Abstract:
Given a smooth manifold, one can ask whether there is a metric of positive scalar curvature on it. The study of this existence question has produced many obstructions. The natural follow-up question is uniqueness of positive scalar curvature metrics. Phrased properly this is the study of the homotopy type of the space of psc metrics. One of the main tools used in its study is the index difference by Hitchin, which provides a map to $K$-theory. This map is known to be non-trivial on homotopy groups if the dimension of the manifold is at least 6.
In my talk I will explain the methods used to prove this non-triviality result and present enhancements, which allow for a simultaneous extension to dimension 5 and incorporation of the fundamental group (yielding an index difference mapping to the $K$-theory of group $C^*$-algebras). These enhancements use the original Madsen–Weiss theory and its extension by Perlmutter in contrast to methods of Galatius–Randal-Williams in earlier results.
The slides are available here.