{"id":{"repo_id":"houston","oai_identifier":"oai:uh-ir.tdl.org:10657/20841"},"canonical_url":"https://search.dev.ndltd.org/etd/houston/oai:uh-ir.tdl.org:10657/20841","repository":{"repo_id":"houston","name":"University of Houston","base_url":"https://uh-ir.tdl.org/server/oai/request"},"display":{"title":"Generalized Rank Invariant : Structure, Möbius Inversion, and Persistence Diagram","abstract":"Topological Data Analysis relies on persistent homology to quantify the shape of data. Unlike the one-parameter case, multiparameter persistence modules lack a complete discrete invariant analogous to barcodes, and their classification requires tools that can capture subtle interactions across the indexing poset. In this thesis we develop and study a generalized rank invariant for $\\mathbf{vec}$-valued functors indexed by arbitrary posets. Our goal is to extend the classical rank invariant so that it reflects the full combinatorial structure of path-connected subposets. We define the rank of a persistence module over a subposet as the image of the canonical morphism between the limit and colimit of the restricted diagram within a category. This construction generalizes existing rank invariants and admits a rigorous formulation for vector-valued and set-valued functors. A pivotal contribution of this work is the derivation of a generalized persistence diagram via Möbius inversion on the lattice of connected subposets. We prove that for interval-decomposable modules, this diagram faithfully recovers the barcode, thereby establishing the generalized rank as a complete invariant in tame settings.Several explicit examples, including zigzag modules and non-interval-decomposable cases, illustrate the expressive power of the invariant and highlight its conceptual advantages over existing rank functions. Furthermore, we demonstrate that this invariant provides meaningful, computable descriptors for non-decomposable modules. Overall, this thesis provides a unified categorical framework for generalized rank invariants, extending classical persistent homology toward broader, structurally rich settings.","abstract_html":"Topological Data Analysis relies on persistent homology to quantify the shape of data. Unlike the one-parameter case, multiparameter persistence modules lack a complete discrete invariant analogous to barcodes, and their classification requires tools that can capture subtle interactions across the indexing poset. In this thesis we develop and study a generalized rank invariant for <span class=\"etd-inline-math\"><strong>vec</strong></span>-valued functors indexed by arbitrary posets. Our goal is to extend the classical rank invariant so that it reflects the full combinatorial structure of path-connected subposets. We define the rank of a persistence module over a subposet as the image of the canonical morphism between the limit and colimit of the restricted diagram within a category. This construction generalizes existing rank invariants and admits a rigorous formulation for vector-valued and set-valued functors. A pivotal contribution of this work is the derivation of a generalized persistence diagram via Möbius inversion on the lattice of connected subposets. We prove that for interval-decomposable modules, this diagram faithfully recovers the barcode, thereby establishing the generalized rank as a complete invariant in tame settings.Several explicit examples, including zigzag modules and non-interval-decomposable cases, illustrate the expressive power of the invariant and highlight its conceptual advantages over existing rank functions. Furthermore, we demonstrate that this invariant provides meaningful, computable descriptors for non-decomposable modules. Overall, this thesis provides a unified categorical framework for generalized rank invariants, extending classical persistent homology toward broader, structurally rich settings.","abstract_has_math":true,"creators":["Dutta, Arnab 1994-"],"institution":"University of Houston","degree_name":"Master of Science","degree_level":null,"degree_discipline":"Mathematics","degree_department":null,"school":null,"contributors":[],"advisors":["Ott, William"],"committee_chairs":[],"committee_members":["Kalantar, Mehrdad","Gunaratne, Gemunu"],"year":2025,"date_issued":"2025-12","date_published":"2025-12","updated_at":"2026-07-24T02:31:42Z","subjects":["GRI","Generalized Persistence Diagrams","Zigzag Persistence","Multiparameter Persistence","Persistence Modules","Category Theory","Poset-Indexed Diagrams","Interval Decomposable Modules","Möbius Inversion","TDA","Topological Data Analysis","Grothendieck Group","Symmetric Monoidal Categories","Generalized Rank Invariant"],"languages":["English"],"rights":[],"rights_urls":[],"identifier_entries":[]},"links":{"outbound_url":"https://hdl.handle.net/10657/20841","outbound_label":"Handle","outbound_source":"dc:identifier.uri"},"metadata_groups":[{"id":"people","label":"People","entries":[{"key":"dc:contributor.advisor","label":"Advisor","values":["Ott, William"]},{"key":"dc:contributor.committeemember","label":"Committee Member","values":["Kalantar, Mehrdad","Gunaratne, Gemunu"]},{"key":"dc:creator","label":"Author","values":["Dutta, Arnab 