Explore recent advances in persistent topological invariants through this mathematical lecture that extends classical notions of cup length and Lusternik–Schnirelmann category to the persistent setting. Learn about persistent analogs of topological complexity and zero-divisor cup length, including their stability properties and behavior under Vietoris–Rips filtrations of compact metric spaces. Discover how these invariants relate to the Gromov–Hausdorff distance and examine their discriminating power compared to other metric invariants such as persistent homology. Understand the development of effective lower bounds for the Gromov–Hausdorff distance through these topological tools. The presentation covers joint research with Ling Zhou from Duke University, offering insights into the intersection of algebraic topology, metric geometry, and persistent homology theory.