Dissertation
Representation theory in persistence and quantum symmetries
University of Iowa
Doctor of Philosophy (PhD), University of Iowa
Spring 2025
DOI: 10.25820/etd.007991
Abstract
The first half of this thesis focuses on the application of quivers and their representations in topological data analysis (TDA), particularly persistent homology. After outlining the TDA pipeline, we analyze the stability of the $p$-presentation distance introduced in \cite{BjerkLes}. In doing so, we contribute a bound for the $p$-presentation distance between two persistence modules. We also give criterion on $p$ and on the persistence modules that ensure this distance is finite.
In the second half of this thesis, we use the language of Hopf actions to explore quantum symmetries. To generalize the special cases studied by Kinser and Oswald in \cite{KO21}, we utilize Hopf-Ore extensions of group algebras to encompass many Hopf algebras, including the special cases already studied. Specifically, we classify the (filtered) Hopf actions of Hopf-Ore extensions of group algebras on path algebras of quivers. Utilizing path algebras of quivers allows us to discuss any Hopf action on a finite-dimensional basic algebra as such algebras are isomorphic to a quotient of the path algebra by an admissible ideal. Having done this, we extend the work to more general Hopf-Ore extensions via a more general comultiplication and iterated Hopf-Ore extensions. We conclude by demonstrating application by specializing our main result to certain Hopf-Ore extensions including some Noetherian Hopf algebras of GK-dimension one and two.
Details
- Title: Subtitle
- Representation theory in persistence and quantum symmetries
- Creators
- Elise Askelsen
- Contributors
- Ryan Kinser (Advisor)Frauke Bleher (Committee Member)Benjamin Cooper (Committee Member)Michail Savvas (Committee Member)
- Resource Type
- Dissertation
- Degree Awarded
- Doctor of Philosophy (PhD), University of Iowa
- Degree in
- Mathematics
- Date degree season
- Spring 2025
- DOI
- 10.25820/etd.007991
- Publisher
- University of Iowa
- Number of pages
- vii, 82 pages
- Copyright
- Copyright 2025 Elise Askelsen
- Language
- English
- Date submitted
- 04/29/2025
- Description illustrations
- illustrations
- Description bibliographic
- Includes bibliographical references (page 78-82).
- Public Abstract (ETD)
Quivers are simply a collection of vertices with arrows between them which forms a directed graph. While their construction is simple, these mathematical structures and objects associated to them prove to be important tools in many areas of mathematics. Two of these areas are topological data analysis (TDA) and quantum symmetries, which are discussed in this thesis in this order.
Topological data analysis is a new field of interdisciplinary mathematics that uses the underlying “shape” of a discrete data set to draw conclusions about the data. By varying the parameters used to equip the data with shape, we get a detailed picture of the data through mathematical objects associated with quivers. Within this area, we study the difference in the mathematical output after slight perturbations of the data.
On the other hand, symmetries are evident all around us. Mathematically, we describe these as transformations that preserve the structure of an object which can be formalized as an action of a group on the object with symmetry. As advancements in quantum mechanics have been made, we desire to understand the symmetries in this setting, called quantum symmetries. Paralleling the use of group actions in the classical setting, we use Hopf actions to study quantum symmetries of path algebras which are built from the arrows of quivers and are closely related to finite-dimensional algebras. To encompass many Hopf algebras, we focus on Hopf actions of Hopf-Ore extensions of group algebras in the second half of this thesis.
- Academic Unit
- Mathematics
- Record Identifier
- 9984831121902771
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