This thesis examines the critical relationship between data, complex models, and other methods to measure and analyze them. As models grow larger and more intricate, they require more data, making it vital to use that data effectively. The document starts with a deep dive into nonconvex functions, a fundamental element of modern complex systems, identifying key conditions that ensure these systems can be analyzed efficiently—a crucial consideration in an era of vast amounts of variables. Loss functions, traditionally seen as mere optimization tools, are analyzed and recast as measures of how accurately a model reflects reality. This redefined perspective permits the refinement of data-sourcing strategies for a better data economy. The aim of the investigation is the model itself, which is used to understand and harness the underlying patterns of complex systems. By incorporating structure both implicitly (through periodic patterns) and explicitly (using graphs), the model's ability to make sense of the data is enhanced. Moreover, online learning principles are applied to a crucial practical scenario: robotic resource monitoring. The results established in this thesis, backed by simulations and theoretical proofs, highlight the advantages of online learning methods over traditional ones commonly used in robotics. In sum, this thesis presents an integrated approach to measuring complex systems, providing new insights and methods that push forward the capabilities of machine learning.

Computable properties of quantum states are given a dual gravitational interpretation via the AdS/CFT correspondence. For holographic states, boundary entanglement entropy is dual to the area of bulk geodesics, known as Ryu-Takayanagi surfaces. Furthermore, the viability of states to admit a holographic dual at all is constrained by their entanglement structure. Entanglement therefore defines a coarse classification of states in the Hilbert space. Similarly, how a state transforms under a group of operators also provides a classification on the Hilbert space. Certain states, e.g. stabilizer states, are invariant under large sets of operations, and consequently can be simulated on a classical computer. Cayley graphs offer a useful representation for a group of operators, where vertices represent group elements and edges represent group generators. In this representation, the orbit of a state under action of the group can also be represented as a ``reachability graph'', defined as a quotient of the group Cayley graph. Reachability graphs can be dressed to encode entanglement information, making them a useful tool for studying entanglement dynamics under quantum operations. Further quotienting a reachability graph by group elements that fix a chosen state property, e.g. entanglement entropy, builds a ``contracted graph''. Contracted graphs provide explicit bounds on state parameter evolution under quantum circuits. In this work, an upper bound on entropy vector evolution under Clifford group action is presented. Another important property of quantum systems is magic, which quantifies the difficulty of classically simulating a quantum state. Magic and entanglement are intimately related, but the two are not equivalent measures of complexity. Nonetheless, entanglement and magic play complementary roles when describing emergent gravitational phenomena in AdS/CFT. This manuscript describes the interplay between entanglement and magic, and offers a holographic interpretation for magic as cosmic brane back-reaction.

We studied binomial edge ideals, which are at the intersection of graph theory and abstract algebra. Our focus was the multidegrees of these ideals, which contain valuable geometric information. We proved algebraic results that allowed us to write a closed formula for the multidegree of the binomial edge ideal of a graph based on combinatorial properties of the graph. Then we discovered methods to make the process more efficient. We concluded our research by using our results to find the multidegrees of the binomial edge ideals of many families of graphs.

Climate change is one of the most pressing issues affecting the world today. One of the impacts of climate change is on the transmission of mosquito-borne diseases (MBDs), such as West Nile Virus (WNV). Climate is known to influence vector and host demography as well as MBD transmission. This dissertation addresses the questions of how vector and host demography impact WNV dynamics, and how expected and likely climate change scenarios will affect demographic and epidemiological processes of WNV transmission. First, a data fusion method is developed that connects non-autonomous logistic model parameters to mosquito time series data. This method captures the inter-annual and intra-seasonal variation of mosquito populations within a geographical location. Next, a three-population WNV model between mosquito vectors, bird hosts, and human hosts with infection-age structure for the vector and bird host populations is introduced. A sensitivity analysis uncovers which parameters have the most influence on WNV outbreaks. Finally, the WNV model is extended to include the non-autonomous population model and temperature-dependent processes. Model parameterization using historical temperature and human WNV case data from the Greater Toronto Area (GTA) is conducted. Parameter fitting results are then used to analyze possible future WNV dynamics under two climate change scenarios. These results suggest that WNV risk for the GTA will substantially increase as temperature increases from climate change, even under the most conservative assumptions. This demonstrates the importance of ensuring that the warming of the planet is limited as much as possible.

