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
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