# 401-3350-68L Introduction to Optimal Transport

Semester | Autumn Semester 2018 |

Lecturers | A. Figalli, further speakers |

Periodicity | non-recurring course |

Language of instruction | English |

Comment | Number of participants limited to 11. |

Abstract | Introductory seminar about the theory of optimal transport. Starting from Monge's and Kantorovich's statements of the optimal transport problem, we will investigate the theory of duality necessary to prove the fundamental Brenier's theorem. After some applications, we will study the properties of the Wasserstein space and we will conclude introducing the dynamical point of view on the problem. |

Objective | |

Content | Given two distributions of mass, it is natural to ask ourselves what is the "best way" to transport one into the other. What are mathematically acceptable definitions of "distributions of mass" and "to transport one into the other"? Measures are perfectly suited to play the role of the distributions of mass, whereas a map that pushes-forward one measure into the other is the equivalent of transporting the distributions. By "best way" we mean that we want to minimize the map in some norm. The original problem of Monge is to understand whether there is an optimal map and to study its properties. In order to attack the problem we will need to relax the formulation (Kantorovich's statement) and to apply a little bit of duality theory. The main theorem we will prove in this direction is Brenier's theorem that answers positively to the existence problem of optimal maps (under certain conditions). The Helmotz's decomposition and the isoperimetric inequality will then follow rather easily as applications of the theory. Finally, we will see how the optimal transport problem gives a natural way to define a distance on the space of probabilities (Wasserstein distance) and we will study some of its properties. |

Literature | "Optimal Transport, Old and New", C. Villani [Link] "Optimal Transport for Applied Mathematicians", F. Santambrogio [Link] |

Prerequisites / Notice | The students are expected to have mastered the content of the first two years taught at ETH, especially Measure Theory. The seminar is mainly intended for Bachelor students. In order to obtain the 4 credit points, each student is expected to give two 1h-talks and regularly attend the seminar. Moreover some exercises will be assigned. Further information can be found at Link |