Introduction to Noncommutative Geometry, SCU Summer School (June 29-July 19, 2025)

Lecturer:

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

Time and Location:

  • From 14:30-17:30, Room 516, Tianyuan Mathematical Center.
  • The lectures are everyday from July 7 to July 11, except July 10 and July 13 (9 lectures).
  • Exam is on July 19 from 14:30-17:30.

Main References:

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

Contents & Lecture Slides:

  1. Banach and Hilbert Spaces. Examples (slides).
  2. C*-algebras (slides). Main Reference: A Short Course on Spectral Theory by W. Arveson (Graduate Texts in Mathematics, Springer, 2002) (e-book).
  3. Operators on Hilbert space (slides; updated July 10). Lecture notes (pdf). Main Reference: Methods of modern mathematical physics. I. Functional analysis by M. Reed and B. Simon (2nd edition, Academic Press, 1980). (Please message me on WeChat to get an e-copy).
  4. Singular Values and Schatten Classes (slides; updated July 10). Lecture notes (pdf).
  5. Connes’ Quantized Calculus (slides; full version). Lectures notes (pdf).
  6. Connes’ trace theorem on tori (handwritten notes pdf) (slides on Part 2 pdf).
  7. Connes’s trace theorem on Euclidean spaces (handwritten notes pdf) (slides on Part 1 pdf) (slides on Part 3 pdf).
  8. Pseudodifferential operators on manifolds and Weyl’s law (slides). Slides on pseudodifferential operators from my 2022 online course on NCG (pdf).
  9. Quantized calculus on manifolds. Noncommutative residue and lower dimensional volumes (slides). Slides on the noncommutative residue from from my 2022 online course on NCG (pdf).
  10. Quantized calculus and semiclassical analysis (slides pdf; full version).
  11. Spectral triples and semiclassical analysis (slides). Additional material on spectral triples and Dirac operators (pdf).
  12. Quantized calculus and semiclassical analysis on quantum tori (slides).