Riemann Surfaces, Dynamics and Hyperbolic Geometry

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Riemann Surfaces, Dynamics and Hyperbolic Geometry-C. McMullen DMS-9806424 This project will study the classification of geometric structures and dynamical systems, in settings that allow for a concerted study of specific examples. Its scope includes iterated rational maps, Kleinian groups, projective structures on Riemann surfaces, and algebraic dynamics on the projective plane. Renormalization, the interaction of spectral theory with complex dynamics, and applications of Teichmuller theory to 3-manifolds are among the main research directions. From meteorology to robotics, the control and prediction of dynamical systems plays a central role in science and technology. This project will include computer explorations of mathematically simple systems that exhibit rich combinatorial structure and universal analytic features. Its aim is to advance the theoretical understanding of dynamics and geometry.