Recurrent oscillations in biological systems, particularly in models of persistent infections, often indicate underlying slow-fast dynamics. Classical approaches such as Geometric Singular Perturbation Methods (GSPM) rely on the existence of critical manifolds and are limited to systems with small perturbation parameters. However, these conditions are not always met in complex or higher-dimensional models. This thesis introduces a set of refined analytical criteria for detecting slow-fast oscillations beyond the scope of traditional perturbation theory. Applying this framework to a three-dimensional SIR model with secondary transmission and intrinsic population growth, we use bifurcation analysis, center manifold theory, and normal form analysis to identify conditions under which Hopf bifurcations and periodic solutions arise, even without asymptotically small growth rates. Numerical simulations support the theoretical findings, demonstrating the robustness and broader applicability of the proposed method in capturing oscillatory behavior across a wide range of parameter regimes.

In this work we study numerical homotopy continuation, a method used for solving nonlinear equations and nonlinear systems of equations. We focus on detecting and resolving error in a fixed step predictor-correct method for numerically solving univariate polynomials. We verify the homotopy paths with a Lipschitz condition to detect path jumping that leads to false multiplicities of roots and missed roots; we show that mutual path jumping cannot occur. At path ends we may obtain spurious roots, which are easy to detect for univariate polynomials when we check for non-zero residuals at the final path points. We show that pseudozeros---roots of nearby polynomials---provide another means of understanding error at path end points and give insight on how to optimize a homotopy continuation algorithm for a specific problem. We demonstrate that a change of basis can improve homotopy continuation results when finding ill-conditioned roots.

Wild turkeys (Meleagris gallopavo) were over-hunted and faced near extinction in the United States in the 1930s. Hunting practices have since adapted, but the US still experiences unexplained turkey population declines. Nest success is a primary driver of annual population change. Female turkeys (hens) are the sole caretakers of the nest, so factors affecting their behaviour influence population dynamics. Little is known about the effect of hunting on hens, as most hunting-related studies of turkeys focus on males. To investigate the effect of hunting on hens during their incubation period, we analyze GPS movement data from over 600 hens in sites across the southeastern US. Hunting occurs in most sites and is prohibited in one. Hen behaviour can be inferred from location data. We are interested in the ways in which hunting affects both the behaviour and physiological state of the hens. We develop a hidden Markov model, treating physiological state as the hidden variable and behaviour as the observed variable. The model assumes that behaviour varies with physiological state and hunting presence. We found that hunting has an effect on hen behaviour, and this effect changes depending on their physiological state. Further, the model predicts that, on average, hens in hunted sites leave their nest for longer consecutive periods than hens in non-hunted sites. Results suggest that hens in hunted sites place their nest at greater risk and prioritize their own survival. These results should be considered when developing hunting policies.

Complex-valued neural networks (cv-NNs) have shown strong potential for tasks that rely on phase-based representations, short-term memory, and dynamic pattern selection — echoing ideas found in oscillator networks and wave patterns phenomena [3, 12]. Building on the previous work, Chapter 2 of this thesis demonstrates how a dynamic cv-NN can reconstruct partially occluded digit images by evolving toward one of several stable states, guided by the system’s oscillatory dynamics and eigenvalue structure [3]. This approach provides an interpretable mechanism for recovering missing information and highlights how complex eigenmodes can drive pattern completion in neural systems. This line of work naturally invites the question: how can such a framework be generalized to operate beyond specific matrix classes such as the HiPPO matrix [11] or other structured normal matrices? To address this, Chapter 3 establishes a new factorization result for diagonaliz-able complex orthogonal matrices [25], making it possible to extend the solvable cv-NN approach to arbitrary real matrices. This factorization bridges synchronization-inspired models with structured state-space representations, offering fresh insights into the spectral properties of HiPPO matrices and suggesting a unified view of oscillatory neural dynamics and modern sequential modeling.

The central objects of algebraic geometry, schemes, are an intricate notion which takes some effort to define - in synthetic algebraic geometry, schemes are just sets with an additional property, but the rules for manipulating sets are different than what is assumed in the classical theory.

More precisely, the language of synthetic algebraic geometry is homotopy type theory together with three axioms. We will explain these axioms and some essential definitions needed to describe schemes and the example of projective space. To show non-trivial facts about projective space, like the classification of line bundles, we will make use of the higher types available in homotopy type theory.