Mathematics Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/2793
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Item HOPF ALGEBRA OF MULTIPLE POLYLOGARITHMS AND ASSOCIATED MIXED HODGE STRUCTURES(2024) Li, Haoran; Zickert, Christian K; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This thesis constructs a variation of mixed Hodge structures based on multiple polylogarithms, and attempts to build candidate complexes for computing motivic cohomology. Firstly, we consider Hopf algebras with generators representing multiple polylogarithms. By quotienting products and functional relations, we get Lie coalgebras whose Chevalley-Eilenberg complexes are conjectured to compute rational and integral motivic cohomologies. We also associate one-forms to multiple polylogarithms, which exhibit combinatorial properties that are easy to work with. Next, we introduce a variation matrix which describes a variation of mixed Hodge structures encoded by multiple polylogarithms. Its corresponding connection form is composed of the one-forms associated to the multiple polylogarithms. Lastly, to ensure the well-definedness of the Hodge structures, we must compute the monodromies of multiple polylogarithms, for which we provide an explicit formula, extending the previous work done for multiple logarithms, a subfamily of multiple polylogarithms.Item Application of Causal Inference in Large-Scale Biomedical Data(2024) Zhao, Zhiwei; Chen, Shuo; Mathematical Statistics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation contains three projects that tackle the challenges in the application of causal inference on large-scale biomedical data. Project 1 proposes a novel mediation analysis framework with the existence of multiple mediators and outcomes. It can extract the mediation pathway efficiently and estimate the mediation effect from multiple mediators simultaneously. The effectiveness of the proposed method is validated through extensive simulation and a real data application focusing on human connectome study. Project 2 introduces a doubly machine learning based method, assisted by algorithm ensemble, for estimating longitudinal causal effects. This approach reduces estimation bias and accommodates high-dimensional covariates. The validity of the proposed method is justified by simulation studies and an application to adolescent brain cognitive development data, specifically evaluating the impact from sleep insufficiency on youth cognitive development. Project 3 develops a new bias-reduction estimation that addresses unmeasured confounding by leveraging proximal learning and negative control outcome techniques. This method can handle a moderate number of exposures and multivariate outcomes in the presence of unmeasured confounders. Both numerical experiment and data application using UK Biobank demonstrate that the proposed method effectively reduces estimation bias caused by unmeasured confounding. Collectively, these three projects introduce innovative methodologies for causal inference in neuroimaging, advancing mediation analysis in neuroimaging, improving longitudinal causal effect estimation, and reducing estimation bias in the presence of unmeasured confounding.Item Special Lagrangians in Milnor Fibers and Almost Lagrangian Mean Curvature Flow(2024) Pinsky, Mirna; Rubinstein, Yanir A.; MathematicsThe focus of this thesis is twofold: (1) We solve the Shapere–Vafa Problem: We construct embedded special Lagrangian spheres in Milnor fibers. We give a necessary and sufficient condition for the existence of embedded special Lagrangian spheres in Milnor fibers. (2) We solve the Thomas–Yau Problem for Milnor fibers: We prove the Thomas–Yau conjecture for the almost Lagrangian mean curvature flow (ALMCF) for Milnor fibers, under the assumption that the initial Lagrangian is an embedded positive Lagrangian sphere satisfying a natural stability condition proposed by Thomas–Yau but adapted to Milnor fibers by us. In addition, we formulate a new approach to resolving the Thomas–Yau conjecture in arbitrary almost Calabi–Yau manifolds. The Thomas–Yau conjecture proposes certain stability conditions on the initial Lagrangian under which the Lagrangian mean curvature flow (LMCF) exists for all time and converges to the unique special Lagrangian in the Hamiltonian isotopy class, and therefore also homology class of the initial Lagranigan. One of the reasons for studying LMCF in Calabi–Yau manifolds (or ALMCF in almost Calabi–Yau manifolds) is that the Lagrangian condition, as well as homotopy and homology classes, are preserved. Therefore, if the flow converges, it converges to a special Lagrangian. We develop a method for finding special Lagrangian spheres in Milnor fibers. We provide examples which illustrate different situations which occur (the total number of special Lagrangian spheres is at least deg f − 1 and at most 1/2 deg f(deg f − 1), where f is the polynomial defining the Milnor fiber). We show that the almost Lagrangian mean curvature flow of Lagrangian spheres in Milnor fibers can be reduced to a generalized mean curvature flow of paths in C. This reduction is different from the one found by Thomas–Yau. We show that the limit of the flow is either a straight line segment or a polygonal line, corresponding to a special Lagrangian sphere or a chain of such spheres. We prove that under certain conditions (more general than the ones achieved by Thomas–Yau) the flow results in a special Lagrangian sphere. Finally, we develop a method for associating a curve in C with a compact Lagrangian in a more general setting of an almost Calabi–Yau manifold. We show that when the Lagrangian flows by ALMCF that the corresponding curve remains convex and shortens its length. The limit is either a straight line segment corresponding to a special Lagrangian or a polygonal line resulting in a decomposition of the original Lagrangian.Item Eventually Stable Quadratic Polynomials over Q(i)(2024) McDermott, Jermain; Washington, Lawrence C; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Let $f$ be a polynomial or a rational function over a field $K$. Arithmetic dynamics studies the algebraic and number-theoretic properties of its iterates $f^n:=f \circ f \circ ... \circ f$.