UMD Theses and Dissertations
Permanent URI for this collectionhttp://hdl.handle.net/1903/3
New submissions to the thesis/dissertation collections are added automatically as they are received from the Graduate School. Currently, the Graduate School deposits all theses and dissertations from a given semester after the official graduation date. This means that there may be up to a 4 month delay in the appearance of a given thesis/dissertation in DRUM.
More information is available at Theses and Dissertations at University of Maryland Libraries.
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Item Classification of Closed Conformally Flat Lorentzian 3-Manifolds with Unipotent Holonomy(2023) Lee, Nakyung; Melnick, Karin; Goldman, William; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)A conformally flat manifold is a manifold that is locally conformally equivalent to a flat affine space. In this thesis, we classify closed conformally flat Lorentzian manifolds of dimension three whose holonomy group is unipotent. More specifically, we show that such a manifold is finitely covered by either $S^2\times S^1$ or a parabolic torus bundle. Furthermore, we show that such a manifold is Kleinian and is essential if and only if it can be covered by $S^2\times S^1$.Item Geometric and Topological Reconstruction(2022) Rawson, Michael G.; Balan, Radu; Robinson, Michael; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The understanding of mathematical signals is responsible for the information age. Computation, communication, and storage by computers all use signals, either implicitly or explicitly, and use mathematics to manipulate those signals. Reconstruction of a particular signal can be desirable or even necessary depending on how the signal manifests and is measured. We explore how to use mathematical ideas to manipulate and represent signals. Given measurements or samples or data, we analyze how to produce, or \emph{reconstruct}, the desired signal and the fundamental limits in doing so. We focus on reconstruction through a geometric and topological lens so that we can leverage geometric and topological constraints to solve the problems. As inaccuracies and noise are present in every computation, we adopt a statistical outlook and prove results with high probability given noise. We start off with probability and statistics and then use that for active reconstruction where the probability signal needs to be estimated statistically from sampling various sources. We prove optimal ways to doing this even in the most challenging of situations. Then we discuss functional analysis and how to reconstruct sparse rank one decompositions of operators. We prove optimality of certain matrix classes, based on geometry, and compute the worst case via sampling distributions. With the mathematical tools of functional analysis, we introduce the optimal transportation problem. Then we can use the Wasserstein metric and its geometry to provably reconstruct sparse signals with added noise. We devise an algorithm to solve this optimization problem and confirm its ability on both simulated data and real data. Heavily under-sampled data can be ill-posed which is often the case with magnetic resonance imaging data. We leverage the geometry of the motion correction problem to devise an appropriate approximation with a bound. Then we implement and confirm in simulation and on real data. Topology constraints are often present in non-obvious ways but can often be detected with persistent homology. We introduce the barcode algorithm and devise a method to parallelize it to allow analyzing large datasets. We prove the parallelization speedup and use it for natural language processing. We use topology constraints to reconstruct word-sense signals. Persistent homology is dependent on the data manifold, if it exists. And it is dependent on the manifold's reach. We calculate manifold reach and prove the instability of the formulation. We introduce the combinatorial reach to generalize reach and we prove the combinatorial reach is stable. We confirm this in simulation. Unfortunately, reach and persistent homology are not an invariant of hypergraphs. We discuss hypergraphs and how they can partially reconstruct joint distributions. We define a hypergraph and prove its ability to distinguish certain joint distributions. We give an approximation and prove its convergence. Then we confirm our results in simulation and prove its usefulness on a real dataset.Item The Effects of Explicit Instruction with Dynamic Geometry Software for Secondary Students with ADHD/Learning Disabilities(2015) Toronto, Allyson Patricia; De La Paz, Susan; Special Education; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The current study examined the effect of an instructional package on the ability of secondary students with mathematics difficulties to solve geometric similarity transformations. The instructional package includes a blend of research-based instructional practices including explicit instruction, the CRA sequence, and Dynamic Geometry Software. A multiple probe design across four participants was used to evaluate the intervention. The participants were four students with a history of mathematics difficulty in a suburban mid-Atlantic high school. Results of the study demonstrated that all four students improved their accuracy on geometric similarity transformations and maintained those skills four weeks after the completion of the intervention. Furthermore, providing multiple visual representations, including technology such as dynamic geometry software, as well as concrete manipulatives, allowed participants to make connections to geometric content and enhanced