have developed graphics software particularly suited to their problems to run on a SUN work- station, and more will be done along these lines also.