Explore the application of Principal Component Analysis (PCA) to persistent homology rank functions in this insightful 57-minute lecture. Delve into topological summaries, rank functions, and persistence concepts while examining the weighted version of these techniques. Learn how to apply PCA to extract meaningful information from complex datasets and understand its stability through practical examples. Gain valuable insights into the intersection of algebraic topology and data analysis, presented by Katharine Turner for the Applied Algebraic Topology Network.