Volume comparison and its generalizations

Guofang Wei

One of the basic problems in Riemannian geometry is to understand the relations between curvature and topology and one of the tools is comparison theorems. For Ricci curvature the Bishop-Gromov volume comparison theorem is very powerful and has many applications. We will discuss this and some generalizations of the volume comparison results to integral Ricci curvature (due to Peter Petersen and myself) and Ricci flow (due to Perelman).