Optimal Transport

At the close of the 1980s, the independent contributions of Yann Brenier, Mike Cullen and John Mather launched a revolution in the venerable field of optimal transport founded by G. Monge in the 18th century, which has made breathtaking forays into various other domains of mathematics ever since. The author presents a broad overview of this area, supplying complete and self-contained proofs of all the fundamental results of the theory of optimal transport at the appropriate level of generality. Thus, the book encompasses the broad spectrum ranging from basic theory to the most recent research results.

C+¬dric Villani has been Professor at the Ecole Normale Sup+¬rieure de Lyon since September 2000 and a junior member of the Institut Universitaire de France since 2006. He was awarded the EMS Prize of the European Mathematical Society in July 2008.

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