©Adriano Córdova
Abstract:
Scalar curvature is a purely geometric notion by definition. But it turns out, that there are topological obstructions to admitting a Riemannian metric of positive scalar curvature (psc). These obstructions can tell you a lot about the question of existence of psc metrics.
In my talk I want to focus on the natural follow-up question of uniqueness. This is correctly phrased as studying the homotopy type of the space of positive scalar curvature metrics. In particular I want to explain how methods of geometric topology pioneered by Madsen–Weiss and Galatius–Randal-Williams enter into a proof of nontriviality of these spaces. This will contain some new improvements by myself, that allow the incorporation of the fundamental group via group $C^*$-algebras and higher index theory, while also dealing with the 5-dimensional case, that escaped previous methods.
The slides are available here.