Saecular Persistence

Explore the concept of saecular persistence in this comprehensive lecture from the Applied Algebraic Topology Network. Delve into a categorically natural method for decomposing persistence modules with non-field coefficients into families of interval modules. Examine how this approach generalizes existing factorizations of 1-parameter persistence modules, leading to persistence diagrams in both homology and homotopy. Discover applications of saecular decomposition, including inverse and extension problems involving filtered topological spaces, generalized persistence diagrams, and the Leray-Serre spectral sequence. Learn about key tools such as modular, semimodular, and distributive order lattices, as well as RE/Puppe exact categories. Gain insights into how this approach extends beyond field coefficients to integer coefficients and homotopy, offering a broader perspective on topological data analysis.