Hopf cyclic cohomology is a cohomological theory that aims to provide some information about the cyclic cohomology of algebras and coalgebras endowed with a symmetry from a Hopf algebra. This information usually appears as certain sub-complex of the cyclic complex of the algebras or coalgebras in question.

The theory initiated by Connes and Moscovici as a byproduct of their study on the long standing computation of the local index formula of the hypo-elliptic operators. What they discovered is the miraculous procedure that allowed them to detect the complicated local index formula in the image of a very simple catachrestic map whose domain was a new complex (Hopf cyclic complex) much simpler than the cyclic complex of the groupoid action algebra.

Roughly speaking the roles of Lie groups and Lie algebras in differential geometry are now given to Hopf algebras in noncommutative geometry. Via this correspondence the Hopf cyclic cohomology replaces Lie algebra cohomology and other cohomology theories associated to Lie groups.

The initial theory was modeled on Hopf algebras with a square identity twisted antipode. Later on it covers Hopf algebras endowed with a pair of character and group-like so called modular pair in involution(MPI). It was observed by Hajac-Khalkhali-R-Sommerha¨user that the theory works perfectly for any Hopf algebras endowed with much more general coefficients so called stable anti Yetter-Drinfeld (SAYD) modules. These higher dimensional modules-comodules then become essential in many occasions such as computations of Hopf cyclic cohomology of Connes Moscovici type Hopf algebras, and also in the definition of cup products in Hopf cyclic cohomology.

1. We start with the motivation and origin of the theory by recalling two most important objects: characteristic classes of foliations and index theory.

2. The second talk deals with the definition of the Hopf cyclic cohomology for general Hopf algebras and general coefficients.

3. The next talk is devoted to the construction of those Hopf algebras that are used (will be used) in the local index

formula of the corresponding Cartan geometries: General, Volume Preserving, Symplectic, Contact, Projective. We call such Hopf algebras the Connes-Moscovici type Hopf algebras.

4. In the fourth talk we develop a general theory by which one can compute Hopf cyclic cohomology of Hopf algebras associated to a general matched pair of Lie algebras or Lie groups.

5. Finally we explain various characteristic maps and cup products that are cornerstones of Hopf cyclic cohomology. We apply these tools to present, for the first time, the characteristic classes of foliations, in codimension higher than 1, as cyclic cocycle over the groupoid action algebra.