Department of Mathematics and Statistics Graduate Theses
Recent Submissions

A Quantification of Long Transient Dynamics
We present a systematic study of transient dynamics starting with a technical definition of transient points which are initial data of an autonomous system of ordinary differential equations that can lead to “long transient ... 
Examining the Burnside Problem on Diff∞ ω (S2)
Let $G$ be a group. $G$ is called \emph{periodic} if, for every element $g \in G$, there is a positive integer $n \in \N$ such that $g^n = \id_G$, and \emph{periodic of bounded exponent} if one such $n$ works for all $g ... 
Marked Length Spectrum Rigidity of Anosov Manifolds
We provide a proof that, for compact Riemannian manifolds with Anosov geodesic flow and nonpositive curvature, two sufficiently close metrics with equal marked length spectrum will necessarily be isometric. To be precise, ... 
A Study of the Minimum pvalue and Related Methods for the Identification of TreatmentSensitive Groups
In clinical practices, a fundamentally important problem is to identify a subgroup of patients who may benefit more in terms of a clinical outcome from a certain treatment based on a specific clinical variable or biomarker, ... 
Tail Asymptotics for the Limiting Distribution of Theta Sums
We define theta sums to be exponential sums of the form S_{N}(x; \alpha, \beta) := \sum_{n =1}^{N} e((\tfrac{1}{2} n^2 + \beta n)x + \alpha n), where e(z) = e^{2 \pi i z}. If \alpha and \beta are fixed rational numbers, ... 
The Exponential Map for TimeVarying Vector Fields
The exponential map that characterises the flows of vector fields is the key in understanding the basic structural attributes of control systems in geometric control theory. However, this map does not exist due to the lack ... 
Comparison and Continuity Properties of Equilibrium Values in Information Structures for Stochastic Games
In stochastic games where players measure a costrelevant exogenous state variable through measurement channels, an information structure is the joint probability measure induced on the state space and player measurement ... 
Online Learning in Control Theory
In this thesis, we study two classes of problems in optimal control theory involving unknown parameters, with focus on LinearQuadraticGaussian systems. In the first problem, the control system is known and linear, and ... 
Optimal Transport in Mechanism Design, Differential Games, and Ensemble Control
In this thesis, we study optimal transport problems arising in game and controltheoretic settings. The thesis has three parts; in part one we study a variation of the matching problem over a continuum of consumers and ... 
Multivariate Hilbert Polynomials For Smooth Projective Toric Varieties
Generalizing Hilbert polynomials for varieties embedded in projective space, we develop the theory of multivariate Hilbert polynomials for varieties embedded in a smooth projective toric variety. These polynomials can be ... 
Theory and Application of Kernel Estimation Methods for CovariateVarying Effects in Survival Analysis
Study of prognostic and predictive biomarkers play an important role in the design and analysis of clinical trials. The development of personalized medicine calls for the investigation of the biomarker effect on the ... 
Numerical Restrictions on Seshadri Curves, with Applications to P^1 x P^1
We study rpoint Seshadri constants on smooth surfaces. The Seshadri constant is bounded above by the Nagata bound, and in cases where it is not maximal this fact is witnessed by a set of curves which we call Nagata curves. ... 
Decentralized Stochastic Control with Many Decision Makers and their MeanField Limit
In this thesis, we study large stochastic team problems (known also as decentralized stochastic control problems) with finite as well as countably infinite number of decision makers. In the first part of the thesis, ... 
Statistical Models for Identification of Treatmentsensitive Subgroups Based on Longitudinal Outcomes in Clinical Trials
In randomized clinical trials, an average effect of a treatment is often evaluated over all patients enrolled. But in the era of personalized medicine, there is an increasing interest in identification of patients who may ... 
ZeroDelay Lossy Coding of Linear Vector Markov Sources with Applications to Networked Control
In this thesis we study the optimal zerodelay coding (quantization) of $\mathbb{R}^d$valued linearly generated Markov sources under quadratic distortion. The structure and existence of deterministic and stationary ... 
Infinitesimal Probability Theory with Amalgamation
We introduce the notion of operatorvalued infinitesimal independence for the free, Boolean, and monotone cases. We show that operatorvalued infinitesimal free (respectively Boolean, monotone) independence is equivalent ... 
Robustness to Approximations and Model Learning in MDPs and POMDPs
This thesis develops a topological approach to robustness to incorrectincomplete models in MDPs and POMDPs, and applications of this study to approximations, model learning and reinforcement learning. In the first part ... 
The Effect of Spectral Line Components on Coefficients in Time Series Regression Models
Inference on the fitted parameters from two time series regressions can be improved by considering their correlation structure. We investigate the performance of an estimator of covariance between two time series regressions ... 
Multitaper Transfer Function Estimates and the Hunt for Coherent Signals Trespassing on Time Series Regression Models
This work provides a test statistic for the detection of coherent harmonic signals underlying time series regression models. Time series regression observes a response series modelled as a function of one or more predictor ... 
Spatial Voting Models: A Dynamical Systems Approach
Spatial voting models have been used extensively to study factors affecting candidate positioning in elections. Most previous research has fixed candidate positioning at one point in time to determine optimal candidate ...