\\ A basic question is, if $f$ is a polynomial, are these iterates irreducible or not? We wish to know what can happen when considering iterates of a quadratic $f= x^2+r\in K[x]$. The most interesting case is when $r=\frac{1}{c}$, which we will focus on, and discuss criteria for irreducibility, i.e. \emph{stability} of all iterates. We also wish to prove that if 0 is not periodic under $f$, then the number of factors of $f^n$ is bounded by a constant independent of $n$, i.e. $f$ is \emph{eventually stable}. This thesis is an extension to $\Qi$ of the paper \cite{evstb}, which considered $f$ over $\mathbb{Q}$. This thesis involves a mixture of ideas from number theory and arithmetic geometry. We also show how eventual stability of iterates ties into the density of prime divisors of sequences.Item Langlands-Kottwitz Method on Moduli Spaces of Global Shtukas(2024) Song, Shin Eui; Haines, Thomas; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We apply the approach of Scholze to compute the trace of Hecke operator twisted by some power of Frobenius on the cohomology of the moduli spaces of global shtukas in the case of bad reduction. We find a formula that involves orbital integrals and twisted orbital integrals which can be compared with the Arthur-Selberg trace formula. This extends the results of Ngo and Ngo Dac on counting points of moduli spaces of global shtukas over finite fields. The main problem lies in finding a suitable compactly supported locally constant function that will be plugged into the twisted orbital integrals. Following Scholze, we construct locally constant functions called the test functions by using deformation spaces of bounded local shtukas. Then we establish certain local-global compatibility to express the trace on the nearby cycle sheaves on the moduli space of global shtukas to the trace on the deformation spaces.Item Variable selection and causal discovery methods with application in noncoding RNA regulation of gene expression(2024) Ke, Hongjie; Ma, Tianzhou; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Noncoding RNAs (ncRNAs), including long noncoding RNAs (lncRNAs), micro RNAs (miRNAs), etc, are critical regulators that control the gene expression at multiple levels. Revealing how the ncRNAs regulate their target genes in disease associated pathways will provide mechanistic insights into the disease and have potential clinical usage. In this dissertation, we developed novel variable selection and causal discovery methods to study the regulatory relationship between ncRNAs and genes. In Chapter 2, we proposed a novel screening method based on robust partial correlation to identify noncoding RNA regulators of gene expression over the whole genome. In Chapter 3, we developed a computationally efficient two-stage Bayesian Network (BN) learning method to construct ncRNA-gene regulatory network from transcriptomic data of both coding genes and noncoding RNAs. We provided a novel analytical platform with a graphical user interface (GUI) which covered the entire pipeline of data preprocessing, network construction, module detection, visualization and downstream analyses to accompany the developed BN learning method. In Chapter 4, we proposed a Bayesian indicator variable selection model with hierarchical structure to uncover how the regulatory mechanism between noncoding RNAs and genes changes over different biological conditions (e.g., cancer stages). In Chapter 5, we discussed about the potential extension and future work. This dissertation presents computationally efficient and statistically rigorous methods that can jointly analyze high-dimensional noncoding RNA and gene expression data to investigate their regulatory relationships, which will deepen our understanding of the molecular mechanism of diseases.Item Advancements in Small Area Estimation Using Hierarchical Bayesian Methods and Complex Survey Data(2024) Das, Soumojit; Lahiri, Partha; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This dissertation addresses critical gaps in the estimation of multidimensional poverty measures for small areas and proposes innovative hierarchical Bayesian estimation techniques for finite population means in small areas. It also explores specialized applications of these methods for survey response variables with multiple categories. The dissertation presents a comprehensive review of relevant literature and methodologies, highlighting the importance of accurate estimation for evidence-based policymaking. In Chapter \ref{chap:2}, the focus is on the estimation of multidimensional poverty measures for small areas, filling an essential research gap. Using Bayesian methods, the