Explore the mathematical framework for featurizing persistence diagrams using template functions in this 52-minute lecture from the Applied Algebraic Topology Network. Delve into the challenges of integrating persistence diagrams with machine learning techniques and discover how template functions can maximize preserved structure when mapping diagrams to Euclidean space. Learn about two exemplar template function families and their applications to synthetic and real datasets. Gain insights into topological data analysis, persistent homology, and the characterization of relatively compact sets. Examine various coordinate systems, including birth-lifetime coordinates, and understand how to combine functions and diagrams to derive numerical values. Investigate specific template systems such as tent functions and Chebychev polynomials, and explore experiments with random diagrams and manifolds. Conclude with an overview of current and future work in adaptive partitioning.