Noncommutative Geometry (Analysis on Manifolds), Fall 2022


  • Prof. Raphaël Ponge.
  • E-mail: ponge[dot]math[at]icloud[dot]com.

Time and Location:

  • Tuesdays, from 14:00-16:35, Room TBD.
  • The lectures can be attended online. Tencent Meeting link. Meeting ID: 951-5797-7972.
  • The lectures are from Week 3 through Week 18. The last class is on December 27, 2022.

Main References:

  • Connes, A.: Noncommutative geometry. Academic Press, 1994 (pdf).
  • Gracia-Bondia, J.M.; Varilly, J.C.; Figueroa, H.: Elements of Noncommutative Geometry. Birkhäuser, 2021 (Springer Link).
  • Lecture notes of my previous graduate courses on noncommutative geometry (here).


  1. Spectrum and duality spaces/algebras (slides)
  2. Examples of noncommutative quotients (slides).
  3. Operators on Hilbert space (slides)
  4. Singular values and Schatten classes (slides; updated 11/10/2022).
  5. Connes’ quantized calculus (slides).
  6. Connes’ integration and Weyl’s laws. Birman-Schwinger principle (slides).
  7. Cwikel estimates. Connes’ integration formula and semiclassical analysis (slides).
  8. Pseudodifferential operators (slides).
  9. Noncommutative residue (slides).
  10. Connes’ trace theorem, integration formula, and lower dimensional volumes (slides). 
  11. Birman-Solomyak’s Weyl’s laws. Applications (slides). 
  12. Spectral triples, Dirac operators, K-theory, and the Atiyah-Singer index theorem (slides). 
  13. Cyclic cohomology, and the local index formula in noncommutative geometry (slides).