An Introduction to the Ribe Program

Contents 1. Introduction 2 2. Metric type 7 2.1. Type for metric spaces 8 2.2. The geometric puzzle 11 2.3. Pisier's argument 12 2.4. Unique obstructions to type 17 3. Metric cotype 19 4. Markov type and cotype 22 4.1. Lipschitz extension via Markov type and cotype 27 5. Markov convexity 29 6. Metric smoothness? 32 7. Bourgain's discretization problem 32 8. Nonlinear Dvoretzky theorems 33 8.1. The structure of ultrametric spaces 34 8.2. Ultrametric spaces are ubiquitous 35 9. Examples of applications 39 9.1. Majorizing measures 39 9.2. Lipschitz maps onto cubes 41 9.3. Approximate distance oracles and approximate ranking 42 9.4. Random walks and quantitative nonembeddability 46 9.4.1. Impossibility results for nonlinear Dvoretzky problems 51 9.4.2. λ n and self mixing 52 9.4.3. The spectral argument in the case of large girth 53 9.4.4. He was an enormously influential mathematician and the founder of the field of research that is surveyed here. This article is dedicated to his memory. A 1932 theorem of Mazur and Ulam [123] asserts that if X and Y are Banach spaces and f : X → Y is an onto isometry then f must be an affine mapping. The assumption that f (X) = Y is needed here, as exhibited by, say, the mapping t → (t, sin t) from R to (R 2 , · ∞). However, a major strengthening of the Mazur-Ulam theorem due to Figiel [60] asserts that if f : X → Y is an isometry and f (0) = 0 then there is a unique linear operator T : span(f (X)) → X such that T = 1 and T (f (x)) = x for every x ∈ X. Thus, when viewed as metric spaces in the isometric category, Banach spaces are highly rigid: their linear structure is completely preserved under isometries, and, in fact, isometries between Banach spaces are themselves rigid. At the opposite extreme to isometries, the richness of Banach spaces collapses if one removes all quantitative considerations by treating them as topological spaces. Specifically, answering a question posed in 1928 by Fréchet [62] and again in 1932 by Banach [18], Kadec [92, 93] proved that any two separable infinite dimensional Banach spaces are homeo-morphic. See [91, 28, 27, 4] for more information on this topic, as well as its treatment in the monographs [29, 55]. An extension of the Kadec theorem to non-separable spaces was …