The last ten years witnessed the rapid development of
algebraic tools to measure scale. The ability to measure
scale is needed whenever we assess the importance of a
feature in a signal. We think of the signal generally
as a dataset we collect and we aim for a measure
that identifies aspects of the data that are sufficiently
important to be considered above the noise level.
The main mathematical idea in this development is the concept of persistent homology of a real-valued function on a space. Think for example of a subset of the plane and consider the function on the plane that maps every point in the plane to the Euclidean distance from that subset. Persistence analyzes which homological features of the subset can be recognized at which distance thresholds. This analysis leads to the concepts of persistence diagrams and proofs of stability of these diagrams. They have ramifications inside and outside mathematics.