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 2: The Kantorovich Problem,We can now introduce Kantorovich’s formulation of the optimal transport problem. It involves the concept of . (also called coupling in the Probability literature) between probability measures. In the discrete setting of Example ., transport plans correspond to bi-stochastic matrices.

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Lecture 9: Analysis on Metric Spaces and the Dynamic Formulation of Optimal Transport,In this section we introduce basic notions and tools of analysis on metric spaces. We begin by defining the property of absolute continuity for curves .?:?[., .]?→?., with (., .) a metric space, and we prove structural properties of this family of curves.

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Lecture 10: Wasserstein Geodesics, Nonbranching and Curvature,Let us now come to the proof of the lower semicontinuity of the action, defined as in (.). The proof could be achieved with more elementary tools, but we prefer to use a general lemma that will play a role also in the sequel.

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