A structure theory for spaces with lower Ricci curvature bounds

Aaron Naber, Northwestern & Minerva Distinguished Visitor
Fine Hall 314

One should view manifolds (M^n,g) with lower Ricci curvature bounds as being those manifolds with a well behaved analysis, a point which can be rigorously stated. It thus becomes a natural question, how well behaved or badly behaved can such spaces be? One studies this by taking noncollapsed sequences (M^n_i,g_i)->(X,d) which limit to some metric space, and asking how badly behaved X may look? This is a nonlinear analogue to asking how degenerate can a subharmonic or plurisubharmonic function look like. In this talk we give an essentially sharp answer to this question. The work discussed is joint with Cheeger, Jiang and with Li.