By Jean Dolbeault, University of Paris-Dauphine

Nonlinear diffusion flows are very interesting tools for the study of sharp functional inequalities, the stability of the optimal functions and for improvements of the inequalities. The lecture will be devoted to a series of recent results which unify some rigidity methods in nonlinear elliptic equations, the Bakry-Emery method and various qualitative features related with functional inequalities in probability theory, information theory and nonlinear analysis.