University of Illinois at Urbana-Champaign

Math 595 Section MTM (Mapping Theory in Metric Spaces) Spring 2011

Course Description

The goal of the course is to introduce modern techniques and results in the study of mappings between metric spaces. A major theme will be the influence of the geometry of both source and target on the prevalence and structure of various classes of mappings. In contrast with the real- or vector-valued case, the theory of metric space-valued mappings is fundamentally nonlinear and geometric. We will use tools from many different subjects, including functional analysis, Fourier analysis, PDE, and differential geometry.

A significant part of the course will be devoted to extension and embedding theorems for finite metric spaces. This subject is characterized by an intricate interplay betweeen geometry, combinatorics and functional analysis. In the latter part of the course we will consider metric spaces with other specific structure arising from synthetic differential geometry or sub-Riemannian geometry. One of the main goals of the course is to discuss a significant recent theorem of Cheeger and Kleiner on the bi-Lipschitz nonembeddability of the sub-Riemannian Heisenberg group into certain Banach spaces, and the application of that theorem to questions from the geometry of finite metric spaces motivated by algorithmic computer science.