This talk introduces a method for characterizing the dynamics of
time-evolving data within the framework of topological data analysis
(TDA), specifically through the lens of persistent homology. Popular
instances of time-evolving data include flocking or swarming behaviors
in animals, and social networks in the human sphere. A natural
mathematical model for such collective behaviors is that of a dynamic
metric space. In this talk I will describe how to extend the well-known
Vietoris-Rips filtration for metric spaces to the setting of dynamic
metric spaces. Also, we extend a celebrated stability theorem on
persistent homology for metric spaces to multiparameter persistent
homology for dynamic metric spaces. In order to address this stability
property, we extend the notion of Gromov-Hausdorff distance between
metric spaces to dynamic metric spaces. This talk will not require any
prior knowledge of TDA. This talk is based on joint work with Facundo
Memoli and Nate Clause.