Persistence spectral sequences

This Doctoral thesis is centered on connections between persistent homology and spectral sequences. We explain some of the approaches in the literature exploring this connection. Our main focus is on Mayer-Vietoris spectral sequences associated to filtered covers on filtered complexes. A particular case of this spectral sequence is used for measuring exact changes on barcode decompositions under small perturbations of the underlying data. On the other hand, these objects allow for a setup to parallelize persistent homology computations, while retaining useful information related to the chosen covers. We explore some generalizations of the traditional setup to diagrams of regular complexes consisting of regular morphisms; these become useful for working with non-sparse complexes. In addition, we explore stability results related to these new invariants, both with respect to local changes and with respect to changes on the chosen covering sets. Finally, we present some computational experiments by the use of PERMAVISS which illustrate some of these ideas