# Research directions in persistent homology

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?

• Are you aware of several recent monographs on persistent homology? – Victor Protsak Jan 20 '17 at 7:04
• @VictorProtsak No, I am not. Could you mention some of them? Thanks – yoyostein Jan 20 '17 at 7:13
• Surely this question should be community-wiki? – HJRW Jan 23 '17 at 7:10
• 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. – Ryan Budney May 13 '20 at 18:26

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.