Séminaire des doctorants
Wasserstein Gradient Flows
18
Feb. 2026
Feb. 2026
| Intervenant : | Quentin Giton |
| Heure : | 14h00 - 15h00 |
| Lieu : | 2L8 |
Le séminaire des doctorants se propose de fournir aux doctorants une occasion de s'ouvrir aux autres domaines des mathématiques que le leur. A chaque séance, un intervenant réalise un exposé sur un fait standard de leur domaine d'étude, de niveau adapté à l'ensemble des doctorants.
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Quadratic optimal transport equips the space of probability measures with the Wasserstein distance $W_2$, a geometry governed by the cost of moving mass rather than pointwise comparison of densities (the $L^2$ geometry). After a brief reminder on Monge and Kantorovich formulations, I will explain heuristically how one can think of "steepest descent" of an energy over the space of probability measures, thus defining a notion of gradient: the heat equation and the Fokker--Planck equation appear as Wasserstein gradient flows of entropy and relative entropy, respectively.
In a second step, I will show that these formulas are not accidental by unveiling the underlying (formal) Riemannian structure hidden in the optimal transport problem. If time allows, I will conclude with a glimpse on the final part of this tetralogy: how convexity of the Boltzmann entropy along Wasserstein geodesics connects to curvature of the base space and functional inequalities.
The PhD students seminar aims to provide PhD students with an opportunity to explore other areas of mathematics beyond their own. At each session, a speaker gives a presentation on a standard topic in their field of study, at a level suitable for all doctoral students.