Explore the groundbreaking research on Kahler-Einstein metrics in Fano manifolds through this 51-minute lecture by Song Sun at BIMSA. Delve into the seminal paper by X. Chen, S. Donaldson, and S. Sun, which proves the equivalence between Kahler-Einstein metrics existence and K-stability on Fano manifolds. Discover the crucial connection between differential geometry and algebraic geometry in understanding singularity formations of Kahler manifolds. Gain insights into recent advancements in this field from both differential and algebro-geometric perspectives, as time allows.