# PhD Course: The Atiyah–Singer Index Theorem

**Time: **
Mon 2022-01-24 15.15 - 17.00

**Location: **
Zoom

**Video link: **
Meeting ID: 657 9577 6516

**Participating: **
Mattias Dahl

### Brief description

The Atiyah–Singer index theorem provides a fundamental connection between differential geometry, partial differential equations, differential topology, operator algebras, and has links to many other fields.

The fundamental observation is that the index of Fredholm operators is a homotopy invariant. Examples of such Fredholm operators are elliptic differential operators on manifolds. The Atiyah–Singer index theorem tells us how the analytic index of an elliptic operator is computed in terms of topological data of the underlying manifold. Since elliptic operators on manifolds often appear related to geometric structures on the manifold, the theorem allows for deep conclusions about the geometry and topology of manifolds. Classical results as The Gauss–Bonnet, Hirzebruch, and Riemann–Roch theorems can be concluded as special cases and will be covered in the course.

The goal of this graduate level course is to understand the Atiyah–Singer index theorem and its applications. The goal is also to give a background in the different areas of mathematics involved, such as functional analysis; elliptic differential operators on manifolds; Clifford algebras, spinors, Dirac operators; vector bundles and characteristic classes.

This is a “broad graduate course” which is supposed to be accessible and meaningful for all graduate students in mathematics.

See course web page for full details.