Optimal Mass Transport on Euclidean Spaces: A Graduate-Level Introduction by Francesco Maggi – Cambridge University Pres

Introduction

Optimal mass transport is one of the most dynamic and rapidly evolving areas in modern mathematics, with deep connections to the calculus of variations, partial differential equations (PDEs), and geometric analysis. Optimal Mass Transport on Euclidean Spaces by Francesco Maggi offers a rigorous yet accessible graduate-level introduction to this fascinating subject. Published by Cambridge University Press, this hardcover volume is an essential resource for Indian students and researchers seeking a solid foundation in the theory and applications of optimal transport.

Book Overview

This book provides a comprehensive treatment of optimal mass transport problems in the familiar setting of Euclidean spaces. By working in this context, the author gradually introduces core concepts with minimal prerequisites, ensuring that readers build geometric intuition and technical proficiency simultaneously. The text explores both the classical Monge problem and the more flexible Kantorovich formulation, solving the former for linear and quadratic transport costs—two cases that are pivotal in geometric and PDE applications respectively. The book is designed to be self-contained, making it ideal for self-study or as a textbook for advanced courses in Indian universities.

Key Highlights

• Gradual, accessible approach to optimal transport theory, starting from Euclidean spaces to build intuition.
• Rigorous treatment of both the Monge and Kantorovich problems, including proofs and key theorems.
• Focus on linear and quadratic transport costs, which are central to geometric analysis and PDE theory.
• Minimal prerequisites—only basic real analysis and measure theory are assumed.
• Complete and self-contained, suitable for graduate students and researchers new to the field.

Inside the Book

The book is structured to guide readers from foundational concepts to advanced results. It begins with a review of measure theory and the geometry of Euclidean spaces, then moves to the Kantorovich problem and its dual formulation. The Monge problem is tackled for linear costs (important in geometric applications like isoperimetric inequalities) and for quadratic costs (central to the theory of Wasserstein distances and gradient flows). Each chapter includes carefully chosen exercises and examples that reinforce learning and encourage independent exploration.

Key Topics

• Measure theory and weak convergence in Euclidean spaces
• The Kantorovich problem: existence, duality, and optimality conditions
• The Monge problem for linear transport cost
• The Monge problem for quadratic transport cost
• Applications to the isoperimetric inequality and geometric inequalities
• Wasserstein distances and their properties
• Connections to the calculus of variations and PDEs

Reader Benefits

• Strong conceptual foundation in optimal transport theory, enabling further study in advanced topics.
• Practical problem-solving skills through worked examples and exercises tailored to Euclidean settings.
• Enhanced geometric intuition for transport problems, which is crucial for applications in analysis and geometry.
• Clear, step-by-step exposition that makes complex ideas accessible to Indian graduate students.
• Up-to-date coverage of a field that is increasingly relevant in data science, economics, and image processing.

Learning Outcomes

By the end of this book, readers will be able to: formulate optimal transport problems in both Monge and Kantorovich frameworks; prove existence and uniqueness of optimal transport maps for linear and quadratic costs; understand the duality theory of optimal transport; apply optimal transport techniques to geometric inequalities; and use Wasserstein distances in analysis and PDE contexts. These skills are invaluable for research in pure and applied mathematics.

Who Should Read

This book is ideal for graduate students in mathematics, physics, and engineering who have completed a first course in real analysis and measure theory. It is also highly recommended for researchers in the calculus of variations, geometric analysis, PDEs, and probability who wish to enter the field of optimal transport. Indian students preparing for advanced studies or research in these areas will find the book particularly beneficial due to its pedagogical clarity.

About the Author

Francesco Maggi is a leading mathematician known for his contributions to geometric analysis and the calculus of variations. He is a professor at the University of Texas at Austin and has authored numerous influential papers on optimal transport, minimal surfaces, and isoperimetric inequalities. His expertise and clear expository style make this book a trusted resource for learners worldwide.

About the Publisher

Cambridge University Press is one of the oldest and most respected academic publishers in the world. With a rich history of publishing groundbreaking mathematical texts, CUP ensures that each book meets the highest standards of scholarship and production quality. This hardcover edition is built to last, making it a valuable addition to any library.

Conclusion

Optimal Mass Transport on Euclidean Spaces is a masterful introduction to a vibrant and important field. Whether you are a student taking your first steps in optimal transport or a researcher seeking a thorough reference, Francesco Maggi's book provides the clarity, depth, and rigor you need. Order your copy today from Bookshops.in and embark on a journey through one of mathematics' most exciting frontiers.