Canonical Metrics in Kähler Geometry

Given a Riemannian manifold $(M^n,g)$, it is natural to ask to whether such a manifold can be equipped a "canonical metric". For Riemann surfaces, i.e., one-dimensional complex manifolds, we have the celebrated uniformisation theorem and we explore whether analogous statements can be made in higher dimensions. This will lead naturally to the subject of Kähler--Einstein metrics and their associated complex Monge--Ampère equations. If time permits, we would like to discuss generalised Kähler--Einstein metrics which have been the subject of current research. Such metrics are of central importance to the analytic minimal model program proposed by Song--Tian, where the program aims to study the bimeromorphic classification of Kähler manifolds using the Kähler--Ricci flow.