Quasiregular Mappings and Analysis in Nonsmooth Spaces

本提案分为三部分。在第一部分中，Heinonen概述了一个研究广义流形之间几何分支覆盖的程序。这是一个与Seppo Rickman合作的项目。建立这样一个理论的动机来自于试图理解度量空间在核聚变空间中允许（可能是分支的）利普希茨或准对称参数化。在第二部分中，Heinonen解释了分支覆盖和度量空间分析的最新进展如何用于处理参数化问题。换句话说，Heinonen提出用拟正则映射理论刻画局域欧几里得度量规。这是和丹尼斯·沙利文合作的项目。在第二部分，Heinonen讨论了拟正则映射的分支集问题。研究了给定拟正则映射是局部同胚的解析条件。这个问题与第二部分有关，也与研究物体变形的非线性弹性问题有关。第三部分是与TeroKilpel“ainen”的联合项目。该提案延续了PI早期的研究，在分析和几何中寻找离散方法，这些方法将适用于传统意义上不光滑的空间。这里的“分析”一词是指从微积分中发展出来的数学分支；它是对无限小数据的连续变化的研究，人们希望从中预测全局结论。“几何学”是对空间的研究，它在许多可能的维度上有许多形式和变化。离散方法与连续方法或微积分方法相反，它涉及更多有限的“计数”思想，例如由计算机完成的思想，尽管在这项工作中仍然具有解析和几何的味道。准共形性的概念以最微妙的方式进入了讨论：它允许空间及其形成在时间和地点是单一的（例如，有弯曲或粗糙），但不是无处不在，也不是一直如此。在扭曲中有一定的控制，40年前在这个领域进行第一次研究时，数学家们可能认为这是人为的。准共形性是数学中普遍存在的现象。从Dennis Sullivan在70年代末的工作中我们知道，每一个局部看起来像欧几里得平面空间的空间都有一个准共形结构，尽管不是一般的光滑结构；除了四维空间，其他维度都是如此。然而，同样在四维空间中，从理论上讲，有无限的空间，人们不能做普通的、光滑的分析或微积分，但可以做拟共形的、非光滑的分析。该建议解决了在非光滑空间中更好地理解这种新微积分的需要以及如何识别准共形欧几里得空间的方法。虽然提案中的主要问题是由理解空间结构的基本探索所激发的，但随之而来的对变换奇点的研究已经应用于材料科学，在材料科学中，人们想知道变形的物体何时在应变下坍缩到自身上。

DMS-9970427ABSTRACTThis proposal has three parts. In Part I, Heinonen outlines a program fora study of geometric branched covers between generalized manifolds. Thisis a joint project with Seppo Rickman. The motivation to build such atheory comes from the attempts to understand what metric spaces admit(possibly branched) Lipschitz or quasisymmetric parametrizations inEuclidean space. In Part II, Heinonen explains how branched covers and therecent advances in analysis on metric spaces could be used to approach the parametrization problem. In different terminology, Heinonen proposes tocharacterize locally Euclidean metric gauge by using the theory ofquasiregular maps. This is a joint project with Dennis Sullivan. In PartIII, Heinonen discusses the problem of the branch set of a quasiregularmap. Analytic conditions are searched that would imply a givenquasiregular map is a local homeomorphism. This problem is relevant toPart II, and to problems in nonlinear elasticity where deformation ofbodies are investigated. Part III is a joint project with TeroKilpel"ainen. The proposal continues the PI's earlier investigations, where discretemethods in analysis and geometry are being searched that would serve inspaces that are not smooth in the classical sense. Here the word``analysis'' means the branch of mathematics that developed out of theCalculus; it is the study of continuous changes in infinitesimal data,from which one wants to predict global conclusions. ``Geometry'' is thestudy of space, its many forms and varieties, in many possible dimensions.Discrete methods mean the opposite to continuous, or calculus methods, andinvolve more finite, ``counting'' ideas, such as those done by a computer,albeit in this work still of analytic and geometric flavor. The concept ofquasiconformality enters the discussion in the most subtle way: it allowsthe space and its formation to be singular (for instance, with kinks or roughness) at times and in places, but not everywhere and not all the time. There is a certain control in the distortion, which perhaps looked artificial to the mathematicians 40 years ago when the first research in the field was conducted. Now quasiconformality is a ubiquitous phenomenon in mathematics. From the work of Dennis Sullivan in the late seventies we know that every spacethat locally looks like a flat space of Euclid has a quasiconformalstructure albeit not a smooth structure in general; and this is true inevery dimension but four. Nevertheless, also in dimension four, there areconjecturally an infinity of spaces where one cannot do ordinary, smoothanalysis, or calculus, but where one can do quasiconformal, nonsmoothanalysis. The proposal addresses both the need to have a betterunderstanding of this new calculus in nonsmooth spaces and methods how torecognize spaces that are quasiconformally Euclidean. Although the mainquestions in the proposal are motivated by a basic quest for understandingthe structure of space, the consequent study of singularities intransformations has had applications to material sciences, where one wantsto know when a deformed body collapses onto itself under a strain.