Introduction to Optimal Transport and Applications - ι

Doctoral course at Università di Roma Tor Vergata.

Practical Information

This a remote course offered on the Teams platform of Roma Tor Vergata, see here for connection details.

Duration: 12h
Schedule: Monday 11am-1pm, Wednesday 4.30pm-6.30pm, Thursday 11.30am-1.30pm, on the weeks of May 17 and May 24. Each class will approximately start 15' after the official starting time (Italian academic quarter).
Examination: students interested in passing an exam should tell me, and schedule and form will be fixed according to the number of students and their level (master, PhD,...).
Prerequisites: some functional analysis and some knowledge of basic PDEs (the main points will be recalled).

Program

There will be 6 classes of 2h each. The program (after small modifications) is more or less the following.

  • Lecture 1 Monge and Kantorovich problems, existence of optimal plans, dual problem, and existence of Kantorovich potentials.
  • Lecture 2 Strong duality (inf sup = sup inf). Cyclical monotonicity and the 1D case. Existence of optimal maps, the quadratic case, and the Monge-Ampère equation. Application to the isoperimetric inequality.
  • Lecture 3 Optimal transport for the distance cost. Wasserstein distances. Curves in the Wasserstein spaces and relation with the continuity equation.
  • Lecture 4 Geodesics in the Wasserstein spaces. Proof of the characterization of AC curves in the Wasserstein spaces. The Benamou-Brenier method for numerical computations.
  • Lecture 5 Introduction to gradient flows in metric spaces; the JKO minimization scheme for some parabolic equation: porous media, chemotaxis, crowd motion...
  • Lecture 6 Heat and Fokker-Planck equations as gradient flows in the Wasserstein space (convergence of the scheme). Geodesic convexity and applications to gradient flows.

    • References:

    All the material covered in the course is contained in the recent book Optimal Transport for Applied Mathematicians (OTAM, see here or here for a non-official version). Precise references for each class will be provided after each lecture.
    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.

    Lecture 1: see Sections 1.1, 1.2 and the beginning of Section 1.3 of OTAM.
    Lecture 2: see Sections 1.3, 1.4, 1.6.1, 1.6.3, 2.2 and 2.5.3 of OTAM.
    Lecture 3: see Sections 3.1.1, 4.1.2, 5.1, 5.2 and the beginning of Section 5.3 of OTAM.
    Lecture 4: see Sections 5.3, 5.4 and some ideas about 6.1 of OTAM.
    Lecture 5: see Sections 8.1, 8.2 and 7.2.2 of OTAM.
    Lecture 6: see Sections 8.3, 7.3.1, 7.3.2 of OTAM. See also Section 4 in the Gradient Flow survey.