**Title:** Collapsing, Differential Forms and Spinors

**Speaker:** Professor John Lott

**Speaker Info:** University of Michigan

**Brief Description:**

**Special Note**:

**Abstract:**

In 1969, Jeff Cheeger gave a lower bound on the smallest positive eigenvalue of the function Laplacian on a Riemannian manifold, in terms of an isoperimetric constant. He posed the problem of extending his bound from the case of the Laplacian on functions, to the case of the Laplacian on differential forms. The Laplacian on differential forms is a more complicated operator, but has interesting connections to topology because of Hodge theory.I will give a general introduction to Cheeger's bound and then talk about recent progress on his problem, using results on collapsing due to Cheeger, Fukaya and Gromov. Their theory separates manifolds into two types : those which can't collapse (and have some rigidity properties), and those which can collapse to a lower-dimensional space. We can apply this to Cheeger's problem by looking at what happens to the differential form Laplacian as a manifold collapses.

I will explain what sort of structures appear on the lower-dimensional space and give applications to Cheeger's problem. If there is time, I will discuss how the methods extend to Dirac operators.

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