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

该奖项是根据2009年美国复苏和再投资法案（公法111-5）资助的。PI将追求与黎曼曲面几何估计有关的三个方向。第一个是继续与Mikhail Lyubich的工作，以证明迭代复二次多项式的重整化界限，最终目标是证明Mandelbrotset的局部连通性。第二个是继续与Vladimir Markovic在有限型Riemann曲面的随机生成覆盖上的工作，希望证明Ehrenpreis猜想。第三是退化复杂结构理论的发展，可能应用于共形可实现的有限细分规则的表征，Ozsvath和Szabo的Heegard-Floer不变量的计算，以及在有理映射的模空间中双曲分量是非紧的必要条件。用一个程序在计算机上迭代一个依赖于两个参数的简单函数，就可以渲染出曼德布洛特集合的漂亮图片。与柳比奇先生的合作将严格地证明许多在这些计算机图像中经验观察到的现象。与V. Markovic在二维几何物体随机生成方面的工作可能会被证明在材料科学、化学和物理方面有应用。黎曼曲面是一种抽象实现的曲面，它可以在表面的任何地方以内部一致的方式绘制小圆。这些表面的形状是弦理论的核心，而弦理论目前预示着高能物理，黎曼表面的退化理论可能会在弦理论和粒子物理的标准模型中找到应用。在计算机上渲染曼德勃罗集合的美丽图片是可能的。这些图片在许多地方广为流传，包括著名的明信片上，许多非数学家都很熟悉。

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.