Explore a novel approach for comparing persistent homology representations of two spaces in this lecture from the Applied Algebraic Topology Network. Delve into a method that goes beyond common numerical summaries like persistence diagrams and landscapes, instead utilizing the full topological information carried by persistent homology. Learn about a correspondence relation between topological representations of different spaces and an efficient computation method based on persistent homology variants. Discover how this framework applies to topological inference, using statistical bootstrap-like techniques to distinguish between signal and noise in point cloud data. Gain insights from the joint work of Yohai Reani and Omer Bobrowski from Technion, presented in this hour-long talk on persistent cycle registration and topological bootstrap.