Mappings and Measures in Sub-Riemannian and Metric Spaces

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The goal of this project is to investigate geometry and analysis in spaces with rough or fractal structure. The project focuses especially on spaces which are poorly described by classical Euclidean language, such as fractals and sub-Riemannian spaces. The latter occur as mathematical models for problems in robotics, neurobiology, celestial mechanics and other physical systems. In each of these models, the underlying geometry is inherently nonsmooth. The proposed research will consequently contribute to developing the mathematical framework which simultaneously underlies a variety of physical applications. On the geometric side, nonsmooth notions of curvature will be studied. Curvature is a fundamental concept in modern geometry. For instance, general relativity predicts that curvature is the primary geometric feature of space responsible for gravity. Nonsmooth generalizations of curvature and their relationship to the structure of measures will also be studied. Measure theory is a wide-ranging mathematical generalization of fundamental concepts of length, area, and volume. Finally, analysis refers to the dynamic properties of transformations acting between spaces. Since the classical Newton--Leibniz theory of the derivative is not well-adapted to transformations of nonsmooth spaces, there is currently significant interest in developing a new analytic toolkit to extend the machinery of calculus beyond its usual context. The principle investigator will continue to train graduate students, postdocs and early-career researchers. Results of this research program will be disseminated via talks at conferences and workshops, publication of journal articles, and exposition for a general mathematical audience. The PI is currently coauthoring a graduate textbook on analysis in nonsmooth spaces. The ongoing preparation of this book will be coordinated with advances in the field, including those obtained as part of this research program. The project concerns the geometry of measures and submanifolds in, and mappings between, sub-Riemannian manifolds. Three coordinating themes will be considered. The first set of problems concerns density of measures and the geometry of submanifolds. The main goal is a sub-Riemannian analog of a celebrated theorem of David Preiss characterizing rectifiability (a measure-theoretic notion of smoothness) via densities. This line of investigation will extend to the sub-Riemannian setting some fundamental aspects of Euclidean geometric measure theory tied to notions of curvature for submanifolds. The primary methodology for this investigation involves the approximation of sub-Riemannian spaces by Riemannian spaces, and an analysis of the limiting behavior of geometric quantities in the Riemannian approximations. The second theme concerns iteration of conformal and quasiregular mappings. Such iterative schemes provide a basic combinatorial model for fractal objects. They arise naturally in both hyperbolic geometry and number theory. Building on his previous work for similarity mappings, the principle investigator will extend the analysis to conformal mappings. The goal is a detailed description of the geometry of self-conformal limit sets in sub-Riemannian spaces and natural measures on such sets. Finally, the principle investigator will study the type problem for sub-Riemannian quasiregular mappings. The goal here is to understand both the flexibility of quasiregular mappings as well as inherent geometric obstructions to their construction. The principle investigator will also investigate new constructions of Sobolev mappings inducing optimal measure and dimension distortion, as well as criteria for the density of Lipschitz mappings in spaces of Sobolev mappings with sub-Riemannian targets. The principle investigator will take advantage of his prior expertise in all of the aforementioned areas, as well as the latest advances in related fields, to pursue answers to the proposed problems. The proposed research will have a broad impact; it addresses a wide range of questions at the intersection of different mathematical areas. The principle investigator will use this broad framework to identify specific problems suitable for beginning graduate students and postdocs.