Positive scalar curvature; lectures at INdAM, Universita La Sapienza, Roma. March 2021

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.

Organisation of the talks

There are 8 lectures of 90 minutes each.

  1. First lecture: Survey I, introducing the main results to be discussed in the course.
  2. Second lecture: Survey II, introducing results on the space of metrics of positive scalar curvature. Introducing the philosophy of primary and secondary index theory (in K-theory of C*-algebras).

    The pdf of the presentation of Talk 2.

  3. Third lecture: The fundamentals of Clifford geometry, the Dirac operator, and its index.

    The pdf of the presentation of Talk 3.

  4. Lecture 4: The spectral picture of K-theory of C*-algebras.

    The pdf of the presentation of Talk 4.

Part 2: September 2021

The second part of the course will take place September 2021. We have 4 additional lectures:

Link to access the meeting online.

Supplementary material