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 L²-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

Related web sites

Bernd Ammann, 29.11.2016 oder später