# Special Topics: Atiyah-Singer Index Theorem (MATH 666, Spring 2023)

## *Lecture Notes*

*Lecture 1*
*Lecture 2*
*Lecture 4*
*Lecture 5*
*Lecture 6*
*Lecture 7*
*Lecture 8*
*Lecture 9*
*Lecture 10*
*Lecture 11*
*Lecture 12*
*Lecture 13*
*Lecture 14*
*Lecture 15*
*Lecture 16*
*Lecture 17*
## *An Unofficial Syllabus (updated 01/11/2023)*

We will give an introduction to the famous Atiyah-Singer index theorem, originally proved by Atiyah and Singer in 1963. This theorem, arguably the most important mathematical theorem proved in the 20th century, provides a bridge between two main fields of mathematics: topology and analysis. It also unifies many celebrated theorems in geometry and topology, such as the Gauss-Bonnet-Chern theorem, Hirzebruch's Signature Theorem, the Hirzebruch-Riemann-Roch theorem, etc. The Atiyah-Singer index theorem also plays an important role in supersymmetric quantum field theories.

### Basic Information

Office Hour: by appointment
Office: Blocker 623A
### Basic Coverage

Basic facts about Fredholm operators and Fredholm index
Elliptic differential operators
A sketch of the K-theory proof of the index theorem (?)
Basic Riemannian geometry
Heat kernel asymptotic expansion
Clifford algebra and Dirac operators
Local index theorem
### References

**Boose and Bleecker** *Topology and Analysis: The Atiyah-Singer Index Formula and Gauge-Theoretic Physics*
**Berline, Getzler, and Vergne** *Heat kernels and Dirac operators*
**Lawson and Michelsohn** *Spin geometry*