Geometric Analysis and Topology Seminar

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Examples and theorems in the collapsing geometry of smooth and non-smooth spaces

Speaker: Ruobing Zhang, University of Wisconsin Madison

Location: Warren Weaver Hall 512

Date: Friday, September 27, 2024, 11 a.m.

Synopsis:

The degenerations of Einstein metrics, in the volume non-collapsed setting, have been extensively studied since the 1990s, which has culminated in a number of profound structure results due to Cheeger, Naber, and their collaborators. In contrast with the successful progress in understanding non-collapsing geometry, the structure of collapsing Einstein manifolds have not been well explored. Except for a pioneering regularity result in dimension four due to Cheeger-Tian, very little is known in general. 

In this talk, we will first summarize some recent progress for collapsing Einstein manifolds from different perspectives (such as metric Riemannian geometry, complex geometry, etc). Also, we will introduce some new observations in classical collapsing geometry and propose a few questions for collapsing metric measure spaces.