Explore persistent homotopy groups of metric spaces in this 46-minute lecture from the Applied Algebraic Topology Network. Delve into the study of persistent homotopy groups of compact metric spaces and their stability properties in the Gromov-Hausdorff sense. Discover how the classical fundamental group exhibits a tree-like structure and an associated ultrametric under mild assumptions. Examine pairs of filtrations that are confounded by persistent homology but distinguished by persistent homotopy groups. Learn about persistent rational homotopy groups and their advantages in providing additional information compared to persistent homology. Cover topics including motivation, previous work, discrete homotopic theory, persistent fundamental groups, stability, dendrograms, computational perspectives, and interleaving distance.