Overview

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.  

Full Product Details

Author:   Cédric Villani
Publisher:   Springer-Verlag Berlin and Heidelberg GmbH & Co. KG
Imprint:   Springer-Verlag Berlin and Heidelberg GmbH & Co. K
Edition:   2009 ed.
Volume:   338
Dimensions:   Width: 15.50cm , Height: 5.20cm , Length: 23.50cm
Weight:   3.450kg
ISBN:  

9783540710493

ISBN 10:   3540710493
Pages:   976
Publication Date:   30 September 2008
Audience:   Professional and scholarly ,  Professional & Vocational
Format:   Hardback
Publisher's Status:   Active
Availability:   Out of print, replaced by POD  
We will order this item for you from a manufatured on demand supplier.

Table of Contents

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.

Reviews

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)

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)

Author Information

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.

Tab Content 6

Author Website:  

Customer Reviews

Recent Reviews

No review item found!

Add your own review!

Countries Available

All regions