Geometric analysis in Carnot groups

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The intellectual core of the proposal combines geometric measure theory, geometric function theory and differential geometry in sub-Riemannian spaces and abstract metric spaces. The proposal consists of three parts. Part I focuses on sub-Riemannian geometric measure theory, specifically dimension comparison theorems for Carnot-Caratheodory and Euclidean Hausdorff measure and dimension. Applications include sharp dimension computations for nonlinear Euclidean iterated function systems of polynomial type. Related projects concern characteristic negligibility for hypersurfaces in Carnot groups. A long-term goal is to classify constant mean curvature surfaces in jet space groups with an eye to identifying candidate extremals for their isoperimetric inequalities. Part II considers sub-Riemannian geometric function theory, specifically Heisenberg analogs of the Tukia-Vaisala quasiconformal extension theorems and a problem of Heinonen-Semmes. In Part III, the PI studies highly regular surjections to metric spaces. This line of research originates in classical point-set topology results of Peano, Lebesgue and Hahn-Mazurkiewicz and is also influenced by recent work on Morse-Sard theory and rectifiability. The PI has constructed highly regular surjections from Euclidean spaces of sufficiently high dimension onto doubling geodesic spaces. Future problems to be considered include borderline regularity, infinite-dimensional analogs and other regularity classes (Holder and Sobolev maps). A problem of Gromov on density of Lipschitz maps in Sobolev spaces with sub-Riemannian target will be studied. The proposed research encompasses a range of topics within nonsmooth geometric analysis, yet remains unified by a common framework. Geometry studies the static structure of spaces of arbitrary complexity and dimension, while analysis studies dynamic properties and functional interrelations of such spaces. The adjective nonsmooth suggests non-Euclidean settings: fractals, stratified (sub-Riemannian) manifolds, and other abstract spaces. Sub-Riemannian geometry is the `geometry of constrained motion?: it models physical situations where motion is subject to a priori geometric constraints. It features in a remarkably broad spectrum of applications including robotic motion, digital image reconstruction, computer vision, neurobiology, and the mathematics of finance. Sub-Riemannian analysis involves an intricate blend of smooth and nonsmooth techniques as these spaces admit both smooth structure (in restricted directions) and fractal structure (in generic directions). The proposal integrates research, teaching, service and outreach on multiple levels. Graduate student training occurs via summer research programs, teaching of graduate core and topics courses, and Ph.D. supervision. Educational opportunities and outreach related to the research are proposed at the undergraduate and secondary school levels. The PI?s collaborators are located across the U.S. and Europe. Visits to and from these institutions by faculty, postdocs and students will generate new opportunities for collaboration and increase the visibility of the area. To this end, the PI will also continue to organize conferences and workshops in sub-Riemannian geometry and analysis.