Analysis and Geometry in Metric Spaces

University Of Tennessee Knoxville, Knoxville TN

Investigators

Abstract

This project seeks to recognize the geometric and topological qualities of metric spaces that allow for the development of a theory of analysis similar to that of Euclidean spaces. While topology, geometry, and analysis are united in the two-dimensional plane, higher dimensional Euclidean spaces or abstract metric spaces lack such powerful tools. Such considerations prompted the development of the field of "analysis on metric spaces," in which first-order differential calculus and geometric measure theory are extended from the classical Euclidean or Riemannian setting to the realm of spaces without a priori smooth structure (such as fractals). Results and techniques in this field have found important applications in geometric group theory, in the structure of manifolds, and in analysis on fractals. Furthermore, besides their mathematical importance, physical applications of these theories range from the reconstruction of missing data in large data sets, to methodologies for data storage and access, and to the study of thin films. This project seeks to develop techniques to address several long-standing questions in the field of analysis on metric spaces and geometric measure theory. The first goal is to relate integral bounds for discrete forms of curvature on non-smooth manifolds with locally Euclidean bi-Lipschitz parameterizations. Such parameterizations are well understood in two dimensions but have so far been elusive in dimensions greater or equal to three. Another goal is to identify sufficient conditions for 2-rectifiability, that is, to understand which sets are contained within the Lipschitz image of a square. Results in this direction will in turn lead to an improved understanding of 2-rectifiable measures. Finally, the project addresses the Euclidean embedding question, namely, to characterize those metric spaces that admit an embedding into a finite-dimensional Euclidean space that does not distort the geometry of the space too much. Apart from providing a better understanding of the geometry and analysis of metric spaces, the existence of such embeddings has been instrumental in recent advances in theoretical computer science and graphic imaging. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.