Mathematical Sciences: Quasiconformal Maps and Nonsmooth Analysis

提案：DMS-9622844 PI: Heinonen提案分为三部分。第一部分是最重要的。在第一部分中，Heinonen概述了度量空间中具有小光滑性的拟共形映射的研究程序。该项目的目标是与Pekka Koskela合作，系统地发展度量空间上的准共形映射理论，这些映射在质量和几何上具有一定的定量界限。发展这种理论的动机部分来自几何学和组合群论，在这些理论中，理解负弯曲空间边界上的拟共形映射是很重要的。Heinonen和Koskela发现空间中庞加莱型不等式的有效性实际上等同于准共形理论的有效性。在第二部分中，Heinonen提出了欧几里得空间中拟共形映射的低维绝对连续性问题。他建立了一个准共形映射，定义在欧几里得空间的单位球上的任何维数大于2，但不等于4，如果像域的边界几乎处处都有相对于表面度量的切线，则具有绝对连续的边界值。讨论了这一结果的清晰度，以及尺寸限制的冗余性。在第三部分中，Heinonen与Seppo Rickman共同提出了关于拟正则映射分支集结构的问题。Heinonen提出了一种构造具有复杂分支集的拟正则映射的方法。在拟共形映射理论中，提出了从分析到几何的离散方法的转变。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，也许是所有数学中最基本的部分，是对空间的研究，它的许多形式和变化，在许多可能的维度上。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讨论：它允许空间及其形成在时间和地点上是单一的，但不是无处不在，也不是一直如此。这种扭曲有一定的控制作用，40年前，当这一领域的研究首次开展时，数学家们可能会认为这是人为的。准共形性是数学中普遍存在的现象。从七十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准共形结构，尽管一般来说不是光滑的结构；除了四维空间，其他维度都是如此。然而，同样在四维空间中，人们推测存在无限的空间，在那里人们不能做普通的、光滑的分析或微积分，但在那里人们可以做拟共形的、非光滑的分析。该提案提出了开发更多技术和对这种新微积分的更多理解的需要。

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.