Abstract

Algebraic structures on Hochschild cochain and chain complexes of algebras are long known to be of considerable interest. On the one hand, some of them arise as standard operations in homological algebra. On the other hand, for the ring of functions on a manifold, Hochschild homology, resp. cohomology, is isomorphic to the space of differential forms, resp. multivector fields, and many algebraic structures from classical calculus can be defined (in a highly nontrivial way) in a more general setting of an arbitrary algebra and its Hochschild complexes. Despite an extensive amount of work on the subject, it appears that large parts of the algebraic structure on the Hochschild complexes of algebras and their tensor products are still unknown. I will review the known results (Kontsevich-Soibelman, Dolgushev, Tamarkin and myself, Keller) and then outline some provisional results and conjectures about new operations. The talk will not assume any prior knowledge beyond the basic facts about complexes and (co)homology.