Projective Structures in Teichmuller Theory and Kleinian Groups

Teichmuller 理论和 Kleinian 群中的射影结构

• 批准号: 0805525
• 负责人: David Dumas
• 金额: $ 15.28万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-09-01 至 2012-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0805525&HistoricalAwards=false
• 关键词: Projective; Structures; Teichmuller; Theory; Kleinian

AbstractAward: DMS-0805525Principal Investigator: David DumasThe principal investigator will explore complex projectivestructures on surfaces and their applications to Teichmullertheory, Kleinian groups, and hyperbolic 3-manifolds. The spaceof all complex projective structures on a fixed surface is acontractible manifold which has two natural but very differentcoordinate systems: A classical analytic approach uses theSchwarzian derivative to identify the moduli space with aholomorphic vector bundle over Teichmuller space, while a morerecent geometric approach builds each projective structure fromhyperbolic and Euclidean pieces in a process known as grafting.The major goals of this project are to understand the relationbetween these two coordinate systems (both theoretically andusing computer experiments), and to use that understanding tostudy Teichmuller spaces and deformations of Kleinian groups.The PI will also explore ways in which the techniques used tostudy complex projective structures could be adapted to otherlow-dimensional geometric structures, such as real projectivestructures.The study of the shapes and configurations of geometric objectshas applications to diverse areas of science and engineering,from understanding the folding of proteins or the formation ofgalaxies to programming autonomous robots that must navigatecomplex terrain. In a mathematical abstraction of this type ofproblem, one studies the space of all possible shapes, or "modulispace", of a geometric object. This project focuses on themoduli space of complex projective Riemann surfaces, a class ofgeometric objects that encode information about 3-dimensionalspaces (hyperbolic manifolds) in 2-dimensional form. Throughboth theoretical study and computational experiments, the PI willdevelop new tools for analyzing these structures, enhance theconnections between 2- and 3-dimensional geometry, and expandapplications of these structures in related fields. The projectwill also produce computer images of the moduli space, displayingits rich structure and complexity in a way that can beappreciated by scientists and non-scientists alike.

项目编号：dms -0805525首席研究员：David dumas首席研究员将探索表面上的复杂投影结构及其在teichmuller理论、Kleinian群和双曲3-流形中的应用。固定曲面上所有复杂射影结构的空间都是可缩流形，具有两种自然但非常不同的坐标系：经典的解析方法使用施瓦兹导数在Teichmuller空间上识别具有自纯向量束的模空间，而最近的几何方法通过双曲和欧几里得块在称为嫁接的过程中构建每个射影结构。这个项目的主要目标是理解这两个坐标系之间的关系（理论上和使用计算机实验），并利用这种理解来研究Teichmuller空间和Kleinian群的变形。PI还将探索如何将用于研究复杂投影结构的技术应用于其他低维几何结构，如真实的投影结构。对几何物体的形状和结构的研究已经应用于科学和工程的各个领域，从理解蛋白质的折叠或星系的形成到编程必须在复杂地形中导航的自主机器人。在这类问题的数学抽象中，人们研究一个几何对象的所有可能形状的空间，或“模空间”。该项目重点研究复杂投影黎曼曲面的模空间，黎曼曲面是一类以二维形式编码有关三维空间（双曲流形）信息的几何对象。通过理论研究和计算实验，PI将开发新的工具来分析这些结构，增强二维和三维几何之间的联系，并扩大这些结构在相关领域的应用。该项目还将产生模空间的计算机图像，以一种科学家和非科学家都能欣赏的方式显示其丰富的结构和复杂性。