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.

PhD students or researchers can read the entire book without any prior knowledge of the field. A comprehensive bibliography with notes that extensively discuss the existing literature underlines the book¿s value as a most welcome reference text on this subject.

Couplings and changes of variables.- Three examples of coupling techniques.- The founding fathers of optimal transport.- Qualitative description of optimal transport.- Basic properties.- Cyclical monotonicity and Kantorovich duality.- The Wasserstein distances.- Displacement interpolation.- The Monge¿Mather shortening principle.- Solution of the Monge problem I: global approach.- Solution of the Monge problem II: Local approach.- The Jacobian equation.- Smoothness.- Qualitative picture.- Optimal transport and Riemannian geometry.- Ricci curvature.- Otto calculus.- Displacement convexity I.- Displacement convexity II.- Volume control.- Density control and local regularity.- Infinitesimal displacement convexity.- Isoperimetric-type inequalities.- Concentration inequalities.- Gradient flows I.- Gradient flows II: Qualitative properties.- Gradient flows III: Functional inequalities.- Synthetic treatment of Ricci curvature.- Analytic and synthetic points of view.- Convergence of metric-measure spaces.- Stability of optimal transport.- Weak Ricci curvature bounds I: Definition and Stability.- Weak Ricci curvature bounds II: Geometric and analytic properties.

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 Sciences

2003: Peccot-Vimont Prize and Cours Peccot of the Collège de France

2007: Jacques Herbrand Prize (French Academy of Sciences)

2008: Prize of the European Mathematical Society

2009: Henri Poincaré Prize

2009: Fermat Prize

2010: Fields Medal

2014: 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