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 N d -indexed multiparameter persistence modules is established and decomposabil-ity and key properties of barcode admissible modules are explored. Relationships between the rank invariant, and the N d -indexed persistence modules that admit bar-codes 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 N d -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 N d -indexed persis-tence module admits a barcode is developed. An extension of Forman’s discrete Morse theory is presented. Some of the exist-ing 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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Ph. D.