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    Vertex models for the product of a permuted-basement Demazure atom and a Schur polynomial
    (University of Waterloo, 2026-01-19) Miller, Timothy
    This thesis is about a manifestly positive combinatorial rule for the expansion of the product of two polynomials: Schur polynomials and permuted-basement Demazure atoms. Special cases of the latter polynomials include Demazure atoms and characters; there are known tableau formulas for their expansions when multiplied by a Schur polynomial, due to Haglund, Luoto, Mason and van Willigenburg (2011). We find a vertex model formula, giving a new rule even in these special cases, extending a technique introduced by Zinn-Justin (2009) for calculating Littlewood–Richardson coefficients. We derive a coloured vertex model for permuted-basement Demazure atoms. This model is inspired by Brubaker, Buciumas, Bump and Gustafsson's model for Demazure atoms (2021) and Borodin and Wheeler's model for permuted-basement nonsymmetric Macdonald polynomials (2022). We make this model compatible with an uncoloured vertex model for Schur polynomials, putting them in a single framework. Unlike previous work on structure coefficients via vertex models, a remarkable feature of our construction is that it relies on a Yang–Baxter equation that only holds for certain boundary conditions. However, this restricted Yang–Baxter equation is sufficient to show our result.
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    Green Space Equity & Environmental Justice: A Comparative Study between North St. James Town & High Park-Swansea Communities in Toronto
    (University of Waterloo, 2026-01-19) Nankisore, Chelsea
    In cities shaped by unequal development and growing environmental pressures, urban green spaces are increasingly recognized not just as aesthetic luxuries but as fundamental components of livable, healthy, and equitable communities. While public parks and naturalized areas linked to a wide range of social, mental, and ecological benefits, the accessibility and distribution of these spaces are often influenced by systemic inequities embedded in urban planning, land use policy, and neighborhood development trajectories. This thesis investigates these disparities by comparing two socioeconomically and spatially distinct communities, High Park-Swansea (HPS) and North St. James Town (NSJ). Guided by social‑ecological systems (SES), political ecology, and community‑based participatory research (CBPR), the study asks how social fabric and planning histories shape the equitable distribution and lived experience of public green space. The research investigates green space quality, accessibility, and use in each neighborhood, focusing on how socio‑economic status, density, infrastructure, and community engagement intersect to produce divergent relationships with urban nature. Drawing on 63 surveys and 24 in‑depth interviews, it employs inductive coding and narrative analysis to identify themes related to accessibility, inclusiveness, safety, ecological quality, and psychological well‑being. Findings show that access to green space is not defined by proximity or quantity alone but is closely tied to perceptions of safety, historical marginalization, and belonging. HPS emerges as a neighborhood with relatively high green space coverage, affluent demographics, and strong stewardship, while NSJ is characterized by dense high‑rise housing, constrained green infrastructure, and heightened social vulnerability. An analysis of Toronto’s green space policies indicates that comprehensive goals are often undermined by weak enforcement, a lack of spatially disaggregated values, and limited community‑oriented design standards, contributing to a spatial politics of exclusion. Moreover, it explores how Toronto’s green space policies, while comprehensive on paper, often lack enforcement mechanisms, spatially disaggregated benchmarks, and community-oriented design standards. By mapping community narratives to broader structural trends, the study reveals how planning practices, past and present, contribute to a spatial politics of exclusion where certain communities are underserved by design. In doing so, it informs concrete recommendations for municipal planners, including the development of neighborhood-level green space equity indicators, integration of community-informed design criteria in development approvals, and policy tools that ensure green infrastructure investment is responsive to local needs. These insights hold relevance beyond the context of Toronto, contributing to global conversations on urban sustainability, environmental justice, and inclusive planning decisions.
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    Considerations for Commons Governance in Chilika Lagoon: New-Commonisation through Codification
    (University of Waterloo, 2026-01-19) Murray, Kaitlin Sarah
    This thesis examines how communities can re-establish governing authority over shared environmental resources (commons) after periods of dispossession (decommonisation), a process described as ‘new-commonisation’. Focusing on Chilika Lagoon, India, it explores how small-scale fishery communities might regain autonomy following decades of externally-driven decommonisation, caused by privatization, encroachment, elite capture, and fragmented state interventions. The central argument is that legally-grounded recognition of commons is helpful for re-gaining rights and essential for protecting communities from renewed external threat. Drawing on process-tracing analysis of three cases; Shimshal Valley in Pakistan, forest governance under India’s Forest Rights Act (2006), and Locally Managed Marine Areas (LMMAs) in Papua New Guinea; the study identifies how community mobilisation and legal codification interact to regain and stabilize self-managed commons. Though the findings are hypothesis generating rather than hypothesis testing, they suggest that enduring governance outcomes emerge when communities achieve de jure recognition of de facto rights, and that their success depends on contextually-dependent enabling conditions, such as equitable enforcement, multi-level support and the mechanism for legal rights. As no two commons are identical, there is no single path to codification; legal arrangements must respond to the specific socio-political and ecological context of each community. This research contributes to commons theory by framing codified legal backing as a critical, yet under-developed, dimension of enduring commons governance, in the face of persistent external pressures.
