- 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.
- the student understands the classification of finite noncommutative metric spaces
- the student knows examples of spectral triples, such as Riemannian spin manifolds
- the student has seen some of the applications of noncommutative geometry, to eg. index theory of gauge theories.
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 algebras modules as the noncommutative analogue of vector bundles. As an application, we will describe how index theory can be described by noncommutative geometry, or how noncommutative manifolds naturally give rise gauge theories.