I have been thinking about this problem for a couple of years, first alone, then with a grad student (Ashleigh Thomas), and then also with a postdoc (Justin Curry, also at this BIRS workshop). As more points of view enter, the investigation becomes more nuanced. The ideas are still under development but are at the point where a group discussion would be valuable and potentially decisive. Beyond providing an opportunity to present background on new phenomena that arise with multiple parameters, and being fodder for substantial active discussion, what I have to say about it falls squarely under the heading of "foundations of other branches of mathematics that may not be used extensively in the field so far but could be relevant" -- the "other branches" being commutative algebra (graded rings), combinatorics (posets), and algebraic geometry (real algebraic sets). These provide instructions regarding not only theoretical considerations, but also algorithmic ones: how do you input a persistence module into a computer so as to carry out statistical inference with it? A lot of this is still open -- a lot remains to be done, especially when it comes to translating the theory into useful statistics. It bears mentioning that all of this is in service to an important question in evolutionary biology for which the goal is to analyze a specific dataset consisting of images of veins in fruit fly wings.