An algebraic action by commuting flows or maps is genuinely higher rank if no algebraic factor is a compact extension of a flow or map. We will discuss conditions under which a genuinely higher rank algebraic action is smoothly rigid, a stronger form of structural stability in which the conjugating map has prescribed smoothness, and is an "honest" dynamical conjugacy, not just an orbit equivalence. Unlike the case of flows or single transformations, smooth rigidity for higher rank actions have no periodic data assumptions (as in the Livsic theorem).

Two principal sources of algebraic actions will be considered: toral automorphisms and Weyl chamber flows. Smooth rigidity for toral automorphisms was established in great generality by Damjanovic and Katok in the late 2000's. The proof of a new result extending smooth rigidity results known only for a handful of Weyl chamber flows to a general setting will be the focus of our attention.