Mathematics Theses and Dissertations

Permanent URI for this collectionhttp://hdl.handle.net/1903/2793

Browse

Now showing 1 - 20 of 619

Special Lagrangians in Milnor Fibers and Almost Lagrangian Mean Curvature Flow
(2024) Pinsky, Mirna; Rubinstein, Yanir A.; Mathematics
The 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.
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.
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.
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.
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.
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.
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.
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.
GOOD POSITION BRAIDS, TRANSVERSAL SLICES AND AFFINE SPRINGER FIBERS
(2024) Duan, Chengze; Haines, Thomas TH; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
In the study of Iwahori-Hecke algebras, Geck and Pfeiffer introduced good elements inCoxeter groups. These elements played a crucial role in the work of He and Lusztig on generalizing Steinberg’s cross-sections and Steinberg slices. This work yields the transversal slices for basic unipotent conjugacy classes in a reductive group G. We improve this result by introducing some more general braid elements called good position braids. We use them to construct transversal slices for any unipotent conjugacy classes in G. On the other hand, these good position braids also correspond to affine Springer fibers via root valuation strata. The correspondence leads to a reformulation of the dimension formula of affine Springer fibers. We also expect these braid elements to help with a conjecture of Goresky, Kottwitz and MacPherson on the cohomology of affine Springer fibers.
CYCLOTOMIC Z2-EXTENSION OF REAL QUADRATIC FIELDS WITH CYCLIC IWASAWA MODULE
(2024) Avila Artavia, Josue David; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
For a number field K and a prime p, let K∞ denote the cyclotomic Zp-extension of K, andAn denote the p-primary part of the class group of its n-th layer Kn. Greenberg conjectured that for a totally real field, the order of An becomes constant for sufficiently large n. Motivated by the work of Mouhib and Movahhedi, we focus on the case where p = 2 and K is a real quadratic field such that the Iwasawa module X∞ = lim←An is cyclic. They determined all such fields and proved that Greenberg’s conjecture holds for some cases. In this dissertation, we provide new examples of infinite families of real quadratic fields satisfying Greenberg’s conjecture which were not covered completely in the work of Mouhib and Movahhedi. To achieve this, we use the theory of binary quadratic forms and biquadratic extensions to determine a fundamental system of units and the class number of the first few layers of the cyclotomic Z2-extension. Additionally, in certain cases, we can determine the size of the module X∞ and the level of the cyclotomic tower where the size of An becomes constant.
Motivic Homotopy Theory and Synthetic Spectra
(2024) Dziedzic, Charles Richard; Ramachandran, Niranjan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Motivic homotopy studies the application of techniques from homotopy theory to algebraic geometry, using A1 as the analogue of the unit interval. Voevodsky found early success in using his constructions to prove Milnor’s conjecture and the Bloch-Kato conjecture. An interesting and deep theory arises when constructing the unstable and stable motivic categories, and it has developed into its own field of study. We begin with a survey of these constructions, detailing the equivalences between the different model used in the construction of H(S) and SH(S). Here, we draw connections between all the constructions one might encounter across the literature, and provide explicit statements on their equivalence. Stable homotopy theorists have also found utility in motivic homotopy, using the stable mo- tivic homotopy category SH to advance computations of the stable homotopy groups of spheres, such as in the work of Isaksen-Wang-Xu. Other work by Bachmann-Kong-Wang-Xu has made great progress in our understanding of motivic homotopy theory. Synthetic spectra are a construction of Pstragrowski which represent a ‘return to form’ ofsome sort, as they are constructed entirely in the ∞−category of spectra. However, they give rise to a natural bigrading and a strong connection to motivic homotopy; one of the main results is an equivalence of ∞−categories with cellular motivic spaces over C, SpCcell. We build up enough of the general theory to establish the connection with motivic homotopy and comment on recent applications.
G-INVARIANT REPRESENTATIONS USING COORBITS
(2024) Tsoukanis, Efstratios; Balan, Radu V; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Consider a finite-dimensional real vector space and a finite group acting unitarily on it. We investigate the general problem of constructing Euclidean stable embeddings of the quotient space of orbits. Our embedding relies on subsets of sorted coorbits with respect to chosen window vectors.Our main injectivity results examine the conditions under which such embeddings are injective. We establish these results using semialgebraic techniques. Furthermore, our main stability result states and demonstrates that any embedding based on sorted coorbits is automatically bi-Lipschitz when injective. We establish this result using geometric function techniques. Our work has applications in data science, where certain systems exhibit intrinsic invariance to group actions. For instance, in graph deep learning, graph-level regression and classification models must be invariant to node labeling.
