A classical rigidity theorem of Martin Ribe (1975) suggests that certain important properties of normed spaces are actually metric properties in disguise, i.e., they can be characterized while using only distances between points and without any reference to the underlying linear structure whatsoever. Ribe's theorem inspired a longstanding research program that aims to uncover an explicit dictionary that translates concepts and phenomena from the structured linear world of normed spaces to the seemingly wild and unstructured world of general metric spaces. Any "entry" in such a dictionary could potentially be used to apply insights from the linear theory to other remote settings, including graphs, groups, Riemannian manifolds and various metric spaces that arise as continuous relaxations of combinatorial optimization problems. Previous advances in the Ribe program involved the use of diverse mathematical tools, and they led to many powerful applications to various areas. Nevertheless, many central mysteries remain, and the hidden dictionary that the Ribe program aims to uncover still contains many missing entries that are needed for further progress.
The purpose of the first lecture is to provide an introduction to the Ribe program, with an indication of examples of key milestones that have already been achieved, examples of applications, and some of the many important problems that remain open. The second lecture will be devoted to a description of one of the abstract mechanisms that underlie the Ribe theorem, as discovered by Bourgain (1986), and its relation to analytic issues such as quantitative versions of differentiation. The third lecture will be devoted to a description of the recent completion of a step in the Ribe program, which discovers the last invariant of metric spaces that was missing from a useful repertoire of invariants that have been discovered over the past three decades that (among their many applications) certify when L_q fails to admit a bi-Lipschitz embedding into L_p. This new invariant yields a metric version of a classical linear theorem of Paley (1936), and the proof of its validity in L_p relies on modern Fourier analytic tools.