The Yau-Tian-Donaldson conjecture is a significant hypothesis in the field of differential geometry and algebraic geometry. It connects the existence of certain types of metrics, known as Kähler metrics, on complex manifolds to the stability of these manifolds in the sense of Geometric Invariant Theory. The conjecture suggests that if a complex manifold is K-stable, then it admits a Kähler-Einstein metric. This conjecture was proposed independently by Shing-Tung Yau, Gao Chen, and Simon Donaldson. It has profound implications for the study of Calabi-Yau manifolds and the broader understanding of the relationship between geometry and algebraic properties. The conjecture remains an active area of research, with many mathematicians working to prove or disprove its claims.