Title: The Tameness of Quantum Field Theories

Abstract: Tameness is a generalized notion of finiteness that is restricting the geometric complexity of sets and functions. The underlying mathematical foundation lies in tame geometry, which is built from o-minimal structures introduced in mathematical logic. In this talk I formalize the connection between quantum field theories and logical structures and argue that the tameness of a quantum field theory relies on its UV definition. I quantify our expectations on the tameness of effective theories that can be coupled to quantum gravity and on CFTs. In particular, I present tameness conjectures about CFT observables and propose universal constraints that render spaces of CFTs to be tame sets. I then highlight the relation of these conjectures to other swampland conjectures, e.g., by arguing that the tameness of CFT observables restricts having parametrical gaps in the operator spectrum.