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Browsing by Author "Moy, Michael, author"

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    Persistence and simplicial metric thickenings
    (Colorado State University. Libraries, 2024) Moy, Michael, author; Adams, Henry, advisor; Patel, Amit, committee member; Peterson, Christopher, committee member; Ben-Hur, Asa, committee member
    This dissertation examines the theory of one-dimensional persistence with an emphasis on simplicial metric thickenings and studies two particular filtrations of simplicial metric thickenings in detail. It gives self-contained proofs of foundational results on one-parameter persistence modules of vector spaces, including interval decomposability, existence of persistence diagrams and barcodes, and the isometry theorem. These results are applied to prove the stability of persistent homology for sublevel set filtrations, simplicial complexes, and simplicial metric thickenings. The filtrations of simplicial metric thickenings studied in detail are the Vietoris–Rips and anti-Vietoris–Rips metric thickenings of the circle. The study of the Vietoris–Rips metric thickenings is motivated by persistent homology and its use in applied topology, and it builds on previous work on their simplicial complex counterparts. On the other hand, the study of the anti-Vietoris–Rips metric thickenings is motivated by their connections to graph colorings. In both cases, the homotopy types of these spaces are shown to be odd-dimensional spheres, with dimensions depending on the scale parameters.
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    Persistence stability for metric thickenings
    (Colorado State University. Libraries, 2021) Moy, Michael, author; Adams, Henry, advisor; King, Emily, committee member; Ben-Hur, Asa, committee member
    Persistent homology often begins with a filtered simplicial complex, such as the Vietoris–Ripscomplex or the Čech complex, the vertex set of which is a metric space. An important result, the stability of persistent homology, shows that for certain types of filtered simplicial complexes, two metric spaces that are close in the Gromov–Hausdorff distance result in persistence diagrams that are close in the bottleneck distance. The recent interest in persistent homology has motivated work to better understand the homotopy types and persistent homology of these commonly used simplicial complexes. This has led to the definition of metric thickenings, which agree with simplicial complexes for finite vertex sets but may have different topologies for infinite vertex sets. We prove Vietoris–Rips metric thickenings and Čech metric thickenings have the same persistence diagrams as their corresponding simplicial complexes for all totally bounded metric spaces. This immediately implies the stability of persistent homology for these metric thickenings.