Optimal Transport Old and New
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 variou
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Series editors M. Berger B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin L. Hörmander A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane A. M. Vershik M. Waldschmidt Editor-in-Chief A. Chenciner J. Coates S.R.S. Varadhan
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Cédric Villani
Optimal Transport Old and New
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Cédric Villani Unité de Mathématiques Pures et Appliquées (UMPA) École Normale Supérieure de Lyon 46, allée d'Italie 69364 Lyon CX 07 France [email protected]
ISSN 0072-7830 ISBN 978-3-540-71049-3
e-ISBN 978 -3 -540 -71050 -9
Library of Congress Control Number: 2008932183 Mathematics Subject Classification Numbers (2000): 49-xx, 53-xx, 60-xx c 2009 Springer-Verlag Berlin Heidelberg This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable to prosecution under the German Copyright Law. The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Cover design: WMX Design GmbH, Heidelberg Printed on acid-free paper
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Preface
When I was first approached for the 2005 edition of the Saint-Flour Probability Summer School, I was intrigued, flattered and scared.1 Apart from the challenge posed by the teaching of a rather analytical subject to a probabilistic audience, there was the danger of producing a remake of my recent book Topics in Optimal Transportation. However, I gradually realized that I was being offered a unique opportunity to rewrite the whole theory from a different perspective, with alternative proofs and a different focus, and a more probabilistic presentation; plus the incorporation of recent progress. Among the most striking of these recent advances, there was the rising awareness that John Mather’s minimal measures had a lot to do with optimal transport, and that both theories could actually be embedded within a single framework. There was also the discovery that optimal transport could provide a robust synthetic approach to Ricci curvature bounds. These links with dynamical systems on one hand, differential geometry on the other hand, were only briefly alluded to in my first book; here on the contrary they will be at the basis of the presentation. To summarize: more probability, more geometry, and more dynamical systems. Of course there cannot be more of everything, so in some sense there is less analysis and less physics, and also there are fewer digressions. So t
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