Preface
Conventions
Introduction
　1　Couplings and changes of variables
　2　Three examples of coupling techniques
　3　The founding fathers of optimal transport
Part Ⅰ　Qualitative description of optimal transport
　4　Basic properties
　5　Cyclical monotonicity and Kantorovich duality
　6　The Wasserstein distances
　7　Displacement interpolation
　8　The Monge-Mather shortening principle
　9　Solution of the Monge problem I: Global approach
　10　Solution of the Monge problem II: Local approach
　11　The Jacobian equation
　12　Smoothness
　13　Qualitative picture
Part Ⅱ　Optimal transport and Riemannian geometry
　14　Ricci curvature
　15　Otto calculus
　16　Displacement convexity I
　17　Displacement convexity II
　18　Volume control
　19　Density control and local regularity
　20　Infinitesimal displacement convexity
　21　Isoperimetric-type inequalities
　22　Concentration inequalities
　23　Gradient flows I
　24　Gradient flows II: Qualitative properties
　25　Gradient flows III: Functional inequalities
Part Ⅲ　Synthetic treatment of Ricci curvature
　26　Analytic and synthetic points of view
　27　Convergence of metric-measure spaces
　28　Stability of optimal transport
　29　Weak Ricci curvature bounds I: Definition and Stability
　30　Weak Ricci curvature bounds II: Geometric and analytic properties
Conclusions and open problems
　References
　List of short statements
　List of figures
　Index
　Some notable cost functions