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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.

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closed as too broad by Will Jagy, max_zorn, Xander Henderson, Parcly Taxel, Shailesh Jul 21 '18 at 9:28

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ I cannot parse what your first question is asking: "So, is it realistic that there are any proper insights about do come in the field around persistent homolgy/TDA?" Specifically, did you intend to write the words I italicized? Or is it a typo, so perhaps that phrase should be about to come? $\endgroup$ – Namaste Jul 20 '18 at 22:01
  • $\begingroup$ @amWhy The phrase would have been "about to come". Rephrased the question. $\endgroup$ – Hans Dieter Jul 22 '18 at 21:47