MATHEMATICS OF THE DYNAMICS AND CONTROL OF THE SARS-COV-2 PANDEMIC
(2024) Pant, Binod; Gumel, Abba B.; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
The pneumonia-like illness that emerged late in 2019, caused by SARS-CoV-2 (and coined COVID-19), became the greatest public health challenge humans have faced since the 1918/1919 influenza pandemic, causing over 670 million confirmed cases and 7 million fatalities globally. This dissertation contributes in providing deep qualitative insights and understanding on the transmission dynamics and control of the pandemic, using mathematical modeling approaches together with data analytics and computation. Specifically, it addresses some of the pertinent challenges associated with modeling the dynamics of the disease, notably the disproportionate effect of the disease on certain (risk and demographic) populations (inducing various heterogeneities) and behavior changes with respect to adherence or lack thereof to interventions. An $m-$group model, which monitors the temporal dynamics of the disease in m heterogeneous populations, was designed and used to study the impact of age heterogeneity and vaccination on the spread of the disease in the United States. For instance, the disease-free equilibrium for the case of the model with m=1 (i.e., the model with a homogeneous population) was shown to be globally-asymptotically stable for two special cases (when vaccine is perfect or when disease-induced mortality is negligible) whenever the associated reproduction number of the homogeneous model is less than one. The homogeneous model has a unique endemic equilibrium whenever the reproduction threshold exceeds unity (this equilibrium was shown to be globally-asymptotically stable for a special case, using a nonlinear Lyapunov function of Goh-Volterra type). The homogeneous model was fitted to the observed cumulative mortality data for the SARS-CoV-2 pandemic in the United States during the period from January to May of 2022 (when Omicron was the predominant variant). It was shown that vaccine-derived herd immunity (needed to eliminate the disease) cannot be attained using the homogeneous model regardless of the proportion of individuals fully vaccinated. Such vaccine-derived immunity can, however, be achieved using the $m$-group heterogeneous model, with $m=2$ (where the total population is split into two groups: those under 65 years of age, and those 65 years and older), if at least 61\% of the susceptible population is fully vaccinated. Thus, this dissertation shows that heterogeneity reduces the level of vaccine coverage needed to eliminate the pandemic (and models that do not account for heterogeneity may be over-estimating the vaccination coverage needed to achieve herd immunity in the community). To quantify the impact of human behavior changes on the spread and control of the pandemic, we designed a novel behavior-epidemiology model which considers numerous metrics for inducing human behavior changes (such as current level of disease burden and intervention adherence fatigue). Unlike the equivalent model without human behavior explicitly incorporated, the behavior-epidemiology model fits the observed cumulative mortality and predicts the observed daily mortality data very well. It was also shown that the behavior metrics related to the level of SARS-CoV-2 mortality and symptomatic transmission were more influential in inducing positive behavior changes than all other behavior metrics considered. Finally, a model was developed to assess the utility of wastewater surveillance to study the transmission dynamics and control of SARS-CoV-2 in a community. Specifically, we developed and calibrated a wastewater-based epidemiology model using wastewater data from Miami-Dade county, Florida, during the third wave of the SARS-CoV-2 pandemic. The model showed a strong correlation between the observed (detected) weekly case data and the corresponding weekly data predicted by the calibrated model. The model's prediction of the week when maximum number of SARS-CoV-2 cases will be recorded in the county during the simulation period precisely matched the time when the maximum observed/reported cases were recorded (August 14, 2021). Furthermore, the model's projection of the maximum number of cases for the week of August 14, 2021 was about 15 times higher than the maximum observed weekly case count for the county on that day (i.e., the maximum case count estimated by the model was 15 times higher than the actual/observed count for confirmed cases). In addition to being in line with other modeling studies, this result is consistent with the CDC estimate that the reported confirmed case data may be 10 times lower than the actual (since the confirmed data did not account for asymptomatic and presymptomatic transmission). Furthermore, the model accurately predicted a one-week lag between the peak in weekly COVID-19 case and hospitalization data during the time period of the study in Miami-Dade, with the model-predicted hospitalizations peaking on August 21, 2021. Detailed time-varying global sensitivity analysis was carried out to determine the parameters (wastewater-based, epidemiological and biological) that have the most influence on the chosen response function (namely, the cumulative viral load in the wastewater). This analysis identified key parameters that significantly affect the value of the response function (hence, they should be targeted for intervention). This dissertation conclusively showed that wastewater surveillance data can be a very powerful indicator for measuring (i.e., providing early-warning signal and current burden) and predicting the future trajectory and burden (e.g., number of cases and hospitalizations) of emerging and re-emerging infectious diseases, such as SARS-CoV-2, in a community.
