Research

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. One way to phrase it is “What is the homotopy type of the space of psc metrics?”. A central tool 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. ...

©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. ...

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. ...

Abstract: This semester we have had several talks on positive scalar curvature already and in this talk we will explore yet another aspect: the question of uniqueness of psc 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 surjective on homotopy groups if the dimension of the manifold is at least 6. ...