Titlebook: Lectures on Optimal Transport; Luigi Ambrosio,Elia Brué,Daniele Semola Textbook 20211st edition The Editor(s) (if applicable) and The Auth

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Lecture 15: Semicontinuity and Convexity of Energies in the Wasserstein Space,We are now interested in the semicontinuity and convexity properties of the following three types of energy functionals defined on measures: internal energy, potential energy and interaction energy.

Junction

Lecture 16: The Continuity Equation and the Hopf-Lax Semigroup,We are now going to explore the connections between the classical ODE Cauchy problem, the continuity equation and the transport equation. These results will be applied to give a differential description (through the continuity equation) of the geodesics in the Wasserstein space.

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endocardium

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Lecture 19: Heat Flow, Optimal Transport and Ricci Curvature,In this final lecture we wish to explore the connections between Heat Flow, Optimal Transport and Ricci curvature on a smooth compact Riemannian manifold. In doing so we will extend and generalise some of the properties we have proved for the Euclidean space.

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Lecture 8: The Metric Side of Optimal Transport,l theory and to some applications. Now our goal is to show how the optimal transport problem can be used to endow . with a natural metric structure. We will see how many metric (and even differential) properties can be “l(fā)ifted” from . to ., as separability, compactness, completeness, geodesic, nonbranching, lower bounds on sectional curvature.

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Lectures on Optimal Transport978-3-030-72162-6Series ISSN 2038-5714 Series E-ISSN 2532-3318

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Textbook 20211st editiond 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 advance