Instructional Conference on Combinatorial

— In these lectures, we present a triple description of the concentration of measure phenomenon, geometric (through Brunn-Minkoswki inequalities), measure-theoretic (through transportation cost inequalities) and functional (through logarithmic Sobolev inequalities), and investigate the relationships between these various viewpoints. Special emphasis is put on optimal mass transportation and the dual hy-percontractive bounds on solutions of Hamilton-Jacobi equations that offer a unified treatment of these various aspects. These notes survey recent developments around the concentration of measure phenomenon through various descriptions, geometric, measure theoretic and functional. These descriptions aim to analyze measure concentration for both product and (strictly) log-concave measures, with a special emphasis on dimension free bounds. Inequalities independent of the number of variables are indeed a key information in the study of a number of models in probability theory and statistical mechanics, with a view towards infinite dimensional analysis. To this task, we review the geometric tool of Brunn-Minkowski inequalities, transportation cost inequalities, and functional logarithmic Sobolev inequalities and semigroup methods. Connections are developed on the basis of optimal mass transportation and dual hypercontractive bounds on solutions of Hamilton-Jacobi equations, providing a synthetic view of these recent developments. Results and methods are only outlined in the simplest and basic setting. References to recent PDE extensions are briefly discussed in the last part of the notes. These notes only collect a few basic results on the topics of these lectures, and only aim to give a flavour of the subject. 1. The concentration of measure phenomenon The concentration of measure phenomenon was put forward in the early seventies by V. Milman [Mi1], [Mi3] in the asymptotic geometry of Banach spaces and the proof of the famous Dvoretzky theorem on spherical sections of convex bodies. Of isoperimetric inspiration, it is of powerful interest in applications, in various areas such as geometry, functional analysis and infinite dimensional integration, discrete mathematics and complexity theory, and probability theory. To introduce to the concept of measure concentration, we first briefly discuss a few examples. A first illustration is suggested by the example of the standard n-sphere S n in R n+1 when dimension n is large. By a standard computation, uniform measure σ n on S n is almost concentrated when the dimension n is large around the (every) equator. Actually, the isoperimetric inequality on S n expresses that spherical caps (geodesic balls) minimize the boundary measure at fixed volume. In its integrated form, given a …