Date/heure
15 avril 2026
16:45 - 17:45
Oratrice ou orateur
Aymeric Martin (Université de Bordeaux)
Catégorie d'évènement Séminaire des doctorants
Résumé
The Wasserstein space $\mathscr{P}(M)$ associated with a closed Riemannian manifold is defined as the space of probability measures on the manifold, endowed with the so-called Otto metric, which provides it with the structure of a formal infinite-dimensional Riemannian manifold. In this talk I will describe the geometric features of this space, emphasizing its connections with optimal transport theory and some classical PDEs. I will then introduce the group of diffeomorphisms $\mathscr{D}(M)$, viewed as an Inverse Limit Hilbert Lie group, and present the Riemannian submersion structure that relates $\mathscr{D}(M)$ and $\mathscr{P}(M)$. The space $\mathscr{P}_\infty(M) \subset \mathscr{P}(M)$ of smooth positive measures is of particular interest. The geodesic convexity of such a space highly depends on the geometry of the base manifold. I will review some significant developments on this topic, mainly due to Ma, Trudinger, Wang, Loeper and Villani. If time permits, I will try to introduce some of the topics of my PhD thesis which focuses on the study of random paths on $\mathscr{P}(M)$ and its tangent bundle $T\mathscr{P}(M)$.