## ABSTRACT

Persistent homology is a technique that has been developed over the last 20 years. Initial ideas developed in the early 1990s [36], but the idea of persistence was introduced by Vanessa Robins in [60], rapidly followed by additional development ([34], [68]), and has been developing rapidly since that time. The original motivation for the method was to extend the ideas of algebraic topology from the category of spaces X to situations where we only have a sampling of the space X. Of course, a sample is a discrete space so there is nothing to be obtained unless one retains some additional information. One assumes the presence of a metric or a more relaxed “dissimilarity measure”, and uses this information restricted to the sample in constructing the algebraic invariant. Over time, persistent homology has been used in other situations, for example where one has a topological space with additional information, such as a continuous real valued function, and the sublevel sets of the function determine a filtration on the space. The output of standard persistent homology (we will discuss some generalizations) 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. Examples of this kind of application are [21], [39], [46], [59], and [25]. Another class of applications uses persistent homology to study data sets where the points themselves are metric spaces, such as databases of molecule structures or images. This second set of applications is developing very rapidly and is exemplified in [15], [16], [66], and [43]. Another direction in which persistent homology is being applied is in the study of coverage and evasion problems arising in sensor net technology [27]. Example of research in this direction are [31], [32], [1], and [38].