1) Prerequisites:
(Introduction to) Functional Analysis (bachelor introductory, or mastermath course); more specifically: Hilbert spaces, bounded linear operators on Hilbert space, spectral theorem for compact self-adjoint operators.
2) Aim of the course:
This course is an introduction to noncommutative geometry. We will start with a "light" version by looking at finite noncommutative metric spaces and their classification. Then, we will introduce spectral triples, as the noncommutative generalization of Riemannian spin manifolds. We will introduce algebra modules as the noncommutative analogue of vector bundles. As an application, we will describe how index theory can be described by noncommutative geometry, or, alternatively, how noncommutative manifolds naturally give rise gauge theories.
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the student can work with the basic concepts in noncommutative geometry, such as spectral triples (aka as noncommutative Riemannian spin varieties), differential calculi, algebra modules, Morita equivalence.
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the student understands the classification of finite noncommutative metric spaces
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the student knows examples of spectral triples, such as Riemannian spin manifolds and the noncommutative torus
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the student has seen some of the applications of noncommutative geometry, to eg. index theory of gauge theories.
3) Homework: Weekly exercises, with four sets of hand-in exercises (distributed over the semester)
3) Exam: The final grade for this course will be determined by the average of the grade for the weekly exercise sheets (30%) and by an essay written by the student on a specialized topic at the end of the semester (70%). The student gives a short presentation on their essay.
4) Lecture notes/Literature:
Draft of second edition of the following book (open access and will be made available online)
W. D. van Suijlekom, Noncommutative Geometry and Particle Physics, Springer, 2014 (second edition expected 2025).
5) Lecturer: Walter van Suijlekom
- Docent: Walter van Suijlekom