Optimal Transport

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.

Includes supplementary material: sn.pub/extras

Autorentext

PIERRE CARTIER is academic by training, emeritus research director at the Centre national de la recherche scientifique (CNRS), and research scientist and a visitor (for indeterminate term) at the Institut des Hautes Etudes Scientifiques (IHES) of Bures-sur-Yvette. He is currently emeritus research director at the University Paris-Diderot. He is former member of the Bourbaki group. Born in 1932 in Sedan, Professor Carier, despite his provincial background, followed the path of excellence in his studies: a secondary school in his home town, followed by the lycée Saint-Louis in Paris and then the École normale superieure, and he obtained an advanced teaching degree and a doctorate in mathematics (1958). He also went to Princeton for two years; there he became acquainted with legendary figures such as Robert Oppenheimer and Andre Weil (brother of philosopher Simone Weil). Last but not least, he carried out a long military service in the marine corps against the backdrop ofthe Algerian war. He was then professor at the University of Strasbourg from 1961 to 1971. The years 1950-1975 were the hey-day of the Bourbaki group, of which he is one of the pillars; there he became friends with famous people like Cartan, Schwartz, Dieudonné, Chevalley, Weil, and with younger ones. A "mathematician without borders", Professor Cartier is active within the framework of the cooperation agency of the French Mathematical Society (Cimpa). He had been working in the Paris region since 1971, and he roamed around between Parisian academic institutions: University Paris-Sud, École polytechnique, École normale supérieure. He is not a member of the French Academy of Sciences (a voluntary choice). His scientific interests are quite diverse (even eclectic), but centered around group theory and mathematical physics. He often cited thesis relates to algebraic geometry, by he also contributed to differential geometry, number theory, combinatorics, numerical analysis, probability and mathematical physics. He wrote a reference book on Feynman Integrals (in collaboration with C. DeWitt-Morette, Functional Integration, Action and symmetries (Cambridge University Press, 2004). He is also one of the editors of Mathematics in the 21st Century: 6th World Conference, Lahore, March 2013 (Springer, 2014). He has supervised more than 40 doctoral theses, and he continues to work with some of my students on multiple zeta values and the Galois theory of differential equations. JEAN DHOMBRES is mathematician and historian of science, emeritus CNRS research director, director of studies at the Alexandre Koyre Centre of the École des hautes études en sciences sociales (EHESS), specialist of the mathematics of functional equations and of their applications, of epistemology of mathematics, and of the history of scientific communities and of the spread of scholarly ideas. Born in Paris in 1942, he studied at the École polytechnique and chose to take a chance with mathematical research instead of graduating as an engineer. He worked in functional analysis and through a series of encounters, he joined a small international group of scientists focused on functional equations. In fact it was in Bangkok that his first book was published as a result of a talk he had given at Chulalongkorn University in 1971. Besides many articles, he is happy to have been able to work with Janos Aczael, the uncontested leader in this field, having published with him a book on the subject, which has now become a reference, in Encyclopaedia of mathematics (University of Cambridge Press, republished in 2008). Probably as a reaction to his family background, as well as to a scientific training centered on France, he wished to see the world, by learning Chinese, by participating in the establishment of experimental classes in Wu Han in China, then by becoming scientific adviser in a French embassy for three years. He became professor at the University of Nantes as early as 1972. Since for a while he

Inhalt
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 MongeMather 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.