Mathematical Sciences: Quasiconformal Maps and Nonsmooth Analysis

提案摘要：DMS-9622844 PI：海诺宁提案由三部分组成。第一部分是最有意义的。在第一部分中，Heinonen概述了度量空间中几乎没有光滑性的拟共形映射的研究方案。该项目的目标是与Pekka Koskela合作，系统地发展度量空间上的拟共形映射理论，该度量空间对其质量和几何具有一定的量化界限。发展这样一个理论的动机部分来自几何和组合群论，在这些理论中，理解负曲线空间边界上的拟共形映射是很重要的。Heinonen和Koskela已经发现，Poincare类型不等式在空间中的有效性实际上等同于一个有效的准共形理论的有效性。在第二部分中，Heinonen提出了欧氏空间中拟共形映射的低维绝对连续性问题。他证明了定义在欧几里得空间单位球上的拟共形映射具有绝对连续的边值，如果像域的边界相对于曲面测度几乎处处都有切线。讨论了这一结果的尖锐性以及尺寸约束的冗余性。在第三部分中，Heinonen与Seppo Rickman一起提出了与拟正则映射的分支集的结构有关的问题。Heinonen提出了一种构造具有复杂分支集的拟正则映射的方法。通过拟共形映射理论中的离散方法，该建议提出了从分析到几何的转变。在这里，“分析”这个词指的是从微积分发展而来的数学分支；它是研究无穷小数据的连续变化，人们想要从这些数据中预测出全球的结论。几何，也许是所有数学中最基本的部分，是研究空间，它的许多形式和种类，在许多可能的维度。离散方法的意思与连续或微积分方法相反，它涉及更有限的“计数”思想，例如由计算机完成的那些，尽管在这项工作中仍然具有解析和几何的味道。准共形的概念以最微妙的方式进入讨论：它允许空间及其形成在时间和地点是单一的，但不是任何地方，也不是所有时间。这种扭曲有一定的可控性，在40年前进行该领域第一次研究时，在数学家看来，这种扭曲可能看起来是人为的。现在，准共形在数学中是一种普遍存在的现象。从七十年代末的工作中我们知道，每个局部看起来像欧几里得平面空间的空间都有准共形结构，尽管总体上不是光滑的结构；除了四个维度外，每个维度都是如此。然而，也是在四维空间中，人们猜想，在无限的空间中，人们不能进行普通的、光滑的分析或微积分，但在那里人们可以进行准共形的、非光滑的分析。该提案解决了开发更多技术和对这种新演算的更多理解的必要性。

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.