The homology of data

Techniques and ideas from topology - the mathematical area that studies shapes - are being applied to the study of data with increasing frequency and success. Persistent homology (PH) is a method that uses homology - a tool in topology that gives a measure of the number of holes of a space - to study qualitative features of data across different values of a parameter, which one can think of as scales of resolution, and provides a summary of how long individual features persist across the different resolution scales. In many applications, data depend not only on one, but several parameters, and to apply PH to such data one therefore needs to study the evolution of qualitative features across several parameters. The theory of one-parameter PH is well-understood, but PH is computationally expensive and ways to speed up the computation are an object of current research. By contrast, the theory of multiparameter PH is hard, and it presents one of the biggest challenges in the topological study of data. In this thesis I address computational and theoretical problems in the application of homology to the study of data: I give a survey of the computation of PH and its challenges, and benchmark all stateof-the-art libraries for the computation of one-parameter PH; I propose new invariants suitable for applications for multiparameter persistent homology; finally, I relate magnitude homology, a homology theory for finite metric spaces that has been recently introduced, to persistent homology.