# Eight or Nine Talks on Contemporary Optimal Transport Problems

Europe/Paris
Salle de conférence (IRMA in Strasbourg)

### Salle de conférence

#### IRMA in Strasbourg

7 rue René Descartes 67084 Strasbourg
Description

Defined by Monge and later by Kantorovich, the mass transportation problem aims at finding a transport plan from a source distribution to a target distribution so that it minimizes a global cost. Since this optimal transport plan is kind of a very specific multi-valued function, no wonder that Optimal Transport Theory has a number of applications in mathematics and in science!

Independently from and due to applications, the original problem was recently enriched by a number of variations among which we find "multi-marginal transport'' problems (for instance, the DFT-OT problem), the "weak transport problem'', the "martingale transport'' problem. In these problems the connection with stochastic optimal transport or other transfer problems (balayage, Skorokhod embedding,...) is more apparent than in the classical transport problem.

During the two days we wish to bring together people working on these particular facets of Optimal Transport.

List of speakers:

• Julio BACKHOFF VERAGUAS (Wien - cancelled)
• Thierry CHAMPION (Toulon)
• Simone DI MARINO (Pisa)
• Nathaël GOZLAN (Paris)
• Nikita GLADKOV (Moskvá)
• Benjamin JOURDAIN (Champs-sur-Marne)
• Anna KAUSAMO (Jyväskylä)
• Victor KLEPTSYN (Rennes)
• Marcel NUTZ (New York)
• Nizar TOUZI (Palaiseau)

Registration
Participants
Participants
• ahmed fadili
• Aleksandr Zimin
• Annemarie Grass
• Gudmund Pammer
• Guillaume Grente
• Jacques Franchi
• Marcel Nutz
• Martin Brückerhoff-Plückelmann
• Mathias Beiglböck
• Nicolas Juillet
• Rafael Coyaud
• Sabri Khadidja
• Simone Di Marino
• Thomas Skill
• Vlada Limic
• William MARGHERITI
• Xiaolin Zeng
• Xiling Zhang
Contact
• Thursday, 4 July
• 10:00 13:00
Arrival & Lunch 3h
• 13:00 13:50
1st Talk: Continuous time Principal Agent and optimal planning 50m

Motivated by the approach introduced by Sanninkov to solve principal-agent problems, we provide a solution approach which allows to address a wider range of problems. The key argument uses a representation result from the theory of backward stochastic differential equations. This methodology extends to the mean field game version of the problem, and provides a connexion with the P.-L. Lions optimal planning problem.

Speaker: Nizar Touzi (Ecole Polytechnique)
• 14:00 14:50
2nd Talk : A counter-example to the Cantelli conjecture 50m

Take two Gaussian independent random variables X and Y, both N(0,1).
The Cantelli conjecture addresses non-linear combinations of the form
Z= X+f(X)*Y, where f is a non-negative function. It states that if Z is Gaussian,
f should be constant almost everywhere.
In a joint work with Aline Kurtzmann, we have constructed a (measurable) counter-example
to this conjecture, with a construction that uses a « Brownian » variation of a transport.
This construction will be the subject of my talk.

Speaker: Victor Kleptsyn (Université de Rennes)
• 15:00 15:30
Break 30m
• 15:30 16:20
3rd Talk: The Monge problem in multi-marginal optimal mass transportation 50m

In this talk I will introduce the concept of Multi-Marginal Optimal Mass Transportation (MOT) with the emphasis on repulsive cost functions. Then I will outline the Monge problem, discuss it's difficulty in the MOT setting, and present some nonexistence results that are joint work Augusto Gerolin and Tapio Rajala

Speaker: Anna Kausamo (University of Jyväskylä)
• 16:30 17:20
4th talk: Weak optimal transport and applications to Caffarelli contraction theorem. 50m

The talk will deal with a variant of the optimal transport problem first considered in a joint paper with C. Roberto, P-M Samson and P. Tetali, where elementary mass transports are penalized through their barycenters. The talk will in particular focus on a recent result obtained in collaboration with N. Juillet describing optimal transport plans for the quadratic barycentric cost. A direct corollary of this result gives a new necessary and sufficient condition for the Brenier map to be 1-Lipschitz. Finally we will present a recent work in collaboration with M. Fathi and M. Prodhomme, where this contractivity criterion is used to give a new proof of the Caffarelli contraction theorem, telling that any probability measure having a log-concave density with respect to the standard Gaussian measure is a contraction of it.