Long-term behavior of randomly perturbed Hamiltonian systems: large deviations and averaging
(2024) Yan, Shuo; Koralov, Leonid; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
This dissertation concerns various asymptotic problems related to the long-term macroscopic behavior of randomly perturbed Hamiltonian systems, with different types of perturbations and on different time scales. Since the perturbations of the systems are assumed to be small while the systems are observed at large times, non-trivial phenomena can arise due to the interplay between the perturbation size and the temporal and spatial scales, and the systems often demonstrate qualitatively distinct types of behavior depending on the subtle quantitative relation between asymptotic parameters. More specifically, on a natural time scale, i.e., in time required for the dynamics to move distance of order one, we investigate the dynamical systems with fast-oscillating perturbations and obtain precise estimates on the distribution. In particular, we calculate the exact asymptotics of the distribution in the case of linear dynamical systems. This problem also inspires the study of the local limit theorem for time-inhomogeneous functions of Markov processes. The local limit theorem is a significant and widely used tool in problems of pure and applied mathematics as well as statistics. This result has been included in [1] and submitted for publication. On the time scale that is inversely proportional to the effective size of the perturbation, we prove that the evolution of the first integral of the Hamiltonian system with fast-oscillating perturbations converges to a Markov process on the corresponding Reeb graph, with certain gluing conditions specified at the interior vertices. The result is parallel to the celebrated Freildin-Wentzell theory on the averaging principle of additive white-noise perturbations of Hamiltonian systems, and provides a description of the long-term behavior of a system when adopting an alternative approach to modeling random noise. Moreover, the current result provides the first scenario where the motion on a graph and the corresponding gluing conditions appear due to the averaging of a slow-fast system. It allows one to consider, for instance, the long-time diffusion approximation for an oscillator with a potential with more than one well. This result has been submitted for publication [2]. In the last part of the dissertation, we return to the more classical case of additive diffusion-type perturbations, combine the ideas of large deviations and averaging, and establish a large deviation principle for the first integral of the Hamiltonian system on intermediate time scales. Besides representing a new step in large deviations and averaging, this result will have important applications to reaction-diffusion equations and branching diffusions. The latter two concepts concern the evolution of various populations (e.g., in biology or chemical reactions). This result has been published in the journal Stochastics and Dynamics [3].
The Twining Character Formula for Split Groups and a Cellular Paving for Quasi-split Groups
(2024) Hopper, Jackson; Haines, Thomas; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
The dissertation contains two main results. The first is on the twisted Weyl character formula for split groups and the second is a cellular paving result for convolution morphisms in partial affine flag varieties of quasi-split groups. Let G^ be a connected reductive group over an algebraically closed field of characteristic 0 with a pinning-preserving outer automorphism σ. Jantzen’s twining character formula relates the trace of the action of σ on a highest-weight representation of G^ to the character of a corresponding highest-weight representation of a related group. This paper extends the methods of Hong’s geometric proof for the case G^ is adjoint, to prove that the formula holds for all split, connected, reductive groups, and examines the role of additional hypotheses. In particular, it is shown that for a disconnected reductive group G, the affine Grassmannian of G is isomorphic to the affine Grassmannian of its neutral component. In the final section, it is explained how these results can be used to draw conclusions about quasi-split groups over a non-Archimedean local field. This paper thus provides a geometric proof of a generalization of the Jantzen twining character formula, and provides some apparently new results of independent interest along the way. Now we turn to the context of Chapter 3. Let G be a tamely ramified, quasi-split group over a Laurent series field K = k((t)), where k is either finite or algebraically closed. If k is finite of order q and the split adjoint form of G contains a factor of type D4, then we also assume either 3 divides q or 3 divides q-1. Given a sequence of Schubert varieties contained in a fixed partial affine flag variety F for G, consider the convolution morphism m that maps the twisted product of those Schubert varieties into the partial affine flag variety F. We show that the fibers of m are paved by finite products of affine spaces and punctured affine spaces. This generalizes a result of Haines, which proves a similar result in the case G is split and defined over k. A consequence for structure constants of parahoric Hecke algebras is deduced.
