# Hodge Theory for Metric Spaces

Tuesday, October 29, 2013 - 3:15pm - 4:05pm

Keller 3-180

Thomas Schick (Georg-August-Universität zu Göttingen)

Hodge theory is a beautiful synthesis of geometry, topology, and analysis which has been developed in the setting of Riemannian manifolds. However, many spaces important in applications do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure.

The goal here is to obtain a theory which is sensitive to features which are present at a given scale. Our approach is based on an Alexander-Spanier replacement of differential forms, with a localization with parameter a.

We discuss that, unfortunately, for completely general metric spaces the theory does not have all nice analytic features one would hope to have.

On the other hand, as the main test case, we provide conditions for the local structure of the metric measure space which implies that the a-scale cohomology is isomorphic with ordinary

homology. In particular, this applies to Riemannian manifolds (if we choose the scale small enough in terms of the injectivity radius and the sectional curvature).

The goal here is to obtain a theory which is sensitive to features which are present at a given scale. Our approach is based on an Alexander-Spanier replacement of differential forms, with a localization with parameter a.

We discuss that, unfortunately, for completely general metric spaces the theory does not have all nice analytic features one would hope to have.

On the other hand, as the main test case, we provide conditions for the local structure of the metric measure space which implies that the a-scale cohomology is isomorphic with ordinary

homology. In particular, this applies to Riemannian manifolds (if we choose the scale small enough in terms of the injectivity radius and the sectional curvature).

MSC Code:

58A14

Keywords: