Network-based modeling of complex systems and data using the language of graphs has
become an essential topic across a range of different disciplines. Arguably, this graph-based …

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 …

An important problem in the field of Topological Data Analysis is defining topological
summaries which can be combined with traditional data analytic tools. In recent work …

The Čech and Rips constructions of persistent homology are stable with respect to
perturbations of the input data. However, neither is robust to outliers, and both can be …

A significant part of modern topological data analysis is concerned with the design and study
of algebraic invariants of poset representations—often referred to as persistence modules …

Homological algebra of modules over posets is developed, as closely parallel as possible to
that of finitely generated modules over noetherian commutative rings, in the direction of finite …

Multigraded Betti numbers are one of the simplest invariants of multiparameter persistence
modules. This invariant is useful in theory—it completely determines the Hilbert function of …

We develop some aspects of the homological algebra of persistence modules, in both the
one-parameter and multi-parameter settings, considered as either sheaves or graded …

Given a finite set A⊂ R d, let Cov r, k denote the set of all points within distance r to at least k
points of A. Allowing r and k to vary, we obtain a 2-parameter family of spaces that grow …

We study the use of the Euler characteristic for multiparameter topological data analysis.
Euler characteristic is a classical, well-understood topological invariant that has appeared in …