Noncommutative Geometry (Analysis on Manifolds), Fall 2022

Lecturer:

  • 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).

Contents:

  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. Birman-Solomyak’s Weyl’s laws.
  11. Dirac operators.
  12. Spectral triples.
  13. Semiclassical Weyl’s laws in noncommutative geometry.
  14. Noncommutative tori: pseudodifferential calculus and noncommutative residue.
  15. Noncommutative tori: Riemannian geometry.
  16. Noncommutative tori: semiclassical analysis.
  17. The Atiyah-Singer index theorem.
  18. K-theory.
  19. Cyclic cohomology.
  20. Connes-Chern character.
  21. The local index formula in noncommutative geometry.