" 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 …

Persistent homology (PH) is a method used in topological data analysis (TDA) to study
qualitative features of data that persist across multiple scales. It is robust to perturbations of …

Persistence theory emerged in the early 2000s as a new theory in the area of applied and
computational topology. This book provides a broad and modern view of the subject …

Voronoi diagrams partition space according to the influence certain sites exert on their
environment. Since the 17th century, such structures play an important role in many areas …

Geometric and topological inference deals with the retrieval of information about a geometric
object using only a finite set of possibly noisy sample points. It has connections to manifold …

Manifold reconstruction has been extensively studied for the last decade or so, especially in
two and three dimensions. Recent advances in higher dimensions have led to new methods …

In this paper we discuss three results. The first two concern general sets of positive reach:
we first characterize the reach of a closed set by means of a bound on the metric distortion …

Comments• page viii, bottom of page: the following names should be added to the
acknowledgements:-Peter Landweber had an invaluable contribution to these notes. First …

It is a well-established fact that the witness complex is closelyrelated to the restricted
Delaunay triangulation in lowdimensions. Specifically, it has been proved that the witness …

We present a novel reconstruction algorithm that, given an input point set sampled from an
object S, builds a one-parameter family of complexes that approximate S at different scales …