Estimating the Geometry of Riemann Surfaces in Dynamical Systems and Hyperbolic Geometry

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This award is funded under the American Recovery and Reinvestment Act of 2009 (Public Law 111-5). The PI will pursue three directions related to the estimation of the geometry of Riemann surfaces. The first is the continuation of the work with Mikhail Lyubich toward proving the bounds for renormalization of iterated complex quadratic polynomials, with the eventual goal of proving the local connectivity of the Mandelbrotset. The second is the continuation of work with Vladimir Markovic on randomly generated covers of finite type Riemann surfaces, with the hope of proving the Ehrenpreis conjecture. The third is thedevelopment of the theory of degenerate complex structures, with possible applications to the characterization of conformally realizable finite subdivision rules, the computation of the Heegard-Floer invariants of Ozsvath and Szabo, and necessary conditions for a hyperbolic component in the moduli space of rationalmaps to be non-compact. It is possible to render beautiful pictures of the Mandelbrot set on the computer with a program that iterates a simple function that depends on two parameters. The work with M. Lyubich will rigorously demonstrate many of the phenomena that have been empirically obsevered in these computer pictures. The work with V. Markovic on the random generation of two-dimensional geometric objects may prove to have applications to material science, chemistry and physics.A Riemann surface is an abstractly realized surface on which small circles can be drawn everywhere on the surface in an internally consistent way. The shapes of these surfaces are central to the theory of strings that currently ominates high-energy physics, and the theory of degeneration of Riemann surfaces may find applications in string theory and the Standard Model of particle physics. It is possible to render beautiful pictures of the Mandelbrot set on the computer. These pictures have been widely circulated in many venues inlcuding on well known post-cards and are familiar to many non-mathematicians.