Explore the concept of stable homology in metric measure spaces through this 58-minute lecture by Washington Mio. Delve into the theoretical framework, including scale-space representation of data, Wasserstein distance, and stability of local homology. Examine practical applications with experiments on sampling distributions and Fréchet functions. Investigate advanced topics such as localization via modulation, metric spaces with diffusion kernels, and taxonomic classification. Gain insights into random local persistence diagrams and their role in applied algebraic topology.