Three applications of topology to physics
(1) What is the classification of invertible phases of lattice systems? (2) Is M-theory time-reversal invariant? (3) How to match topological features (spin structures, global anomalies) of QCD with the effective theory of pions? Question (1) lies in condensed matter theory and question (2) in string theory, but we can make a conceptual jump and reformulate them in quantum field theory. In this talk I sketch how an Axiom System for Wick-rotated quantum field theory, coupled with ideas and techniques from geometry and topology, let us address each question. The recent work on (1) and (2) is joint with Mike Hopkins.