Ricci curvature occurs in the Einstein equation, Ricci flow, optimal transport, and is important both in mathematics and physics. Comparison method is one of the key tools in studying the Ricci curvature. We will start with Bochner formula and derive Laplacian comparison, Bishop-Gromov volume comparison, first eigenvalue and heat kernel comparison and some application. Then we will discuss some of its generalizations to Bakry-Emery  Ricci curvature and integral Ricci curvature

We consider X a compact Kaehler manifold and D a divisor in X.  We study the regularity of solutions of degenerate complex Monge-Ampère equations where the right hand-side is smooth just outside D, establishing uniform a priori estimates which generalize both Yau's and Kolodziej's celebrated estimates.
This is a joint work with Hoang-Chinh Lu.