Geometry and Dynamics of Teichmuller Space

摘要奖：DMS-1007811首席研究员：卡斯拉·拉菲这些项目追求两个主要主题。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。这个类比是对所提出的一些问题，即刚性问题和计数问题的推论。另一个主题是理解Teichmueller空间上不同度量之间的关系。在Teichmueller空间上有几个不同的令人感兴趣的度量，许多问题对一个度量得到了回答，而对另一个度量没有得到回答。研究了Teichmueller空间上的Lipschitz度量下测地线的行为，它的特点是在自由群的外自同构的模型空间--外层空间中，Lipschitz度量可以作为Teichmueller度量和Lipschitz度量之间的桥梁.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，因为它们在数学中发挥着核心作用，并用于分析物理系统的形状相关特征，包括MRI图像中识别的大脑解剖。其中一条较新的研究路线通过同时考虑几个几何图形，旨在利用每个几何图形的良好特征，弥补了在TeichMueller空间上强加真正令人满意的几何图形或度量的长期困难。这个项目旨在进行这项研究，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.