## Kähler-Einstein metric, K-stability and moduli spaces

Abstract: A complex variety with a positive first Chern class is called a Fano variety. The question of whether a Fano variety has a Kähler-Einstein metric has been a major topic in complex geometry since the 1980s. In the last decade, algebraic geometry, or more specifically higher dimensional geometry has played a surprising role in advancing our understanding of this problem. In fact, the algebraic part of this question is one step of a larger project, namely constructing projective moduli spaces that parametrize Fano varieties satisfying the K-stability condition. The latter is exactly the algebraic characterization of the existence of a Kähler-Einstein metric. In the lecture, I will explain the main ideas behind the recent progress of the field.