Exact Results in Quantum Theory & Gravity

In the talk, I discuss how the renormalization group stems from the S-matrix in the perturbative quantum field theory. Then I turn to the complementary Wilsonian approach of integrating out the high momenta modes in the path integral. I illustrate it on the 1/r2 potential model in quantum mechanics. I briefly review the functional renormalization group techniques in the context of asymptotic safety in quantum gravity. I discuss when the renormalization scale can be identified with the energy scale of the process. I propose criteria for running in terms of amplitudes. Finally I show that string theory satisfies those criteria.