The Serre-Swan theorem extending the Gelfand-Naimark equivalence of unital C*-algebras and compact Hausdorff spaces to the case of finitely generated projective modules and topological vector bundles lead to the celebrated Chern-Connes pairing between the cyclic cohomology and K-theory. This pairing substitutes the theory of characteristic classes for quantum spaces and encompasses the Noncommutative Index Theorem generalizing the famous Atiyah-Singer Index Theorem. The noncommutative index theory extends in various directions the Atiyah-Singer Theorem of classical differential geometry and finds numerous applications well beyond the scope of classical geometry. A recent fascinating development (due to A.Connes and H.Moscovici) concerns the application of Hopf symmetry in computing numerical invariants given by the Chern-Connes pairing. This symmetry served as a crucial organizing principle in computing characteristic classes for foliations.

Syllabus: Chern-Connes pairing of cyclic cohomology and K-theory, Noncommutative Index Theorem, categorical approach to Hochschild and cyclic cohomology (pre-cosimplicial and pre-cocyclic objects in Abelian categories, cyclic cohomology as a derived functor), modular pairs in involution, Hopf-cyclic cohomology, characteristic map from Hopf-cyclic cohomology to usual cyclic cohomology, examples (e.g., universal enveloping Hopf algebra of a Lie algebra, Connes-Moscovici Hopf algebra of foliation, modular square Hopf algebra from a locally compact quantum group).