Explore emerging connections between applied topology and quantitative topology in this 58-minute lecture by Henry Adams. Delve into the history and applications of Vietoris-Rips complexes, from their invention by Vietoris for defining (co)homology theory in metric spaces to their use in geometric group theory by Rips. Discover their recent applications in computational topology for dataset shape approximation. Learn about Michael Moy's recent findings on the equivalence of persistence diagrams in Vietoris-Rips simplicial complexes and metric thickenings. Examine speculations on the homotopy types of Vietoris-Rips complexes of n-spheres. Access accompanying slides for visual aid and gain insights from this talk, which was part of the "Topological Data Analysis - Theory and Applications" workshop supported by the Tutte Institute and Western University.