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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The Langevin diffusion process is obtained by perturbating an euclidean gradient flow with a random noise. On a compact set, its law converges to the uniform law, but at what speed ? 

The framework of $\Gamma$-calculus introduced by Bakry and Emery is a powerful tool to study diffusion processes, such as the Langevin process. It introduces the "carré du champ" operator, which roughly speaking measures how far the infinitesimal generator of the process is from being a derivation.

Under curvature-dimension conditions, this framework allows the derivation of Poincaré inequalities and logarithmic-Sobolev inequalities and their consequences, such as the speed of convergence of diffusion processes. Even though curvature-dimension conditions are not necessary to obtain these inequalities, they are significant in the Wasserstein space. Indeed, they characterize the convexity of the relative entropy in this geometry, as shown by Sturm and Von-Renesse. A striking consequence for functional inequalities is the HWI inequality, proven by Otto and Villani.

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.