Analysis and Potential Theory in Metric Spaces

度量空间中的分析和势论

• 批准号: 0228807
• 负责人: Jeremy Tyson
• 金额: $ 9.92万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0228807&HistoricalAwards=false
• 关键词: Analysis; Potential; Theory; Metric; Spaces

Professor Tyson's proposed research concerns linear and nonlinear potential theory in nonsmooth (i.e., non-Riemannian) environments. It consists of three parts. Part I is a joint project with Ilkka Holopainen and Nageswari Shanmugalingam. Certain conformally invariant compactification operations arise naturally in the context of nonlinear potential theory. Understanding the structure of these compactifications should have important consequences for the boundary behavior of quasiconformal maps. These questions will be explored in the setting of metric spaces of bounded geometry, which is a general framework encompassing both smooth and nonsmooth examples. In Part II, Tyson will consider quasiconformal geometry and analysis on Dirichlet spaces. The goal is to relate the already well-developed theory of Dirichlet forms to the recently developed theory in the bounded geometry case. Work of Kigami, Strichartz and others has shown that Dirichlet spaces include various nonsmooth, fractal-type objects which are not covered by previous developments in quasiconformal analysis on metric spaces. Part II is joint work with Pekka Koskela and Shanmugalingam. In Part III, Tyson will consider specific applications of nonlinear potential theory in sub-Riemannian spaces, specifically, Carnot groups. These applications include sharp constant questions for geometric inequalities as well as strong A-infinity deformations of geometry. Part III is a joint project with Zoltan Balogh.This proposal is part of a larger investigation into nonsmooth analysis which is being carried out by a number of research groups worldwide. In informal terms, analysis is the mathematical study of motion and change; its historical roots lie in the development of the Calculus by Newton and Leibniz. The modern subject of analysis can be traced back to the pioneering work of Laplace, Cauchy and Poincare (among others). The classical setting for analysis is flat Euclidean spaces; this is the subject typically covered in multi-variable calculus. The Euclidean theory serves in turn as a model for analysis on curved spaces (surfaces and higher-dimensional manifolds); here the smooth structure of the underlying Riemannian space permits one to transport the Euclidean theory directly. In contrast, the proposed research focuses on nonsmooth and fractal-type settings. In extending the theory to this more general context the principal difficulties are twofold: first, the relevant concepts and definitions must be reformulated in an intrinsic manner suitable for such an extension, and second, new techniques and ideas must be introduced to prove basic results in the absence of the usual ambient Euclidean structure. The motivation for carrying out such an extension stems from the desire for better mathematical models for the nonsmooth and disordered media which arise in applications. Put simply, although classical smooth calculus has served for many years and throughout the sciences as an essential tool in the mathematical study of physical processes, it is reasonable to expect that further insight will be gained if the underlying mathematics is developed a priori on spaces of minimal inherent smoothness.

泰森教授提出的研究涉及非光滑（即，非黎曼环境。 它由三部分组成。第一部分是与Ilkka Holopainen和纳格斯瓦里Shanmugalingam的联合项目。某些共形不变的紧化运算在非线性势理论中自然出现。理解这些紧化的结构对拟共形映射的边界行为有重要的影响。这些问题将在有界几何的度量空间中进行探讨，这是一个包含光滑和非光滑例子的一般框架。在第二部分中，泰森将考虑拟共形几何和Dirichlet空间的分析。我们的目标是与已经发展良好的理论狄利克雷形式最近发展的理论在有界几何的情况下。Kigami、Kighartz和其他人的工作表明狄利克雷空间包括各种非光滑的、分形型的对象，这些对象在度量空间的拟共形分析中没有被涵盖。第二部分是与Pekka Koga和Shanmugalingam的联合工作。在第三部分中，泰森将考虑非线性势理论在次黎曼空间中的具体应用，特别是卡诺群。这些应用包括尖锐的常数问题的几何不等式以及强A-无穷变形的几何。第三部分是与Zoltan Balogh的联合项目。该提案是全球许多研究小组正在进行的一项更大规模的非光滑分析调查的一部分。在非正式的术语中，分析是对运动和变化的数学研究;它的历史根源在于牛顿和莱布尼茨对微积分的发展。现代的分析学科可以追溯到拉普拉斯、柯西和庞加莱（以及其他人）的开创性工作。分析的经典背景是平坦的欧几里得空间;这是多变量微积分中通常涉及的主题。欧几里得理论又作为分析弯曲空间（曲面和高维流形）的模型;在这里，基本黎曼空间的光滑结构允许人们直接传输欧几里得理论。相比之下，拟议的研究重点是非光滑和分形类型的设置。在扩展理论到这个更一般的情况下，主要的困难是双重的：第一，有关的概念和定义必须重新制定的内在方式适合这样的扩展，第二，新的技术和思想必须引进，以证明基本结果在缺乏通常的环境欧几里德结构。进行这种扩展的动机源于对应用中出现的非光滑和无序介质的更好的数学模型的期望。简单地说，虽然经典光滑微积分已经作为物理过程的数学研究中的一个重要工具服务了很多年，并且贯穿整个科学，但是如果基础数学是在最小固有光滑度的空间上先验地发展的，那么我们有理由期望获得进一步的洞察力。