Geometry and Dynamics of Teichmuller Space

项目编号：dms -1007811项目负责人：Kasra rafi。首先，我们可以把映射类群在Teichmueller空间上的作用看作是格在李群上的作用的类比。这种类比是提出一些问题的动机，即刚性和计数问题。另一个主题是理解Teichmueller空间中不同指标之间的关系。在Teichmueller空间中有几个不同的感兴趣的指标，许多问题对一个指标有答案，但对另一个却没有。本文提出了在Teichmuellerspace上的Lipschitz度量中测大地线的行为，该度量具有在自由群外部自同构的模型空间——outer space中充当Teichmueller度量和Lipschitz度量之间的桥梁的特点。在这个研究项目中研究的数学结构提供了完全确定一组二维几何形状的坐标。这些方向上最古老的问题和结构已经有150多年的历史了，但仍然至关重要，因为它们在数学中的核心作用，以及它们在分析物理系统的形状依赖特征方面的应用，包括在MRI图像中识别的大脑解剖学。一种新的研究方法弥补了长期以来在teichmueller空间上施加真正令人满意的几何或度量的困难，方法是同时考虑几种几何，旨在利用每种几何的良好特征。本项目将继续进行这项研究，以提高对这些重要空间的基本数学理解。

AbstractAward: DMS-1007811Principal Investigator: Kasra RafiThese projects pursue two major themes. First, we can think of theaction of the mapping class group on Teichmueller space as an analogueof the action of a lattice on a Lie group. This analogy is themotivation of some of the proposed problems, namely the rigidity andthe counting problems. The other theme is to understand the relationbetween different metrics on Teichmueller space. There are severaldistinct metrics of interest on Teichmueller space and many questionsare answered for one metric but not for another. We propose the studyof the behavior of geodesics in the Lipschitz metric on Teichmuellerspace, which has the feature that the Lipschitz metric can act as abridge between the Teichmueller metric and the Lipschitz metric inOuter space, the model space for outer automorphisms of a free group.The mathematical structures studied in this research program providecoordinates that determine completely a family of two-dimensionalgeometries. The oldest questions and constructions in these directionsare more than 150 years old but remain vital because of their centralrole in mathematics and their use in analyzing shape-dependent featuresof physical systems, including brain anatomy identified in MRI images.One of the newer lines of investigation compensates for the longstandingdifficulty of imposing a truly satisfactory geometry or metric onTeichmueller spaces by considering several geometries simultaneously,aiming to take advantage of the good features of each. This projectaims to pursue this investigation to improve fundamental mathematicalunderstanding of these important spaces.