Analysis and Potential Theory in Metric Spaces

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

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.