Topics in Optimal Transportation

by Cédric Villani

This is the first comprehensive introduction to the theory of mass transportation with its many―and sometimes unexpected―applications. In a novel approach to the subject, the book both surveys the topic and includes a chapter of problems, making it a particularly useful graduate textbook. In 1781, Gaspard Monge defined the problem of “optimal transportation” (or the transferring of mass with the least possible amount of work), with applications to engineering in mind. In 1942, Leonid Kantorovich applied the newborn machinery of linear programming to Monge's problem, with applications to economics in mind. In 1987, Yann Brenier used optimal transportation to prove a new projection theorem on the set of measure preserving maps, with applications to fluid mechanics in mind. Each of these contributions marked the beginning of a whole mathematical theory, with many unexpected ramifications. Nowadays, the Monge-Kantorovich problem is used and studied by researchers from extremely diverse horizons, including probability theory, functional analysis, isoperimetry, partial differential equations, and even meteorology. Originating from a graduate course, the present volume is intended for graduate students and researchers, covering both theory and applications. Readers are only assumed to be familiar with the basics of measure theory and functional analysis.

Genres: Mathematics

370 pages, Paperback

First published March 1, 2003

About the author

Cédric Villani is a French mathematician who has received many international awards for his work including the Jacques Herbrand Prize, the Prize of the European Mathematical Society, the Fermat Prize and the Henri Poincaré Prize.

In 2010 he was awarded the Fields Medal, the International Medal for Outstanding Discoveries in Mathematics, for his work on Landau damping and the Boltzmann equation. Often called ‘the mathematicians’ Nobel Prize’, it is awarded every four years and is viewed by some as the highest honour a mathematician can achieve.

He is a professor at Lyon University and Director of the Institut Henri Poincaré in Paris, working primarily on partial differential equations and mathematical physics.