Rigidity of Lipschitz and Related Mappings on Metric Spaces

Ball State University, Muncie IN

Investigators

Abstract

The most basic setting for differential calculus is the study of smoothly changing functions on the real line. In this project, more general, non-smooth, functions that are defined on geometric objects (metric spaces) that may be very different from the real line will be investigated. These geometries may be abstract objects with no reasonable embedding into any Euclidean geometry, or they may be fractal, or they may otherwise admit complicated behavior at all scales. Far from being a technical curiosity, non-smooth analysis and geometry have now become important tools in many areas of pure and applied mathematics and computer science, where non-Euclidean geometries may arise. For example, from discrete groups or dynamical systems, as limits of smooth objects, as large data sets, or in computational problems. In more precise terms, the goal of this project is to study the relationship between, on the one hand, the infinitesimal and global geometry of non-smooth spaces and, on the other hand, the analysis of Lipschitz and related classes of mappings defined on these spaces. One specific goal is to further understand the spaces which allow for differentiation of real-valued Lipschitz functions (in the sense of Cheeger): what topological and geometric properties can such spaces possess, and can we construct new examples? Another goal is to understand rigidity theorems for Lipschitz mappings and rectifiable curves in metric spaces. For example, when must Lipschitz mappings between metric spaces have more rigid (e.g., bi-Lipschitz) behavior on large subsets of their domains? Can we characterize rectifiable curves in metric spaces via local flatness conditions, as in the "Analyst's Traveling Salesman Theorem" of Jones in the plane? Understanding these questions involves combining techniques from classical geometric measure theory with those of the newer field of "analysis on metric spaces". Analytic investigations like these have also provided, and should continue to provide, insights into the geometry of the non-smooth.