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][1] and [Rob Ghrist][2].

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][3], 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. 


  [1]: http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/
  [2]: http://www.math.upenn.edu/~ghrist/preprints/barcodes.pdf
  [3]: http://www.cs.dartmouth.edu/~afra/papers/socg04/persistence.pdf