Generalized Rank Invariant : Structure, Möbius Inversion, and Persistence Diagram

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 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.

Name thesis:degree_name

Master of Science

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Houston

Year dc:date.issued

2025

Author dc:creator

• Dutta, Arnab 1994-

Advisor dc:contributor.advisor

• Ott, William

Committee members dc:contributor.committeemember

• Kalantar, Mehrdad
• Gunaratne, Gemunu

• 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

Language dc:language.iso

English

Handle dc:identifier.uri

https://hdl.handle.net/10657/20841

OAI identifier oai:identifier

oai:uh-ir.tdl.org:10657/20841