  We show that there is an hierarchy of intersection rigidity properties of
sets in a closed symplectic manifold: some sets cannot be displaced by
symplectomorphisms from more sets than the others. We also find new examples of
rigidity of intersections involving, in particular, specific fibers of moment
maps of Hamiltonian torus actions, monotone Lagrangian submanifolds (following
the works of P.Albers and P.Biran-O.Cornea), as well as certain, possibly
singular, sets defined in terms of Poisson-commutative subalgebras of smooth
functions. In addition, we get some geometric obstructions to semi-simplicity
of the quantum homology of symplectic manifolds. The proofs are based on the
Floer-theoretical machinery of partial symplectic quasi-states.
