Explore the algebraic foundations of persistent homology in this 1-hour 27-minute lecture from the Hausdorff Trimester Program on Applied and Computational Algebraic Topology. Delve into the concept of persistence diagrams (barcodes) and their role in describing filtration homology. Examine the fundamental theorem stating that small input data changes result in minor persistence barcode alterations. Investigate a constructive algebraic approach to this theorem, offering a high level of generality and deeper insights into the field of computational topology.