Preface ....................................................... VII
Conventions .................................................. XVII
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 1 .......................................... 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
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