CAREER: Complex Projective Structures, Teichmuller Theory, and Character Varieties

职业：复杂射影结构、Teichmuller 理论和性格多样性

• 批准号: 0952869
• 负责人: David Dumas
• 金额: $ 50万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-05-15 至 2016-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0952869&HistoricalAwards=false
• 关键词: CAREER; Complex; Projective; Structures; Teichmuller

CAREER: Complex Projective Structures, Teichmuller Theory, and Character VarietiesIn this project, the PI will explore complex projective structures on surfaces and their applications to other areas of analysis and geometry. The space of all complex projective structures on a surface is a contractible manifold which has two natural but very different coordinate systems; one involves a classical complex-analytic construction (the Schwarzian derivative), while the other comes from the operation of grafting, which assembles a projective surface from Euclidean and hyperbolic pieces. The major research goals of this project are to understand the relationship between the two coordinate systems for the moduli space of projective structures and to use this understanding to solve problems in related areas. The project will also include a significant educational component focusing on undergraduate mathematics at the University of Illinois at Chicago (UIC) through two targeted programs: First, the PI will organize a yearly undergraduate research symposium, which will include lectures by senior mathematicians and by undergraduates reporting on their own research projects. Second, the PI will create a new course at UIC for undergraduate mathematics and mathematical computer science majors on "experimental mathematics", discussing the way computer software (both custom and off-the-shelf) is used as an exploratory tool in mathematical research. The PI will teach the course for the first time in the 2010-2011 academic year, emphasizing computer exploration of the geometry of curves and surfaces.The study of the shapes and configurations of geometric objects has applications to diverse areas of science and engineering, from understanding the folding of proteins or the formation of galaxies to programming autonomous vehicles that must navigate complex terrain. In a mathematical abstraction of this type of problem, one studies the space of all possible shapes, or "moduli space", of a geometric object. This project focuses on the moduli space of complex projective Riemann surfaces, a class of geometric objects that encode information about $3$-dimensional spaces (hyperbolic manifolds) in $2$-dimensional form. Through both theoretical study and computational experiments, the PI will develop new tools for analyzing these structures, enhance the connections between $2$- and $3$-dimensional geometry, and expand applications of these structures in related fields of mathematics. The project will also produce computer images of the moduli space, displaying its rich structure and complexity in a way that can be appreciated by scientists and non-scientists alike.

职业：复杂的投影结构，Teichmuller理论和字符变化在这个项目中，PI将探索表面上的复杂投影结构及其在分析和几何其他领域的应用。 曲面上的所有复射影结构的空间是一个可收缩流形，它有两个自然但非常不同的坐标系;一个涉及经典的复解析构造（施瓦茨导数），而另一个来自嫁接操作，它将欧几里得和双曲片段组装成一个射影曲面。 本计画的主要研究目标为了解射影结构模空间的两个坐标系之间的关系，并利用此了解来解决相关领域的问题。 该项目还将包括一个重要的教育组成部分，侧重于本科数学在伊利诺伊大学芝加哥（UIC）通过两个有针对性的计划：首先，PI将组织一个年度本科研究研讨会，其中将包括讲座由资深数学家和本科生报告自己的研究项目。 第二，PI将在UIC为本科数学和数学计算机科学专业开设一门关于“实验数学”的新课程，讨论如何将计算机软件（定制和现成）用作数学研究中的探索工具。 PI将在2010-2011学年首次教授这门课程，强调计算机对曲线和曲面几何的探索。几何物体形状和配置的研究可应用于科学和工程的各个领域，从理解蛋白质折叠或星系形成到编程必须在复杂地形中导航的自动驾驶汽车。 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“模空间”。 这个项目的重点是复射影黎曼曲面的模空间，这是一类几何对象，以2维形式编码有关3维空间（双曲流形）的信息。 通过理论研究和计算实验，PI将开发用于分析这些结构的新工具，增强2 $和3 $维几何之间的联系，并扩展这些结构在相关数学领域的应用。 该项目还将产生模空间的计算机图像，以科学家和非科学家都能欣赏的方式展示其丰富的结构和复杂性。