ABSTRACT
This chapter explains how to recover topological properties about the shape from which one's dataset is sampled. Topology is a way to quantify the number of holes, or topological features of each dimension in a space. The chapter describes example applications of topology to data analysis and introduces simplicial complexes, which are the data structure to store topological spaces on a computer. It discusses homology which is a way to count the number of holes of each dimension in a space. The chapter describes persistent homology, which was developed in the late 1990s and early 2000s as a way to compute topological properties when one is given only a dataset, i.e., a finite sampling from some unknown underlying space. The chapter provides a discussion on sublevelset persistence, a survey of software packages for topological data analysis, and a brief pointer to related papers and expository articles.
