Mappings with Little Smoothness

摘要非光滑现象的研究是当代几何函数理论的主要课题之一。本课题的目的是研究空间上的度量结构和这些空间之间的适当的非光滑类映射，如bilipschitz映射、拟对称映射、拟共形映射和拟正则映射。这个领域的许多基本问题都是开放的。例如，对于准对称于标准二维球体的二维度量球体的令人满意的表征是未知的。这个问题与瑟斯顿的夸张猜想有关。该领域的进展依赖于最近在度量空间分析中建立的几何和解析方法的结合。如何绘制表面的特殊问题有着悠久的历史。地图学的目的是在适当的映射下保留地表的特定特征。这导致了数学的重大发展。在19世纪，测量和大地测量学的基本问题促使高斯建立了一个系统的曲面理论，奠定了现代微分几何的基础。对于褶皱和分形物体的研究需要新的数学工具。对描述非光滑现象的数学概念的清晰理解将有利于其他领域的研究并导致实际应用。例如，研究球体的准共形参数化理论问题的方法也被用于寻找映射人脑表面的算法。该项目的目的之一是让研究生参与这个有前途和重要的数学研究领域。

Proposal Number: DMS-0200566PI: Mario BonkABSTRACTThe investigation of non-smooth phenomena is one of the main subjects of contemporary Geometric Function Theory.The purpose of this project is to study metric structures on spaces and appropriate nonsmooth classes of mappingsbetween these spaces such as bilipschitz, quasisymmetric,quasiconformal, and quasiregular mappings. Many basic questions in this area are open. For example, a satisfactory characterization of two-dimensional metric spheres that are quasisymmetric to the standardtwo-dimensional sphere is unknown. This problem is relevantin connection to Thurston's hyperbolization conjecture.Progress in the field relies on a combination of geometric and analytic methods that were recently established in the analysis of metric spaces.The particular question of how to map surfaces has a long history. Cartography aims to preserve particular features of a surface under suitable mappings. This has lead to major mathematical developments. In the nineteenth century basic problems of surveying and geodesy motivated Gauss to build up a systematic theoryof curved surfaces which laid the foundation of modern differential geometry. For the investigation of wrinkledand fractal objects new mathematical tools arerequired. A clear understanding of the mathematical concepts for describing nonsmooth phenomena will benefitresearch in other areas and lead to practical applications.For example, methods for attacking the theoretical questions of quasiconformal parametrizations of sphereswere also used for finding algorithms for mapping the surface of the human brain. One of the aims of the project is to involve graduate students in this promising and important area of mathematical research.