Generalized Branched Coverings and Parameterizations

本课题主要研究黎曼流形的整体共形几何和非光滑流形的局部拟共形几何。问题和建议的方法来自分析、几何和拓扑学。主要的主题是扭曲无穷小共形几何的映射的整体刚性现象和非光滑拟共形几何的自然局部参数化。在整体刚性问题的背景下，将引入超越非线性位势理论范围的新方法，并将探索拟共形流形上的新的几何结构。拟共形方法同时给出空间在所有尺度上的几何性质的信息。这些方法植根于复数分析和复数平面的几何。然而，这些方法既可以用于更高的维度，也可以用于基于传统微积分的分析不可用的空间。这些方法的数学应用领域包括TeichMuller理论、流形的拓扑和几何、几何群论、非线性几何分析和流形上的非光滑微积分。拟共形映射最近也开始在应用领域发挥重要作用。其中包括流体动力学、弹性，甚至纳米结构的分析。这个项目集中在以下基本问题上：什么时候一个空间的给定几何可以被理解为可能高度扭曲的欧几里德几何？

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?