The Dynamical Alekseevskii Conjecture states that if the
universal cover of a homogeneous manifold is not contractible, then all
homogeneous Ricci flow solutions on it have finite extinction time. Böhm
showed that the conjecture is true for compact homogeneous spaces. In
this talk, we will discuss the conjecture in the context of noncompact
homogeneous manifolds. We will be especially concerned in testing the
conjecture in special families of homogeneous metrics on noncontractible
spaces that satisfy some compatibility condition between the Lie
algebraic structure and the geometry and that, in particular, contain
Ricci negative ones.