# Geometric Analysis and Topology Seminar

#### Gromov-Hausdorff Convergence and the Cut-Off Covering Spectrum

**Speaker:**
Christina Sormani, CUNY

**Location:**
Warren Weaver Hall 1013

**Date:**
Friday, April 6, 2007, 3 p.m.

**Synopsis:**

The covering spectrum of a length space (or Riemannian manifold) roughly measures the size of its one dimensional holes. On compact length spaces, it has been shown to be closely related to the length spectrum and useful for determining when the space has a universal cover. It is also continuous under Gromov-Hausdorff convergence of compact spaces.

On complete spaces, I will demonstrate that the covering spectrum is not so well behaved. I will then introduce the R cut-off covering spectrum which only detects a localized collection of the holes in the space. It is closely related to the length spectrum and is continuous with respect to pointed Gromov-Hausdorff convergence when the spaces are locally compact. I then build the cut-off covering spectrum which is not localized and examine its properties. The talk will close with applications to spaces with curvature bounds. The concepts and results are new even for Riemannian manifolds. This is joint work with Guofang Wei.