These lectures do not assume previous knowledge of the subject, and will be accessible to a diverse audience of graduate students. The specific topics to be addressed are the following:

Generalities on the deformation theory of associative algebras (Hochschild complex, statement of Kontsevich's formality theorem)

Deformations of functions on symplectic manifolds (Fedosov quantization)

Cyclic cohomology

Lie algebra cohomology and Gelfand-Fuchs construction

The algebraic index theorem of Nest and Tsygan

Relation to the Atiyah-Singer index theorem