Geometry of canonical metrics on Kähler manifolds

The aim of this talk is to report recent trends in Kähler geometry. Kähler geometry consists of two aspects: the one is algebraic geometry and the other is metric geometry.The first one hour is an introduction for non-mathematicians.
I begin with a simple example of algebraic variety from ancient Greek, which I believe is the simplest example illustrating motivation for compact complex manifolds.
On the other hand, I explain the first motivation for canonical metrics in Kähler geometry via Riemann’s uniformization theorem.The last one hour is an introduction to recent trends in Kähler geometry, especially Kähler-Einstein metrics.
The existence of Kähler-Einstein metrics turns out to be related to geometry of degenerations of space, which is so called Yau-Tian-Donaldson conjecture.
I explain various aspects of this topic. We encounter deep studies in metric geometry, birational geometry and non-archimedean geometry.
I finally explain recent breakthrough on Kähler-Ricci flow.The goal of this talk is the starting point of my study. I briefly explain my study if time permits.