On the $A_\infty$-Formality conjecture
  It is proved that the associative differential graded algebra of (polynomial) polyvector fields on a vector space (may be infinite- dimensional) is quasi-isomorphic to the corresponding cohomological Hochschild complex of (polynomial) functions on this vector space as an associative differential graded algebra. This result is an $A_\infty$-version of the Formality conjecture of Maxim Kontsevich [K]. 
Boris Shoikhet (Independent University of Moscow)