Abstract.

Day 1: Introduction to Ricci curvature. Basic analysis tools, volume comparison, Gromov-Hausdorff convergence, tangent cones. If time we will outline the rest of the course.

Day 2: Rigidity and Almost Rigidity Theorems. Cheeger-Gromoll Splitting Theorem. Hessian Estimates for Harmonic functions, almost splitting theorem. If time allows we will discuss segment inequality.

Day 3: Stratification and Quantitative Stratification. Almost Volume Cone=>Almost Metric Cone, definition of (quantitative) stratification, outline of proof. Applications to regularity of Einstein Manifolds.

Day 4: Holder Continuity of Tangent Cones, outline of proof + applications (constant dimension conjecture, convexity of regular set, isometry group is a lie group). Will focus on new analysis estimates, in particular sharp estimates for the hessian and excess function.

Day 5: Bounded Ricci Curvature on Smooth and Nonsmooth spaces. Will discuss new estimates for bounded Ricci curvature on smooth spaces. If time allows we will discuss how these may be used to define bounded Ricci curvature on metric-measure spaces.