Given a sufficiently dense sample of points in a Riemannian manifold, the manifold can be reconstructed up to homotopy equivalence as the Cech or Vietoris--Rips complex of the sample. I will discuss extensions of this beyond Riemannian manifolds and how optimal transport seems to be the correct point of view for metric reconstruction of metric spaces in addition to recovering the homotopy type. Time permitting, I will also try to explain how understanding homotopy types of Cech complexes of circular point sets has applications in harmonic analysis.

This is joint work in progress with Michal Adamaszek and Henry Adams.