**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.

- First lecture: Survey I, introducing the main results to be discussed in the course.
- 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).
- Third lecture: The fundamentals of Clifford geometry, the Dirac operator, and its index.
- Lecture 4: The spectral picture of K-theory of C*-algebras.

- Lecture 5. Monday, Sep 13, 14:00-16:00, Aula C Universita La Sapienza (and online)
- Lecture 6. Thursday, Sep 16, 14:00-16:00, Aula C Universita La Sapienza (and online)
- Lecture 7. Monday, Sep 20, 14:00-16:00, Aula C Universita La Sapienza (and online)
- Lecture 8. Thursday, Sep 23, 10:30-12:30, Sala Consiglio Universita La Sapienza (and online)

Link to access the meeting online.

- Notes of a 4-lecture course on index theory given in Cortona
- Very incomplete and preliminary typed notes for the present course
- Program for a related course in Teheran 2008