The theory of Optimal Transport finds its roots in the work of Gaspard Monge in 1781, whose interest was in finding the cheapest possible way to transfer some resources from producers to consumers.

After some major contributions due to Leonid Kantorovich (who was later awarded the Nobel Prize in Economics, partly for his work on the subject) in the 1940s, Optimal Transport has been blooming in the last thirty years, with applications covering partial differential equations, geometric analysis, mathematical finance, and machine learning, among the others.

The goal of this course will be to introduce the basic theory of Optimal Transport and to hint at some of its more recent developments and applications.

The main topics covered will include:

• classical formulations of the problem;

• duality theory;

• optimality conditions;

• existence of optimal transport maps (Brenier’s theorem);

• the Monge-Ampère equation;

• applications to isoperimetric inequalities;

• the Wasserstein distance and the Wasserstein space;

References

[1] Ambrosio, Luigi; Gigli, Nicola; Savaré, Giuseppe: Gradient flows in metric spaces and in the space of probability measures. 2nd ed. Lectures in Mathematics, ETH Zürich. Basel: Birkhäuser vii, 334 p. (2008).

[2] Figalli, Alessio; Glaudo, Federico: An invitation to optimal transport, Wasserstein distances, and gradient flows. EMS Textbooks in Mathematics. Berlin: European Mathematical Society (EMS) vi, 136 p. (2021).

[3] Villani, Cédric: Topics in optimal transportation. Graduate Studies in Mathematics 58. Providence, RI: American Mathematical Society (AMS) xvi, 370 p. (2003).