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Fall 2021

Jiang, Yangdi

The demand for large dataset and demand of privacy protection are in constantly conflicts as the balance between the two is hard to keep. Differential privacy is a mathematical rigor definition that provides the balance bewteen these two opposite sides. It's developed with the purpose of making...

Spring 2021

Budd, Valerie Gayle

Let G be a semi-simple algebraic group over the complex numbers, B a Borel subgroup, and T a maximal torus contained in B. In the first part of this thesis, we examine the singular loci of rationally smooth T-orbit closures X (of some point x) in the flag variety G/B in types A and D. In type A,...

Fall 2021

Hongxi, Wan

Imposing a constraint on the initial wealth may cause the perfect hedging impossible. In this case, the goal of an investor is to find a strategy that minimize the shortfall under a certain measure, which leads to the concept of partial hedging. In this thesis, the shortfall risk is measured by...

Fall 2021

Ge, Yan

This thesis consists of the following two parts: In the first part, we investigate the relationship between positive definite functions on the unit sphere and in the Euclidean space with the same dimensions. For the dimension d is odd, a new technique is developed to establish the inheritance of...

Fall 2021

Fang, Xi

The Covid pandemic has lasted for over a year influencing everyone's physical and emotional well-beings. Our work is aimed at exploring the capability of various types of functional data clustering methods on the complex Covid data. We collect the Covid data from the Our World in Data website,...

Fall 2021

Xie, Haihan

This thesis proposes a novel Gaussian copula function-on-scalar regression, which is more flexible to characterize the relationship between functional or image response and scalar predictors and is able to relax the linear assumption in traditional function-on-scalar linear regression. Estimation...

Spring 2021

Lu, Ran

Generalizing wavelets by adding desired redundancy and flexibility, framelets (a.k.a. wavelet frames) are of interest and importance in many applications such as image processing and numerical algorithms. Several key properties of framelets are high vanishing moments for sparse multi-scale...

Fall 2021

Moore, Kenneth

A classic result in the field of Riemannian Geometry is the Splitting Theorem of Cheeger and Gromoll. Since this result there have been numerous alternate versions under a variety of different conditions. Continuing in this vein, we prove structure results on manifolds with boundary components...

Fall 2021

Heggerud, Christopher M.

​​In recent decades freshwater lakes have seen an increase in human presence. A common byproduct of this human presence is anthropogenic nutrient pollution resulting in eutrophication, a term that is becoming all too synonymous with harmful algal blooms. It is well known that phytoplankton...

Fall 2021

Pak, Andrey

This thesis is dedicated to the study of the general class of random processes, called optional processes, and their various applications in Mathematical Finance, Risk Theory, and Statistics. First, different versions of a comparison theorem and a uniqueness theorem for a general class of...