  We show that most homogeneous Anosov actions of higher rank Abelian groups
are locally smoothly rigid (up to an automorphism). This result is the main
part in the proof of local smooth rigidity for two very different types of
algebraic actions of irreducible lattices in higher rank semisimple Lie groups:
(i) the Anosov actions by automorphisms of tori and nil-manifolds, and (ii) the
actions of cocompact lattices on Furstenberg boundaries, in particular,
projective spaces. The main new technical ingredient in the proofs is the use
of a proper "non-stationary" generalization of the classical theory of normal
forms for local contractions.
