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