Hodge Theory on Metric Spaces

TL;DR

This paper extends Hodge theory to metric spaces with probability measures, aiming to bridge geometry and vision, and includes an example of infinite dimensional homology in such spaces.

Contribution

It introduces a novel version of Hodge theory applicable to metric measure spaces, broadening the scope beyond Riemannian manifolds.

Findings

Development of Hodge theory on metric spaces with measures

Construction of a separable, compact metric space with infinite dimensional homology

Potential implications for understanding the geometry of vision

Abstract

Hodge theory is a beautiful synthesis of geometry, topology, and analysis, which has been developed in the setting of Riemannian manifolds. On the other hand, spaces of images, which are important in the mathematical foundations of vision and pattern recognition, do not fit this framework. This motivates us to develop a version of Hodge theory on metric spaces with a probability measure. We believe that this constitutes a step towards understanding the geometry of vision. The appendix by Anthony Baker provides a separable, compact metric space with infinite dimensional \alpha-scale homology.