Explore a 46-minute lecture on substitutions in non-commutative multivariate power series presented by Frédéric Patras from CNRS/Université Côte d'Azur at the Institut des Hautes Etudes Scientifiques (IHES). Delve into the description of a group law on formal power series in non-commuting variables, induced by their interpretation as linear forms on a Hopf algebra of sentences. Examine the corresponding structures and discover how they can be applied to reframe various identities and transformations on formal power series in a group theoretic form, which have been crucial in non-commutative probability theory. Learn about the joint work with K. Ebrahimi-Fard, N. Tapia, and L. Zambotti that forms the basis of this presentation.