Topics in Geometry: Metric Geometry and Degenerations of Einstein Manifolds

The first part of the course introduces basic tools in the geometry of the Ricci curvature, including volume comparison and splitting theorems, structure theorems of nonnegative Ricci, elliptic PDEs and lower Ricci bound, etc. The second part focuses on the degenerations of Einstein metrics, such as the structure of Einstein limit spaces, quantitative differentiation theory, the Codimension-4 Conjecture, regularity theory for collapsed Einstein manifolds, etc. Time permitting, we also introduce the collapsing geometry of special holonomy spaces.