Abstract: In this sequence of talks, I will describe our work with Alexander Frenkel and Stephen Randall, in which we presented a novel topological quantum gravity, relating three previously unrelated fields:  Topological quantum field theories (of the cohomological type), the theory of Ricci flows on Riemannian manifolds, and nonrelativistic quantum gravity.  The remarkable richness of results produced in the recent decades by mathematicians studying the Ricci flow promises to shed new light on the physics of the path integral in quantum gravity (at least in the topological regime).  In the opposite direction, the techniques of quantum field theory and path integrals may end up putting some of the mathematical results in the Ricci flow theory in a new perspective as well.