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Peer-reviewedCC BY-NCFoundations of Computational Mathematics

Hodge theory on metric spaces

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

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Cite

BibTeXRIS
BibTeX
@article{bartholdi2026hodge,
  title = {{Hodge theory on metric spaces}},
  author = {Laurent Bartholdi and Thomas Schick and Nat Smale and Steve Smale},
  journal = {Foundations of Computational Mathematics},
  year = {2026},
  month = nov,
  doi = {10.1007/s10208-011-9107-3},
  eprint = {0912.0284},
  archivePrefix = {arXiv},
  url = {https://hal.science/hal-05849587v1},
  note = {CC BY-NC},
}
RIS
TY  - JOUR
TI  - Hodge theory on metric spaces
AU  - Laurent Bartholdi
AU  - Thomas Schick
AU  - Nat Smale
AU  - Steve Smale
PY  - 2026
DA  - 2026/11/03
JO  - Foundations of Computational Mathematics
DO  - 10.1007/s10208-011-9107-3
UR  - https://hal.science/hal-05849587v1
AB  - 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.
SN  - 1615-3375
SN  - 1615-3383
ER  -