Explore a 51-minute lecture by Jorge Lauret from Universidad Nacional de Cordoba, Argentina, examining the existence problem for Einstein metrics on compact homogeneous spaces. Delve into the fascinating world of Lie groups and homogeneous spaces, where closed subgroups of real matrices G form differentiable manifolds, and quotients M=G/K create homogeneous spaces. Learn how inner products on tangent spaces at the origin of M, when K-action invariant, define Riemannian metrics with G as an isometry subgroup. Investigate three fundamental open problems in this field: the conditions under which homogeneous spaces M=G/K admit Einstein G-invariant metrics with constant Ricci curvature, the strength of such existence, and the possibility of infinite admissions for a given M=G/K. Discover the rich interplay between algebra, geometry, and topology that has driven mathematical advancement for over a century.