Persistent homology is a well-developed tool which allows topological analysis of large data sets. From a topological perspective, the input is a filtered complex, and the output is a sequence of collections of intervals (one for each dimension) called a *persistence barcode*. The barcode gives information about homology classes which are born and die as you vary the scale (filtration parameter).

This is a very brief, non-expert summary. By now there are several good survey articles on the subject by experts in the field, for example by Gunnar Carlsson and Rob Ghrist.

On the other hand, given a filtered complex $X_\bullet$, one obtains a spectral sequence converging to the homology $H_\ast(X_\bullet)$ (or at least its associated graded object) . It is natural to ask how the persistence barcode relates to this spectral sequence. In a formative paper on the subject by Carlsson and Zomorodian, the authors ask exactly this question in section 1.4 of the introduction, claiming that a persistence interval of length $r$ in the barcode corresponds to a differential $d_{r+1}$. Thus, in principle, any algorithm for computing persistent homology should give an algorithmic way of computing the differentials in a spectral sequence. So persistent homology, which already has many applications outside of topology, becomes potentially applicable to topology itself.

Has anyone ever pursued this approach, and used algorithms for persistent homology to compute the differentials in a spectral sequence? Does this lead to any new theoretical insights?

I am imagining that by knowing the values of a differential in a given situation one might guess at a description of the differential (eg, in terms of cohomology operations) which applies more generally.

**Edit:** The book Computational Topology: An Introduction by Edelsbrunner and Harer states a Spectral Sequence Theorem in Chapter VII.4, which says roughly that the total rank of the $E^r_{\ast,\ast}$ page of the spectral sequence equals the number of homology classes of persistence $r$ or larger. Here coefficients are taken mod 2. This makes precise the claim made by Carlsson and Zomorodian.