Optimal Transport

Cedric Villani has been Professor at the Ecole Normale Superieure 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.

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)

• 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.