  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.

Quantum Algebra;Operator Algebras