One of the most important open problems on Einstein homogeneous manifolds is the *Alekseevskii conjecture*. This conjecture says that any connected homogeneous Einstein space of negative scalar curvature is diffeomorphic to a Euclidean space. Due to recent results, this conjecture is equivalent to the analogous statement for expanding algebraic solitons, which we call *Generalized Alekseevskii's conjecture*.

The aim of this talk is to present the classification of expanding algebraic soliton up to dimension 5; to verify that the Generalized Alekseevskii conjecture holds in these dimensions; and to show that the Alekseevskii conjecture holds up to dimension 8 (excluding 5 possible exceptions), and also in dimensions 9 and 10 provided the transitive group is not semisimple. This talk is based on two joint works with Ramiro Lafuente.

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