Geometry of Teichmuller Space and Mapping Class Group

本研究计划主要致力于研究Teichmueller空间、模空间和映射类群的几何和拓扑。一些具体的项目包括解决映射类群的共轭问题的一些推广，研究模空间上收缩长度函数的性质，以及在Teichmueller空间上寻找Thurston度量中测地线的粗略描述。本文还提出了在有限覆盖上研究自由群的不可约自同构的同调增长。表面是二维空间，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一直吸引着数学家的注意。表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。理解表面的一种方法是研究各种自然的表面。表面上的几何结构以及不同结构之间的关系。在给定曲面上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。这通常是一个高维空间，它本身就自然地带有许多丰富而有趣的几何结构。模空间的研究是指导数学原理的一部分，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元空间的结构。PI的工作致力于扩展我们对模空间和一些相关空间的认识。

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.