Topological ideas have at various times played an important role in condensed matter physics. This year's Nobel Prize recognized the origins of a particular application of great current interest: the classification of phases of a quantum mechanical system. Mathematically, we would like describe them as path components of a moduli space, but that is not rigorously defined as of now. In joint work with Mike Hopkins we apply
stable homotopy theory (Adams spectral sequence) to compute the group of
topological phases of "invertible" systems. We posit a continuum field
theory and then use the Axiom System for field theory initiated by Segal and
Atiyah, and various refinements, to prove a theorem which underlies the
computations.