We will discuss recent developments on how geometric PDEs can perform canonical geometric surgery. We propose the analytic minimal model program with Ricci flow to classify algebraic varieties via geometric surgeries in Gromov-Hausdorff topology equivalent to birational surgeries such as contractions and flips. This approach can also be applied to the study of degeneration of Calabi-Yau metrics. As an application, we prove a conjecture of Candelas and de la Ossa for conifold flops and transitions.

We establish various stability results for solutions of complex Monge-Ampère equations in big cohomology classes, generalizing results that were known in the context of Kähler classes. (This is a joint work with Vincent Guedj).

I will discuss the existence of classical and weak solutions to the complex Hessian equation on compact Kähler manifolds. In particular I will present a gradient a priori estimate for the solutions. By a standard blow-up argument it is linked to a certain Liouville type theorem for maximal m-subharmonic functions.

One of the key problems in geometric analysis is to understand limits of sequences of Riemannian manifolds which collapse (in the sense that their volume goes to zero). We will discuss this problem in the case when the metrics are solutions of complex Monge-Ampère equations on compact Kähler manifolds. This applies for example to the case of limits of Ricci-flat Calabi-Yau metrics or to certain solutions of the Kähler-Ricci flow. This is partly joint work with M.Gross and Y.Zhang.