# 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.

### Program

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.

### Streaming

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

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).