Statistics with bars in persistent homology

Sayan and his collaborators pioneered the formalization of statistics in persistent homology by answering fundamental questions like whether there are spaces of persistence diagrams that admit reasonable probability distributions, and how to compute basic summaries like Fréchet means in such situations. These advances rest on the meaning, stability, and completeness of bars as features in one parameter. What happens in multiple parameters? Various ways of thinking about "bars" diverge for nonlinear posets, giving rise to a multitude of options for how to attempt statistics. All of the options are complicated by the much richer algebra of modules over posets and, crucially, how the algebra interacts with statistical considerations such as metrics and variation. This talk is an opportunity to take stock of where we have been and how we might move forward. It is dedicated to Sayan's memory, influence, and friendship.