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  • © 2021

Lectures on Optimal Transport

  • Book suitable for a Phd course in Optimal transport and applications

  • Contents refined on the basis of the 20 years teaching experience of the first author

  • Hints at the most recent developments in the research field

Part of the book series: UNITEXT (UNITEXT, volume 130)

Part of the book sub series: La Matematica per il 3+2 (UNITEXTMAT)

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  • ISBN: 978-3-030-72162-6
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Table of contents (19 chapters)

  1. Front Matter

    Pages i-ix
  2. Lecture 1: Preliminary Notions and the Monge Problem

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 1-11
  3. Lecture 2: The Kantorovich Problem

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 13-22
  4. Lecture 3: The Kantorovich–Rubinstein Duality

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 23-34
  5. Lecture 4: Necessary and Sufficient Optimality Conditions

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 35-41
  6. Lecture 5: Existence of Optimal Maps and Applications

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 43-52
  7. Lecture 6: A Proof of the Isoperimetric Inequality and Stability in Optimal Transport

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 53-63
  8. Lecture 7: The Monge-Ampére Equation and Optimal Transport on Riemannian Manifolds

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 65-76
  9. Lecture 8: The Metric Side of Optimal Transport

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 77-86
  10. Lecture 9: Analysis on Metric Spaces and the Dynamic Formulation of Optimal Transport

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 87-94
  11. Lecture 10: Wasserstein Geodesics, Nonbranching and Curvature

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 95-107
  12. Lecture 11: Gradient Flows: An Introduction

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 109-124
  13. Lecture 12: Gradient Flows: The Brézis-Komura Theorem

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 125-135
  14. Lecture 13: Examples of Gradient Flows in PDEs

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 137-146
  15. Lecture 14: Gradient Flows: The EDE and EDI Formulations

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 147-159
  16. Lecture 15: Semicontinuity and Convexity of Energies in the Wasserstein Space

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 161-181
  17. Lecture 16: The Continuity Equation and the Hopf-Lax Semigroup

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 183-197
  18. Lecture 17: The Benamou–Brenier Formula

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 199-209
  19. Lecture 18: An Introduction to Otto’s Calculus

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 211-228
  20. Lecture 19: Heat Flow, Optimal Transport and Ricci Curvature

    • Luigi Ambrosio, Elia Brué, Daniele Semola
    Pages 229-243

About this book

This textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations.

Keywords

  • Optimal Transport
  • Gradient Flows
  • Partial Differential Equations
  • Analysis on Metric Spaces
  • Calculus of Variations

Reviews

“This book is particularly suited for students who desire to learn from a text which closely follows the organization of a course, as well as for researchers and professors looking for inspiration for their own lecturers on the topic. … The exposition is clear and mostly self-contained, with a nice list of examples that show the necessity of the assumptions of some classical results of the theory.” (Nicolò De Ponti, zbMATH 1485.49001, 2022)

“This book is very well written and will be accessible to graduate students without background on optimal transport … . All in all, this textbook is recommended to graduate students and researchers who want to discover the fundamental theory of optimal transport and its ramifications to several areas of mathematics. It can also easily be used by professors who want to teach a graduate course on the topic.” (Hugo Lavenant, Mathematical Reviews, June, 2022)

Authors and Affiliations

  • Scuola Normale Superiore, Pisa, Italy

    Luigi Ambrosio

  • Institute for Advanced Study, School of Mathematics, Princeton, USA

    Elia Brué

  • Mathematical Institute, University of Oxford, Oxford, UK

    Daniele Semola

About the authors

Prof. Luigi Ambrosio is a Professor of Mathematical Analysis, a former student of the Scuola Normale Superiore and presently its Director. His research interests include calculus of variations, geometric measure theory, optimal transport and analysis in metric spaces. For his scientific achievements, he has been awarded several prizes, in particular the Fermat prize in 2003 and the Balzan Prize in 2019.

Dr. Elia Brué is a postdoctoral member at the Institute for Advanced Studies in Princeton. He earned his PhD degree at the Scuola Normale Superiore in 2020. His research interests include geometric measure theory, optimal transport, non-smooth geometry and PDE.

Dr. Daniele Semola is a postdoctoral research assistant at the Mathematical Institute of the University of Oxford. He was a student in Mathematics at the Scuola Normale Superiore, where he earned his PhD degree in 2020.  His research interests lie at the interface between geometric analysis and analysis on metric spaces, mainly with a focus on lower curvature bounds.

Bibliographic Information

  • Book Title: Lectures on Optimal Transport

  • Authors: Luigi Ambrosio, Elia Brué, Daniele Semola

  • Series Title: UNITEXT

  • DOI: https://doi.org/10.1007/978-3-030-72162-6

  • Publisher: Springer Cham

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021

  • Softcover ISBN: 978-3-030-72161-9

  • eBook ISBN: 978-3-030-72162-6

  • Series ISSN: 2038-5714

  • Series E-ISSN: 2532-3318

  • Edition Number: 1

  • Number of Pages: IX, 250

  • Number of Illustrations: 1 illustrations in colour

  • Topics: Analysis, Calculus of Variations and Optimization, Measure and Integration

Buying options

eBook USD 44.99
Price excludes VAT (USA)
  • ISBN: 978-3-030-72162-6
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 59.99
Price excludes VAT (USA)