On Lefschetz exceptional collections and quantum cohomology of Grassmannians

Given a Lefschetz exceptional collection on a variety $X$ one defines its residual subcategory as
the orthogonal to the rectangular part of the collection. In this talk we will discuss some conjectural
relations between the quantum cohomology of $X$ and the structure of the residual subcategory
motivated by homological mirror symmetry. We give examples of this relation when $X$ is an ordinary
or a symplectic isotropic Grassmannian.