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The lecture will be organized in a series of questions and answers:
1) What is a noncommutative differential form? On a ring? On a manifold? On a finite set!
2) What is the relation between a connection on a vector bundle and the theory of noncommuative differential forms? What is the interpretation of characteristic classes in this language?
3) What is the formal definition of cyclic homology and how this is related with differential geometry?
4) Examples. Calculations. Relation with K-theory...