Explore the algorithmic complexity of computing persistent homology for random Vietoris-Rips filtrations in this 52-minute conference talk by Barbara Giunti. Delve into the first theoretical study of its kind, examining upper bounds for the average fill-up of the boundary matrix after reduction. Discover how these bounds suggest a significantly sparser reduced matrix compared to general worst-case predictions. Learn about the connection between expected first Betti numbers of Vietoris-Rips complexes and boundary matrix fill-up. Understand the broader applicability of these results to clique filtrations that "become acyclic fast enough" and their potential expansion to other degrees and filtrations. Examine benchmarks demonstrating the asymptotic tightness of the bound for Vietoris-Rips complexes. If time allows, explore the construction of a clique filtration achieving worst-case fill-up and complexity.