Homogeneous Ricci solitons and the Alekseevskii conjecture in low dimensions.

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.