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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