The topology of positive scalar curvature
Mathematisch-Physikalisches Kolloquium, 19.12.2017
We discuss aspects of a classical question in geometry: given a fixed topological space (manifold) M, which geometric shape can it take.
Concretely for us: can we equip M with a metric of positive scalar curvature (the easiest curvature expresseion which exists)? This is also the curvatureexpression which determines the cosmological constant in Einsteins generalrelativity.
There is a classical answer to this question for surfaces by the Gauss-Bonnettheorem. This theorem says that the integral of the scalar curvature gives (uptoa positive constant factor) the Euler characteristic. Therefore the onlysurfaces with positive scalar curvarture are those with positive Eulercharacteristic, i.e. the spheres.
We discuss methods to analyze higher dimensional manifolds, and we also addressthe question: if there are metrics of positive scalar curvature, how many ofthem do exist? Does the space of such metrics have different components ...
The main tool will use the analysis of the Dirac operator and of variants from non-commutative geometry.
Prof. Dr. Thomas Schick/ Universität Göttingen
Fakultät für Mathematik und Physik
17:15 Uhr - 19:00 Uhr