Speaker: Yoann Dabrowski (Université Lyon 1)

Title: New applications of random matrices and free processes to free entropy

Abstract: First, we will explain how the use of classical entropy of random matrices instead of volumes of microstates enables us to solve old problems for microstate free entropy. Especially we produce a concavification of Voiculescu's free entropy and we extend orbital free entropy to not necessarily hyperfinite multivariables. As a consequence we solve an old problem for microstate free entropy, in proving that additivity of microstate free entropy implies freeness (in the block multivariable case). This part of the talk comes from a joint work with P. Biane.

Second, we will explain how several problems for non-microstate free entropy can be attacked in thinking in terms of conjugate variables of processes and in terms of the free analogues of relative entropy of a process with respect to Wiener measure. These problems contain changes of variables for non-microstate free entropy, free LSI with respect to convex potentials, inequalities with mutual information or microstate variants. We will explain our progresses in completing this program.