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 …

This paper introduces a new data structure, called simplex tree, to represent abstract
simplicial complexes of any dimension. All faces of the simplicial complex are explicitly …

Scientific data is often in the form of a finite set of noisy points, sampled from an unknown
space, and embedded in a high-dimensional space. Topological data analysis focuses on …

We seek to determine a real algebraic variety from a fixed finite subset of points. Existing
methods are studied and new methods are developed. Our focus lies on aspects of topology …

Morse theory offers a natural and mathematically‐sound tool for shape analysis and
understanding. It allows studying the behavior of a scalar function defined on a manifold …

The quest for low-dimensional models which approximate high-dimensional data is
pervasive across the physical, natural, and social sciences. The dominant paradigm …

Natural product and synthetic macrocycles are chemically and topologically diverse. An
efficient, accurate, and general method for generating macrocycle conformations would …

Suppose that ball-shaped sensors wander in a bounded domain. A sensor does not know its
location but does know when it overlaps a nearby sensor. We say that an evasion path …

Given a sample of points X in a metric space M and a scale r>0, the Vietoris--Rips simplicial
complex VR(X;r) is a standard construction to attempt to recover M from X up to homotopy …

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 …