Lecture

Optimal Transport: Heat Equation and Metric Spaces

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In course

MATH-476: Optimal transport

The first part is devoted to Monge and Kantorovitch problems, discussing the existence and the properties of the optimal plan. The second part introduces the Wasserstein distance on measures and develops applications of optimal transport to PDEs, functiona ...

Description

This lecture covers the application of optimal transport to the heat equation in metric spaces, discussing the work of Ambrosio, Gigli, and Savaré. The lecture explores the concept of optimal transport, the heat equation, and the properties of metric spaces.

Instructors (2)

Maria Colombo

Teaching & PhD PhD Students Roberto Colombo, Antonio Tirotta, Carl Johan Peter Johansson, Giulia Mescolini Past EPFL PhD Students Luigi De Rosa, Silja Noëmi Aline Haffter, Massimo Sorella Courses Analysis IV - Lebesgue measure, Fourier analysis MATH-205 Learn the basis of Lebesgue integration and Fourier analysis Topics on the Euler and Navier-Stokes equations MATH-647 This topics course focuses on recent and classical fundamental results on the Euler and Navier-Stokes equa

Xavier Fernandez-Real Girona

Teaching & PhD PhD Students Roberto Colombo Courses Optimal transport MATH-476 The first part is devoted to Monge and Kantorovitch problems, discussing the existence and the properties of the optimal plan. The second part introduces the Wasserstein distance on measures and develops applications of optimal transport to PDEs, functional/geometric inequalities, traffic models.

Official source

Related concepts (32)

Metric space

In mathematics, a metric space is a set together with a notion of distance between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general setting for studying many of the concepts of mathematical analysis and geometry. The most familiar example of a metric space is 3-dimensional Euclidean space with its usual notion of distance. Other well-known examples are a sphere equipped with the angular distance and the hyperbolic plane.

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Shoot 'em up

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Complete metric space

In mathematical analysis, a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M. Intuitively, a space is complete if there are no "points missing" from it (inside or at the boundary). For instance, the set of rational numbers is not complete, because e.g. is "missing" from it, even though one can construct a Cauchy sequence of rational numbers that converges to it (see further examples below).

Outer measure

In the mathematical field of measure theory, an outer measure or exterior measure is a function defined on all subsets of a given set with values in the extended real numbers satisfying some additional technical conditions. The theory of outer measures was first introduced by Constantin Carathéodory to provide an abstract basis for the theory of measurable sets and countably additive measures.

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Optimal Transport: In-Depth Evaluation