Date of Award

1-1-2021

Language

English

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

College/School/Department

Department of Mathematics and Statistics

Content Description

1 online resource (x, 111 pages) : illustrations (some color)

Dissertation/Thesis Chair

Justin M Curry

Committee Members

Boris Goldfarb, Matthew Zaremsky, Justin Curry

Keywords

Topology, Shape theory (Topology), Information visualization, Trees (Graph theory), Homology theory

Subject Categories

Physical Sciences and Mathematics

Abstract

In the field of applied topology, or more specifically, topological data analysis, we usetools such as the persistent homology of a filtered space to give shape to data. Alongside the persistent homology groups of a space is the barcode invariant. In this thesis, we explore the failure of injectivity of the barcode invariant for merge trees, Reeb graphs, circle embeddings in R2 and S2 embeddings in R3 up to different types of equivalence. We first extend the results of Justin Curry in [25] to give a connection between merge trees and the lattice of set partitions. This allows us to count the total number of merge trees that are realized by barcodes with n half-open intervals, up to what we call combinatorial equivalence. We then mimic those results to reconstruct Reeb graphs from their level set barcodes. We show that for untwisted Reeb graphs, the level set barcode embeds onto the Reeb graph, which allows us to construct both an upper and lower bound for the number of untwisted simply connected Reeb graphs realizing a given level set barcode. We also provide bounds for the number of height equivalence classes of circles embedded in R2 as well as exact calculations for the number of unbraided height equivalence classes of embeddings of S2 in R3. The different notions of equivalence used are all motivated by the definitions and results of V. Arnold and L. Nicolaescu. Here, we also provide further comparisons of their notions of equivalence with ours for the circle and sphere embeddings.

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