# Introduction to Optimal Transport and Applications - ι

### Practical Information

**Duration:** 16h

** Schedule:** 2h on Monday afternoon (4-6pm), from Feb 6th to Mar
20th, 2016.

**Where:** Imperial College, room 6m42. Classes will be live
broadcasted and accessible in Bath, Bristol, Oxford and Warwick.

**Examination:** according to the number of students who want to
take it, schedule and form to be fixed.

** Enrollment:** students should register by emailing
graduate.studies@maths.ox.ac.uk regardless of whether they intend on
taking the course for assessment. Also, writing me an email is
appreciated, in order to have a list of attending student.

**Prerequisites:** some functional analysis and some knowledge of
basic PDEs (the main points will be recalled).

**News:** The program has been rescheduled on a total of 8 lessons.

### Program

There will be 8 classes of 2h each, tentatively organized as follows.

* Lecture 1 (6/2)* Monge and Kantorovich problems, existence
of optimal plans, dual problem, and existence of Kantorovich potentials.
* Lecture 2 (13/2)* The example of the marriage
market. Strong duality (inf sup = sup inf). Existence of optimal maps and the Monge-Ampère equation. The quadratic and the distance cases.
* Lecture 3 (20/2)* Optimal flow formulation of the optimal transport problem. Traffic congestion variants.
* Lecture 4 (27/2)* Wasserstein distances, curves in the Wasserstein spaces and connections with the continuity equation.
* Lecture 5 (6/3)* Geodesics in the Wasserstein spaces and the Benamou-Brenier method for numerical computations.
* Lecture 6 (13/3)* Introduction to gradient flows in metric
spaces; the JKO minimization scheme for some parabolic
equation. Geodesically convex functionals.
* Lecture 7 (20/3)* Heat and Fokker-Planck equations as
gradient flows in the Wasserstein space (convergence of the
scheme).
* Lecture 8 (27/3)* Porous media, chemotaxis and other
population dynamics PDEs. Estimates by using the JKO scheme.
### 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), and in particular Chapter 1, Sections 4.2 and 4.4.1, Chapter 5, Section 6.1, Chapter 8.

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).

For the last lectures, on gradient flows, useful material is also
contained in a survey I recently wrote, accessible here.

Precise references class by class:

6/2: OTAM, sections 1.1 +1.2

13/2: OTAM, sections 1.3, 1.6.1, 1.6.3, 1.7.3

20/2: OTAM, sections 4.1, 4.2, 4.3, 4.4.1

27/2: OTAM, sections 5.1, 5.2, 5.3.1, 5.3.2, 5.3.3

6/3: OTAM, sections 5.3.4, 5.4, 6.1

13/3: Survey, sections 2.1, 4.3, 4.5

### Exercises

Exercises are available at the end of the book OTAM. A list of
exercises related to each class will be given later on.