Une journée de conférence organisée par les deux groupes thématiques SIGMA (Signal-Image) et MODE (Optimisation) de la SMAI aura lieu à INRIA Paris mardi 30 janvier 2024: huit exposés de recherche sur des thèmes communs. Inscription gratuite et obligatoire.

A five days conference with introductory lectures in the format of Oberwolfach seminars, 19 – 25 November 2023, Mathematisches Forschungsinstitut Oberwolfach.

A two days conference centered on computational optimal transport and its applications, September 14th-15th, University Gustave Eiffel, Paris area.

In this talk, we insist on the concept of nonnegative cross-curvature and its synthetic definition for a general cost on a product space. Then, by using a formal argument we show why one can expect that such a property should be also true for the Wasserstein space. Then, we give examples of cost satisfying this synthetic nonnegative cross-curvature, in particular a new one with the Bures-Wasserstein case. We extend the result to the case of unbalanced optimal transport and show some potential applications.

In this talk, we study the gradient flow with respect to the Wasserstein metric of the Maximum Mean Discrepancy associated with the Coulomb kernel. In this context, we present several sufficient conditions for global convergence of the gradient flow to the unique global minimum. For instance, on closed Riemannian manifolds, we prove that the so-called Polyak-Lojasiewicz condition holds in some cases, resulting in an exponential convergence. To obtain this result, we use standard estimates from potential theory. An other result is the fact that there is no local minimum apart from the global one. This result is proven using flow interchange techniques.

In this talk, we present our use of the conic formulation first proposed by Liero, Mielke and Savaré, in different applications related to unbalanced optimal transport: the first one is regularity of optimal transport, the second is the Camassa-Holm equation which is the fluid dynamic model directly related to unbalanced optimal transport and the metric on metric measures spaces with unnormalized measures.