The notion of K-stability for a Fano varieties was introduced by differential geometers in late 90s, to capture the existence of a Kähler-Einstein metric. In the last decade, it has gradually become clear to algebraic geometers that K-stability provides a rich algebraic theory in higher dimensional geometry. In particular, it can be used to solve the longstanding question of constructing moduli spaces for Fano varieties.
I will survey the background of K-stability and how algebraic geometers’ understanding of it has evolved. In particular, I will explain algebraic geometry plays a key role of establishing the equivalence between K-stability and the existence of a Kähler-Einstein metric, i.e. the Yau-Tian-Donaldson Conjecture, for all Fano varieties. If time permits, I want to also discuss the construction of K-moduli spaces parametrizing Fano varieties, and how the recipe given by K-stability can be used to resolve the issues that mystify people for a long time.