Introduction to Noncommutative Geometry, Spring 2025

Lecturer:

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

Time and Location:

  • Fridays, from 14:00-15:40, Room 403, Graduate School Bldg.
  • From Week 3 onward the lectures will be at Room 403, Graduate School Bldg.

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).

Homework:

  • Homework 1 (pdf). Due date: May 15, 2025.

Contents & Lecture Slides:

  1. Spectrum and duality spaces/algebras (slides). Lecture notes (pdf). Main Reference: A Short Course on Spectral Theory by W. Arveson (Graduate Texts in Mathematics, Springer, 2002) (e-book).
  2. Examples of noncommutative quotients (slides). Lecture notes (pdf).
  3. Operators on Hilbert space (slides). 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). Lecture notes (pdf).
  5. Connes’ Quantized Calculus (slides). Lectures notes (pdf).
  6. Pseudodifferential Operators (slides; updated May 29, 2025). Slides on pseudodifferential operators from my 2022 course (pdf).
  7. Noncommutative Residue and Zeta Functions of Elliptic Operators (slides; updated May 29, 2025). Slides from my 2022 course (pdf).
  8. Connes’s Trace Theorem and Lower Dimensional Volumes (slides; final version, posted May 30, 2025).
  9. Spectral Triples and Dirac Operators (slides). Notes on Clifford algebras (pdf). Notes on spin structures and Dirac operators (pdf). (The notes on the Atiyah-Singer index theorem below contains some details on the examples of spectral triples mentioned in this chapter.)
  10. K-Theory and Atiyah-Singer Index Theorem (slides). Notes on the Atiyah-Singer index theorem (pdf).
  11. Cyclic Cohomology (slides).
  12. The Local Index Formula in Noncommutative Geometry (slides).