Topic on complex analysis Spring 2025

This topic course will (tentatively) cover the following topics, and we will update the detailed materials as the course progresses.

1. Complex and differential geometry: Riemannian/Kahler metrics, curvatures, vector bundles…
2. basic PDE theory: maximum principle, Schauder estimates, regularity theory for linear equations
3. Complex Monge-Ampere equations: a complete proof of Calabi conjecture
4. Hormander’s L^2 estimates for solving d-bar equations: Kodaira’s embedding theorem, partial C0 estimates, expansion of Bergman kernel
5. Existence of special Kahler metrics on Kahler manifolds: KE metrics, constant scalar curvature Kahler (cscK) metrics, Kahler-Ricci solitons

If time permits, we will explain briefly:

1. Metric geometry: Gromov-Hausdorff topology, introductory Cheeger-Colding theory
2. Recent development on the geometry of Kahler metrics from the works of Guo-Phong-Song-Sturm