  This is a review paper on recent work about the connections between rough
path theory, the Connes-Kreimer Hopf algebra on rooted trees and the analysis
of finite and infinite dimensional differential equation. We try to explain and
motivate the theory of rough paths introduced by T. Lyons in the context of
differential equations in presence of irregular noises. We show how it is used
in an abstract algebraic approach to the definition of integrals over paths
which involves a cochain complex of finite increments. In the context of such
abstract integration theories we outline a connection with the combinatorics of
rooted trees. As interesting examples where these ideas apply we present two
infinite dimensional dynamical systems: the Navier-Stokes equation and the
Korteweg-de-Vries equation.
