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.

该研究建议的大部分都致力于研究Teichmueller空间，Moduli空间和映射课程组的几何学和拓扑。一些特定的项目涉及解决映射课程组的共轭问题的一些概括，研究收缩期函数在模量空间上的特性，并在Teichmueller空间上对Thurston指标中的大地测量学进行粗略描述。 PI还建议研究有限覆盖物中自由群体不可约合的汽车的同源性增长。表面是二维空间，例如球的表面或甜甜圈。几个世纪以来，他们的拓扑和几何特性吸引了数学家的思想。对表面的研究不仅有趣又美丽，而且还产生了数学的整个领域，并且可以说是数学领域的交集。理解表面的一种方法是研究各种自然？表面上的几何结构以及不同结构之间的相互关系。参数空间给定表面上的所有几何结构都是称为模量空间的基本数学对象。这通常是一个高维空间，本身自然会带有许多丰富而有趣的几何结构。模量空间的研究是指导数学原理的一部分，该原理为了理解空间上的一个特定结构，必须尝试了解所有结构组成的元空间的结构。 PI的工作致力于扩大我们对模量空间和一些相关空间的了解。

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