Interacting UV fixed points — from quantum field theory to quantum gravity

It is a fascinating idea that all the physics as we know it could arise as a small perturbation of some UV conformal field theory. This necessitates that the running couplings of the underlying quantum field theory achieve a fixed point in the UV, such as in asymptotic freedom. In this talk, I discuss the prospects for QFTs to develop interacting UV fixed points, covering weakly and strongly coupled 4d QFTs, perturbatively non-renromalisable theories such as 3d fermionic QFTs, and 4d quantum gravity. I will give an overview of rigorous results and methodologies, and address open challenges for quantum gravity.