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Quantum persistent homology
Persistent homology is a powerful mathematical tool that summarizes useful information about the shape of data allowing one to detect persistent...
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Homotopy, homology, and persistent homology using closure spaces
We develop persistent homology in the setting of filtrations of (Čech) closure spaces. Examples of filtrations of closure spaces include metric...
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Persistent Homology
We’re ready to reap the fruits of our labors in the previous chapters, and attack the motivating problem from the preface: given point cloud data ,...
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A directed persistent homology theory for dissimilarity functions
We develop a theory of persistent homology for directed simplicial complexes which detects persistent directed cycles in odd dimensions. We relate...
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The extended persistent homology transform of manifolds with boundary
A shortcoming of persistent homology is that when two domains have different numbers of components or holes the persistence diagrams of any...
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Stable volumes for persistent homology
This paper proposes a stable volume and a stable volume variant, referred to as a stable sub-volume, for more reliable data analysis using persistent...
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Distributing Persistent Homology via Spectral Sequences
We set up the theory for a distributed algorithm for computing persistent homology. For this purpose we develop linear algebra of persistence...
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Rigidity of terminal simplices in persistent homology
Given a filtration function on a finite simplicial complex, stability theorem of persistent homology states that the corresponding barcode is...
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Multiparameter Persistent Homology
In Carlsson and Zomorodian (Discr Comput Geom 42, 71–93, 2009), Carlsson–Zomorodian introduced an extension of persistent homology to the setting of...
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Hochschild homology, and a persistent approach via connectivity digraphs
We introduce a persistent Hochschild homology framework for directed graphs. Hochschild homology groups of (path algebras of) directed graphs vanish...
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Field Choice Problem in Persistent Homology
This paper tackles the problem of coefficient field choice in persistent homology. When we compute a persistence diagram, we need to select a...
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Stability of 2-Parameter Persistent Homology
The Čech and Rips constructions of persistent homology are stable with respect to perturbations of the input data. However, neither is robust to...
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An Efficient Algorithm for 1-Dimensional (Persistent) Path Homology
This paper focuses on develo** an efficient algorithm for analyzing a directed network (graph) from a topological viewpoint. A prevalent technique...
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The Persistent Homology of Dual Digital Image Constructions
To compute the persistent homology of a grayscale digital image one needs to build a simplicial or cubical complex from it. For cubical complexes,...
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Multiparameter Persistent Homology via Generalized Morse Theory
We define a class of multiparameter persistence modules that arise from a one-parameter family of functions on a topological space and prove that...
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Correspondence modules and persistence sheaves: a unifying perspective on one-parameter persistent homology
We develop a unifying framework for the treatment of various persistent homology architectures using the notion of correspondence modules. In this...
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