x Preface These notes are definitely not intended to be exhaustive, and should rather be seen as an introduction to the subject. Their reading can be com- plemented by some of the reference texts which have appeared recently. In particular, I should mention the two volumes of Mass transportation problems, by Rachev and R¨ uschendorf, which depict many applications of Monge-Kantorovich distances to various problems, together with the classi- cal theory of the optimal transportation problem in a very abstract setting the survey by Evans, which can also be considered as an introduction to the subject, and describes several applications of the L1 theory (i.e., when the cost function is a distance) which I did not cover in these notes the ex- tremely clear lecture notes by Ambrosio, centered on the L1 theory from the point of view of calculus of variations and also the lecture notes by Urbas, which are a marvelous reference for the regularity theory of the Monge- Amp` ere equation arising in mass transportation. Also recommended is a very pedagogical and rather complete article recently written by Ambrosio and Pratelli, and focused on the L1 theory, from which I extracted many remarks and examples. The present volume does not go too deeply into some of the aspects which are very well treated in the above-mentioned references: in particular, the L1 theory is just sketched, and so is the regularity theory developed by Caf- farelli and by Urbas. Several topics are hardly mentioned, or not at all: the application of mass transportation to the problem of shape optimization, as developed by Bouchitt´ e and Buttazzo the fascinating semi-geostrophic system in meteorology, whose links with optimal transportation are now un- derstood thanks to the amazing work of Cullen, Purser and collaborators or applications to image processing, developed by Tannenbaum and his group. On the other hand, I hope that this text is a good elementary reference source for such topics as displacement interpolation and its applications to functional inequalities with a geometrical content, or the differential view- point of Otto, which has proven so successful in various contexts (like the study of rates of equilibration for certain dissipative equations). I have tried to keep proofs as simple as possible throughout the book, keeping in mind that they should be understandable by non-expert students. I have also stated many results without proofs, either to convey a better intuition, or to give an account of recent research in the field. In the end, these notes are intended to serve both as a course, and as a survey. Though the literature on the Monge-Kantorovich problem is enormous, I did not want the bibliography to become gigantic, and therefore I did not try to give complete lists of references. Many authors who did valuable work on optimal transportation problems (Abdellaoui, Cuesta-Albertos, Dall’Aglio, Kellerer, Matr´ an, Tuero-D´ ıaz, and many others) are not even cited within

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