Workshop on Minimal Surfaces, Sub-Elliptic PDE's and Geometric Analysis

Dartmouth College, Hanover NH

Investigators

Abstract

Workshop on Minimal Surfaces, Sub-Elliptic PDE's and Geometric Analysis. The study of PDE's, harmonic analysis, and geometric analysis in the sub-Riemannian setting has reached a critical juncture: recently, researchers from disparate fields have made significant progress in this area and have opened up many new avenues of research. The conference will focus on contemporary developments in the study of several problems from analysis and geometry in the setting of Carnot-Carath\'eodory metrics. Most of the invited lecturers will address a variety of interrelated topics, such as: ``best-constant'' type problems concerning Sobolev and isoperimetric inequalities; the study of minimal and constant-curvature submanifolds; rectifiability and geometric measure theory; quasiconformal maps and potential theory; geometric flows and applications. Analysis in Carnot-Carath\'eodory spaces is an important component in the general theory of abstract, non-smooth analysis which has seen extensive development in recent years. The conference, as envisioned by the PI's, will foster the collaboration of different research groups and provide a ground for discussion. The study of systems whose dynamics is subject to physical constraints has been a focus of attention for a long time, both from the point of view of pure mathematics and from the point of view of engineering and physics. Motivation for these inquiries stems from the wide variety of applications to problems in control theory, robotic planning, the structure of crystalline materials, image reconstruction, nonholonomic mechanics, and others. In mathematical terms such systems are represented by Carnot-Carath\'eodory (sub-Riemannian) spaces. These are manifolds with a preferred set of directions at every point. These preferred directions represent the constraints; motion is only allowed in these directions. The study of geometry and analysis on CC spaces is based on techniques from several mathematical disciplines: several complex variables, contact geometry, partial differential equations, harmonic analysis and geometric function theory. In turn, new results in the sub-Riemannian context often yield important progress in these areas.