Quantum Algorithms for Nonconvex Optimization: Theory and Implementation
(2024) Leng, Jiaqi; Wu, Xiaodi XW; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
Continuous optimization problems arise in virtually all disciplines of quantitative research. While convex optimization has been well-studied in recent decades, large-scale nonconvex optimization problems remain intractable in both theory and practice. Quantum computers are expected to outperform classical computers in certain challenging computational problems. Some quantum algorithms for convex optimization have shown asymptotic speedups, while the quantum advantage for nonconvex optimization is yet to be fully understood. This thesis focuses on Quantum Hamiltonian Descent (QHD), a quantum algorithm for continuous optimization problems. A systematic study of Quantum Hamiltonian Descent is presented, including theoretical results concerning nonconvex optimization and efficient implementation techniques for quantum computers. Quantum Hamiltonian Descent is derived as the path integral of classical gradient descent algorithms. Due to the quantum interference of classical descent trajectories, Quantum Hamiltonian Descent exhibits drastically different behavior from classical gradient descent, especially for nonconvex problems. Under mild assumptions, we prove that Quantum Hamiltonian Descent can always find the global minimum of an unconstrained optimization problem given a sufficiently long runtime. Moreover, we demonstrate that Quantum Hamiltonian Descent can efficiently solve a family of nonconvex optimization problems with exponentially many local minima, which most commonly used classical optimizers require super-polynomial time to solve. Additionally, by using Quantum Hamiltonian Descent as an algorithmic primitive, we show a quadratic oracular separation between quantum and classical computing. We consider the implementation of Quantum Hamiltonian Descent for two important paradigms of quantum computing, namely digital (fault-tolerant) and analog quantum computers. Exploiting the product formula for quantum Hamiltonian simulation, we demonstrate that a digital quantum computer can implement Quantum Hamiltonian Descent with gate complexity nearly linear in problem dimension and evolution time. With a hardware-efficient sparse Hamiltonian simulation technique known as Hamiltonian embedding, we develop an analog implementation recipe for Quantum Hamiltonian Descent that addresses a broad class of nonlinear optimization problems, including nonconvex quadratic programming. This analog implementation approach is deployed on large-scale quantum spin-glass simulators, and the empirical results strongly suggest that Quantum Hamiltonian Descent has great potential for highly nonconvex and nonlinear optimization tasks.
CUR Matrix Approximation Through Convex Optimization
(2024) Linehan, Kathryn; Balan, Radu V; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
In this dissertation we present work on the CUR matrix approximation. Specifically, we present 1) an approximation of the proximal operator of the L-infinity norm using a neural network, 2) a novel deterministic CUR formulation and algorithm, and 3) a novel application of CUR as a feature selection method to determine discriminant proteins when clustering protein expression data in a self-organizing map (SOM). The proximal operator of the L-infinity norm arises in our CUR algorithm. Since the computation of the proximal operator of the L-infinity norm requires a sort of the input data (or at least a partial sort similar to quicksort), we present a neural network to approximate the proximal operator. A novel aspect of the network is that it is able to accept vectors of varying lengths due to a feature selection process that uses moments of the input data. We present results on the accuracy of the approximation, feature importance, and computational efficiency of the approach, and present an algorithm to calculate the proximal operator of the L-infinity norm exactly, relate it to the Moreau decomposition, and compare its computational efficiency to that of the approximation. Next, we present a novel deterministic CUR formulation that uses convex optimization to form the matrices C and R, and a corresponding algorithm that uses bisection to ensure that the user selected number of columns appear in C and the user selected number of rows appear in R. We implement the algorithm using the surrogate functional technique of Daubechies et al. [Communications on Pure and Applied Mathematics, 57.11 (2004)] and extend the theory of this approach to apply to our CUR formulation. Numerical results are presented that demonstrate the effectiveness of our CUR algorithm as compared to the singular value decomposition (SVD) and other CUR algorithms. Last, we use our CUR approximation as a feature selection method in the application by Higuera et al. [PLOS ONE, 10(6) (2015)] to determine discriminant proteins when clustering protein expression data in an SOM. This is a novel application of CUR and to the best of our knowledge, this is the first use of CUR on protein expression data. We compare the performance of our CUR algorithm to other CUR algorithms and the Wilcoxon rank-sum test (the original feature selection method in the work).
