Conference

©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. ...

This was a byproduct of my Master’s thesis. Abstract Atiyah’s $KR$-theory from his “$K$-Theory and Reality” paper (1966) establishes a unified view of complex and real $K$-theory and in particular gives a quite elegant proof of the periodicity theorem for $KO$. In this expository talk I will try to give a quick overview of the basic properties of this theory and afterwards focus on the proof of Bott periodicity in the real case via $KR$-theory. Contrary to the approach of the original paper we will also encounter a proof that the Atiyah–Bott–Shapiro homomorphism is an isomorphism during this discussion. ...