**Guofang Wei, UC Santa Barbara **

**Title:**
Relative volume comparison for integral Ricci curvature and some applications.

**Abstract:** We give several Bishop-Gromov relative volume
comparisons in terms of integral Ricci curvature which improve the
results in Petersen-Wei. Using one of these volume comparisons, we
derive an optimal volume entropy estimate for integral Ricci curvature
bound. We also obtain an almost minimal volume rigidity for integral
Ricci curvature. This is a joint work with Lina Chen.