Department of Mathematics and Statistics Graduate Theses
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Noncommutative Independences for Pairs of Faces
In noncommutative probability theories, many different notions of independence arise in various contexts. According to the early classification work, there are only five such notions that are universal/natural in the sense ... 
Information and Estimation Theoretic Approaches to Data Privacy
Warner [145] in 1960s proposed a simple mechanism, now referred to as the randomized response model, as a remedy for what he termed “evasive answer bias” in survey sampling. The randomized response setting is as follows: ... 
Statistical Methods For Biomarker Threshold Models in Clinical Trials
In clinical trials, the main objective is to investigate the treatment effects on patients. However, many molecularly targeted drugs or treatments tend to benefit a subset of patients more, identified by a certain biomarker. ... 
An impulsive differential equation model for Marek's disease
(20160922)Many dynamical processes are subject to abrupt changes in state. Often these perturbations can be periodic and of short duration relative to the evolving process. These types of phenomena are described well by what are ... 
Irreducibility of Random Hilbert Schemes
(20160913)We prove that a random Hilbert scheme that parametrizes the closed subschemes with a fixed Hilbert polynomial in some projective space is irreducible and nonsingular with probability greater than $0.5$. To consider the ... 
Codomain Rigidity of the Dirichlet to Neumann Operator for the Riemannian Wave Equation
(20160823)We study the Dirichlet to Neumann operator for the Riemannian wave equation on a compact Riemannian manifold. If the Riemannian manifold is modelled as an elastic medium, this operator represents the data available to an ... 
Geometry of Dirac Operators
(20160705)Let $M$ be a compact, oriented, even dimensional Riemannian manifold and let $S$ be a Clifford bundle over $M$ with Dirac operator $D$. Then \[ \textsc{Atiyah Singer: } \quad \text{Ind } \mathsf{D}= \int_M ... 
Computationally Intensive Methods for Spectrum Estimation
(20160427)Spectrum estimation is an essential technique for analyzing time series data. A leading method in the field of spectrum estimation is the multitaper method. The multitaper method has been applied to many scientific fields ... 
Higher rank sieves and applications
(20160425)This thesis focuses on some of the key sieve theoretic ideas behind recent progress on bounded gaps between the primes. One such idea is the notion of higher rank sieve weights, first proposed by Atle Selberg and applied ... 
On the role of regularity in mathematical control theory
(20160408)In this thesis, we develop a coherent framework for studying timevarying vector fields of different regularity classes and their flows. This setting has the benefit of unifying all classes of regularity. In particular, ... 
Arithmetic and Intermediate Jacobians of CalabiYau threefolds
(20150910)This thesis is centered around particular CalabiYau threefolds. Borcea \cite{Borcea} and Voisin \cite{Voisin} construct CalabiYau threefolds using elliptic curves and K3 surfaces with nonsymplectic involutions. This ... 
Distributed Online Optimization on timevarying networks
(20150901)This thesis introduces two classes of discretetime distributed online optimization algorithms, with a group of agents which communicate over a network. At each time, a private convex objective function is revealed to ... 
Curves of low genus on surfaces and applications to Diophantine problems
(20150831)We describe in detail a technique due to Vojta for finding the explicit set of curves of low genus on certain algebraic surfaces of general type, and refine some of its aspects. We then provide applications of this method ... 
Optimality of WalrandVaraiya Type Policies and Approximation Results for ZeroDelay Coding of Markov Sources
(20150728)Optimal zerodelay coding of a finite state Markov source through quantization is considered. Building on previous literature, the existence and structure of optimal policies are studied using a stochastic control problem ... 
Optimal Quantization and Approximation in Source Coding and Stochastic Control
(20150623)This thesis deals with nonstandard optimal quantization and approximation problems in source coding and stochastic control. The first part of the thesis considers randomized quantization. Adapted from stochastic ... 
Mathematics Problems and Thinking Mathematically in Undergraduate Mathematics
(20150501)Mathematics is much more than a formal system of procedures and formulae; it is also a way of thinking built on creativity, precision, reasoning, and representation. I present a model for framing the process of doing ... 
Applications of Multitaper Spectral Analysis to Nonstationary Data
(20141015)This thesis is concerned with changes in the spectrum over time observed in Holocene climate data as recorded in the Burgundy grape harvest date series. These changes represent nonstationarities, and while spectral estimation ... 
Nonparametric and Parametric Methods for Solar Oscillation Spectra
(20140927)The study of the systematic oscillations of the Sun has led to better understanding of the Sun’s inner structure and dynamics, and may help to resolve inconsistencies between observations and the standard solar model. ... 
On Malliavin Calculus and Concentration Inequalities
(20140707)We prove new abstract results concerning concentration inequalities and density estimates for Malliavin differentiable random variables. The efficacy of these results are demonstrated by practical computations, such as the ... 
Arithmetic problems around the ABC conjecture and connections with logic
(20140428)The main theme in this thesis is the ABC conjecture. We prove some partial results towards it and we find new applications of this conjecture, mainly in the context of B\"uchi's n squares problem (which has consequences ...