Using a modern rigidity theory result of Gortler, Theran, and Thurston, we also proved that if G is globally rigid, then P is identifiable. G being globally rigid means you cannot teleport points in P without preserving edge lengths in G, unless you teleport via isometry.

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G being rigid means we can't wiggle the points in P without changing the edge lengths of G, unless we wiggle the points in P via isometry.

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We can quantify how big a level set is by computing its dimension. If D is a family of barcodes (one for each homological degree) with k distinct endpoint values, we showed the following inequalities hold (see picture)

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As many of my faithful followers will be aware, using either Čech or Vietoris-Rips persistent homology (PH) we get a barcode in each homological degree from any point cloud. So we have a mapping PH:
Point clouds with n points in d dimensions
->
Barcodes

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I posted about this paper this a while ago on another website, but figured I'd post it here now that I have a few followers. (Joint with @haharrington.bsky.social, Jacob Leygonie, @uzulim.bsky.social, and Louis Theran).

arxiv.org/abs/2411.08201

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