Homological mirror symmetry asserts that the connection, discovered by physicists, between a count of rational curves in a Calabi-Yau manifold and period integrals of its mirror should follow from an equivalence between the derived Fukaya category of the first manifold and the derived category of coherent sheaves on the second one. Physicists' observation can be reformulated as, or rather upgraded to, a statement about an isomorphism of certain Hodge-like data attached to both manifolds, and a natural first step towards proving the above assertion would be to try to attach similar Hodge-like data to abstract derived categories. The aim of the talk is to report on some recent progress in this direction and illustrate the approach in the context of what physicists call Landau-Ginzburg B-models.[-]