Eight or Nine Talks on Contemporary Optimal Transport Problems
de
jeudi 4 juillet 2019 (10:00)
à
vendredi 5 juillet 2019 (16:00)
lundi 1 juillet 2019
¶
mardi 2 juillet 2019
¶
mercredi 3 juillet 2019
¶
jeudi 4 juillet 2019
¶
10:00
Arrival & Lunch
Arrival & Lunch
10:00 - 13:00
Room: Salle de conférence
13:00
1st Talk: Continuous time Principal Agent and optimal planning
-
Nizar Touzi
(
Ecole Polytechnique
)
1st Talk: Continuous time Principal Agent and optimal planning
Nizar Touzi
(
Ecole Polytechnique
)
13:00 - 13:50
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.
14:00
2nd Talk : A counter-example to the Cantelli conjecture
-
Victor Kleptsyn
(
Université de Rennes
)
2nd Talk : A counter-example to the Cantelli conjecture
Victor Kleptsyn
(
Université de Rennes
)
14:00 - 14:50
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.
15:00
Break
Break
15:00 - 15:30
Room: Salle de conférence
15:30
3rd Talk: The Monge problem in multi-marginal optimal mass transportation
-
Anna Kausamo
(
University of Jyväskylä
)
3rd Talk: The Monge problem in multi-marginal optimal mass transportation
Anna Kausamo
(
University of Jyväskylä
)
15:30 - 16:20
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
16:30
4th talk: Weak optimal transport and applications to Caffarelli contraction theorem.
-
Nathaël Gozlan
(
Université Paris 5
)
4th talk: Weak optimal transport and applications to Caffarelli contraction theorem.
Nathaël Gozlan
(
Université Paris 5
)
16:30 - 17:20
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.
vendredi 5 juillet 2019
¶
08:30
5th talk : Optimal transport planning with a non linear cost
-
Thierry Champion
(
Université de Toulon
)
5th talk : Optimal transport planning with a non linear cost
Thierry Champion
(
Université de Toulon
)
08:30 - 09:20
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.
09:30
6th talk : Monge-Kantorovich problem for n-dimensional measures with fixed k-dimensional marginals
-
Nikita Gladkov
(
University of Moscow
)
6th talk : Monge-Kantorovich problem for n-dimensional measures with fixed k-dimensional marginals
Nikita Gladkov
(
University of Moscow
)
09:30 - 10:20
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.
10:30
Break
Break
10:30 - 10:50
Room: Salle de conférence
10:50
7th talk : Fine Properties of the Optimal Skorokhod Embedding Problem.
-
Marcel Nutz
(
Columbia U. New York
)
7th talk : Fine Properties of the Optimal Skorokhod Embedding Problem.
Marcel Nutz
(
Columbia U. New York
)
10:50 - 11:40
Room: Salle de conférence
We study the problem of stopping a Brownian motion at a given distribution
ν
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
(
ν
)
of stopping times embedding
ν
is weakly dense in the set
R
(
ν
)
of randomized embeddings. In particular, the optimal Skorokhod embedding problem over
T
(
ν
)
has the same value as the relaxed one over
R
(
ν
)
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.
11:50
Lunch
Lunch
11:50 - 13:10
Room: Salle de conférence
13:10
8th talk : The inverse transform martingale coupling
-
Benjamin Jourdain
(
Université Paris-Est
)
8th talk : The inverse transform martingale coupling
Benjamin Jourdain
(
Université Paris-Est
)
13:10 - 14:00
Room: Salle de conférence
We exhibit a new martingale coupling between two probability measures
μ
and
ν
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
μ
and
ν
. The integral of
|
y
−
x
|
with respect to this coupling is smaller than twice the Wasserstein distance with index one between
μ
and
ν
. When the comonotonous coupling between
μ
and
ν
is given by a map
T
, it minimizes the integral of
|
y
−
T
(
x
)
|
among all martingales couplings.
14:10
9th talk : Seidl conjecture in Density Functional Theory: results and counterexamples
-
Simone Di Marino
(
Indam
)
9th talk : Seidl conjecture in Density Functional Theory: results and counterexamples
Simone Di Marino
(
Indam
)
14:10 - 15:00
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.