If TDA wants to move away from the study of persistence diagrams a posteriori and towards the construction of analytical models or predictions of persistence diagrams from calculus-based physics-type models, an analytical (rather than algorithmic) study of the persistence map must be undertaken. Beyond stability, remarkably few things are known about the persistence map. In this talk, I will provide a complete characterization of the image and the fiber of the persistence map for time series (functions from the interval to the real line) as detailed in a recent arXiv preprint. This talk will require very little background knowledge and will contain several open problems.