# Applications of the Atiyah and Singer index theorem

Prof. Bernd Ammann, Zimmer 119, and Karsten Bohlen

## Content of the lecture

This 2-hour lecture continues the lecture from last semester. In the winter term we have proved the Atiyah-Singer theorem using the heat kernel method.
In the summer term we will consider applications of the theorem. We will discuss the Gauss-Bonnet-Chern theorem, consequences in the Kähler case, and other facts associated to spin geometry potentially reaching to recent research projects.
If time admits, we will also discuss several generalisations, e.g.: The index theorem for elliptic operators of arbitrary order by using the "reduction to Dirac" by Baum and Douglas. The L^{2}-index theorem. Enlargeability obstruction to positive scalar curvature. KO-valued index and obstructions in dim 1 and 2 mod 8. The family index theorem and applications to the topology of the space of metrics with positive scalar curvature. The positive mass theorem of general relativity for spin manifolds.
## Recommeded previous knowledge

Atiyah-Singer index theorem in the Chern-Weil formalism.
## Literature

### Books

- John Roe: Elliptic operators, topology and asymptotic methods, first edition, Pitman Research Notes in Mathematics Series 179, Longman Scientific and Technical, MathSciNet link.
- John Roe: Elliptic operators, topology and asymptotic methods, second edition, CRC Research Notes in Mathematics 395, Chapman and Hall, MathSciNet link.
- Lawson, Michelsohn: Spin Geometry; Princeton Math. Series, 38. Princeton University Press,
MathSciNet link
- Berline, Getzler, Vergne: Heat kernels and Dirac operators, Springer Verlag
- O. Hijazi: Spectral properties of the Dirac operator and Geometrical
structures, Geometric methods for quantum field theory (Villa de Leyva, 1999),
116–169, World Scientific, 2001,
MathSciNet link.
- T. Friedrich: Dirac-Operatoren in der Riemannschen Geometrie, Vieweg,
MathSciNet link.
- J.-P. Bourguignon, O. Hijazi, J.-L. Milhorat, A. Moroianu, S.Moroianu,
A spinorial approach to Riemannian and conformal geometry,
EMS Monographs in Mathematics, 2015,
MathSciNet-Link.
Available as E-Book of our library.
- P. Gilkey: Invariance Theory, the heat equation and the
Atiyah-Singer index theorem
MathSciNet link to the second edition and MathSciNet link to the first edition

### Lecture notes (Diverse Vorlesungsskripte)

## Place and Time

Friday, 8-10, M009

## Recommended Links

## Related web sites

Bernd Ammann, 29.11.2016 oder später