# Department of Mathematics and Statistics Graduate Theses

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Item The Orthogonal Weingarten Calculus and Free Probability(2024-09-05) Gawlak, Dylan; Mathematics and Statistics; Mingo, JamesWe provide a self-contained derivation of the Weingarten calculus for compact matrix groups. We also provide a self-contained introduction to the combinatorics of the symmetric group needed for computations in free probability. We introduce partitioned pairings, an analogue of partitioned permutations. Using partitioned pairings, we will derive expressions for the leading and subleading terms of the Weingarten function for the orthogonal group. Using the combinatorial theory that we have developed, as well as what is called second-order real freeness, we will determine the first and second-order limiting distributions of the random matrix ensemble Xm = O1+Ot1+· · +·Om+Otm where {Oi} are independent Haar-distributed orthogonal matrices.Item Curves of genus 2 and quadratic forms(2024-09-05) Kir, Harun; Mathematics and Statistics; Kani, ErnstOur approach focuses on a special associated integral quadratic form qC, which is intrinsically attached to a curve C of genus 2. This form, known as the refined Humbert invariant, was introduced by Kani (1994). Our first main result is the classification of those imprimitive ternary quadratic forms which can occur as one of the forms qC, for some curve C of genus 2. We use this classification to classify the qC’s for curves with a prescribed group of automorphisms, and/or with some additional structure such as having an elliptic subcover of prescribed degree. This study, along with other results, has concrete applications on understanding the nature of intersections of Humbert surfaces. For instance, we prove that the intersection of two Humbert surfaces is not empty, and we determine the intersection of infinitely many prescribed Humbert surfaces in terms of ternary qC’s. To obtain these results, and many others, it is useful to study generalized Humbert sets. A generalized Humbert set H(q), associated with a given quadratic form q, was introduced by Kani in order to understand the nature of curves of genus 2. Regarding the structure of these sets, we find all ternary quadratic forms q for which H(q) is irreducible by relying on SAGE. Additionally, in our joint work with Kani, we give formulas for the cardinality of H(q) for any ternary form q. As an application of this formula, we establish formulas for the number of isomorphism classes of genus 2 curves with certain specified properties, such as curves with a prescribed automorphism group and having an elliptic subcover of a prescribed degree. We examine Shimura curves following the approaches of Lin and Yang, who studied these curves by associating positive binary quadratic forms with them. We clarify the connection between these forms and the refined Humbert invariant. By doing so, we obtain interesting results, such as improving a criterion for the existence of CM points on Shimura curves given by Lin and Yang.Item Arithmetic Dynamics and Higher Direct Images(2024-08-12) Zotine, Alexandre; Mathematics and Statistics; Roth, Mike; Smith, Gregory G.This thesis consists of two separate papers. In the first, we prove the Kawaguchi--Silverman conjecture for projective bundles on elliptic curves, thereby completing the conjecture for all projective bundles over curves. Our approach is to use the transition functions of the bundles. This allows us to further prove the conjecture for projective split bundles over a smooth projective variety with finitely generated Mori cone. In the second paper, we give an algorithm for computing higher direct images of line bundles for toric morphisms. The state-of-the-art prior to our work only allows for computation when the toric morphism is between a product of projective spaces. We provide a new combinatorial framework for computing sheaf cohomology using cell complexes. Our method for computing higher direct images makes use of this construction. The algorithm developed is implemented explicitly in the mathematical software Macaulay2.Item Disease Transmission on Random Graphs Using Edge‐Based Percolation and its Application to Syphilis Control in KFL&A Area(2024-07-29) Zhao, Sicheng; Mathematics and Statistics; Magpantay, FeliciaBond percolation methods can be used to model disease transmission on complex networks and accommodate social heterogeneity while keeping tractability. We review the seminal works on this field by Newman (2002, 2003, 2010), and Miller, Slim & Volz (2011) and present a more clear and systematic discussion about the theoretical background, assumptions, derivation and development of the percolation method. We also present a new R package based on these results that take epidemic and network parameters as input and generates estimates of the epidemic trajectory and final size. Such theoretical framework and calculation tools allow us to apply the edge-based percolation model to solving real world public health emergencies. With syphilis rates continue rising in Ontario at an alarming rate, an ongoing project is collaborating with KFL&A public health to model Syphilis transmissions within the underserved high-risk community from data. The analysis and prediction of the model could provide scientific evidence to optimize implementation strategy based on community structure, thus help public health professionals to better response to the urgent crisis.Item Optimal Zero-Delay Transmission of Markov Sources: Reinforcement Learning and Approximations(2024-07-15) Cregg, Liam Barry; Mathematics and Statistics; Yuksel, Serdar; Alajaji, FadyWe study the problem of zero-delay coding for the transmission a Markov source over a noisy channel with feedback and present rigorous finite model approximations and reinforcement learning solutions which are guaranteed to achieve near-optimality. To this end, we formulate the problem as a Markov decision process (MDP) where the state is a probability-measure valued predictor/belief and the actions are quantizer maps. This MDP formulation has been used to show the optimality of certain classes of encoder policies in prior work. Despite such an analytical approach in determining optimal policies, their computation is prohibitively complex due to the uncountable nature of the constructed state space and the lack of minorization or strong ergodicity results which are commonly assumed for average cost optimal stochastic control. These challenges invite rigorous reinforcement learning methods, which entail several open questions addressed in our paper. We present two complementary approaches for this problem. In the first approach, we approximate the set of all beliefs by a finite set and use nearest-neighbor quantization to obtain a finite state MDP, whose optimal policies become near-optimal for the original MDP as the quantization becomes arbitrarily fine. In the second approach, a sliding finite window of channel outputs and quantizers together with a prior belief state serve as the state of the MDP. We then approximate this state by marginalizing over all possible beliefs, so that our policies only use the sliding finite window term to encode the source. Under an appropriate notion of predictor stability, we show that such policies are near-optimal for the zero-delay coding problem as the window length increases. We give sufficient conditions for predictor stability to hold. For each scheme, we propose a reinforcement learning algorithm to compute near-optimal policies. We provide a detailed comparison of the two coding policies in terms of their approximation bounds and reinforcement learning implementation, in terms of their performance, as well as conditions for reinforcement learning convergence to near-optimality. We include key differences between the noisy and noiseless channel cases, as well as supporting simulation results.