An Invitation To Optimal Transport Wasserstein Distances And Gradient Flows
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An Invitation to Optimal Transport, Wasserstein Distances, and Gradient Flows
Author: Alessio Figalli
language: en
Publisher: European Mathematical Society
Release Date: 2023-05-15
This book provides a self-contained introduction to optimal transport, and it is intended as a starting point for any researcher who wants to enter into this beautiful subject. The presentation focuses on the essential topics of the theory: Kantorovich duality, existence and uniqueness of optimal transport maps, Wasserstein distances, the JKO scheme, Otto's calculus, and Wasserstein gradient flows. At the end, a presentation of some selected applications of optimal transport is given. Suitable for a course at the graduate level, the book also includes an appendix with a series of exercises along with their solutions. The second edition contains a number of additions, such as a new section on the Brunn–Minkowski inequality, new exercises, and various corrections throughout the text.
Conversations on Optimal Transport
This work is closely tied to the renowned mathematics textbook series known as UNITEXT, tailored for university students pursuing bachelor’s or master’s degrees. What sets this particular book apart in the Springer collection is its unique origin: it has been crafted through a meticulous process involving interviews handled with and by world-class mathematicians. The content featured in this book revolve around a highly relevant and engaging topic: Optimal Transport. These conversations involve not only authors from the UNITEXT series, but also members of the series’ Editorial Board. Additionally, they feature prominent figures in the field, including a Field Medalist. This work provides readers with a snapshot of remarkable vitality and freshness, guaranteed to captivate and engage anyone with an interest in mathematics. It’s important to note that these interviews were initially shared as podcasts and originally broadcasted as online events on the Cassyni platform. Subsequently, advanced AI tools were employed under human supervision to transcribe the audios and edit them for better readability. A human copy-editor was involved during the whole process, and the authors revised the final copy-edited texts before publication. The content in each format – the interviews, the PODCASTS and the book – is self-contained and not a mere adaptation from one medium to another. Instead, it represents an independent exploration of the subject matter.
Optimal Transport on Quantum Structures
The flourishing theory of classical optimal transport concerns mass transportation at minimal cost. This book introduces the reader to optimal transport on quantum structures, i.e., optimal transportation between quantum states and related non-commutative concepts of mass transportation. It contains lecture notes on classical optimal transport and Wasserstein gradient flows dynamics and quantum optimal transport quantum couplings and many-body problems quantum channels and qubits These notes are based on lectures given by the authors at the "Optimal Transport on Quantum Structures" School held at the Erdös Center in Budapest in the fall of 2022. The lecture notes are complemented by two survey chapters presenting the state of the art in different research areas of non-commutative optimal transport.