Oscar Randal-Williams: Infinite loop spaces and positive scalar curvature

It is well known that there are topological obstructions to a manifold M admitting a Riemannian metric of everywhere positive scalar curvature (psc): if M is Spin and admits a psc metric, the Lichnerowicz–Weitzenböck formula implies that the Dirac operator of M is invertible, so the vanishing of the A-hat genus is a necessary topological condition for such a manifold to admit a psc metric. If M is simply-connected as well as Spin, then deep work of Gromov--Lawson, Schoen--Yau, and Stolz implies that the vanishing of (a small refinement of) the A-hat genus is a sufficient condition for admitting a psc metric. For non-simply-connected manifolds, sufficient conditions for a manifold to admit a psc metric are not yet understood, and are a topic of much current research.
I will discuss a related but somewhat different problem: if M does admit a psc metric, what is the topology of the space of all psc metrics on it? Recent work of V. Chernysh and M. Walsh shows that this problem is unchanged when modifying M by certain surgeries, and I will explain how this can be used along with work of Galatius and myself to show that the algebraic topology of psc metrics on a manifold of dimension at least 6 is "as complicated as can possibly be detected by index-theory". This is joint work with Boris Botvinnik and Johannes Ebert.