Optimal Transport (eBook)

After attending the Lycée Louis-le-Grand, Villani was admitted to the École normale supérieure in Paris and studied there from 1992 to 1996. He was later appointed assistant professor in the same school. He received his doctorate at Paris-Dauphine University in 1998, under the supervision of Pierre-Louis Lions, and became professor at the École normale supérieure de Lyon in 2000. He is now professor at Lyon University. He has been the director of Institut Henri Poincaré in Paris since 2009. Prizes:2001: Louis Armand Prize of the Academy of Sciences2003: Peccot-Vimont Prize and Cours Peccot of the Collège de France2007: Jacques Herbrand Prize (French Academy of Sciences)2008: Prize of the European Mathematical Society2009: Henri Poincaré Prize2009: Fermat Prize2010: Fields Medal2014: Joseph L. Doob Prize of the American Mathematical Society for his book [Optimal Transport: Old and New (Springer 2009)] Extra-academic distinctions:2009: Knight of the National Order of Merit (France)2011: Knight of the Legion of Honor (France)2013: Member of the French Academy of Sciences 

Preface 6
Contents 11
Conventions 14
Introduction 20
1 Couplings and changes of variables 22
2 Three examples of coupling techniques 38
3 The founding fathers of optimal transport 46
Part I Qualitative description of optimal transport 55
4 Basic properties 57
5 Cyclical monotonicity and Kantorovich duality 64
6 The Wasserstein distances 106
7 Displacement interpolation 125
8 The Monge–Mather shortening principle 175
9 Solution of the Monge problem I: Global approach 216
10 Solution of the Monge problem II: Local approach 225
11 The Jacobian equation 283
12 Smoothness 290
13 Qualitative picture 342
Part II Optimal transport and Riemannian geometry 361
14 Ricci curvature 364
15 Otto calculus 428
16 Displacement convexity I 441
17 Displacement convexity II 454
18 Volume control 498
19 Density control and local regularity 510
20 Infinitesimal displacement convexity 530
21 Isoperimetric-type inequalities 549
22 Concentration inequalities 570
23 Gradient flows I 632
24 Gradient flows II: Qualitative properties 696
25 Gradient flows III: Functional inequalities 721
Part III Synthetic treatment of Ricci curvature 732
26 Analytic and synthetic points of view 735
27 Convergence of metric-measure spaces 742
28 Stability of optimal transport 772
29 Weak Ricci curvature bounds I: Definition and Stability 793
30 Weak Ricci curvature bounds II: Geometric and analytic properties 845
Conclusions and open problems 900
References 910
List of short statements 952
List of figures 960
Index 962