In topological data analysis (TDA), one often studies the shape of data by constructing a
filtered topological space, whose structure is then examined using persistent homology …

" In this chapter, we introduce some of the very basics that are used throughout the book.
First, we give the definition of a topological space and related notions of open and closed …

The algebraic stability theorem for persistence modules is a central result in the theory of
stability for persistent homology. We introduce a new proof technique which we use to prove …

We show that computing the interleaving distance between two multi-graded persistence
modules is NP-hard. More precisely, we show that deciding whether two modules are 1 …

Topological data analysis and its main method, persistent homology, provide a toolkit for
computing topological information of high-dimensional and noisy data sets. Kernels for one …

The matching distance is a pseudometric on multi-parameter persistence modules, defined
in terms of the weighted bottleneck distance on the restriction of the modules to affine lines. It …

Characterizing the dynamics of time-evolving data within the framework of topological data
analysis (TDA) has been attracting increasingly more attention. Popular instances of time …

The classical persistence algorithm computes the unique decomposition of a persistence
module implicitly given by an input simplicial filtration. Based on matrix reduction, this …

One approach to understanding complex data is to study its shape through the lens of
algebraic topology. While the early development of topological data analysis focused …

Motivated by applications to topological data analysis, we give an efficient algorithm for
computing a (minimal) presentation of a bigraded Kx,y-module M, where K is a field. The …