Dissertation
Hom-size identifiable subcategories of quiver representations for use in multi-parameter persistence theory
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Summer 2023
DOI: 10.25820/etd.006937
Abstract
Persistence theory draws on quiver representations theory and homology to generate an algebraic summary, called a persistence diagram, which details the appearance and disappearance of topological features in a filtration of a topological space. It has applications in topological data analysis, where topological spaces may be be inferred from a sampling of data points. Once the information about the life of topological features is computed, their meaning can be inferred within the setting from which the original data was drawn. Introductions to the topic can be found in \cite{Oudot, BLintro2022, BOOsigned2023}. Topological data analysis has applications to a breadth of subjects and disciplines, including music theory, urban planning, machine learning, microbiology, physics, and economics.
When filtering a topological space by a single parameter, the theory of quiver representations completely describes how to decompose the resulting representation. The complexity increases significantly when filtering by two or more parameters. In particular, multi-parameter persistence yields quivers whose indecomposable representations are more complicated to describe. The theme of this work is to provide examples of subcategories of quiver representations (which will be determined by their indecomposable objects) whose objects can be distinguished from one another in a computationally feasible manner.
Details
- Title: Subtitle
- Hom-size identifiable subcategories of quiver representations for use in multi-parameter persistence theory
- Creators
- Yariana Diaz
- Contributors
- Ryan Kinser (Advisor)Frauke Bleher (Committee Member)Benjamin Cooper (Committee Member)Isabel Darcy (Committee Member)Colleen Mitchell (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Summer 2023
- Publisher
- University of Iowa
- DOI
- 10.25820/etd.006937
- Number of pages
- vii, 101 pages
- Copyright
- Copyright 2023 Yariana Diaz
- Language
- English
- Date submitted
- 07/25/2023
- Description illustrations
- illustrations, tables
- Description bibliographic
- Includes bibliographical references (pages 99-101).
- Public Abstract (ETD)
- Persistence theory is a method for capturing and tracking the important topological (or geometric) features of a topological space (or shape). It is a subfield of topological data analysis, where it is used to analyze data sets whose shape is meaningful to the interpretation of the collected information. Topological data analysis has applications to a breadth of subjects and disciplines, including music theory, urban planning, machine learning, microbiology, physics, and economics. To use persistence theory, the first step is to filter a data set by one or more parameters (e.g. time) and then to associate algebraic structures to each snapshot of data at the various filtration steps (e.g. different times). Collectively, these algebraic structures provide information which communicates the importance of certain geometric features of the data. When filtering by a single parameter, the structure of the resulting persistence diagram is well-understood. The complexity increases significantly when filtering by two or more parameters. The theme of this work is to provide a framework to more easily compute and understand algebraic summaries in the multiparameter setting.
- Academic Unit
- Mathematics
- Record Identifier
- 9984454318702771
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