Multiparameter persistence modules
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In this dessertation, Hausdorff and Gromov-Hausdorff distances are discussed. Building on higher dimensional partitions and their Young diagrams, a framework for Nd-indexed multiparameter persistence modules is established and decomposability and key properties of barcode admissible modules are explored. Relationships between the rank invariant, and the Nd-indexed persistence modules that admit barcodes are investigated and, a necessary and sufficient condition for two persistence modules admitting barcodes to share the same rank invariant is shown. A connection between the barcode admissibility of an Nd-indexed persistence module and those of the persistence sub-modules associated to the connected components of its Hasse graph is discussed. Lastly, an algorithm which verifies whether an Nd-indexed persistence module admits a barcode is developed. An extension of Forman's discrete Morse theory is presented. Some of the existing theorems are corrected and the collapse theory under group actions is explored. Additionally, in connection to persistence homology, a Morse-theoretic proof of the structure theorem is offered.
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