More particularly, I was interested in the log-Sobolev inequality on the simplest non-trivial Carnot group: the Heisenberg group. Since the standard technique of -calculus could not work and that direct computations do not lead to an optimal constant, I adapted an argument of L. Gross based on an exact log-Sobolev inequality on the two-points space. The non-commutativity yields a weighted gradient.
N. Gozlan and G. Peccati offered me to keep on working in the field of functional inequalities as a PhD student co-supervised by the two of them. Roughly speaking, I try to understand the geometry of (eventually non-smooth) spaces (graphs, infinite dimensional spaces from probability theory, sub-Riemannian manifolds). To that extend, I study:
the classical (Gaussian) and less classical (Poisson, Rademacher) infinite dimensional stochastic analysis;
the geometry of Markov generators and their chaos;
optimal transport and functional inequalities (concentration, log-Sobolev, isoperimetric, super-concentration);
geometric measure theory and metric differential calculus (heat equation on metric spaces).
Since I arrived in Luxembourg, I also gained interest, at the heuristic level, in free probability and the geometry used in physical mathematics.