最优输运

PrefaceConventionsIntroduction1 Couplings and changes of variables2 Three examples of coupling techniques3 The founding fathers of optimal transportPart Ⅰ Qualitative description of optimal transport4 Basic properties5 Cyclical monotonicity and Kantorovich duality6 The Wasserstein distances7 Displacement interpolation8 The Monge-Mather shortening principle9 Solution of the Monge problem I： Global approach10 Solution of the Monge problem II： Local approach11 The Jacobian equation12 Smoothness13 Qualitative picturePart Ⅱ Optimal transport and Riemannian geometry14 Ricci curvature15 Otto calculus16 Displacement convexity I17 Displacement convexity II18 Volume control19 Density control and local regularity20 Infinitesimal displacement convexity21 Isoperimetric-type inequalities22 Concentration inequalities23 Gradient flows I24 Gradient flows II： Qualitative properties25 Gradient flows III： Functional inequalitiesPart Ⅲ Synthetic treatment of Ricci curvature26 Analytic and synthetic points of view27 Convergence of metric-measure spaces28 Stability of optimal transport……Conclusions and open problems

When I was first approached for the 2005 edition of the Saint-Flour Probability Summer School， I was intrigued， flattered and scared. l Apart from the challenge posed by the teaching of a rather analytical　　subject to a probabilistic audience， there was the danger of producing a remake of my recent book Topics in， Optimal Transportation.