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
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Approximate Schauder Frames for Banach Sequence Spaces
(Georgia Institute of Technology, 2021-04-29)The main topics of this thesis concern two types of approximate Schauder frames for the Banach sequence space l_1^n. The first main topic pertains to finite-unit norm tight frames (FUNTFs) for the finite-dimensional real ... -
Mathematical and Data-driven Pattern Representation with Applications in Image Processing, Computer Graphics, and Infinite Dimensional Dynamical Data Mining
(Georgia Institute of Technology, 2021-04-30)Patterns represent the spatial or temporal regularities intrinsic to various phenomena in nature, society, art, and science. From rigid ones with well-defined generative rules to flexible ones implied by unstructured data, ... -
ON SCALABLE AND FAST LANGEVIN-DYNAMICS-BASED SAMPLING ALGORITHMS
(Georgia Institute of Technology, 2021-04-28)Langevin dynamics-based sampling algorithms are arguably among the most widelyused Markov Chain Monte Carlo (MCMC) methods. Two main directions of the modern study of MCMC methods are (i) How to scale MCMC methods to big ... -
On the stationary and uniformly-rotating solutions of active scalar equations
(Georgia Institute of Technology, 2021-04-13)In this thesis, we study qualitative and quantitative properties of stationary/uniformly- rotating solutions of the 2D incompressible Euler equation and the generalized Surface Quasi- Geostrophic (SQG) equations. The main ... -
Numerical Estimation of Several Topological Quantities of the First Passage Percolation Model
(Georgia Institute of Technology, 2021-04-13)In this thesis, our main goal is to use numerical simulations to study some quantities related to the growing set B(t). Motivated by prior works, we mainly study quantities including the boundary size, the hole size, and ... -
Interaction energies, lattices, and designs
(Georgia Institute of Technology, 2020-05-21)This thesis has four chapters. The first three concern the location of mass on spheres or projective space, to minimize energies. For the Columb potential on the unit sphere, this is a classical problem, related to arranging ... -
Rayleigh-Taylor instability with heat transfer
(Georgia Institute of Technology, 2020-05-13)In this thesis, the Rayleigh-Taylor instability effect with heat transfer in the setting of the Navier-Stokes equations, given three-dimensional and incompressible fluids, is investigated. Under suitable initial and boundary ... -
Legendrian large cables and non-uniformly thick knots
We define the notion of a knot type having Legendrian large cables and show that having this property implies that the knot type is not uniformly thick. Moreover, there are solid tori in this knot type that do not thicken ... -
Finding and certifying numerical roots of systems of equations
(Georgia Institute of Technology, 2020-05-01)Numerical algebraic geometry studies methods to approach problems in algebraic geometry numerically. Especially, finding roots of systems of equations using theory in algebraic geometry involves symbolic algorithm which ... -
Small torsion generating sets for mapping class groups
(Georgia Institute of Technology, 2020-04-27)A surface of genus g has many symmetries. These form the surface’s mapping class group Mod(S_g), which is finitely generated. The most commonly used generating sets for Mod(S_g) are comprised of infinite order elements ... -
Numerical estimates for arm exponents and the acceptance profile in two-dimensional invasion percolation
(Georgia Institute of Technology, 2020-05-05)The main object of this thesis is to numerically estimate some conjectured arm exponents when there exist a number of open paths and closed dual paths that extend to the boundary of different sizes of boxes centering at ... -
The Maxwell-Pauli equations
(Georgia Institute of Technology, 2020-03-19)We study the quantum mechanical many-body problem of N ≥ 1 non-relativistic electrons with spin interacting with their self-generated classical electromagnetic field and K ≥ 0 static nuclei. We model the dynamics of the ... -
Randomness as a tool for modeling and uncovering structure
(Georgia Institute of Technology, 2020-03-12)This thesis contains four main research directions, united by the themes of using randomness to (i) construct structure and (ii) uncover structure. Randomness has long been used for these tasks. Random models are defined ... -
Coloring graphs with no k5-subdivision: disjoint paths in graphs
(Georgia Institute of Technology, 2019-03-27)The Four Color Theorem states that every planar graph is 4-colorable. Hajos conjectured that for any positive integer k, every graph containing no K_{k+1}-subdivision is k-colorable. However, Catlin disproved Hajos conjecture ... -
The proxy point method for rank-structured matrices
(Georgia Institute of Technology, 2019-11-06)Rank-structured matrix representations, e.g., $\mathcal{H}^2$ and HSS, are commonly used to reduce computation and storage cost for dense matrices defined by interactions between many bodies. The main bottleneck for their ... -
Quantum torus methods for Kauffman bracket skein modules
(Georgia Institute of Technology, 2019-08-22)We investigate aspects of Kauffman bracket skein algebras of surfaces and modules of 3-manifolds using quantum torus methods. These methods come in two flavors: embedding the skein algebra into a quantum torus related to ... -
6-connected graphs are two-three linked
(Georgia Institute of Technology, 2019-11-11)Let $G$ be a graph and $a_0, a_1, a_2, b_1,$ and $b_2$ be distinct vertices of $G$. Motivated by their work on Four Color Theorem, Hadwiger's conjecture for $K_6$, and J\o rgensen's conjecture, Robertson and Seymour asked ... -
Topics on the length of the longest common subsequences, with blocks, in binary random words
(Georgia Institute of Technology, 2019-08-27)The study of LIn, the length of the longest increasing subsequences, and of LCIn, the length of the longest common and increasing subsequences in random words is classical in computer science and bioinformatics, and has ... -
Lattice points, oriented matroids, and zonotopes
(Georgia Institute of Technology, 2019-07-26)The first half of this dissertation concerns the following problem: Given a lattice in R^d which refines the integer lattice Z^d, what can be said about the distribution of the lattice points inside of the half-open unit ... -
The polaron hydrogenic atom in a strong magnetic field
(Georgia Institute of Technology, 2019-07-19)It is shown that: (1) The ground-state electron density of a polaron bound in a Coulomb potential and exposed to a homogeneous magnetic field of strength B–with its transverse electron coordinates integrated out and when ...
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