  One approach to understanding complex data is to study its shape through the
lens of algebraic topology. While the early development of topological data
analysis focused primarily on static data, in recent years, theoretical and
applied studies have turned to data that varies in time. A time-varying
collection of metric spaces as formed, for example, by a moving school of fish
or flock of birds, can contain a vast amount of information. There is often a
need to simplify or summarize the dynamic behavior. We provide an introduction
to topological summaries of time-varying metric spaces including vineyards
[19], crocker plots [56], and multiparameter rank functions [37]. We then
introduce a new tool to summarize time-varying metric spaces: a crocker stack.
Crocker stacks are convenient for visualization, amenable to machine learning,
and satisfy a desirable continuity property which we prove. We demonstrate the
utility of crocker stacks for a parameter identification task involving an
influential model of biological aggregations [58]. Altogether, we aim to bring
the broader applied mathematics community up-to-date on topological summaries
of time-varying metric spaces.
