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How much geometry is in a truncated spectral triple?

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A spectral triple consists of an algebra, a Hilbert space and a Dirac operator, and if these three fulfill certain relations to each other they contain the entire information of a compact Riemannian manifold.
Using the language of spectral triples makes it possible to generalize the concept of a manifold to include non-commutativity.

While it is possible to write down finite spectral triples, often categorized as fuzzy spaces, that describe discretized geometries, classical geometries are encoded in infinite dimensional spectral triples. However working in numerical systems (and maybe ultimately in physical systems), only a finite part of this information can be encoded, which opens the question; If we know a part of the spectrum, how clearly can we characterise a geometry.
In this talk I will present first steps towards answering this question.