Explore a practical framework for dimensionality reduction, comparison, and optimization across various mathematical objects in this 41-minute conference talk. Delve into the world of vector-space valued representations of quivers, encompassing single and multiparameter persistence modules, cellular sheaves, and connection matrices. Learn about ear decomposition, terminal edges, acyclification, and augmentation as key concepts in this approach. Discover how flow spaces, maps, and the space of sections contribute to understanding principal components in quiver representations. Gain insights into alternate perspectives and spectral interpretations of this framework. Join Vidit Nanda as he presents joint work with Anna Seigal and Heather Harrington, offering a comprehensive look at this innovative mathematical approach.