QLunch: Charles-Philippe Diez

Speaker: Charles-Philippe Diez, University of Luxenbourg

Abstract: In a recent breakthrough, Fathi (2018) established a sharp symmetrized improvement of Talagrand’s transport–entropy inequality for the Gaussian measure, unveiling a deep connection with the functional Blaschke–Santaló inequality in convex geometry. This duality resonates with the structure of Kähler–Einstein equations and the moment measure problem studied by Cordero-Erausquin and Klartag (2013), where convexity, optimal transport, and complex geometry converge.

In this talk, I present a free probabilistic analogue of Fathi’s result: a sharp symmetrized free transport–entropy inequality for the semicircular law. Along the way, several inequalities and formulations inspired by convex geometry naturally emerge in the free setting. The one-dimensional case is treated via a novel approach relying on the theory of free moment maps introduced by Bahr and Boschert (2023), and a reverse free log-Sobolev inequality, obtained through a free analogue of the Blaschke–Santaló inequality. We are also able to show that our free symmetrized inequality implies the free Blaschke–Santaló inequality, though the converse direction remains open.

In higher dimensions, the inequality is established for the microstates version of free entropy via random matrix approximations and large deviation techniques, combined with Fathi’s inequality on Euclidean spaces. If time allows, we will also discuss an inverse form of the Blaschke–Santaló inequality and its free counterparts, which are closely connected to the famous Mahler conjecture in convex geometry.