Explore the fundamentals and applications of topological data analysis (TDA) in matrix analysis through this informative lecture by Nicole Sanderson from Penn State University. Delivered at IPAM's Mathematical Approaches for Connectome Analysis Workshop, the talk introduces persistent homology as a powerful TDA algorithm for analyzing matrix structure. Learn how Betti curves can be used to compare data to null models and see practical applications in analyzing neural correlation matrices from calcium imaging data of zebrafish larvae. Gain insights into potential uses for connectivity matrices in connectome data analysis. The lecture includes a demonstration of open-source TDA software and provides a comprehensive overview of this emerging subfield of applied mathematics and its relevance to neuroscientific research.