The Witt ring of a field gives the structure of the isometry classes of quadratic forms over that field. In particular, the Witt ring provides an algebraic invariant for fields away from characteristic $2$, which also allow us to study the orderings we can put on that field. During the latter quarter of the 20th century, abstract Witt rings, a wider class of rings that had the structure of a Witt ring but constructed independently from fields, were introduced. In this thesis, we will use what is known about the structure of Witt rings over quadratic extensions of fields in order to come up with an analog that extends over to abstract Witt rings.

The field of topological data analysis (TDA) uses tools from algebraic topology to capture quantitative structural properties in a data set. Perhaps the most popular tool in TDA is persistent homology (PH), which leverages homology theory to provide insights into the structure of data by quantifying ``holes'' across different scales. PH has been applied to many domains, including neuroscience, materials science, image processing, and social systems. In this thesis, we study PH in both theory and application.

On the theory side, we build upon the foundations of PH by studying persistence modules (which are algebraic objects that are fundamental to PH) and their interval decompositions.The existence of interval decompositions of a persistence module underlies much of PH. Interval decompositions are guaranteed to exist for persistence modules with coefficients in a field. However, interval decompositions may not exist in more general settings, such as when the coefficients are not in a field. We prove necessary and sufficient conditions for persistence modules with coefficients in a principal ideal domains to have an interval decomposition. We also formulate an algorithm to compute an interval decomposition when one exists.

We then use PH to quantify and assess the accessibility of resources in a geographic area. This allows us to identify regions that have poor resource access. Our work focuses on the accessibility of polling sites across six geographic regions (five cities and Los Angeles County). We adapt traditional approaches in PH to incorporate important factors of polling-site accessibility, including travel times to and from polling sites and waiting times at those sites.

Finally, we discuss the modelling of opinion dynamics on networks. We consider bounded-confidence models (BCMs), which are models of opinion dynamics in which agents are receptive only to agents whose opinions are sufficiently similar (i.e., within a confidence bound). We formulate and analyze two BCMs with adaptive confidence bounds; our BCMs generalize the Hegselmann--Krause and Deffuant--Weisbuch models. Using mathematical analysis and numerical simulations, we demonstrate that our adaptive BCMs exhibit quantitatively and qualitatively different behavior than the associated baseline (i.e., nonadaptive) HW and DW BCMs. This includes fewer major opinion clusters, longer convergence times, and adjacent nodes that converge to the same opinion but are not receptive to each other.