ASYMPTOTIC AND NUMERICAL ANALYSIS IN KAHLER GEOMETRY: EDGE METRICS, EINSTEIN METRICS AND SOLITONS
(2024) Ji, Yuxiang; Rubinstein, Yanir A; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
This thesis is divided into two parts. The first part focuses on theoretical problems arising from the study of K\"{a}hler edge geometry. The second part introduces a practical method for finding K\"{a}hler--Einstein and soliton metrics that exist on toric Fano surfaces. This method is based on a numerical implementation of the Ricci iteration, which essentially solves a sequence of Monge--Amp\`{e}re equations. Let $M$ be a compact \K manifold and $D=D_1+\dots D_r$ be a simple normal crossing divisor. Given that $K_M+D$ is ample, there exist \KE crossing edge metrics on $M$ with edge singularities of cone angle $\beta_i$ along each component $D_i$ for small $\beta_i$. The first result in the thesis shows that such negatively curved K\"{a}hler--Einstein crossing edge metrics converge to the K\"{a}hler--Einstein mixed cusp and edge metrics smoothly away from the divisor, as some of the cone angles approach $0$. We further show that, near the divisor, a family of appropriately renormalized K\"{a}hler--Einstein crossing edge metrics converges to a mixed cylinder and edge metric in the pointed Gromov--Hausdorff sense as some of the cone angles approach $0$ at (possibly) different speeds. Generalizing $\mathbb{P}^1$, Calabi--Hirzebruch manifolds are constructed by adding an infinite section to the total space of a tensor product of the hyperplane bundle over the projective space, leading to two disjoint divisors: the zero and infinite sections. The second main result in the thesis is the discovery of K\"{a}hler--Einstein edge metrics with singularities along the two divisors on Calabi--Hirzebruch manifolds, and the study on Gromov--Hausdorff limits of these metrics when either cone angle tends to its extreme value. As a very special case, we show that the Eguchi--Hanson metric arises in this way naturally as a Gromov--Hausdorff limit. We also completely describe all other (possibly rescaled) Gromov--Hausdorff limits which exhibit a wide range of behaviors, resolving in this setting a conjecture of Cheltsov--Rubinstein. This gives a new interpretation of both the Eguchi--Hansonspace and Calabi’s Ricci flat spaces as limits of compact singular Einstein spaces. The second part of the thesis focuses on numerical implementation of the Ricci iteration on toric del Pezzo surfaces: $\mathbb{P}^2$, $\mathbb{P}^1\times \mathbb{P}^1$, and blow-up of $\mathbb{P}^2$ at one, two or three distinct points in general position. The Ricci iteration on these surfaces can be reduced to solving a sequence of real Monge--Amp\`{e}re equations in $\mathbb{R}^2$ with the second boundary value condition. As the third contribution of the thesis, we successfully conduct the Ricci iteration on the aforementioned surfaces. We find that the resulting solutions numerically converge to either the unique K\"{a}hler--Einstein metric on $\mathbb{P}^2, \mathbb{P}^1\times \mathbb{P}^1$, and $\dPThr$, or the unique K\"{a}hler--Ricci soliton metric on $\dPOne$ and $\dPTwo$. This provides a novel numerical approach to finding K\"{a}hler--Einstein and soliton metrics on these manifolds. Our numerical results also provide evidence that, in the toric case, the Ricci iteration may converge and produce canonical metrics without the necessity of modifying the metrics obtained during the iterations by automorphisms.
