Donaldson's program to solve the Yau-Tian-Donaldson conjecture from an algebraic viewpoint

Longer than 50 years ago, Eugenio Calabi consider which projective manifolds accept a Kahler metric whose Ricci tensor is constant. In the early 80s, Aubin and Yau proved that we always can find such a metric when the manifold is of general type or Calabi-Yau. The Fano case was open. Tian and Yau conjectured that, for Fanos, the existence should be equivalent to some sort of stability known as K-stability, a completely algebraic concept. A few years ago Donaldson sketched a programme to prove this conjecture. Recently Chen, Donaldson and Sun announced a proof and, independently, Tian uploaded a complete proof in the ArXiv. In this talk I will sketch the main ideas behind their proof, illustrating the use of a modified version of Tian's alpha-invariant, for the case of del Pezzo surfaces.