This dissertation centers on the development of Bayesian methods for learning differ- ent types of variation in switching nonlinear gene regulatory networks (GRNs). A new nonlinear and dynamic multivariate GRN model is introduced to account for different sources of variability in GRNs. The new model is aimed at more precisely capturing the complexity of GRN interactions through the introduction of time-varying kinetic order parameters, while allowing for variability in multiple model parameters. This model is used as the drift function in the development of several stochastic GRN mod- els based on Langevin dynamics. Six models are introduced which capture intrinsic and extrinsic noise in GRNs, thereby providing a full characterization of a stochastic regulatory system. A Bayesian hierarchical approach is developed for learning the Langevin model which best describes the noise dynamics at each time step. The trajectory of the state, which are the gene expression values, as well as the indicator corresponding to the correct noise model are estimated via sequential Monte Carlo (SMC) with a high degree of accuracy. To address the problem of time-varying regulatory interactions, a Bayesian hierarchical model is introduced for learning variation in switching GRN architectures with unknown measurement noise covariance. The trajectory of the state and the indicator corresponding to the network configuration at each time point are estimated using SMC. This work is extended to a fully Bayesian hierarchical model to account for uncertainty in the process noise covariance associated with each network architecture. An SMC algorithm with local Gibbs sampling is developed to estimate the trajectory of the state and the indicator correspond- ing to the network configuration at each time point with a high degree of accuracy. The results demonstrate the efficacy of Bayesian methods for learning information in switching nonlinear GRNs.

In image classification tasks, images are often corrupted by spatial transformationslike translations and rotations. In this work, I utilize an existing method that uses the Fourier series expansion to generate a rotation and translation invariant representation of closed contours found in sketches, aiming to attenuate the effects of distribution shift caused by the aforementioned transformations. I use this technique to transform input images into one of two different invariant representations, a Fourier series representation and a corrected raster image representation, prior to passing them to a neural network for classification. The architectures used include convolutional neutral networks (CNNs), multi-layer perceptrons (MLPs), and graph neural networks (GNNs). I compare the performance of this method to using data augmentation during training, the standard approach for addressing distribution shift, to see which strategy yields the best performance when evaluated against a test set with rotations and translations applied. I include experiments where the augmentations applied during training both do and do not accurately reflect the transformations encountered at test time. Additionally, I investigate the robustness of both approaches to high-frequency noise. In each experiment, I also compare training efficiency across models. I conduct experiments on three data sets, the MNIST handwritten digit dataset, a custom dataset (QD-3) consisting of three classes of geometric figures from the Quick, Draw! hand-drawn sketch dataset, and another custom dataset (QD-345) featuring sketches from all 345 classes found in Quick, Draw!. On the smaller problem space of MNIST and QD-3, the networks utilizing the Fourier-based technique to attenuate distribution shift perform competitively with the standard data augmentation strategy. On the more complex problem space of QD-345, the networks using the Fourier technique do not achieve the same test performance as correctly-applied data augmentation. However, they still outperform instances where train-time augmentations mis-predict test-time transformations, and outperform a naive baseline model where no strategy is used to attenuate distribution shift. Overall, this work provides evidence that strategies which attempt to directly mitigate distribution shift, rather than simply increasing the diversity of the training data, can be successful when certain conditions hold.

The pandemic has not only increased economic inequities within variouscommunities, but it has also exacerbated the social, emotional, and math achievement inequities of middle school students, creating an environment that increases the potential for heightened anxiety and peer conflict. Now, more than ever, it is imperative that educators not only understand the existence and impact of these social and emotional inequities but have the knowledge and skills to effectively address them. Within this study, I facilitated a 10-week online community of practice with three middle school math teachers, entitled The More than Math Collective (MTMC), with the purpose of improving participant self-efficacy with SEL, developing their professional capital, discussing various strategies to address the social and emotional skill needs of students in their classrooms, and providing time for implementation of the discussed strategies. At the conclusion of the study, most participants reported an increase in self-efficacy, human capital, and decisional capital while only one out of three participants reported an increase in social capital. All participants described a positive impact on their students and their professional growth due to their participation in the MTMC and the various strategies that were learned and implemented in their classrooms. Given the small sample size, more research can be done to determine if the results of this study may be transferable to other educational settings.

