Formality of cyclic cochains

We prove M. Kontsevich's Cyclic Formality Conjecture. This conjecture is the cyclic analog of the Formality Theorem for Hochschild chains. Concretely, it states that the differential graded Lie algebra of polydifferential cyclic cochains of the algebra of smooth functions on a manifold M is L∞-quasi-isomorphic to the second term in the spectral sequence computing its cohomology. As an application, we can classify all closed star products on M.