1st Talk: Continuous time Principal Agent and optimal planning

• Nizar Touzi (Ecole Polytechnique)

Room: Salle de conférence 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.

2nd Talk : A counter-example to the Cantelli conjecture

• Victor Kleptsyn (Université de Rennes)

Room: Salle de conférence 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.

3rd Talk: The Monge problem in multi-marginal optimal mass transportation

• Anna Kausamo (University of Jyväskylä)

Room: Salle de conférence 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

4th talk: Weak optimal transport and applications to Caffarelli contraction theorem.

• Nathaël Gozlan (Université Paris 5)

Room: Salle de conférence 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.

5th talk : Optimal transport planning with a non linear cost

• Thierry Champion (Université de Toulon)

Room: Salle de conférence 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.

6th talk : Monge-Kantorovich problem for n-dimensional measures with fixed k-dimensional marginals

• Nikita Gladkov (University of Moscow)

Room: Salle de conférence 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.

7th talk : Fine Properties of the Optimal Skorokhod Embedding Problem.

• Marcel Nutz (Columbia U. New York)

Room: Salle de conférence 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.

8th talk : The inverse transform martingale coupling

• Benjamin Jourdain (Université Paris-Est)

Room: Salle de conférence 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.

9th talk : Seidl conjecture in Density Functional Theory: results and counterexamples

• Simone Di Marino (Indam)

Room: Salle de conférence 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.