《Topics in Optimal Transportation》是法國數學家Cédric Villani於2003年由美國數學學會出版的數學理論專著，隸屬於《數學研究生教程》叢書。作為質量運輸理論領域的首部綜合性著作，該書系統梳理了該理論的發展脈絡與跨學科套用背景。
全書圍繞蒙日、康托羅維奇和布倫尼爾的奠基性研究展開：蒙日1781年提出最優傳輸問題的工程學套用，康托羅維奇1942年引入線性規劃的經濟學模型，布倫尼爾1987年通過測度保留映射定理拓展流體力學研究。內容涵蓋Kantorovich對偶性、Brenier極分解定理及Monge-Ampère方程等核心理論，同時涉及機率論、泛函分析與等周不等式等交叉領域專題。書中設有習題章節，適用於研究生教學場景。該書要求讀者具備測度論與泛函分析基礎，主要面向相關領域科研人員與高階學習者。
基本介紹
- 作者：Cedric Villani
- 出版時間：2003年3月1日
- 出版社：American Mathematical Society
- 頁數：370 頁
- ISBN：9780821833124
- 定價：65.00 美元
- 裝幀：Hardcover
- 叢書：Graduate Studies in Mathematics
內容簡介
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.
