Noncommutative geometry and physics

In this very short essay we shall describe a "spectral" point of view on geometry which allows to start taking into account the lessons from both renormalization and of general relativity. We shall first do that for renormalization and explain in rough outline the content of our recent collaborations with Dirk Kreimer and Matilde Marcolli leading to the universal Galois symmetry of renormalizable quantum field theories provided by the renormalization group in its cosmic Galois group incarnation. As far as general relativity is concerned, since the functional integral cannot be treated in the traditional perturbative manner, it relies heavily as a "sum over geometries" on the chosen paradigm of geometric space. This will give us the occasion to discuss, in the light of noncommutative geometry, the issue of "observables" in gravity and our joint work with Ali Chamseddine on the spectral action, with a first attempt to write down a functional integral on the space of noncommutative geometries.