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    Advancing the measurement and use of daily life heart rate using wearable technology: applications to Parkinson's disease
    (University of Waterloo, 2026-01-19) Godkin, Florence Elizabeth
    Measures of heart rate (HR), including those conducted during resting conditions and in response to physical activity, have been shown to relate to risk of adverse health outcomes and can provide insight into the function of the autonomic nervous system (ANS). However, HR measures have conventionally been conducted at a single point in time and in controlled settings, such as during clinical or laboratory visits. Alternatively, when HR has been measured in uncontrolled, free-living settings, there has been a lack of standardization in the methods used and most have not provided sufficient behavioral context to support the interpretation of HR in daily life. Use of wearable technology, and specifically, a multi-sensor and -device approach, presents a unique opportunity to measure HR continuously in the context of daily life behaviors, and advance understanding of ANS function and context-dependent HR responses. This may be particularly relevant to individuals living with Parkinson’s disease (PD), who can experience ANS dysfunction that may manifest with altered diurnal HR profiles and who can have diverse movement characteristics that may affect the HR response to physical activity. The overarching goals of this thesis work were to 1) advance the development of standard approaches to assess HR from continuously collected data over multiple days and behavioral contexts, and 2) apply these approaches to advance understanding of ANS function and characterization of daily life physical activity profiles among individuals with PD. All studies were based on a collection protocol that instrumented participants with a portable, chest-worn electrocardiography device and two limb-worn inertial measurement units (wrist and ankle) that continuously collected data across a week-long period in daily life. In the first study, HR was measured across accumulated periods of sedentary behavior and sleep within a cohort of adults without PD (“control”), and the high levels of within-participant variability in HR that were observed within a given resting condition led to recommendations being put forth for standardizing the measurement of HR outcomes during resting conditions in daily life. In the second study, standards suggested in Study 1 were applied and extended upon in a cohort of adults with and without PD, and revealed no differences between cohorts in resting condition-specific HR outcomes (e.g., minimum HR during sleep) or in measures of nocturnal HR dipping in daily life. However, a significant association was observed between nocturnal HR dipping and a clinical measure of ANS dysfunction in PD. In the third study, resting HR was used as a reference when quantifying the relative intensity of the HR response to daily life walking across the same cohort of adults with and without PD. While no between-cohort differences were observed in the average relative intensity of daily life walking, there was considerable variability in the between- and within-participant HR response to continuous, longer walking bouts (≥ 30 seconds), which were not consistently associated with walking cadence or bout duration. This dissertation provides recommendations for, and illustrates the importance of, adopting standardized approaches to measure resting and activity-dependent HR in daily life. Findings from the studies in this thesis suggest that context-based ambulatory HR monitoring has the potential to improve characterization of daily life behaviors and provide insight into clinically relevant outcomes in individuals with and without PD. Continued research in larger, more heterogenous populations, which includes robust clinical profiling and continues to attend to the importance of high data quality, is necessary to determine generalizability and advance understanding of within- and between-participant drivers of variability in context-dependent HR measures. This type of future work is required before context-dependent HR monitoring can be integrated and evaluated as a tool to support self-management or clinical health care decision making.
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    New Methods for Analyzing the Properties of Automatic Sequences
    (University of Waterloo, 2026-01-19) Khodier, Mazen
    Automatic sequences and morphic words lie at the intersection of automata theory, logic, and combinatorics on words. Many of their structural properties can be formulated as logical predicates over integer representations and decided using automata. This thesis presents automata-based methods for efficiently constructing and verifying deterministic finite automata corresponding to such predicates, and builds on this foundation to analyze key combinatorial properties of morphic words, including the critical exponent and subword complexity. In the first part of this thesis, Chapters 2 to 4, we introduce the notion of self-verifying predicates, which are logical predicates capable of verifying their own correctness. We show how this property enables verification of candidate automata through a small set of inductive conditions and allows the corresponding automata to be constructed deterministically rather than through heuristic guessing. Building on Angluin’s L* learning algorithm, we demonstrate that for such predicates, the associated minimal automata can be generated in time polynomial in the size of both the automaton for the underlying sequence and the resulting automaton, thereby avoiding potentially extremely large intermediate automata that sometimes arise in Walnut. In particular, we give effective constructions for the equality-of-factors predicate, which is used extensively in the second half of the thesis, as well as for other self-verifying predicates, including periodicity of factors, addition relations for numeration systems, and summation of synchronized sequences. The second part, Chapters 5 to 7, applies the previously constructed equality-of-factors predicate to investigate two central combinatorial measures of infinite words: the critical exponent and the subword complexity. Although binary 3-uniform morphisms are used as illustrative examples, the methods generalize naturally to all binary uniform morphisms. For the critical exponent, we present a decision procedure implemented in Walnut that detects whether the exponent is infinite and computes its exact rational value when finite. For subword complexity, we propose two complementary approaches: a constructive method that combines established concepts to produce exact formulas for ρ(n), and a fully deterministic procedure that implements Frid’s approach using Walnut. The new results include explicit subword-complexity formulas for twelve morphisms, and critical-exponent values for ten morphisms. All algorithms and implementations developed in this thesis are made publicly available on the Github repository Cashew as open-source code to support and facilitate further research in combinatorics on words, and automata theory.