Through the use of examples, we explain one way in which applied topology has evolved since the birth of persistent homology in the early 2000s. The first applications of topology to data emphasized the global shape of a dataset, such as the three-circle model for 3 × 3 pixel patches from natural images, or the configuration space of the cyclo-octane molecule, which is a sphere with a Klein bottle attached via two circles of singularity. In these studies of global shape, short persistent homology bars are disregarded as sampling noise. More recently, however, persistent homology has been used to address questions about the local geometry of data. For instance, how can local geometry be vectorized for use in machine learning problems? Persistent homology and its vectorization methods, including persistence landscapes and persistence images, provide popular techniques for incorporating both local geometry and global topology into machine learning. Our meta-hypothesis is that the short bars are as important as the long bars for many machine learning tasks. In defense of this claim, we survey applications of persistent homology to shape recognition, agent-based modeling, materials science, archaeology, and biology. Additionally, we survey work connecting persistent homology to geometric features of spaces, including curvature and fractal dimension, and various methods that have been used to incorporate persistent homology into machine learning.

A metric thickening of a given metric space X is any metric space admitting an isometric embedding of X. Thickenings have found use in applications of topology to data analysis, where one may approximate the shape of a dataset via the persistent homology of an increasing sequence of spaces. We introduce two new families of metric thickenings, the p-Vietoris-Rips and p-Čech metric thickenings for all 1 ≤ p ≤ ∞, which include all measures on X whose p-diameter or p-radius is bounded from above, equipped with an optimal transport metric. The p-diameter (resp. p-radius) of a measure is a certain p relaxation of the usual notion of diameter (resp. radius) of a subset of a metric space. These families recover the previously studied Vietoris-Rips and Čech metric thickenings when p = ∞. As our main contribution, we prove a stability theorem for the persistent homology of p-Vietoris-Rips and p-Čech metric thickenings, which is novel even in the case p = ∞. In the specific case p = 2, we prove a Hausmanntype theorem for thickenings of manifolds, and we derive the complete list of homotopy types of the 2-Vietoris-Rips thickenings of the n-sphere as the scale increases.

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