Algebraic Combinatorics, Renormalization, Free probability and Operads – CARPLO

The aim of the project is to explore relations between combinatorial Hopf algebras (CHAs) and problems in physics (renormalization), algebra and topology (operads), control theory (Fliess operators) and probability (free probability, random walks). After the seminal work ofConnes and Kreimer the theory of CHAs and Rota-Baxter algebras experienced a profound renewal. Recently, CHAs arised in the study of stochastic differential and partial differential equations. Hairer's theory of regularity structures has revealed various CHAs and renormalization groups that are still partially understood. CHAs also appeared in Control Theory (Fliess operators, Faà di Bruno type groups). Interactions with dynamical systems include linearization of vector fields with the help of Connes-Kreimer Hopf algebras. Connections with probability theory include the combinatorial approach to free probability and the works of Diaconis et al. on random walks on Hopf algebras.