Preface This set of notes grew from a graduate course that I taught at Georgia Tech, in Atlanta, during the fall of 1999, on the invitation of Wilfrid Gangbo. It is a great pleasure for me to thank Georgia Tech for its hospitality, and all the faculty members and students who attended this course, for their interest and their curiosity. Among them, I wish to express my particular gratitude to Eric Carlen, Laci Erd¨os, Michael Loss, and Andrzej Swiech. It was Eric and Michael who first suggested that I make a book out of the lecture notes intended for the students. Three years passed by before I was able to complete these notes of course, I took into account as much as I could of the mathematical progress made during those years. Optimal mass transportation was born in France in 1781, with a very famous paper by Gaspard Monge, emoire sur la th´ eorie des eblais et des remblais. Since then, it has become a classical subject in probability theory, economics and optimization. Very recently it gained extreme popularity, because many researchers in different areas of mathematics understood that this topic was strongly linked to their subject. Again, one can give a precise birthdate for this revival: the 1987 note by Yann Brenier, D´ecomposition polaire et earrangement des champs de vecteurs. This paper paved the way towards a beautiful interplay between partial differential equations, fluid mechanics, geometry, probability theory and functional analysis, which has developed over the last ten years, through the contributions of a number of authors, with optimal transportation problems as a common denominator. ix
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