dissertation demonstrates how multidimensional poverty rates and the relative contributions of different dimensions can be estimated for small areas. The proposed approach can be extended to various definitions of multidimensional poverty, including counting or fuzzy set methods. Chapter \ref{chap:3} introduces a novel hierarchical Bayesian estimation procedure for finite population means in small areas, integrating primary survey data with diverse sources, including social media data. The approach incorporates sample weights and factors influencing the outcome variable to reduce sampling informativeness. It demonstrates reduced sensitivity to model misspecifications and diminishes reliance on assumed models, making it versatile for various estimation challenges. In Chapter \ref{chap: 4}, the dissertation explores specialized applications for survey response variables with multiple categories, addressing the impact of biased or informative sampling on assumed models. It proposes methods for accommodating survey weights seamlessly within the modeling and estimation processes, conducting a comparative analysis with Multilevel Regression with Poststratification (MRP). The dissertation concludes by summarizing key findings and contributions from each chapter, emphasizing implications for evidence-based policymaking and outlining future research directions.Item Metastable Distributions for Semi-Markov Processes(2024) Mohammed Imtiyas, Ishfaaq Ahamed; Koralov, Leonid; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)In this work, we consider semi-Markov processes whose transition times and transitionprobabilities depend on a small parameter ε. Understanding the asymptotic behavior of such processes is needed in order to study the asymptotics of various randomly perturbed dynamical and stochastic systems. The long-time behavior of a semi-Markov process Xε t depends on how the point (1/ε, t(ε)) approaches infinity. We introduce the notion of complete asymptotic regularity (a certain asymptotic condition on transition probabilities and transition times), originally developed for parameter-dependent Markov chains, which ensures the existence of the metastable distribution for each initial point and a given time scale t(ε). The result may be viewed as a generalization of the ergodic theorem to the case of parameter-dependent semi-Markov processes.Item OUT OF DISTRIBUTION EVALUATION OF NATURAL LANGUAGE PROCESSING SYSTEMS: GENERALIZATION TO LOW-RESOURCE AND DISTANT LANGUAGES AND HUMAN-AI COLLABORATIVE WRITING(2024) Richburg, Aquia; Carpuat, Marine; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Large language models have revolutionized natural language processing with their capabilities in text generation and understanding. Their rich contextual representations learned from training on diverse text datasets have lead LLMs to be used across a variety of settings. However this increases the chance of models being used in unintended use cases and causing harm to users. This dissertation delves into empirical studies of out-of-distribution issues in text generation (machine translation) and text classification (authorship analysis) tasks, examining how LLMs perform in settings distant from their training distributions.In our first work, the goal is to understand the characteristics of the training distribution of LLMs by visualizing the roles of samples during the training of a machine translation model. Our results indicate that sample contributions are not uniform and play complex roles throughout the training process. This highlights the difficulty of describing samples that are representative of the training distribution and motivates thorough evaluation of models in diverse settings. Our second and third works turn to the evaluation of LLMs in out-of-distribution settings to better understand their strengths and limitations for generalization on unseen tasks. We evaluate LLMs in machine translation tasks, focusing on how translation quality is affected by the presence or absence of specific language pairs in the training data. Our findings show that while finetuning improves translation for unseen languages, the impact varies across different language pairs. This emphasizes the need for further research to enable effective massively multilingual translation with LLMs. In text classification, we explore out-of-distribution generalization for authorship analysis in the context of human-AI collaborative writing. Our studies reveal that traditional AI detection models underperform when distinguishing between human and AI cowritten text. Simpler n-gram techniques are more robust than LLM for authorship identification, suggesting the need for adapted authorship analysis tools. In summary this dissertation advances our understanding of LLM generalization and provides insights for improving the robustness and adaptability of NLP systems.Item Polynomials with Equal Images over Number Fields(2024) Hirsh, Jordan; Washington, Lawrence C; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Chapman and Ponomarenko [1] characterized when two polynomials f, g ∈ Q[x] have thesame image f(Z) = f(Z). We extend this result to rings of integers in number fields. In particular, if K is a finite extension of Q and O is the ring of algebraic integers in K, we characterize when polynomials f, g ∈ K[x] satisfy f(O) = g(O). As part of our proof, we give a variant of Hilbert’s irreducibility theorem.