Geometry of Teichmuller Space and Mapping Class Group

Teichmuller空间的几何和映射类群

• 批准号: 1311834
• 负责人: Jing Tao
• 金额: $ 14.14万
• 依托单位: University of Oklahoma Norman Campus
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-08-01 至 2016-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1311834&HistoricalAwards=false
• 关键词: Geometry; Teichmuller; Space; Mapping; Class

Most of this research proposal is dedicated to the study of the geometry and topology of Teichmueller space, moduli space, and mapping class groups. Some specific projects involve solving some generalizations of the conjugacy problem for mapping class groups, investigating the properties of the systole-length function on moduli space, and finding a coarse description of the geodesics in the Thurston metric on Teichmueller space. The PI also proposes to study the homology growth of irreducible automorphisms of a free group in finite covers. Surfaces are two-dimensional spaces like the surface of a ball or a doughnut. Their topological and geometric properties have captured the minds of mathematicians for centuries. The study of surfaces is not only interesting and beautiful, but has generated entire fields of mathematics and arguably lies at the intersection of all fields of mathematics. One way to understand a surface is to study the various ?natural? geometric structures on the surface and how the different structures relate to each other. The parameter space all geometric structures on a given surface is a fundamental mathematical object called the moduli space. This is usually a higher-dimensional space that itself naturally carries many rich and interesting geometric structures. The study of moduli space is part of the guiding mathematical principle that, in order to understand one particular structure on a space, one must try to understand the structure of the meta space comprised of all structures. The work of the PI is dedicated to expanding our knowledge about moduli space and some related spaces.

本研究计划的大部分内容致力于研究Teichmueller空间、模空间和映射类群的几何和拓扑。一些具体项目涉及解决映射类群的共轭问题的一些推广，研究模空间上系统长度函数的性质，以及找到Teichmueller空间上Thurston度量中测地线的粗略描述。PI还建议研究有限覆盖中自由群的不可约自同构的同调增长。曲面是二维空间，就像球或甜甜圈的表面。几个世纪以来，它们的拓扑和几何性质吸引了数学家的注意。曲面的研究不仅有趣和美丽，而且产生了整个数学领域，可以说是所有数学领域的交叉点。了解一个表面的一种方法是研究不同的表面。自然的？表面上的几何结构以及不同结构之间的关系。给定曲面上的所有几何结构的参数空间是一个基本的数学对象，称为模空间。这通常是一个更高维的空间，它本身自然带有许多丰富而有趣的几何结构。模空间的研究是数学指导原则的一部分，为了理解空间上的一个特定结构，必须尝试理解由所有结构组成的Meta空间的结构。PI的工作致力于扩展我们对模空间和一些相关空间的知识。

项目成果

Veech surfaces and simple closed curves

Veech 曲面和简单闭合曲线

• DOI: 10.1007/s11856-017-1617-5
• 发表时间: 2018
• 期刊: Israel Journal of Mathematics
• 影响因子: 1
• 作者: Forester, Max;Tang, Robert;Tao, Jing
• 通讯作者: Tao, Jing