Persistent homology and applied homotopy theory

The output of standard persistent homology is represented in two ways, via persistence barcodes and persistence diagrams. Initially persistent homology was used, as homology is used for topological spaces, to obtain a large scale geometric understanding of complex data sets, encoded as finite metric spaces. Persistent homology has as its output a diagram of complexes, parametrized by the partially ordered set of real numbers, on which algebraic computations are performed so as to produce barcodes. Since persistent homology is used to analyze data sets, and data sets are often noisy in the sense that one does not want to assign meaning to small changes, it is important to analyze the stability of persistent homology outputs to small changes in the underlying data. Persistent homology gives ways of assessing the shape of a finite metric space. One situation where this is very useful is in problems in evolution.