Multi-scale geometry of bi-Lipschitz and quasiconformal maps

Syracuse University, Syracuse NY

Investigators

Abstract

The notion of distance is fundamental both to human perception of the physical space around us, and to the idealization of this space in geometry. The distillation of the most salient properties of distance leads to the concept of a metric space. Applications of this concept abound in computer science, machine learning, mathematical genetics, statistics and other disciplines where quantitative dissimilarity of objects is of interest. A proper way to understand metric spaces is to consider their transformations that shrink or stretch the distance on the space by a bounded amount. These are called bi-Lipschitz transformations, and they are at the focus of the proposed investigation. As an example of an application of such transformations, one can mention that accurate representations of a given, possibly quite irregular, metric space within a Euclidean space is an essential aspect of data visualization and analysis. Normed linear spaces provide a convenient model to which less regular metric spaces can be compared. Thus, it is of interest to know when a given metric space is bi-Lipschitz equivalent to a normed space, or can be embedded into one. This is one of the principal problems addresses in the proposal. Even for two-dimensional spaces (metric planes) our understanding of bi-Lipschitz equivalence is far from complete. One approach to this problem is to introduce and study numeric invariants of metric spaces, such as dimension-type concepts which capture the size or degree of connectivity within the space. A different, but related approach involves analytic properties of functions supported on the space: Poincare inequality and various extension properties (Sobolev, Lipschitz, etc). The proposal involves both of these approaches. In particular, the quantitative aspects of the bi-Lipschitz extension problem will be investigated.