Topological Hopf Algebras and Their cyclic cohomology

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Hopf algebras (quantum groups) and their cohomology provides invariants for algebras. Usually algebras are complicated objects to deal with even in the case of classical ones that mostly appear as the coordinates of geometric spaces. One of the main duties of Hopf algebras is to act on algebras and define a part of algebra at which we can control easier. Hopf cyclic cohomology was defined by Alain Connes and Henri Moscovici. They observed that a certain Hopf algebra plays a bookkeeping role in their celebrated index formula. They also calculated the cohomology of their Hopf algebra and identified it with the Gelfand-Fuks cohomology of the algebra of formal vector fields. Later on it was observed by the author and his collaborators that what Connes and Moscovici observed is the tip of an iceberg. It was enlarged to encompass coalgberas endowed with symmetry from Hopf algebras and also the coefficients was added to the theory. In this proposal we extend Hopf cyclic cohomology to the level of topological Hopf algebras. This allows us to solve many of the open questions that has raised naturally in the algebraic cases. For instance justification to the annihilation of Godbillon-Vey classes in case of the corresponding algebraic Hopf algebra act on the type III algebra associated to general foliations on the Euclidean space. As another improvement one observes that in the case of topological Hopf algebras the correspondence between classical and nonclassical coefficients are perfect. This was missing in the algebraic case.We also try to solve the long standing problem in characteristic classes of foliation: is the Gelfand-Fuks cohomology are the only source of characteristic classes of foliations. We observe that there is a natural set of coefficients produced by Godbillon when he tried to answer the question on his very last published paper. We try to compute the cohomology of the complex he left uncalculated at the degree of greater than 2.Two PhD students and one postdoctoral fellow will be trained to involve in project. The main collaborates are Henri Moscovici, Serkan Sutlu, and Fereshteh Yazdani.