Lecure 1: Dualities in Non Commutative Geometry
Date and Time: Sunday, August 5, 2012 at 10:00-11:00

Lecure 2: Cyclic Homology and Applications
Date and Time: Sunday, August 5, 2012 at 11:15-12:15

Slide Show 1

Lecure 3: Hopf Algebras and Hopf Cyclic Cohomology
Date and Time: Monday, August 6, 2012 at 10:00-11:00

Lecure 4: Hopf-Galios Extensions
Date and Time: Monday, August 6, 2012 at 11:15-12:15

Slide Show 2

Abstract

Noncommutative geometry asserts that the category of algebras (commutative or noncommutative)of various flavors is operationally the category of spaces (commutative or noncommutative) of various flavors. This duality leads to build a dictionary to translate important concepts from one category to the other. In the first session we shortly review some of these dualities such as Gelfand-Neimark, Serre-Swan and Connes reconstruction theorems. In the second talk we study an important subject in noncommutative geometry called cyclic cohomology which carries out interesting geometrical, topological and algebraic information. Next we survey the basics of Hopf cyclic cohomology which was discovered by Connes and Moscovici in 1998 in the course of their breakthrough analysis of transverse index theory on foliated spaces. Finally in the last lecture, we study Hopf Galois extensions of algebras which generalizes the classical Galois extensions of fields and also the notion of principal bundles. We introduce some new results in this subject.



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