The uniformization theorem classifies simply-connected Riemann surfaces into a trichotomy, which can be described in many ways, such as Euler characteristic, existence of special metric, or ampleness of its canonical bundle.

In higher dimensions, uniformization is much more complicated, and Miyaoka-Yau inequalities is an algebro-geometric approach towards this goal. In particular, when equality is achieved, we have an analog of the 1-dimensional situation.

We will introduce Chern classes of a line bundle, Kähler-Einstein (KE) metrics, and sketch a proof of the inequality in the smooth case using KE metrics. Half of the talk will be introducing notions and tools in complex differential geometry.