In this talk we will describe basic topological properties of the space of commuting unitary matrices. In particular we will show that they can be assembled to form a space which classifies commutativity for vector bundles and which has very interesting homotopy-theoretic properties.

We will continue with the following topics:

---Operators of Dirac type and its main examples, ---Clifford algebras, Clifford modules, spin structures, Dirac operators, Weitzenbock formula.