Mathematical Sciences: Quasiconformal Maps and Nonsmooth Analysis

数学科学：拟共形映射和非光滑分析

• 批准号: 9622844
• 负责人: Juha Heinonen
• 金额: $ 10.07万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1996
• 资助国家: 美国
• 起止时间: 1996-06-01 至 2000-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9622844&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Quasiconformal; Maps; Nonsmooth

ABSTRACT Proposal: DMS-9622844 PI: Heinonen The proposal has three parts. Part I is the most significant. In Part I, Heinonen outlines a program of study of quasiconformal maps in metric spaces with little smoothness. The goal of the project, joint with Pekka Koskela, is to systematically develop a theory of quasiconformal maps on metric spaces that possess certain quantitative bounds on their mass and geometry. The motivation to develop such a theory comes partly from geometry and combinatorial group theory, where it is important to understand quasiconformal maps on boundaries of negatively curved spaces. Heinonen and Koskela have discovered that the validity of a Poincare type inequality in a space is practically tantamount to the validity of a working quasiconformal theory. In Part II, Heinonen proposes problems on the lower dimensional absolute continuity properties of quasiconformal maps in Euclidean space. He has established that a quasiconformal map, defined on the unit ball of Euclidean space of any dimension larger than two, but not equal to four, has absolutely continuous boundary values if the boundary of the image domain has tangents almost everywhere with respect to the surface measure. The sharpness of this result is discussed, as well as the redundancy of the dimensional restriction. In Part III, Heinonen proposes problems, jointly with Seppo Rickman, related to the structure of the branch set of a quasiregular map. Heinonen suggests a way to construct quasiregular maps with complicated branch set. The proposal presents a shift from analysis to geometry via discrete methods in the theory of quasiconformal mappings. Here the word "analysis" means the branch of mathematics that developed out of the Calculus; it is the study of continuous changes in infinitesimal data, from which one wants to predict global conclusions. Geometry, perhaps the most fundamental part of all mathematics, is the study of space, its many forms and varieties, in many possible dimensions. Discrete methods mean the opposite to continuous, or calculus methods, and involve more finite, "counting" ideas, such as those done by a computer, albeit in this work still of analytic and geometric flavor. The concept of quasiconformality enters the discussion in the most subtle way: it allows the space and its formation to be singular 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 mathematicians 40 years ago when the research first in the field was conducted. Now quasiconformality is a ubiquitous phenomenon in mathematics. From work in the late seventies we know that every space that locally looks like a flat space of Euclid has quasiconformal structure, albeit not a smooth structure in general; and that this is true in every dimension but four. Nevertheless, also in dimension four, it is conjectured that there is an infinity of spaces where one cannot do ordinary, smooth analysis, or calculus, but where one can do quasiconformal, nonsmooth analysis. The proposal addresses the need to develop more techniques, and more understanding, of this new calculus.

摘要建议：DMS-9622844 PI：Heinonen该提案有三个部分。第一部分是最重要的。在第一部分中，Heinonen概述了一项研究计划的计划，在公制空间中的准信息图很少。 该项目与Pekka Koskela的关节的目标是系统地在度量空间上建立一个准文化图的理论，这些图在其质量和几何形状上具有某些定量界限。发展这种理论的动机部分来自几何学和组合群体理论，其中重要的是要了解呈负弯曲空间界限的准形式图。 Heinonen和Koskela发现，空间中庞加罗类型不平等的有效性实际上符合工作的准文献理论的有效性。在第二部分中，Heinonen提出了欧几里得空间中准形式图的较低维绝对连续性特性的问题。他已经确定，如果图像域的边界几乎在表面度量方面，几乎到处都有切线，则在大于两个（但不等于四个）的欧几里得空间的单位球上定义了一个准文化图，绝对连续的边界值。讨论了该结果的清晰度，以及维度限制的冗余。在第三部分中，Heinonen提出了与Seppo Rickman共同提出的问题，该问题与准图映射的分支集合有关。 Heinonen提出了一种用复杂的分支集构建准图形图的方法。 该提案通过离散方法在准文献映射理论中提出了从分析到几何的转变。在这里，“分析”一词是指从微积分开发的数学分支；这是对无限数据的持续变化的研究，人们希望从中预测全球结论。几何形状，也许是所有数学中最基本的部分，是在许多可能的维度上对空间，其多种形式和品种的研究。 离散的方法与连续或微积分方法相反，并且涉及更多有限的“计数”思想，例如计算机所做的思想，尽管在这项工作中仍然具有分析性和几何风味。准梦的概念以最微妙的方式进入讨论：它使空间及其形成有时和在某些地方，但不是所有时间，而不是所有时间。失真中有一定的控制权，这对于数学家来说可能是人为的，40年前，当时进行了该领域的研究。现在，准独立是数学中普遍存在的现象。从七十年代后期的工作开始，我们就知道，当地看起来像欧几里得的平坦空间的每个空间都具有准文献结构，尽管通常不是平滑的结构。这在每个维度中都是正确的，只有四个。然而，在第四维度中，也可以猜想，有很多空间无法进行普通，平滑的分析或微积分，但是可以进行列符，非平滑分析。该提案解决了对这种新的演算的更多技术，更多的理解的需求。