Thomas Schick, Mathematisches Institut, Universität Göttingen
In this course, we address the question if a given smooth manifold M admits a Riemannian metric with positive scalar curvature. If so, we also introduce the tools to classify all such metrics.
One principal tool to invistigate these problems is the Schrödinger-Lichnerowicz formula, which connects these problems to index theory — more specifically to the index of the Dirac operator on a spin manifold. A decisive role in the calculation of these invariants is played by the Atiya-Singer index theorem and its higher variants, and also by its relative version. The latter involves also ideas around the Atiyah-Patodi-Singer index theorem. It is crucial to use “higher” variants of these invariants to get optimal information In particular, we will use the K-theory of C*-algebras and index obstructions in these K-theory groups.
For the classification results we rely on the Atiyah-Patodi-Singer index theorem and the calculation of eta-invariants.
If time permits, we will also discuss briefly obstructions to positive scalar curvature due to Schoen and Yau, using the geometry of stable minimal hypersurfaces and geometric measure theory.
There are 8 lectures of 90 minutes each.
Link to access the meeting online.