Mappings with Little Smoothness

对非光滑现象的研究是当代几何函数论的主要课题之一。本项目的目的是研究空间上的度量结构以及这些空间之间适当的非光滑映射类，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悬而未决的。例如，对与标准二维球体准对称的二维度量球的令人满意的表征是未知的。这个问题与瑟斯顿的双曲猜想有关。这一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。如何映射曲面的特殊问题由来已久。地图学的目的是在适当的地图绘制下保留表面的特定特征。这导致了重大的数学发展。在十九世纪，测量学和大地测量学的基本问题促使高斯建立了一套系统的曲面理论，奠定了现代微分几何的基础。为了研究起皱和分形的物体，需要新的数学工具。清楚地理解描述非光滑现象的数学概念将有助于其他领域的研究和实际应用。例如，解决球的拟共形参数化的理论问题的方法也被用来寻找绘制人脑表面的算法。该项目的目的之一是让研究生参与这一前景光明且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.