Explore a rigorous framework for tracking noise in persistent homology computations in this 1-hour 5-minute talk from the Applied Algebraic Topology Network. Delve into recent advancements that allow for precise monitoring of errors introduced during barcode and persistence diagram calculations. Examine various examples, including sub-sampling and discretization techniques, and compare this approach to traditional uniform error measurements using Bottleneck distance. Discover how this framework addresses an open problem related to non-uniform sub-level set filtrations. Cover topics such as two-dimensional fluid flows, persistent homology, stability, visualization, persistence modules, and digital image analysis.