-Einstein homogeneous manifolds. Invariant geometric structures on Lie groups. Geometry of nilmanifolds. Ricci soliton metrics.

-Nilpotent Lie algebras. Variety of Lie algebras. Degenerations, rigidity and rational forms of Lie algebras.

-Real geometric invariant theory.

-Anosov diffeomorphisms on nilmanifolds.

-Weakly symmetric and commutative spaces. Gelfand pairs. Symmetric-like Riemannian manifolds.