Speaker: Nathaël Gozlan (Université Paris 5)
• Friday, 5 July
• 08:30 09:20
5th talk : Optimal transport planning with a non linear cost 50m

In this talk, I consider optimal transport problems that involve non-linear transportation costs which favour optimal plans non associated to a single valued transport map. I will describe some results concerning this type of problem (existence, duality principle, optimality conditions) and focus on specific examples in a finite dimensional compact setting. I will consider in particular the case where the cost involves the opposite of the variance or the indicator of a constraint on the barycenter of $p$ (martingale transport).
This is from a joined work with J.J. Alibert and G. Bouchitté.

Speaker: Thierry Champion (Université de Toulon)
• 09:30 10:20
6th talk : Monge-Kantorovich problem for n-dimensional measures with fixed k-dimensional marginals 50m

The classical Monge-Kantorovich (transportation) problem deals with measures on a product of two spaces with two independent fixed marginals. Its natural generalization (multimarginal Monge-Kantorovich problem) deals with the products of n spaces X_1, ..., X_n with n independent marginals. We study the Monge-Kantorovich problem on X_1 \times X_2 ... \times ... X_n with fixed projections onto the products of X_{i_1} , ... X_{i_k} for all k-tuples of indices (k<n). On the language of descriptive geometry this can be called "k-dimensional Monge's protocols for n-dimensional bodies". There are both similarities and differences from the classical problem concerning feasibility, uniqueness, smoothness, duality theorem, existence of the dual solution.

Speaker: Nikita Gladkov (University of Moscow)
• 10:30 10:50
Break 20m
• 10:50 11:40
7th talk : Fine Properties of the Optimal Skorokhod Embedding Problem. 50m

We study the problem of stopping a Brownian motion at a given distribution $\nu$ while optimizing a reward function that depends on the (possibly randomized) stopping time and the Brownian motion. Our first result establishes that the set $T(\nu)$ of stopping times embedding $\nu$ is weakly dense in the set $R(\nu)$ of randomized embeddings. In particular, the optimal Skorokhod embedding problem over $T(\nu)$ has the same value as the relaxed one over $R(\nu)$ when the reward function is semicontinuous, which parallels a fundamental result about Monge maps and Kantorovich couplings in optimal transport. A second part studies the dual optimization in the sense of linear programming. While existence of a dual solution failed in previous formulations, we introduce a relaxation of the dual problem and establish existence of solutions as well as absence of a duality gap, even for irregular reward functions. This leads to a monotonicity principle which complements the key theorem of Beiglbock, Cox and Huesmann. These results can be applied to characterize the geometry of optimal embeddings through a variational condition. (Joint work with Mathias Beiglbock and Florian Stebegg)s over the years.

Speaker: Marcel Nutz (Columbia U. New York)
• 11:50 13:10
Lunch 1h 20m
• 13:10 14:00
8th talk : The inverse transform martingale coupling 50m

We exhibit a new martingale coupling between two probability measures $\mu$ and $\nu$ in convex order on the real line. This coupling is explicit in terms of the integrals of the positive and negative parts of the difference between the quantile functions of $\mu$ and $\nu$. The integral of $|y-x|$ with respect to this coupling is smaller than twice the Wasserstein distance with index one between $\mu$ and $\nu$. When the comonotonous coupling between $\mu$ and $\nu$ is given by a map $T$, it minimizes the integral of $|y-T(x)|$ among all martingales couplings.

Speaker: Benjamin Jourdain (Université Paris-Est)
• 14:10 15:00
9th talk : Seidl conjecture in Density Functional Theory: results and counterexamples 50m

The Seidl conjecture in Density Functional Theory is the equivalent of the Monge Ansatz for the classical optimal transport problem with the cost $c(x,y)=|x-y|$, in the multimarginal case with the Coulomb cost. We provide positive results in the one dimensional case as well as both positive and negative results in the radial 2-dimensional case.

Speaker: Simone Di Marino (Indam)