School of Mathematics Theses and Dissertations
Original work in partial fulfillment of the requirements for a graduate degree from the School of Mathematics.
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Recent Submissions

On difference graph covers and the local dimension of the Boolean lattice
(Georgia Institute of Technology, 20220503)N/A 
Dual representations of polynomial modules with applications to partial differential equations
(Georgia Institute of Technology, 20220502)In 1939, Wolfgang Gröbner proposed using differential operators to represent ideals in a polynomial ring. Using Macaulay inverse systems, he showed a onetoone correspondence between primary ideals whose variety is a ... 
Duality Between Deep Learning And Algorithm Design
(Georgia Institute of Technology, 20220502)This thesis introduces 'Duality Between Deep Learning And Algorithm Design'. Deep learning is a datadriven method, whereas conventional algorithm design is a knowledgedriven method. Based on their connections and ... 
A Selflimiting Hawkes Process: Interpretation, Estimation, and Use in Modeling
(Georgia Institute of Technology, 20220428)Many real life processes that we would like to model have a selfexciting property, i.e. the occurrence of one event causes a temporary spike in the probability of other events occurring nearby in space and time. Examples ... 
Heat kernel and geometry of metric measure spaces with Ricci curvature lower bounds
(Georgia Institute of Technology, 20220428)The thesis is a study of geometric properties of noncollapsed metric measure spaces with Ricci curvature lower bounds. We establish some characterizations of noncollapsed spaces and as a consequence, solve the De ... 
Numerical computation and analysis related to optimal transport theory
(Georgia Institute of Technology, 20220428)In this thesis we apply the optimal transport (OT) theory to various disciplines of applied and computational mathematics such as scientific computing, numerical analysis, and dynamical systems. The research consists of ... 
NONPARAMETRIC ANALYSIS FOR TIME SERIES GAP DATA WITH APPLICATIONS IN ACUTE MYOCARDIAL INFARCTION DISEASE
(Georgia Institute of Technology, 20210215)Gap data problems are very popular recently, since scientists are more curious about what occurs during a period where information might be missing or unrecorded. Here, a nonparametric method called Imputed Empirical ... 
Decay of Entropy and Information for multidimensional Kac models
(Georgia Institute of Technology, 20211214)We study the approach to equilibrium in relative entropy of systems of gas particles modeled via the Kac master equation in arbitrary dimensions. First, we study the Kac system coupled to a thermostat, and secondly connected ... 
The Extremal Function for K10 Minors
(Georgia Institute of Technology, 20211215)We prove that every graph on n >= 8 vertices and at least 8n35 edges either has a K10 minor or is isomorphic to some graph included in a few families of exceptional graphs. 
Domains of analyticity and Gevrey estimates of tori in weakly dissipative systems
(Georgia Institute of Technology, 20211207)We consider the problem of following quasiperiodic tori in perturbations of Hamiltonian systems which involve friction and external forcing. In the first part, we study a family of dissipative standard maps of the cylinder ... 
Persistence of Invariant Objects under Delay Perturbations
(Georgia Institute of Technology, 20210730)In this dissertation, we investigate functional differential equations which come from adding delayrelated perturbations to ODEs or evolutionary PDEs. Despite the singular nature of the perturbations, when the perturbations ... 
Algorithmic approaches to problems in probabilistic combinatorics
(Georgia Institute of Technology, 20210727)The probabilistic method is one of the most powerful tools in combinatorics; it has been used to show the existence of many hardtoconstruct objects with exciting properties. It also attracts broad interests in designing ... 
Multiscale Problems in Mechanics: Spin Dynamics, StructurePreserving Integration, and DataDriven Methods
(Georgia Institute of Technology, 20210728)This thesis focuses on analyzing the physics and designing multiscale methods for nonlinear dynamics in mechanical systems, such as those in astronomy. The planetary systems (e.g., the Solar System) are of great interest ... 
A combinatorial approach to biological structures and networks in predictive medicine
(Georgia Institute of Technology, 20210809)This work concerns the study of combinatorial models for biological structures and networks as motivated by questions in predictive medicine. Through multiple examples, the power of combinatorial models to simplify problems ... 
Polyhedral and tropical geometry in nonlinear algebra
(Georgia Institute of Technology, 20210726)This dissertation consists of four chapters on various topics in nonlinear algebra. Particularly, it focuses on solving algebraic problems and polynomial systems through the use of combinatorial tools. Chapter one gives a ... 
Physical Billiards and Open Dynamical Systems
(Georgia Institute of Technology, 20210714)This thesis consists of four works in dynamical systems with a focus on billiards. In the first part, we consider open dynamical systems, where there exists at least a hole of positive measure in the phase space which some ... 
Fast Algorithm for Invariant Circle and their Stable Manifolds: Rigorous Results and Efficient Implementations
(Georgia Institute of Technology, 20210804)In this thesis, we present, analyze, and implement a quadratically convergent algorithm to compute the invariant circle and the foliation by stable manifolds for 2dimensional maps. The 2dimensional maps we are considering ... 
Applications of monodromy in solving polynomial systems
(Georgia Institute of Technology, 20210714)Polynomial systems of equations that occur in applications frequently have a special structure. Part of that structure can be captured by an associated Galois/monodromy group. This makes numerical homotopy continuation ... 
Spectrum Reconstruction Technique and Improved Naive Bayes Models for Text Classification Problems
(Georgia Institute of Technology, 20200421)This thesis studies two topics. In the first part, we study the spectrum reconstruction technique. As is known to all, eigenvalues play an important role in many research fields and are foundation to many practical techniques ... 
Classification fo tight contact structures on the Weeks manifold
(Georgia Institute of Technology, 20210818)One of important problems in 3dimensional contact geometry is to figure out which 3manifolds admit tight contact structures and classify tight contact structures on the manifolds which admit tight contact structures. It ...