their metacognition, self-efficacy, and disposition toward geometry. This study supports the use of integrated instruction utilizing explicit instruction and visual representations for high school students with MD on grade-level geometry content.Item Teaching For Inclusion: The Effects Of A Professional Development Course For Secondary General And Special Education Mathematics Teachers For Increasing Teacher Knowledge And Self-Efficacy In Geometry(2015) Wright, Kenneth; Leone, Peter; Special Education; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)The current study examined the effects of a co-teacher professional development course for increasing the knowledge and self-efficacy of special education and general education geometry teachers in an inclusion setting. The professional development course included instruction on Universal Design for Learning instructional strategies as well as similarity and congruence in geometry. The course was presented in a blended learning format and included in-person and online activities. The online activities used animated scripts of teacher instruction for collaborative discussion and decision-making. A multiple probe design across three sets of two teachers for a total of 6 participants was used in this study to demonstrate a functional relationship between the independent and dependent variables. The participants were six special education and general education geometry teachers from public charter schools in Washington, DC. Results of the study demonstrated that participants were able to improve their content and pedagogical content knowledge in geometry as well as their self-efficacy for teaching in an inclusion setting. Specifically, special education teachers demonstrated a greater increase in content knowledge while general education teachers demonstrated a greater increase in self-efficacy for teaching students with disabilities. The study suggests that providing professional development for co-teachers can enhance collaboration as well as increase content knowledge and teacher self-efficacy.Item Progress Toward Classifying Teichmueller Disks with Completely Degenerate Kontsevich-Zorich Spectrum(2012) Aulicino, David; Forni, Giovanni; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We present results toward resolving a question posed by Eskin-Kontsevich-Zorich and Forni-Matheus-Zorich. They asked for a classification of all $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measures with completely degenerate Kontsevich - Zorich spectrum. Let $\mathcal{D}_g(1)$ be the subset of the moduli space of Abelian differentials $mathcal{M}_g$ whose elements have period matrix derivative of rank one. There is an $\text{SL}_2(\mathbb{R})$-invariant ergodic probability measure $nu$ with completely degenerate Kontsevich-Zorich spectrum, i.e. $lambda_1 = 1 > lambda_2 = cdots = lambda_g = 0$, if and only if $nu$ has support contained in $\mathcal{D}_g(1)$. We approach this problem by studying Teichm"uller disks contained in $\mathcal{D}_g(1)$. We show that if $(X,omega)$ generates a Teichm"uller disk in $\mathcal{D}_g(1)$, then $(X,omega)$ is completely periodic. Furthermore, we show that there are no Teichm"uller disks in $\mathcal{D}_g(1)$, for $g = 2$, and the known example of a Teichm"uller disk in $mathcal{D}_3(1)$ is the only one. Finally, we present an idea that might be able to fully resolve the problem.Item Length Spectral Rigidity of Non-Positively Curved Surfaces(2011) Frazier, Jeffrey Russell; Goldman, William M.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)Length spectral rigidity is the question of under what circumstances the geometry of a surface can be determined, up to isotopy, by knowing only the lengths of its closed geodesics. It is known that this can be done for negatively curved Riemannian surfaces, as well as for negatively curved cone surfaces. Steps are taken toward showing that this holds also for flat cone surfaces, and it is shown that the lengths of closed geodesics are also enough to determine which of these three categories a geometric surface falls into. Techniques of Gromov, Bonahon, and Otal are explained and adapted, such as topological conjugacy, geodesic currents, Liouville measures, and the average angle between two geometric surfaces.Item Error Control for the Mean Curvature Flow(2002) Lakkis, Omar; Nochetto, Ricardo H.; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We study the equation describing the motion of a nonparametric surface according to its mean curvature flow. This is a nonuniformly parabolic equation that can be discretized in space via a finite element method. We conduct an aposteriori error analysis of the semidiscrete scheme and derive upper bounds to the error in terms of computable quantities called estimators. The reliability of the estimators is practically tested through numerical simulations.Item Local Rigidity of Triangle Groups in Sp(4,R)(2009) Hoban, Ryan F.; Goldman, William; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)This paper studies configurations of Lagrangians in a four dimensional real symplectic vector space. We develop a generalized cross ratio as an invariant for quadruples of Lagrangians. This invariant is then used to study representations of triangle groups into the symplectic group. The main theorem is a local rigidity result for a certain representations factoring through the isometry group of the hyperbolic plane.Item Twisted Cohomology Groups(2006-08-18) Watson, Toni Aliza; Rosenberg, Jonathan M; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)We introduce the notion of a twisted cohomology group. Additionally, certain examples and implications are discussed.