New directions in TDA

 Conference topic

This post is informal, meant as a collection of (personally) new things from the workshop "Topological data analysis: Developing abstract foundations" at the Banff International Research Station, July 31 - August 4, 2017. New actual questions:

1. Does there exist a constructible sheaf valued in persistence modules over $\Ran^{\leqslant n}(M)$?
   • On the stalks it should be the persistence module of $P\in \Ran^{\leqslant n}(M)$. What about arbitrary open sets?
   • Is there such a thing as a colimit of persistence modules?
   • Uli Bauer suggested something to do with ordering the elements of the sample and taking small open sets.
2. Can framed vector spaces be used to make the TDA pipeline functorial? Does Ezra Miller's work help?
   • Should be a functor from $(\R,\leqslant)$, the reals as a poset, to $\text{Vect}$ or $\text{Vect}_{fr}$, the category of (framed) vector spaces. Filtration function $f:\R^n\to \R$ is assumed to be given.
   • Framed perspective should not be too difficult, just need to find right definitions.
   • Does this give an equivalence of categories (category of persistence modules and category of matchings)? Is that what we want? Do we want to keep only specific properties?
   • Ezra's work is very dense and unpublished. But it seems to have a very precise functoriality (which is not the main thrust of the work, however).
3. Can the Bubenik-de Silva-Scott interleaving categorification be viewed as a (co)limit? Diagrams are suggestive.
   • Reference is 1707.06288 on the arXiv.
   • Probably not a colimit, because that would be very large, though the arrows suggest a colimit.
   • Have to be careful, because the (co)limit should be in the category of posets, not just interleavings.

New things to learn about:

1. Algebraic geometry / homotopy theory: the etale space of a sheaf, Kan extensions, model categories, symmetric monoidal categories.
2. TDA related: Gromov-Hausdorff distance, the universal distance (Michael Lesnick's thesis and papers), merge trees, Reeb graphs, Mapper (the program).