Generalized Branched Coverings and Parameterizations

广义分支覆盖和参数化

基本信息

批准号：
0757732
负责人：
Richard Canary
金额：
$ 10.1万
依托单位：
Regents of the University of Michigan - Ann Arbor
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2008
资助国家：
美国
起止时间：
2008-07-01 至 2012-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=0757732&HistoricalAwards=false
关键词：
Generalized Branched Coverings Parameterizations

项目摘要

This project involves investigations into the global conformal geometry of Riemannian manifolds and the local quasiconformal geometry of nonsmooth manifolds. Questions and proposed methods arise from analysis, geometry, and topology. The main topics are the global rigidity phenomena of mappings distorting the infinitesimal conformal geometry and natural local parameterizations of nonsmooth quasiconformal geometries. New methods that go beyond the range of nonlinear potential theory in the context of global rigidity questions will be introduced, and new geometric structures on quasiconformal manifolds will be explored.Quasiconformal methods give information about the geometrical properties of spaces simultaneously at all scales. These methods have their roots in complex analysis and in the geometry of the complex plane. The methods, however, can be used both in higher dimensions and in spaces where analysis based on traditional calculus is not available. Mathematical areas of application for these methods include such subjects as Teichmuller theory, topology and geometry of manifolds, geometric group theory, nonlinear geometric analysis, and nonsmooth calculus on manifolds. Quasiconformal mappings have recently begun to play a serious role in applied areas as well. These include fluid dynamics, elasticity, and even the analysis of nanostructures. This project focuses on the following fundamental question: When can a given geometry of a space be understood as a possibly highly distorted Euclidean geometry?

该项目涉及对Riemannian歧管的全球保形几何形状和非滑膜歧管的局部准文献几何形状进行的研究。问题和提出的方法来自分析，几何和拓扑。主要主题是映射的全球刚度现象，它扭曲了无限的共形几何形状和非平滑形式形式几何形状的自然局部参数化。将引入超出非线性潜在理论的新方法，将引入全球刚性问题的背景下，并且将探讨有关准文献流形的新几何结构。Quasiconformal方法在所有尺度上同时同时提供有关空间的几何特性的信息。这些方法具有复杂分析和复杂平面的几何形状的根源。但是，这些方法既可以在较高的维度和基于传统演算的分析的空间中使用。这些方法的数学应用领域包括诸如Teichmuller理论，歧管的拓扑和几何形状，几何组理论，非线性几何分析和非歧管上的非平滑彩学。准信息映射最近也开始在应用领域发挥重要作用。这些包括流体动力学，弹性，甚至包括纳米结构的分析。该项目的重点是以下基本问题：何时可以将给定的空间几何形状理解为可能高度扭曲的欧几里得几何形状？