Lipschitz Analysis in Normed and Metric Spaces

Syracuse University, Syracuse NY

Investigators

Abstract

The notion of distance, or dissimilarity of objects, is ubiquitous in geometry, data analysis, image and signal processing, and other disciplines. Naturally associated to it is the notion of transformations that distort the distances by at most a fixed amount. For example, in order to represent high-dimensional data visually, one needs to reduce the number of dimensions while keeping the distortion of distances low. Another example is image recovery, where the goal is to produce a geometrically natural image (with crisp lines and low noise) that has a small measure of dissimilarity from a given noisy image. Signal recovery by error-correcting codes depends on the concept of distance minimization as well: a corrupted sequence of bits is replaced by the nearest sequence that conforms to the specifications of the encoding. The principal investigator will introduce graduate and undergraduate students to these concepts and associated research problems. The project aims to advance the understanding of the Lipschitz geometry of metric spaces, focusing on their embeddings, retractions, and bi-Lipschitz equivalence. It aims to solve open problems concerning the possibility of retracting finite subset spaces onto each other; the process of symmetrizing and extending bi-Lipschitz maps with linear control of their distortion, the feasibility of robust recovery of a vector from the absolute values of its frame coefficients, the geometry of removable sets and quasi-convex sets, and the relation of low-dimensional representations of a metric space with the geometry of its tight span. An array of methods from analysis, geometry, and geometric measure theory will be employed, such as gradient flows on metric spaces and probability measures supported on them, comparison estimates on spaces with curvature bounds, constructive methods of geometric mapping theory, and the calculus of variations as applied to geometrically motivated problems. 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.