One common problem that occurs to students during breaks is the retrogression of knowledge due to lack of practice. This problem occurs for students at all levels of education but is especially harmful to students who are taking sequential classes such as Calculus for Engineers I and Calculus for Engineers II where the retention of topics taught in Calculus for Engineers I are required for students to succeed. One solution to this problem is the Keep in School Shape (KiSS) program. The KiSS program is a very efficient and easily accessible program that allows students to stay warmed up and ready to go when they start a sequential course by having daily review material during academic breaks. During an academic break, students who are signed up for the KiSS program are sent a link through text message or email every day that allows them to access a multiple choice review problem. The review problem that they are given is a problem that presents material from the previous course that will be needed in the upcoming course. At the beginning of the review, students have the option to choose between a Level 1 or a Level 2 problem, where a Level 2 problem is related to its Level 1 counterpart but slightly more difficult. Before the students are permitted to solve the problem, they must first use a five point scale that indicates their confidence in their ability to solve the problem. After they complete either the Level 1 or Level 2 daily problem, those that got it wrong have the option to view a hint and try again or view a solution. The students that got the Level 1 daily problem right are also allowed to view the solution but will be permitted to go onto the next level right away whereas the students that got the Level 1 problem incorrect will need to try a similar problem before being able to move onto Level 2. For students who chose to do the Level 2 problem and were not very confident, they were given the option to solve a level 1 problem instead. Students who chose level 2 and got it wrong are given the options to view a hint and try again or simply view the solution before moving on to flashcard versions of the daily problems. Students who get the Level 2 problem correct are also given the option to continue practicing using the flashcards if they choose to. Once a week, there is also a trivia day where students have the choice to complete solely a mathematical trivia question or complete both the trivia question along with a daily review problem. This feature allows students to take a day off from doing mathematics if they choose, but still stay engaged by doing a related activity. Through this program, there is a lot to learn about whether doing Level 1 problems can help students improve their understanding of a concept enough to correctly solve a Level 2 problem. There are many factors to consider such as which question the student chose to answer first, student confidence, and student perseverance. Through the Summer Break 2023 KiSS program, there was data collected for every student answer for each day they accessed the daily KiSS activity. This thesis presents an analysis of the data showing how having two levels of problems is beneficial for students and the correlation between students’ results in Level 1 problems and Level 2 problems for students who chose to attempt both problems.

Imitation Learning, also known as Learning from Demonstration (LfD), is a field of study dedicated to aiding an agent's learning process by providing access to expert demonstrations. Within LfD, Movement Primitives is a particular family of algorithms that have been widely studied and implemented in complex robot scenarios. Interaction Primitives, a subset of Movement Primitives, have demonstrated notable success in learning single interactions between humans and robots. However, literature addressing the extension of these algorithms to support multiple variations of an interaction is limited. This thesis presents a physical human-robot interaction algorithm capable of predicting appropriate robot responses in complex interactions that involve a superposition of multiple interactions. The proposed algorithm, Blending Bayesian Interaction Primitives (B-BIP), achieves responsive motions in complex hugging scenarios and can reciprocate and adapt to the motion and timing of a hug. B-BIP generalizes prior work, where the original formulation reduces to the particular case of a single interaction. The performance of B-BIP is evaluated through an extensive user study and empirical experiments. The proposed algorithm yields significantly better quantitative prediction error and more favorable participant responses concerning accuracy, responsiveness, and timing compared to existing state-of-the-art methods.

This study investigates the impact and experiences of students designated as English Language Learners (ELLs) as they engage with student-centered worked example videos (WEVs). Students from two southwestern high schools collaborated and provided their experiences as they watched WEVs and worked through four slope calculation problems. Although high school ELLs are placed in appropriate mathematics classes, the WEVs they engage with, by design, do not consider their diverse educational needs, one of which is the amount of cognitive load experienced when watching the videos. Through this Multi-Phase Mixed Methods study, I begin to understand inclusive design practices for WEVs, in which ELLs will not experience cognitive over-load, and as a result, will receive the needed remediation and/or instruction and develop concept proficiency through active learning as they engage with the videos. The research finds that specific design principles, closed captioning, conversational narration, and music, reduce cognitive load and provide ELLs a familiar and safe space from which to engage with mathematical content.