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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

Patterson, Devin C.D.

Topological Recursion began its life as a series of recursive equations aimed at solving constraints which occur in matrix models of Quantum Field Theory. After its inception, Topological Recursion was given a more abstract formulation in terms of Quantum Airy Structures and has since been of...

Fall 2021

O'Brien, Michael John

The theory of convergence structures delivers a promising foundation on which to study general notions of convergence. However, that theory has one striking feature that stands out against all others: it is described using the language of filters. This is contrary to how convergence is used in...

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

Zhang, Xueying

Kernel methods are often used for nonlinear regression and classification in machine learning because they are computationally cheap and accurate. Fourier basis and wavelet basis are the bases that can efficiently approximate the kernel functions. In previous research, Bayesian approximate kernel...

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

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

Kurt S. MacKay

In this note we present a construction which improves the best known bound on the minimal dispersion of large volume boxes in the unit cube. The dispersion of a subset of the cube is the supremal volume over all axis parallel boxes in the cube which do not intersect the given subset. The minimal...

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...

Fall 2021

Niu, Yeli

This thesis includes 2 parts. Part 1 is: Discretization on High Dimensional Compact Do￾mains. Part 2 is: Polynomial Approximation on High Dimensional Spheres. A special example for part 1 is the result on the unit sphere of high-dimensional Euclidean spaces. In chapter 2, we obtained general...