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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Statistical Inference for Large Matrices
(Georgia Institute of Technology, 20160729)This thesis covers two topics on matrix analysis and estimation in machine learning and statistics. The first topic is about density matrix estimation with application in quantum state tomography. The density matrices are ... 
Transverse Surgery on Knots in Contact ThreeManifolds
(Georgia Institute of Technology, 20160519)We study the effect of surgery on transverse knots in contact $3$manifolds by examining its effect on open books, the Heegaard Floer contact invariant, and tightness in general. We first compare surgery on transverse ... 
A study of stochastic differential equations and FokkerPlanck equations with applications
(Georgia Institute of Technology, 20160415)FokkerPlanck equations, along with stochastic differential equations, play vital roles in physics, population modeling, game theory and optimization (finite or infinite dimensional). In this thesis, we study three topics, ... 
Open book decompositions in high dimensional contact manifolds
(Georgia Institute of Technology, 20160527)In this thesis, we study the open book decompositions in high dimensional contact manifolds. We focus on the results about open book decomposition of manifolds and their relationship with contact geometry. 
Numerical algorithms based on the back and forth error compensation and correction
(Georgia Institute of Technology, 20141218)In this thesis we carry out a further study of the back and forth error compensation and correction (BFECC) method. The first part discusses the time reversibility of numerical schemes. Motivated by the BFECC method, a ... 
Combinatorial problems for graphs and partially ordered sets
(Georgia Institute of Technology, 20151113)This dissertation has three principal components. The first component is about the connections between the dimension of posets and the size of matchings in comparability and incomparability graphs. In 1951, Hiraguchi proved ... 
Thermostated Kac models
(Georgia Institute of Technology, 20151116)We consider a model of N particles interacting through a Kacstyle collision process, with m particles among them interacting, in addition, with a thermostat. When m = N, we show exponential approach to the equilibrium ... 
The Filippov moments solution on the intersection of two and three manifolds
(Georgia Institute of Technology, 20151118)In this thesis, we study the Filippov moments solution for differential equations with discontinuous righthand side. In particular, our aim is to define a suitable Filippov sliding vector field on a codimension $2$ ... 
A numerical study of vorticityenhanced heat transfer
(Georgia Institute of Technology, 20140626)In this work, we have numerically studied the effect of the vorticity on the enhancement of heat transfer in a channel flow. In the first part of the work, we focus on the investigation of a channel flow with a vortex ... 
Multifold sums and products over R, and combinatorial problems on sumsets
(Georgia Institute of Technology, 20150722)We prove a new bound on a version of the sumproduct problem studied by Chang. By introducing several combinatorial tools, this expands upon a method of Croot and Hart which used the TarryEscott problem to build distinct ... 
Symmetric ideals and numerical primary decomposition
(Georgia Institute of Technology, 20150727)The thesis considers two distinct strategies for algebraic computation with polynomials in high dimension. The first concerns ideals and varieties with symmetry, which often arise in applications from areas such as algebraic ... 
Minimization problems involving polyconvex integrands
(Georgia Institute of Technology, 20150602)This thesis is mainly concerned with problems in the areas of the Calculus of Variations and Partial Differential Equations (PDEs). The properties of the functional to minimize with respect to the given topology play an ... 
Complete nonnegatively curved spheres and planes
(Georgia Institute of Technology, 20150602)We study the space of complete Riemannian metrics of nonnegative curvature on the sphere equipped with C^{k+\alpha} topology. We show the space is homogenous for k>=2. If k is infinite, we show that the space is homeomorphic ... 
Smalltime asymptotics of call prices and implied volatilities for exponential Lévy models
(Georgia Institute of Technology, 20150108)We derive atthemoney callprice and implied volatility asymptotic expansions in time to maturity for a selection of exponential Lévy models, restricting our attention to assetprice models whose log returns structure is ... 
Method of evolving junctions: a new approach to path planning and optimal control
(Georgia Institute of Technology, 20140407)This thesis proposes a novel and efficient method (Method of Evolving Junctions) for solving optimal control problems with path constraints, and whose optimal paths are separable. A path is separable if it is the concatenation ... 
Some results on sums and products
(Georgia Institute of Technology, 20141117)We demonstrate new results in additive combinatorics, including a proof of a conjecture by J. Solymosi: for every epsilon > 0, there exists delta > 0 such that, given n² points in a grid formation in R², if L is a set of ... 
Accelerated algorithms for composite saddlepoint problems and applications
(Georgia Institute of Technology, 20141113)This dissertation considers the composite saddlepoint (CSP) problem which is motivated by realworld applications in the areas of machine learning and image processing. Two new accelerated algorithms for solving composite ... 
A nonasymptotic study of lowrank estimation of smooth kernels on graphs
(Georgia Institute of Technology, 20140723)This dissertation investigates the problem of estimating a kernel over a large graph based on a sample of noisy observations of linear measurements of the kernel. We are interested in solving this estimation problem in the ... 
Linear systems on metric graphs and some applications to tropical geometry and nonarchimedean geometry
(Georgia Institute of Technology, 20140702)The divisor theories on finite graphs and metric graphs were introduced systematically as analogues to the divisor theory on algebraic curves, and all these theories are deeply connected to each other via tropical geometry ... 
Stein fillings of contact structures supported by planar open books
(Georgia Institute of Technology, 20140620)In this thesis we study topology of symplectic fillings of contact manifolds supported by planar open books. We obtain results regarding geography of the symplectic fillings of these contact manifolds. Specifically, we ...