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Explore the intricacies of reconstructing hidden geometric structures in data from distance matrices in this comprehensive lecture. Delve into the classical problem of reconstructing metric measure spaces using information on distances between points from large subsets. Examine techniques for embedding these spaces in Euclidean or Hilbert spaces, with a focus on applications to smooth Riemannian manifolds. Investigate partial answers to open questions in the field, bridging the gap between pure theory and applied folklore. Gain insights into the challenges and limitations of current approaches, and discover potential avenues for future research in this fascinating area of applied algebraic topology.
