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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.

Are there any other research or generalizations of persistent homology?

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    $\begingroup$ Are you aware of several recent monographs on persistent homology? $\endgroup$
    – Victor Protsak
    Jan 20, 2017 at 7:04
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    $\begingroup$ @VictorProtsak No, I am not. Could you mention some of them? Thanks $\endgroup$
    – yoyostein
    Jan 20, 2017 at 7:13
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    $\begingroup$ Multi-parameter persistent homology is a fairly active branch of the subject, as of late. There's the Carlsson paper from the arXiv, Apr 1st "Persistent Homology and Applied Homotopy Theory". I imagine that's one of the papers Victor refers to. $\endgroup$
    – Ryan Budney
    May 13, 2020 at 18:26

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Persistent homology is related to spectral sequences. There is already a discussion 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).

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Persistent homology has applications in symplectic geometry. For one striking example, see the recent paper Autonomous Hamiltonian flows, Hofer's geometry and persistence modules by Polterovich and Shelukhin. They use persistent homology to give very explicit examples of area-preserving diffeomorphisms of closed surfaces which cannot be included into $1$-parameter subgroups of such.

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