Prerequisite courses: MA 333 - Riemannian Geometry
Bochner formula, Laplace comparison, Volume comparison, Heat kernel estimates, Cheng-Yau gradient estimates, Cheeger-Gromoll splitting theorem, Gromov-Haudorff convergence, epsilon regularity, almost rigidity, quantitative structure theory of Riemannian manifolds with Ricci curvature bounds. If time permits, we will discuss the proof of the co-dimension four conjecture due to Cheeger and Naber.
Suggested books and references:
Peter Petersen, Riemannian geometry, Graduate Texts in Mathematics, 171. Springer-Verlag, New York, 1998.
Richard Schoen and ST Yau, Lectures of Differential Geometry, International Press, 1997.
Jeff Cheeger, Degenerations of Riemannian metrics under Ricci curvature bounds, Publications of the Scuola Normale Superiore, Birkhauser, 2001.