Introduction 1
1 Couplings and changes of variables 5
2 Three examples of coupling techniques 21
3 The founding fathers of optimal transport 29
Part Ⅰ Qualitative description of optimal transport 39
4 Basic properties 43
5 Cyclical monotonicity and Kantorovich duality 51
6 The Wasserstein distances 93
7 Displacement interpolation 113
8 The Monge-Mather shortening principle 163
9 Solution of the Monge problem Ⅰ:Global approach 205
10 Solution of the Monge problem Ⅱ:Local approach 215
11 The Jacobian equation 273
12 Smoothness 281
13 Qualitative picture 333
Part Ⅱ Optimal transport and Riemannian geometry 353
14 Ricci curvature 357
15 Otto calculus 421
16 Displacement convexity Ⅰ 435
17 Displacement convexity Ⅱ 449
18 Volume control 493
19 Density control and local regularity 505
20 Infinitesimal displacement convexity 525
21 Isoperimetric-type inequalities 545
22 Concentration inequalities 567
23 Gradient flows Ⅰ 629
24 Gradient flows Ⅱ:Qualitative properties 693
25 Gradient flows Ⅲ: Functional inequalities 719
Part Ⅲ Synthetic treatment of Ricci curvature 731
26 Analytic and synthetic points of view 735
27 Convergence of metric-measure spaces 743
28 Stability of optimal transport 773
29 Weak Ricci curvature bounds Ⅰ:Definition and Stability 795
30 Weak Ricci curvature bounds Ⅱ:Geometric and analytic properties 847
Conclusions and open problems 903
References 915
List of short statements 957
List of figures 965
Index 967
Some notable cost functions 971