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The existence of Kahler-Einstein metrics on Kahler manifolds is a basic problem in complex differential geometry. This problem has connections to other fields: complex algebraic geometry, partial differential equations and several complex variables. I will discuss the existence of Kahler-Einstein metrics on Fano manifolds and its relation to K-stability. I will mainly focus on the analytic part of the theory, discuss how to solve the related complex Monge-Ampere equations and provide concrete examples both in the smooth and conical setting. If time permits, I will also say something about the algebraic part of the theory, including the study of K-stability using Minimal Model Program (joint with Chenyang Xu) and the existence of proper moduli space of smoothable K-polystable Fano varieties (recent joint work with Xiaowei Wang and Chenyang Xu). Apart from this last topic, the talk is mostly a survey of my recent research results.

There will be a dinner in honor of the speaker.