I am interested in what are the possible directions for new research in persistent homology (more of the mathematical theoretical aspects rather than the computer algorithm aspects).


So far from googling, proving stability (not affected by small changes) seems to be one direction.


Another direction I have seen is the inverse problem of persistent homology: to what extent can the original space be reconstructed from the persistent homology.



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Research directions in persistent homology
Persistent homology is related to spectral sequences. There is already a discussion [Persistence barcodes and spectral sequences](https://mathoverflow.net/questions/208406/persistence-barcodes-and-spectral-sequences) about if this leads to new theoretical insides and there are stated further references (for example, there is stated the book of Edelsbrunner and Harer, which is a standard book for Persistent homology and can find further discussions about some research directions in Edelsbrunner and Harer's works as well).


Furthermore, Persistent homology leads to discrete Morse theory, what can be used for computing cellular sheaf cohomology (if you want to know more about it, i.e. Justin Curry's work could be interesting for you). 