1994-"]}]},{"id":"academic_context","label":"Academic Context","entries":[{"key":"dc:date.accessioned","label":"Dc Date Accessioned","values":["2026-02-05T16:46:41Z"]},{"key":"dc:date.issued","label":"Date","values":["2025-12"]},{"key":"dc:type","label":"Dc Type","values":["Thesis"]},{"key":"thesis:degree_discipline","label":"Discipline","values":["Mathematics"]},{"key":"thesis:degree_name","label":"Degree Name","values":["Master of Science"]},{"key":"thesis:institution_name","label":"Thesis Institution Name","values":["University of Houston"]}]},{"id":"subjects_keywords","label":"Subjects and Keywords","entries":[{"key":"dc:subject","label":"Dc Subject","values":["GRI","Generalized Persistence Diagrams","Zigzag Persistence","Multiparameter Persistence","Persistence Modules","Category Theory","Poset-Indexed Diagrams","Interval Decomposable Modules","Möbius Inversion","TDA","Topological Data Analysis","Grothendieck Group","Symmetric Monoidal Categories","Generalized Rank Invariant"]}]},{"id":"language_rights","label":"Language and Rights","entries":[{"key":"dc:language.iso","label":"Language (ISO)","values":["English"]}]},{"id":"identifiers","label":"Identifiers","entries":[{"key":"dc:identifier.uri","label":"Identifier URI","values":["https://hdl.handle.net/10657/20841"]}]},{"id":"additional","label":"Additional Metadata","entries":[{"key":"dc:description.abstract","label":"Abstract","values":["Topological Data Analysis relies on persistent homology to quantify the shape of data. Unlike the one-parameter case, multiparameter persistence modules lack a complete discrete invariant analogous to barcodes, and their classification requires tools that can capture subtle interactions across the indexing poset. In this thesis we develop and study a generalized rank invariant for $\\mathbf{vec}$-valued functors indexed by arbitrary posets. Our goal is to extend the classical rank invariant so that it reflects the full combinatorial structure of path-connected subposets. We define the rank of a persistence module over a subposet as the image of the canonical morphism between the limit and colimit of the restricted diagram within a category. This construction generalizes existing rank invariants and admits a rigorous formulation for vector-valued and set-valued functors. A pivotal contribution of this work is the derivation of a generalized persistence diagram via Möbius inversion on the lattice of connected subposets. We prove that for interval-decomposable modules, this diagram faithfully recovers the barcode, thereby establishing the generalized rank as a complete invariant in tame settings.Several explicit examples, including zigzag modules and non-interval-decomposable cases, illustrate the expressive power of the invariant and highlight its conceptual advantages over existing rank functions. Furthermore, we demonstrate that this invariant provides meaningful, computable descriptors for non-decomposable modules. Overall, this thesis provides a unified categorical framework for generalized rank invariants, extending classical persistent homology toward broader, structurally rich settings."]},{"key":"dc:format.mimetype","label":"Dc Format Mimetype","values":["application/pdf"]},{"key":"dc:title","label":"Title","values":["Generalized Rank Invariant : Structure, Möbius Inversion, and Persistence Diagram"]}]}],"canonical_facts":{"dc:contributor.advisor":["Ott, William"],"dc:contributor.committeemember":["Kalantar, Mehrdad","Gunaratne, Gemunu"],"dc:creator":["Dutta, Arnab 1994-"],"dc:date.accessioned":["2026-02-05T16:46:41Z"],"dc:date.issued":["2025-12"],"dc:description.abstract":["Topological Data Analysis relies on persistent homology to quantify the shape of data. Unlike the one-parameter case, multiparameter persistence modules lack a complete discrete invariant analogous to barcodes, and their classification requires tools that can capture subtle interactions across the indexing poset. In this thesis we develop and study a generalized rank invariant for $\\mathbf{vec}$-valued functors indexed by arbitrary posets. Our goal is to extend the classical rank invariant so that it reflects the full combinatorial structure of path-connected subposets. We define the rank of a persistence module over a subposet as the image of the canonical morphism between the limit and colimit of the restricted diagram within a category. This construction generalizes existing rank invariants and admits a rigorous formulation for vector-valued and set-valued functors. A pivotal contribution of this work is the derivation of a generalized persistence diagram via Möbius inversion on the lattice of connected subposets. We prove that for interval-decomposable modules, this diagram faithfully recovers the barcode, thereby establishing the generalized rank as a complete invariant in tame settings.Several explicit examples, including zigzag modules and non-interval-decomposable cases, illustrate the expressive power of the invariant and highlight its conceptual advantages over existing rank functions. Furthermore, we demonstrate that this invariant provides meaningful, computable descriptors for non-decomposable modules. 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