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势理论。它由三部分组成。第一部分是与Ilkka Holopainen和Nageswari Shanmugalingam的联合项目。在非线性势理论的背景下，自然产生了某些共形不变紧化运算。理解这些紧化的结构对于拟共形映射的边界行为具有重要的意义。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，泰森将考虑拟共形几何和狄利克雷空间的分析。目的是把已经发展得很好的狄利克雷形式理论与最近发展起来的有界几何理论联系起来。Kigami、Strichartz等人的工作表明，Dirichlet空间包括各种非光滑的分形对象，这些对象在度量空间的拟共形分析中没有被以前的发展所涵盖。第二部分是与Pekka Koskela和Shanmugalingam的合作。在第三部分中，Tyson将考虑非线性势理论在亚黎曼空间中的具体应用，特别是卡诺群。这些应用包括几何不等式的尖锐常数问题以及几何的强a无穷大变形。第三部分是与Zoltan Balogh的联合项目。这一提议是一项针对非光滑分析的更大调查的一部分，该调查正在由世界各地的一些研究小组进行。通俗地说，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、柯西和庞加莱等人的开创性工作。经典的分析背景是平坦的欧几里德空间；这是多变量微积分中典型的主题。欧几里得理论反过来又作为分析弯曲空间（曲面和高维流形）的模型；在这里，底层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，我们提出的研究侧重于非光滑和分形类型的设置。在将理论扩展到更一般的环境中，主要的困难是双重的：首先，相关的概念和定义必须以一种适合于这种扩展的内在方式重新表述，其次，必须引入新的技术和思想来证明在缺乏通常的欧几里得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。简单地说，尽管经典光滑微积分作为物理过程的数学研究的重要工具已经服务了许多年，在整个科学中，如果在最小固有平滑度的空间上先验地发展基础数学，则有理由期望获得进一步的见解。

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.