Résumé : Distances in Noncommutative Geometry.
We shall give an overview of the metric aspect of Noncommutative
Geometry, underlining the link between
the distance formula in Connes' theory of spectral triple and other
distances in mathematics, in particular:
- the horizontal distance in sub-riemannian geometry,
- the Wasserstein distance in the theory of optimal transport.
We will also propose several physical interpretation of this distance,
regarding the Higgs field in the standard
modelof elementary particles, or the distance in some model of quantum
spacetimes.
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