Hodge theory on metric spaces. Appendix by Anthony W. Baker. (English) Zbl 1366.58001
Summary: 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 \(\alpha \)-scale homology.
Appendix B by Anthony Baker discusses a separable, compact metric space with infinite-dimensional \(\alpha \)-scale homology.
MSC:
| 58A14 | Hodge theory in global analysis |
| 55P55 | Shape theory |
| 57M50 | General geometric structures on low-dimensional manifolds |
| 58E05 | Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces |
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