Projective Structures in Teichmuller Theory and Kleinian Groups

摘要奖：DMS-0805525首席研究员：David Dumas首席研究员将探索曲面上的复杂射影结构及其在TeichMiller理论、Kleinan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是一个可压缩的流形，它有两个自然但非常不同的坐标系：经典的分析方法使用Schwarzian导数来识别TeichMuller空间上具有全纯向量丛的模空间，而最近的几何方法则通过一个称为嫁接的过程从双曲片和欧几里德片断建立每个射影结构。这个项目的主要目的是了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Teichmiller空间和Klein群的变形。PI还将探索用于研究复杂射影结构的技术如何适用于其他低维几何结构，即Klein群对几何物体形状和构型的研究应用于科学和工程的不同领域，从理解蛋白质的折叠或星系的形成，到为必须在复杂地形中导航的自主机器人编程。在这类问题的数学抽象中，人们研究几何对象的所有可能形状的空间，或称“模空间”。这个项目主要研究复射影黎曼曲面的模空间，这是一类以二维形式编码关于三维空间(双曲流形)的信息的几何对象。通过理论研究和计算实验，PI将开发分析这些结构的新工具，加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。该项目还将产生模数空间的计算机图像，以一种科学家和非科学家都能理解的方式展示其丰富的结构和复杂性。

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.