Homological Units

Let X and Y be smooth projective varieties (over the field of complex numbers). A conjecture attributed to Kontsevich asserts that if the derived categories of X and Y are equivalent then the Hodeg numbers of X and Y are the same.
This conjecture is wide open when the dimension of X and Y is bigger than 5. In this talk, I will focus on a slightly less ambitious question : is there a canonically defined subalgebra of the Hochschild cohomology of X which is invariant by derived equivalent? I have a very simple candidate for this algebra and I will explain that this candidate can be defined for non-commutative Calabi-Yau varieties. In particular, it gives a natural Hodge-type filtration on the Hochschild homology of non-commutative Calabi-Yau threefolds. I will give applications to the study of the non-commutative mirror of the rigid Z-threefold studied by Candelas, Derrick and Parkes.