Topics in optimal transportation /
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| Author / Creator: | Villani, Cédric, 1973- |
|---|---|
| Imprint: | Providence, R.I. : American Mathematical Society, c2003. |
| Description: | xvi, 370 p. : ill. ; 27 cm. |
| Language: | English |
| Series: | Graduate studies in mathematics, 1065-7339 ; v. 58 |
| Subject: | |
| Format: | Print Book |
| URL for this record: | http://pi.lib.uchicago.edu/1001/cat/bib/4864398 |
Table of Contents:
- Preface
- Notation
- Introduction
- 1.. Formulation of the optimal transportation problem
- 2.. Basic questions
- 3.. Overview of the course
- Chapter 1.. The Kantorovich Duality
- 1.1.. General duality
- 1.2.. Distance cost functions
- 1.3.. Appendix: A duality argument in C[subscript b](X x Y)
- 1.4.. Appendix: {{0, 1}}-valued costs and Strassen's theorem
- Chapter 2.. Geometry of Optimal Transportation
- 2.1.. A duality-based proof for the quadratic cost
- 2.2.. The real line
- 2.3.. Alternative arguments
- 2.4.. Generalizations to other costs
- 2.5.. More on c-concave functions
- Chapter 3.. Brenier's Polar Factorization Theorem
- 3.1.. Rearrangements and polar factorization
- 3.2.. Historical motivations: fluid mechanics
- 3.3.. Proof of Brenier's polar factorization theorem
- 3.4.. Related facts
- Chapter 4.. The Monge-Ampere Equation
- 4.1.. Informal presentation
- 4.2.. Regularity
- 4.3.. Open problems
- Chapter 5.. Displacement Interpolation and Displacement Convexity
- 5.1.. Displacement interpolation
- 5.2.. Displacement convexity
- 5.3.. Application: uniqueness of ground state
- 5.4.. The Eulerian point of view
- Chapter 6.. Geometric and Gaussian Inequalities
- 6.1.. Brunn-Minkowski and Prekopa-Leindler inequalities
- 6.2.. The Alesker-Dar-Milman diffeomorphism
- 6.3.. Gaussian inequalities
- 6.4.. Sobolev inequalities
- Chapter 7.. The Metric Side of Optimal Transportation
- 7.1.. Monge-Kantorovich distances
- 7.2.. Topological properties
- 7.3.. The real line
- 7.4.. Behavior under rescaled convolution
- 7.5.. An application to the Boltzmann equation
- 7.6.. Linearization
- Chapter 8.. A Differential Point of View on Optimal Transportation
- 8.1.. A differential formulation of optimal transportation
- 8.2.. Differential calculus
- 8.3.. Monge-Kantorovich induced dynamics
- 8.4.. Time-discretization
- 8.5.. Differentiability of the quadratic Wasserstein distance
- 8.6.. Non-quadratic costs
- Chapter 9.. Entropy Production and Transportation Inequalities
- 9.1.. More on optimal-transportation induced dissipative equations
- 9.2.. Logarithmic Sobolev inequalities
- 9.3.. Talagrand inequalities
- 9.4.. HWI inequalities
- 9.5.. Nonlinear generalizations: internal energy
- 9.6.. Nonlinear generalizations: interaction energy
- Chapter 10.. Problems
- List of Problems
- Bibliography
- Table of Short Statements
- Index
