Freie Wahrscheinlichkeit Lehre

Dr. Tobias Mai

Introduction to Noncommutative Differential Geometry

(Summer Semester 2019)

News

Lecture

Monday, 10 -- 12, Seminar Room 10, Building E2 4

In 1982, the French mathematician Alain Connes was awarded the Fields medal for his numerous contributions to the theory of operator algebras, and in particular for applications of operator algebraic methods to differential geometry. The latter he developed further intensively to an influential theory at the interface of various fields of mathematics and with deep connections to physics, which became known as Noncommutative (Differential) Geometry.
The underlying idea is that there are interesting “spaces” (such as the space of leaves of a foliation, or the orbit space of the action of a discrete group on a manifold) which are badly behaved as point sets, but become accessible via canonically associated (often noncommutative) operator algebras. In such situations, the usual methods of differential geometry cannot be applied directly, but they can be imitated in an operator theoretic framework using the notion of so-called spectral triples. This opens the door to the noncommutative world and extends differential geometry far beyond its traditional setting.

In this lecture, we want to give an introduction to this circle of ideas from an operator algebraic perspective. Participants are thus required to have a solid background in functional analysis, especially on C*- and von Neumann algebras. Prior knowledge in differential geometry and homological algebra might be helpful, but is not required; this can be discussed when necessary.

Lecture announcement

For further information, please contact Tobias Mai.

Lecture notes

Lecture notes, April 8
Lecture notes, April 15
Lecture notes, April 29
Lecture notes, May 6
Lecture notes, May 13
Lecture notes, May 20
Lecture notes, June 3
Lecture notes, June 11
Lecture notes, June 24
Lecture notes, July 1
Lecture notes, July 8
Lecture notes, July 15
Lecture notes, July 16 (Appendix on Homology and Cohomology)

A typed version of the lecture notes, kindly provided by Friedrich Günther, is available here (version: June 24).

Tutorials

The tutorials (problem sessions) take place every second week on

Tuesday, 10 -- 12, Seminar Room 10, Building E2 4.

There will be no further problem session.

Assignment 1A (for the tutorial on April 23)    Solution Assignment 1A
Assignment 1B (for the tutorial on April 23)    Solution Assignment 1B
Assignment 2A (for the tutorial on May 7)    Solution Assignment 2A
Assignment 2B (for the tutorial on May 7)    Solution Assignment 2B
Assignment 3A (for the tutorial on May 21)    Solution Assignment 3A
Assignment 3B (for the tutorial on May 21)    Solution Assignment 3B
Assignment 4A & B (for the tutorial on June 4)    Solution Assignment 4A & B
Assignment 5A (for the tutorial on June 25)    Solution Assignment 5A
Assignment 5B (for the tutorial on June 25)    Solution Assignment 5B
Assignment 6A (for the tutorial on July 9)    Solution Assignment 6A
Assignment 6B (for the tutorial on July 9)    Solution Assignment 6B

You do not have to hand in written solutions. We will discuss the problems in
the tutorials, but you should be prepared then to present your solution.

References

Connes, A. (1999). Noncommutative geometry.
Connes, A., Doplicher, S. (2004). Noncommutative geometry: Lectures given at the CIME Summer
School held in Martina Franca, Italy, Sept. 3 - 9, 2000
Consani, C., Marcolli, M. (2006). Noncommutative geometry and number theory: Where arithmetic
meets geometry and physics.
Eckstein, M., Iochum, B. (2018). Spectral Action in Noncommutative Geometry.
Gracia-Bondía, J., Várilly, J., Figueroa, H. (2001). Elements of Noncommutative Geometry.
Khalkhali, M. (2013). Basic noncommutative geometry.
Madore, J. (1999). An introduction to noncommutative differential geometry and its physical applications.
Marcolli, M. (2005). Arithmetic noncommutative geometry.
Scheck, F., Upmeier, H., Werner, W. (2002). Noncommutative Geometry and the Standard Model
of Elementary Particle Physics.
Van Suijlekom, W. (2015). Noncommutative Geometry and Particle Physics.
Várilly, J. (2006). An Introduction to Noncommutative Geometry.

These books can be found in the Course Reference Shelve that is kindly
provided by the Campusbibliothek für Informatik und Mathematik.

updated: 6 August 2019 Tobias Mai