Singularity Analysis on Manifolds with Ricci Curvature Bounds

Singularity Analysis on Manifolds with Ricci Curvature Bounds

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Aaron Naber, Minerva Distinguished Visitor
Fine Hall 110

We begin this talk by discussing the ideas and methods involved in the codimension four conjecture, a joint work with Jeff Cheeger. The conjecture states that a metric space M_i->X which is a limit of manifolds with two sided Ricci curvature bounds must be smooth away from a codimension four set. The proof of this conjecture really comes down to understanding codimension two singularities of X well enough to rule them out. The goal of the second half of this talk will be to discuss the apriori existence of L^2 curvature bounds on manifolds with Ricci curvature bounds, joint with Wenshuai Jiang. This result involves a much more refined analysis requiring several new ideas as one needs to push the analysis all the way to the codimension four part of the singular set itself. The result in particular implies the n-4 finiteness conjecture of Cheeger and Colding, though in the process we prove much more, including rectifiable structure on the singular set.