Geometric Mapping Theory in Sub-Riemannian and Metric Spaces

University Of Illinois At Urbana-Champaign, Urbana IL

Investigators

Abstract

The PI will study mappings in and between sub-Riemannian spaces and more general metric spaces. A recurring theme is the study of generic dimension distortion properties of mappings. Several problems are posed concerning precise, quantitative statements of the frequency with which given Hausdorff dimension bounds hold relative to a specified parameter space: such frequency is measured with respect to measures on the parameter space (in many cases, these parameterizing measures are themselves Hausdorff measures). The preceding themes will be developed in two contexts: (i) metric space-valued Sobolev mappings, and (ii) linear and nonlinear projection-type mappings defined on the Heisenberg group and other Carnot groups. Another theme covered by the proposed research program concerns new aspects of the analytic theory of quasiconformal and quasiregular mappings on the Heisenberg group. This includes both novel constructions of such mappings as well as emerging connections to partial differential equations and sub-Riemannian exterior differential calculus as formulated by Rumin. Additional topics included in the proposal include density of Lipschitz mappings in spaces of Heisenberg group-valued Sobolev mappings, and Hausdorff measure negligibility of highly characteristic points on submanifolds in sub-Riemannian Carnot groups. This research program lies at the intersection of geometry and analysis. Geometry is the study of the structure of space, both the familiar Euclidean spaces of classical antiquity as well as more recent, exotic spaces. Analysis is the study of change and motion: its origins lie in the theory of differential and integral calculus developed by Newton and Leibniz in the 17th century, clarified and rigorously formulated by Cauchy, Weierstrass, Riemann and others in the second half of the 19th century, and finally extended to its modern form in the 20th century in work of Lebesgue, Hilbert, Hardy and (later) Sobolev. The two subjects interact extensively: geometric structure both influences and is influenced by the dynamic, analytic behavior of mappings between spaces. Sobolev spaces provide a framework for quantifying the analytic properties of `non-smooth' functions and mappings. They are a foundational tool in modern approaches to partial differential equations. Understanding analysis on such spaces is important for several reasons. First, such understanding provides new perspectives on the classical theory, highlighting relevant and necessary features of the underlying geometry. At the same time, this abstract approach has a greater range of applicability. Techniques and results from analysis on metric spaces have found application in the study of metric graphs and networks, fractals and other geometric environments such as sub-Riemannian spaces. Sub-Riemannian geometries arise in `external' applications to mathematical models of control theory, robotic path planning and neurobiology, as well as `internal' applications to other branches of pure mathematics such as several complex variables, symplectic and contact geometry and geometric group theory. One aspect of the proposed research involving sub-Riemannian mapping theory relates to mathematical models in impedance tomography and image reconstruction.