  The purpose of this report is to acknowledge the influence of M. Gromov's
vision of geometry on our own works. It is two-fold: in the first part we aim
at describing some results, in dimension 3, around the question: which open
3-manifolds carry a complete Riemannian metric of positive or non negative
scalar curvature? In the second part we look for weak forms of the notion of
"lower bounds of the Ricci curvature" on non necessarily smooth metric measure
spaces. We describe recent results some of which are already posted in
[arXiv:1712.08386] where we proposed to use the volume entropy. We also attempt
to give a new synthetic version of Ricci curvature bounded below using
Bishop-Gromov's inequality.
