Metric Dichotomies

These are notes from talks given at ICMS, Edinburgh, 4/2007 ("Geometry and Algorithms workshop") and at Bernoulli Center, Lausanne 5/2007 ("Limits of graphs in group theory and computer science"). We survey the following type of dichotomies exhibited by certain classes X of finite metric spaces: For every host space H, either all metrics in X embed almost isometrically in H, or the distortion of embedding some metrics of X in H is unbounded.