Optimal Transport and Applications

Doctoral course in Trento and Verona

Practical Information

Duration: 20h
Schedule: 10h in Trento, March 8-11 + 10h in Verona, April 18-22.
For the week in Trento : Tue and Wed 2.30pm-5pm, Thur and Fri 9am-11.30am (aula A102, polo Ferrari).
For the week in Verona : Mon and Wed 4.30pm-6.30pm (aula G), Tue and Thur 1.30pm-3.30pm and Fri 3.30pm-5.30pm (aula M).
Examination: different forms of examinations (and cominations of them) are possible: written exam with exercises; written dissertation or oral presentation on a related topic; numerical implementation of an alorithm oresented in class (in groups).
Language: the classes have finally been given in Italian in Trento and in english in Verona.
Prerequisites: some functional analysis, in particular existence criteria for minimization problems (semicontinuity, compactness), measure theory and weak compactness for measures, a little bit of convex analysis (Legendre transform) and some basic PDEs. F. Serra Cassano and G. Orlandi gave some preparatory classes on these topics.


A tentative program for the course is the following.

Lecture TN-1 Monge and Kantorovich problems, existence of optimal plans, duality, Kantorovich potentials.
Lecture TN-2 Existence of optimal maps and the Monge-Ampère equation. Cyclical monotonicity, analysis of the 1D case.
Lecture TN-3 Discrete and semi-discrete numerical methods.
Lecture TN-4 Optimal flow formulation of the optimal transport problem. Traffic congestion variants.

Lecture VR-1 Wasserstein distances and Wasserstein spaces.
Lecture VR-2 Curves in the Wasserstein spaces and connections with the continuity equation.
Lecture VR-3 Geodesics in the Wasserstein spaces and the Benamou-Brenier method for numerical computations.
Lecture VR-4 Introduction to gradient flows in metric spaces; the minimization scheme for the Heat and Fokker-Planck equation.
Lecture VR-5 Heat and Fokker-Planck equations as gradient flows in the Wasserstein space (convergence of the scheme), porous media, chemotaxis and other population dynamics PDEs.


Due to the bi-location of this course, streaming videos are available for people interested.


All the material covered in the course is contained in the recent book Optimal Transport for Applied Mathematicians (see here or here for a non-official version).
TN-1 : Sections 1.1, 1.2 and 1.6
TN-1 : Sections 1.3, 2.1 and 2.2
TN-3 : Sections 6.4.1 and 6.4.2
TN-4 : Sections 4.2 and 4.4.1
VR-1 : Sections 5.1 and 5.2
VR-2 : Section 5.3
VR-3 : Sections 5.4 and 6.1
VR-4 : Sections 8.1, 8.2 and 8.3
VR-5 : Sections 8.3 and 8.4.2
Of course, there are more classical references: the first book by Cédric Villani Topics in Optimal Transportation (Am. Math. Soc., GSM, 2003) on the general theory, and Gradient Flows in Metric Spaces and in the Space of Probabiliy Measures, by Luigi Ambrosio, Nicola Gigli and Giuseppe Savaré (Birkhäuser, 2005).