Explore the mathematical concepts of persistence diagrams and their associated metrics in this 28-minute lecture. Delve into the universality of persistence diagrams and gain a deeper understanding of bottleneck and Wasserstein distances. Learn how these tools are applied in topological data analysis and their significance in capturing the shape and structure of data. Discover the theoretical foundations and practical applications of these concepts as presented by Alex Elchesen from the Applied Algebraic Topology Network.