Explore the connections between Vietoris-Rips persistent homology, injective metric spaces, and the filling radius in this comprehensive lecture. Delve into a geometric approach for generating persistent homology of metric spaces by embedding them into larger ambient metric spaces. Discover how this method relates to the standard Vietoris-Rips simplicial filtration, and learn about the isomorphism between these approaches when the ambient space is injective. Examine applications of this isomorphism, including characterizing intervals in persistence barcodes, analyzing products and metric gluings of metric spaces, and establishing bounds on barcode interval lengths. Investigate the relationship between geometric persistent homology and Gromov's filling radius concept, exploring implications for the homotopy type of Vietoris-Rips complexes of spheres and rigidity results for spheres based on their Vietoris-Rips persistence barcodes.