Optimal Transport

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

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.

From the reviews:

"The book is aimed to old and new problems of optimal transport. ... This meticulous work is based on very large bibliography ... that is converted into a very valuable monograph that presents many statements and theorems written specifically for this approach, complete and self-contained proofs of the most important results, and extensive bibliographical notes." (Mihail Voicu, Zentralblatt MATH, Vol. 1156, 2009)

"This book wins the challenge to give a new and broad perspective on the multifacet topic of the optimal mass transport. ... Besides extensive and accurate references therein the reader will find comments on related questions barely touched upon in the main text as well as lively presentations on how ideas and results have developed. This book should prove useful both to the expert and to the beginner looking for a reference text on the subject." (Dario Cordero Erausquin, Mathematical Reviews, Issue 2010 f)

"The book is an in-depth, modern, clear exposition of the advanced theory of optimal transport, and it tries to put together in a unified way almost all the recent developments of the theory. ... the book is extremely well written and very pleasant to read. ... I strongly recommend this excellent book to every researcher or graduate student in the field of optimal transport. ... of interest to many mathematicians in different areas, who are simply interested in having an overview of the subject." (Alessio Figalli, Bulletin of the American Mathematical Society, Vol. 47 (4), February, 2010)