Explore a 49-minute lecture on metric reconstruction using optimal transport theory. Delve into the Vietoris-Rips simplicial complex and its limitations in recovering metric information from sample points in a metric space. Learn about a novel approach to address this issue through the Vietoris-Rips thickening, defined using optimal transport theory. Discover how this method improves upon traditional techniques for recovering homotopy type, particularly for complete Riemannian manifolds. Examine the simplified proof of Hausmann's theorem and its implications for metric reconstruction. Investigate the homotopy type of the Vietoris-Rips thickening of the n-sphere at critical scale parameters. Access accompanying slides for visual aids and further understanding of this joint work by Henry Adams, Michal Adamaszek, and Florian Frick.