• Approximate Schauder Frames for Banach Sequence Spaces 

  Cheng, Yam-Sung (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 

  He, Yuchen (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 

  Li, Ruilin (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 

  Park, Jaemin (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 

  Ma, Yuanzhe (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 

  Park, Josiah (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 

  Hu, Qianli (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 

  McCullough, Andrew

  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 

  Lee, Kisun (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 

  Lanier, Justin Dale (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 

  Li, Jiaheng (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 

  Kieffer, Thomas Forrest (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 

  Petti, Samantha N. (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 

  Xie, Qiqin (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 ...

• Detecting gerrymandering with mathematical rigor 

  Pegden, Wesley (2020-02-06)

  In recent years political parties have more and more expertly crafted political districtings to favor one side or another, while at the same time, entirely new techniques to detect and measure these efforts are being ...

• The proxy point method for rank-structured matrices 

  Xing, Xin (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 

  Paprocki, Jonathan Michael (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 

  Xie, Shijie (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 

  Zhang, Yuze (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 ...

• Concentration and Convexity - Part 3 

  Paouris, Grigoris (Georgia Institute of Technology, 2019-12-13)

  The Concentration of measure phenomenon is a fundamental tool of high dimensional probability and of Asymptotic Geometric Analysis. Independence or Isoperimetry are two typical reasons for the appearance of this phenomenon. ...

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