The goal of this special week is to illustrate the recent progress in the understanding of optimal transportation as well as the diversity of their applications to PdE's, optimization, functional analysis, geometry, probability or statistics. The workshop features two mini-courses by L. Ambrosio and N. Gozlan, lectures by J. Bertrand, Y. Brenier, G. Carlier, T. Del Barrio, E. Oudet and a colloquium talk by L. Ambrosio. It will also provide plenty of free time for informal discussions.

Week partially supported by the ANR GeMeCod.

L'objectif ce cette semaine thématique est d'illustrer les progrès récents de la compréhension du transport optimal et la variété de leurs applications en EdP, optimisation, analyse fonctionnelle, géométrie, probabilités ou statistiques. La semaine est organisée autour de mini-cours de L. Ambrosio et de N. Gozlan. Elle proposera aussi des exposés de J. Bertrand, Y. Brenier, G. Carlier, T. Del Barrio, E. Oudet, un exposé de colloquium de L. Ambrosio, ainsi que des plages de temps libre pour les discussions informelles.

The welcome coffee will take place in the lobby of 1R3 and the talks in the seminar room (First floor 1R3).

### Speakers

- Luigi Ambrosio (SNS Pisa)
- Jérôme Bertrand (IMT, Université Paul-Sabatier)
- Yann Brenier (École Polytechnique)
- Guillaume Carlier (CEREMADE, Université Paris-Dauphine)
- Eustasio Del Barrio Tellado (Dpt de Estadística e Investigación Operativa, Univ. de Valladolid)
- Nathael Gozlan (LAMA, Université Paris-Est)
- Edouard Oudet (LJK, Université Joseph Fourier)

### Mini-courses

#### Luigi Ambrosio (Scuola Normale Superiore Pisa) - Sobolev vector fields and well posedness of the continuity equation and flows in abstract metric measure spaces

I will illustrate, with a reasonable number of details, my recent work with D.Trevisan. In this work we revisit the Di Perna-Lions theory of almost sure well posedness for ODE's associated to Sobolev Euclidean vector fields, providing a counterpart of the theory also in metric measure spaces. In this context, of course, the notions ``Sobolev vector field'' and ''solution to the ODE'' have to be properly understood. For the first one we achieve the goal using Gamma-calculus tools (and so the natural context will be Dirichlet forms, Markov semigroups, Carre du Champ operators) for the second one we use ideas coming from Optimal Transport and Geometric Measure Theory, the so-called superposition principle.Slides.

#### Nathael Gozlan (LAMA, Université Paris Est Marne-la-Vallée) - Some applications of optimal transport in functional inequalities

This course is devoted to some applications of optimal transport in the field of functional inequalities. After having recalled some now classical transport proofs of different well known inequalities (log-Sobolev, HWI) we will present recent developpements around Poincaré type and transport-entropy inequalities.First talk

Second talk

Third talk

### Talks

#### Jérôme Bertrand (IMT, Université Paul-Sabatier) - Prescribing Gauss curvature using mass transport

In this talk, I will give a proof of Alexandrov’s theorem on the Gauss curvature prescription of Euclidean convex body. The proof is mainly based on mass transport. In particular, it doesn’t rely on pdes method nor convex polyhedra theory. To proceed, I will discuss generalizations of well-known results for the quadratic cost to the case of a cost function which assumes infinite values.#### Yann Brenier (CNRS, CMLS, Ecole Polytechnique) - Gradient flows based on optimal transport of closed differential forms

Gradient flows based on optimal transport of volume forms in dimension $n$ are now well understood. We will consider some possible generalizations to closed $k‹n$ differential forms. We will focus on the particular cases $k=1$ and $k=n-1$ in connection with fluid mechanics and magnetohydrodynamics.#### Guillaume Carlier (CEREMADE, Université Paris-Dauphine) - Multi-population matching, Wasserstein barycenters, theory, numerics and applications

Minimizing a weighted sum of squared Wasserstein distances to prescribed probabilities is a natural way to interpolate between them, generalizing the classical McCann's interpolation.This "Wasserstein barycenter" problem naturally finds applications to image processing, it is also a particular case of multi-population matching arising in mathematical economics. I will explain some properties of such barycenters, give some examples and discuss their numerical computation. This will be based on joint works with I. Ekeland, M. Agueh and A. Oberman and E.Oudet.

Slides

#### Eustasio del Barrio (IMUVA, Universidad de Valladolid) - The empirical cost of optimal incomplete transportation

I consider the problem of optimal incomplete transportation between the empirical measure on an i.i.d. uniform sample on the $d$-dimensional unit cube, $[0, 1]^d$ , and the true measure. This is a family of problems lying in between classical optimal transportation and nearest neighbor problems. It will be shown that the empirical cost of optimal incomplete transportation vanishes at rate $O (n^{1/d} )$, where $n$ denotes the sample size. In dimension $d \ge 3$ the rate is the same as in classical optimal transportation, but in low dimension it is (much) higher than the classical rate. An equivalent problem for the Komogorov metric will be also discussed#### Edouard Oudet (LJK, Université Joseph Fourier) - Approximation of solutions of optimal transport problems: A hybrid method

Both continuous and discrete approaches made considerable progresses last decades in approximating solutions of optimal transportation problems. Several recent improvements are strongly link to a better comprehension of problems under convexity constraint. After recalling some recent approaches for the discretization of this type of constraints, we describe a new hybrid method using both advantages of continuous and discrete formulations to solve efficiently optimal transportation problems. This is joint work with J-D. Benamou , G. Carlier and Q. Mérigot.