Towards a Classification of Almost Complex and Spin^h Manifolds
(2024) Mills, Keith; Rosenberg, Jonathan; Mathematics; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
We show that all homotopy CP^ns, smooth closed manifolds with the oriented homotopy type of CP^n, admit almost complex structures for 3 ≤ n ≤ 6, and classify these structures by their Chern classes for n=4, 6. Our methods provide a new proof of a result of Libgober and Wood on the classification of almost complex structures on homotopy CP^4s. We also show that all homotopy RP^(2k+1)s admit stably almost complex structures. Spin^h manifolds are the quaternionic analogue to spin^c manifolds. At the prime 2 we compute the spin^h bordism groups by proving a structure theorem for the cohomology of the spin^h bordism spectrum MSpin^h as a module over the mod 2 Steenrod algebra. This provides a 2-local splitting of MSpin^h as a wedge sum of familiar spectra. We also compute the decomposition of H^*(MSpin^h; Z/2Z) explicitly in degrees up through 30 via a counting process.
EFFICIENT COMPUTATIONAL ALGORITHMS FOR MAGNETIC EQUILIBRIUM IN A FUSION REACTOR
(2024) Liang, Jiaxing; Elman, Howard C.; Sanchez-Vizuet, Tonatiuh; Applied Mathematics and Scientific Computation; Digital Repository at the University of Maryland; University of Maryland (College Park, Md.)
In a magnetic confinement fusion reactor, like a Tokamak, hydrogen isotopes are injected into the chamber and heated to form a plasma. The hot plasma tends to expand, so it's crucial to confine it within the center of the reactor to prevent contact with the reactor walls. This confinement is achieved through magnetic fields generated by currents running through external coils surrounding the reactor. However, these currents may suffer from uncertainties, which could arise from factors like temperature fluctuations and material impurities. These variables introduce a level of unpredictability in the plasma's behavior and the overall stability of the fusion process. This thesis aims to investigate the impact that stochasticity in the current intensities has on the confinement properties of the plasma and to estimate the expected behavior of the magnetic field. While the focus is on the variability in current intensities, the tools developed can be applied to other sources of uncertainty, such as the positioning of coils and the source term parameters. To quantify the variability in model predictions and to evaluate the statistical properties of solutions over a range of parameter values, traditional sampling methods like Monte Carlo, often require intensive and expensive nonlinear computations. To tackle this challenge, we propose three approaches. Firstly, we focus on the development and application of a surrogate function, constructed via a stochastic collocation approach on a sparse grid in the parameter space. This surrogate function is employed to replace the nonlinear solution in Monte Carlo sampling processes. For our numerical experiments, we evaluate the efficiency and accuracy of the outcomes produced by the surrogate, in comparison with those obtained through direct nonlinear solutions. Our findings indicate that a carefully selected surrogate function reduces the sampling cost -- achieving acceleration factors ranging from 7 to over 30 -- while preserving the accuracy of the results. The second part of the thesis explores the multilevel Monte Carlo approach, investigating its potential for cost savings compared to simple Monte Carlo. This method involves conducting the majority of computations on a sequence of coarser spatial grids compared to what a simple Monte Carlo simulation would typically use. We examine this approach with non-linear computation, using both uniformly refined meshes and adaptively refined grids guided by a discrete error estimator. Numerical experiments reveal substantial cost reductions achieved through multilevel methods, typically ranging from a factor of 60 to exceeding 200. Adaptive gridding results in more accurate computation of relevant geometric parameters. In the last part of this work, we explore hybridmethods that integrate surrogates with multilevel Monte Carlo to further reduce the sampling cost. We establish the optimal construction and sampling costs for the surrogate-based multilevel Monte Carlo. Numerical results demonstrate that surrogate-based multilevel Monte Carlo remarkably reduces the computational burden, requiring only 0.1 to 14 seconds for a target relative mean square error ranging from $8\times 10^{-3}$ to $2\times10^{-4}$, reducing the cost of direct computation by factors of 50 to 300. In terms of accuracy, the surrogate-based sampling results exhibit close congruence with those obtained via direct computation, both in plasma boundary and geometric descriptors. The primary contributions of our work entail the application of stochastic collocation techniques and multilevel Monte Carlo methods to analyze plasma behavior under uncertainties in current within fusion reactors. Furthermore, we establish the universal sampling cost for the surrogate-enhanced multilevel Monte Carlo approach. Our methodology presents a paradigm in how we approach and manage computational challenges in this field.

Browse

Recent Submissions