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

估计动力系统和双曲几何中黎曼曲面的几何形状

• 批准号: 0905812
• 负责人: Jeremy Kahn
• 金额: $ 11.39万
• 依托单位: SUNY at Stony Brook
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2009
• 资助国家: 美国
• 起止时间: 2009-06-15 至 2013-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0905812&HistoricalAwards=false
• 关键词: Estimating; Geometry; Riemann; Surfaces; Dynamical

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.

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相关的三个方向。第一个是继续米哈伊尔·柳比奇的工作，证明迭代复二次多项式的重整化的界限，最终目标是证明曼德尔布罗特集的局部连通性。第二个是继续工作与弗拉基米尔Markovic随机产生的覆盖有限型黎曼曲面，希望证明Escherichpreis猜想。第三是退化复结构理论的发展，可能应用于共形可实现有限细分规则的表征，Ozsvath和Szabo的Heegard-Floer不变量的计算，以及有理映射模空间中双曲分量为非紧的必要条件。通过一个程序迭代一个依赖于两个参数的简单函数，可以在计算机上绘制出Mandelbrot集的美丽图像。与M的工作。柳比奇将严格地证明在这些计算机图像中已经经验性地观察到的许多现象。与V. Markovic在二维几何物体的随机生成方面的工作可能被证明在材料科学、化学和物理学中有应用。黎曼曲面是一种抽象实现的曲面，在其上可以以内部一致的方式在曲面上的任何地方画出小圆圈。这些表面的形状是弦理论的核心，目前忽略了高能物理学，黎曼表面的退化理论可能会在弦理论和粒子物理学的标准模型中找到应用。这是可能的曼德尔布罗特集在计算机上渲染美丽的图片。这些图片已被广泛流传在许多场馆inlcuding著名的明信片和熟悉的许多非数学家。