Analysis and Potential Theory in Metric Spaces

泰森教授提出的研究涉及非光滑(即非黎曼)环境中的线性和非线性位势理论。它由三个部分组成。第一部分是与Ilkka Holopainen和Nageswari Shanmugalingam的联合项目。某些共形不变紧致运算是在非线性位势理论的背景下自然产生的。理解这些紧致化的结构应该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的设置中进行探索，这是一个包括光滑和非光滑例子的一般框架。在第二部分中，泰森将考虑Dirichlet空间上的拟共形几何和分析。其目的是将已经发展得很好的狄利克雷形式理论与最近在有界几何情况下发展起来的理论联系起来。Kigami，Strichartz等人的工作表明，Dirichlet空间包括各种非光滑的、分形型的对象，这些对象没有被度量空间上的拟共形分析的以前的发展所覆盖。第二部分是与Pekka Koskela和Shanmugalingam的联合工作。在第三部分中，泰森将考虑非线性位势理论在次黎曼空间，特别是卡诺群中的具体应用。这些应用包括几何不等式的尖锐常量问题以及几何的强A-无穷变形。第三部分是与Zoltan Balogh的联合项目。这项建议是由世界各地的一些研究小组正在进行的对非光滑分析的更大调查的一部分。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。现代分析的主题可以追溯到拉普拉斯、柯西和庞加莱等人的开创性工作。分析的经典背景是平坦的欧几里德空间；这是多变量微积分通常涉及的主题。欧几里德理论反过来作为分析弯曲空间(曲面和高维流形)的模型；在这里，潜在的黎曼空间的光滑结构允许人们直接传递欧几里德理论。相反，所提出的研究侧重于非光滑和分形型背景。把这个理论推广到这个更一般的背景下，主要有两个困难：第一，相关的概念和定义必须以一种适合这种扩展的内在方式重新表述，第二，必须引入新的技术和思想来证明在没有通常的环境欧几里德结构的情况下的基本结果。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渴望。简而言之，尽管经典光滑微积分多年来一直作为物理过程的数学研究的基本工具在整个科学中服务，但如果基本的数学是在最小固有光滑性的空间上先验地发展出来的，那么我们有理由期待得到进一步的洞察。

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.