Averaging, renormalization group and criticality in cosmology

Several problems in physics, in particular the averaging problem in gravity, can be described in a formalism derived from the real-space Renormalization Group (RG) methods. It is shown that the RG flow is provided by the Ricci-Hamilton equations which are thereby provided with a {\it physical} interpretation. The connection between a manifold deformation according to these equations and Thurston’s conjecture is exhibited. The significance of criticality which naturally appears in this framework is briefly discussed. This article summarizes also recent work with M. Carfora. Moreover, a report on some work in progress is given and some open issues in the averaging problem pointed out.