xii Preface investment for any student in probability, analysis or partial differential equations. This book owes a lot to many people. I was lucky enough to learn the subject of optimal mass transportation directly from two of those researchers who have most contributed to turn it into a fascinating area: Yann Brenier and Felix Otto it is a pleasure here to express my enormous gratitude to them. I first encountered optimal transportation in Tanaka’s work about the Boltzmann equation, and my curiosity about it increased dramatically from discussions with Yann but it was only after hearing a beautiful and enthusiastic lecture given in Paris by Craig Evans, that I made up my mind to study the subject thoroughly. My involvement in the study of functional inequalities related to mass transportation was partly triggered by interac- tions with Michel Ledoux, whose influence is gratefully acknowledged. The present manuscript profited a lot from numerous discussions with Luigi Am- brosio, Eric Carlen, Dario Cordero-Erausquin, Wilfrid Gangbo and Robert McCann. Both Robert and Luigi taught the material of this book, made many suggestions and pointed out numerous misprints and mistakes in the first version of these notes. The most serious one concerned the “proof” of Theorem 1.3, as given in the first version of these notes the gap was fixed thanks to the kind help of Luigi again, and of Bernd Kirchheim, with the final result of an improved statement. Some of my students at the Ecole normale sup´ erieure also spotted and repaired a gap in the proof of Theo- rem 2.18. Fran¸ cois Bolley, Jean-Fran¸ cois Coulombel and Maxime Hauray should be thanked for the time they spent hunting for mistakes and mis- prints in various parts of the book, and testing many of the exercises and problems. Richard Dudley was kind enough to give a quick but thorough look at Chapters 1 and 7. Chapter 4 would not have existed without the ex- planations which I received from Luis Caffarelli and Andrzej Swiech. Most of the material in Chapter 6 was taught to me by Franck Barthe. Chapter 8 was reshaped by the exchanges which I had with Luigi Ambrosio, Nicola Gigli and Etienne Ghys during the last stages of preparation of the manu- script. Finally, Mike Cullen corrected some mistakes in the presentation of the physical model in Problem 9 of Chapter 10. All comments, suggestions and bug reports will be extremely welcome and can be sent to me by electronic mail at cvillani@umpa.ens-lyon.fr. I will maintain a list of errata on my Internet home-page, accessible via the Internet server of the Mathematics Department at Ecole normale sup´erieure de Lyon, http://www.umpa.ens-lyon.fr/ edric Villani Lyon, January 2003
Previous Page Next Page