Symetries Galoisiennes et Renormalisation
  We first explain our joint work with Dirk Kreimer on the Hopf and Lie algebras of Feynman graphs. The conceptual meaning of the concrete computations of perturbative renormalisation is obtained from the Birkhoff decomposition in the Riemann-Hilbert problem. The relation of the Hopf algebra of graphs with the group of formal diffeomorphisms of complexified coupling constants allows for a geometric interpretation of the renormalisation procedure. We then discuss the relation between the above occurrence of the Riemann-Hilbert problem, the interpretation of the renormalisation group as a group of ambiguity of physical theories and the still mysterious Galois theory that should account at Archimedian places for the connected component of identity in the Idele class group of class field theory. This paper is the content of a talk given in the memory of Louis Michel in January 2001, and has appeared as such in the Poincare seminar of October 2002 in Paris. 
Alain Connes (College de France and IHES)