Hodge Theory on Metric Spaces

  • Authors:

  • Laurent Bartholdi;Thomas Schick;Nat Smale;Steve Smale

  • Affiliations:

  • Georg-August-Universität, Göttingen, Germany;Georg-August-Universität, Göttingen, Germany;University of Utah, Salt Lake City, USA;City University of Hong Kong, Pokfulam, P.R. China

  • Venue:
  • Foundations of Computational Mathematics
  • Year:
  • 2012

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Abstract

Hodge theory is a beautiful synthesis of geometry, topology, and analysis which has been developed in the setting of Riemannian manifolds. However, 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 toward understanding the geometry of vision. Appendix B by Anthony Baker discusses a separable, compact metric space with infinite-dimensional α-scale homology.