Explore the Vietoris-Rips complex of the circle in this 57-minute lecture from the Applied Algebraic Topology Network. Delve into the behavior of Vietoris-Rips complexes for varying distance thresholds, starting with Jean-Claude Hausmann's theorem on small thresholds for Riemannian manifolds. Discover how the homotopy type of the Vietoris-Rips complex of the circle evolves as the distance threshold increases, progressing through various spheres before becoming contractible. Learn about the applications of these complexes in persistent homology and gain insights from the joint work with Michal Adamaszek. Access accompanying slides for visual support of the concepts presented.