Currently, I am occupying myself with (non-applied perspectives on) Persistent Homology, having a background in non-applied maths (representation theory).
After having spent some time on the topic however, it seems to me that this field is already very grazed, leading to the following:
Question: Which new mathematical insights can be attributed to this field, if any? Should one expect that there will be any? Or should I predominantly expect applications on actual data-driven problems?
Concerns:
By a persistent module $M$, it seems that one understands a graded $k[x]$-module (or, more general, a graded module over the ring $\bigoplus_{\mathbf R_{\geq 0}} k$ with grading in a monoid). The fundamental theorem of persistent homology states that $M$ decomposes as a direct sum of interval modules $k_{[i, j)}$, which is the module whose degree-$l$-part is $k$ if $l\in[i, j)$ and zero otherwise.
This is just the classification of f.g. modules over a PID; haven't they been aware of this?
A neat generalisation is to consider general posets for the grading, and not only $\mathbf Z$ or $\mathbf R$. To this end, one considers the $k[x]$-module $M$ as a representation of the category $\mathbf Z$ with morphisms $i\leq j$; Bubenik/Scott and Bauer/Lesnick takes some effort to develop a categorical view on persistent homology.
Are there any researchers pursuing this further? Until now, I have only found articals that employ this neat formalism of abstraction, without going beyond the basics in persistent homology.