Explore persistent homotopy groups and cohomology rings in this 45-minute lecture from the Applied Algebraic Topology Network. Delve into persistent fundamental groups, their description via dendrograms, and the generalization of hierarchical clustering. Examine persistent rational homotopy groups and discover how they differentiate filtrations that persistent homology cannot. Investigate the persistent graded ring structure induced by the cup product and learn about the efficiently computable persistence cup-length diagram. Cover topics including stable fundamental groups, couplings functions and diagrams, stability, and algorithms, complementing the information captured by persistent homology.