Index theory and spin geometry.  Masters degree Ruhr-Universität Bochum, Winter semester  2025/26

News: I've created also a Moodle page of the course for RUB students.

Lecturer

Roberto Ladu 

Abstract

The Atiyah-Singer index theorem is arguably one of the most important mathematical achievements of the last century. It establishes a deep relation between differential operators on manifolds and the topology of the underlying vector bundles/manifold. The aim of this course is to present the heat kernel proof, due to Getzler, of the Atiyah-Singer index theorem. Much of the course will be devoted to cover the necessary analytical, geometrical and topological background.

Syllabus 

• Elliptic operators on manifolds, Hodge theory.

• Principal bundles, connections and curvature, characteristic classes via Chern-Weil theory.

• Spin structures on manifolds, Dirac operators, Bochner technique. 

• The Atiyah-Singer index theorem for Dirac operators proved via heat kernel method.

• Applications of the index theorem, depending on time and student interest we can discuss the K-theoretic proof. 

Bibliography

I will provide my notes covering what we see in class, the source material are the following books:

1. Lawson,Michelsohn. Spin geometry.

2. Beline,Getzler,Vergne. Heat Kernels and Dirac Operators.

3. Roe. Elliptic operators topology and asymptotic methods.

4. Friedrich. Dirac Operators in Riemannian Geometry.

5. Kobayashi-Nomizu. Fundations of Differential Geometry vol 1 and 2.

Prerequisites 

It is necessary to have followed an introductory course in Differential Geometry such as Differentialgeometrie I. 

We will introduce in itinere (without proofs) what we need from elementary Algebraic Topology and Functional Analysis so it's not mandatory to have taken a class on such topics.