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.