Mathematical Sciences: Quasiconformal Maps and Nonsmooth Analysis

摘要提案：DMS-9622844 PI：Heinonen 该建议有三个部分。第一部分是最重要的部分。在第一部分中，Heinonen概述了度量空间中几乎不光滑的拟共形映射的研究计划。 该项目的目标，联合 Pekka Koga，系统地发展了度量空间上的拟共形映射理论，这些度量空间在质量和几何上具有一定的定量界。发展这样一个理论的动机部分来自几何学和组合群论，其中理解负弯曲空间边界上的拟共形映射是很重要的。Heinonen和Kokora发现，空间中的庞加莱型不等式的有效性实际上等同于工作拟共形理论的有效性。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。他已经建立了一个拟共形映射，定义在单位球的欧几里德空间的任何维度大于2，但不等于4，具有绝对连续的边界值，如果边界的图像域切线几乎无处不在的表面措施。讨论了这一结果的尖锐性，以及维数限制的冗余性。在第三部分中，海诺宁提出的问题，连同塞波里克曼，有关的结构的分支集的准正规映射。Heinonen提出了一种构造具有复杂分支集的拟正则映射的方法。 该建议提出了一个转移，从分析几何通过离散方法在理论的拟共形映射。这里的“分析”一词是指从微积分发展而来的数学的分支;它是对无穷小数据连续变化的研究，人们希望从中预测全局结论。几何学，也许是所有数学中最基本的部分，是研究空间，它的许多形式和品种，在许多可能的维度。 离散方法意味着与连续或微积分方法相反，并且涉及更多有限的“计数”思想，例如由计算机完成的那些，尽管在这项工作中仍然具有分析和几何风味。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是单一的，但不是在任何地方，也不是所有时间。在扭曲中有一定的控制，这可能看起来人为的数学家40年前，当研究第一次在该领域进行。拟共形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，每个局部看起来像欧几里得平坦空间的空间都具有拟共形结构，尽管一般来说不是光滑结构;而且这在除了四维之外的每个维度上都是正确的。然而，同样在四维空间中，也有无穷多的空间不能做普通的光滑分析或微积分，但可以做拟共形的非光滑分析。该提案解决了需要开发更多的技术，更多的理解，这种新的微积分。

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.