Optimal Transport: Old and New

Front Cover
Springer Science & Business Media, Oct 26, 2008 - Mathematics - 976 pages

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.

 

Selected pages

Contents

Couplings and changes of variables
5
Three examples of coupling techniques
21
The founding fathers of optimal transport
29
Basic properties
43
Cyclical monotonicity and Kantorovich duality 51
50
The Wasserstein distances
93
Displacement interpolation
113
The MongeMather shortening principle
163
Volume control
493
Density control and local regularity
505
Infinitesimal displacement convexity
525
Isoperimetrictype inequalities
545
Concentration inequalities 567
566
Gradient flows I
629
Qualitative properties
693
Functional inequalities
719

Global approach 205
204
Local approach
215
The Jacobian equation
273
Smoothness 281
280
Qualitative picture
333
Optimal transport and Riemannian geometry 353
354
Ricci curvature
357
Otto calculus
421
Displacement convexity I
435
Displacement convexity II 449
445
Synthetic treatment of Ricci curvature 731
730
Analytic and synthetic points of view
735
Convergence of metricmeasure spaces 743
742
Stability of optimal transport
773
Geometric
847
References
915
List of short statements
957
Some notable cost functions
971
Copyright

Other editions - View all

Optimal Transport: Old and New
Cedric Villani
No preview available - 2016
Optimal Transport: Old and New
Cédric Villani
No preview available - 2008
Optimal Transport: Old and New
Cédric Villani
No preview available - 2013

Common terms and phrases

absolutely continuous action-minimizing curves argument assume assumption bibliographical notes Borel c-convex c-cyclically monotone CD(K Chapter compact set constant convergence convex function Corollary cost function cs,t curvature-dimension d(xo defined Definition density dimension displacement convexity displacement interpolation equivalent Euclidean finite follows formula geodesic gradient flows Gromov-Hausdorff Hamilton-Jacobi holds true implies infimum Jacobian Lagrangian Lemma lim inf lim sup Lipschitz locally logarithmic Sobolev inequality lower semicontinuous McCann metric space minimizing curves Monge problem nonnegative notation optimal coupling optimal transference plan optimal transport particular path Poincaré inequality Polish space probability measures proof of Theorem Proposition prove quadratic reference measure regularity Remark Ricci curvature Riemannian manifold satisfies sectional curvature semiconcave semiconvex sequence smooth Sobolev inequality solution subdifferentiable subset t₁ tangent theory tion uniformly unique variables velocity Wasserstein distance Wasserstein space

References to this book

About the author (2008)

After attending the Lycée Louis-le-Grand, Villani was admitted to the École normale supérieure in Paris and studied there from 1992 to 1996. He was later appointed assistant professor in the same school. He received his doctorate at Paris-Dauphine University in 1998, under the supervision of Pierre-Louis Lions, and became professor at the École normale supérieure de Lyon in 2000. He is now professor at Lyon University. He has been the director of Institut Henri Poincaré in Paris since 2009.

Prizes:
2001: Louis Armand Prize of the Academy of Sciences
2003: Peccot-Vimont Prize and Cours Peccot of the Collège de France
2007: Jacques Herbrand Prize (French Academy of Sciences)
2008: Prize of the European Mathematical Society
2009: Henri Poincaré Prize
2009: Fermat Prize
2010: Fields Medal
2014: Joseph L. Doob Prize of the American Mathematical Society for his book [Optimal Transport: Old and New (Springer 2009)]

Extra-academic distinctions:
2009: Knight of the National Order of Merit (France)
2011: Knight of the Legion of Honor (France)
2013: Member of the French Academy of Sciences


Bibliographic information

TitleOptimal Transport: Old and New
Volume 338 of Grundlehren der mathematischen Wissenschaften
AuthorCédric Villani
Editionillustrated
PublisherSpringer Science & Business Media, 2008
ISBN3540710507, 9783540710509
Length976 pages
Subjects ›  › 

Mathematics / Calculus
Mathematics / Differential Equations / General
Mathematics / Functional Analysis
Mathematics / Geometry / Differential
Mathematics / Mathematical Analysis
  
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