Mathematics
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The numerical algebraic geometry approach to polynomial optimization
Numerical algebraic geometry (NAG) consists of a collection of numerical algorithms, based on homotopy continuation, to approximate the solution sets of systems of polynomial equations arising from applications in science ... 
Topological techniques for characterization of patterns in differential equations
Complex data can be challenging to untangle. Recent advances in computing capabilities has allowed for practical application of tools from algebraic topology, which have proven to be useful for qualitative and quantitative ... 
First year graduate teaching assistants : fostering successful teaching
The importance of effective graduate teaching assistant (GTA) training is often greatly under appreciated. However, it is imperative that GTAs receive optimal professional development because they are often responsible for ... 
Pattern formation in reaction diffusion systems and ion bombardment of surfaces
We have analyzed pattern formation in two different systems: (1) Vaportoparticle reaction diffusion systems and (2) Highly ordered square arrays in ion bombardment. The vaportoparticle reaction exhibits oscillatory ... 
A geometric data analysis approach to dimension reduction in machine learning and data mining in medical and biological sensing
Geometric data analysis seeks to uncover and leverage structure in data for tasks in machine learning when data is visualized as points in some dimensional, abstract space. This dissertation considers data which is high ... 
On automorphism groups of pgroups
We provide the necessary framework to use filters in computational settings, in particular for finitely generated nilpotent groups. The main motivation for this is to construct automorphisms of the group from derivations ... 
Grassmann, Flag, and Schubert varieties in applications
This dissertation develops mathematical tools for signal processing and pattern recognition tasks where data with the same identity is assumed to vary linearly. We build on the growing canon of techniques for analyzing and ... 
Avoiding singularities during homotopy continuation
In numerical algebraic geometry, the goal is to find solutions to a polynomial system F(x1,x2,...xn). This is done through a process called homotopy continuation. During this process, it is possible to encounter areas of ... 
General modelbased decomposition framework for polarimetric SAR images
Polarimetric synthetic aperture radars emit a signal and measure the magnitude, phase, and polarization of the return. Polarimetric decompositions are used to extract physically meaningful attributes of the scatterers. Of ... 
Mathematical modeling of groundwater anomaly detection
Public concerns about groundwater quality have increased in recent years due to the massive exploitation of shale gas through hydraulic fracturing which raises the risk of groundwater contamination. Groundwater monitoring ... 
An integrated mathematics/science activity for secondary students : development, implementation, and student feedback
Mathematics teachers are often challenged by their students to give reasoning for why learning mathematics is necessary. An approach to address this question is to show students the value in learning mathematics by ... 
On the formulation and uses of SVDbased generalized curvatures
In this dissertation we consider the problem of computing generalized curvature values from noisy, discrete data and applications of the provided algorithms. We first establish a connection between the FrenetSerret Frame, ... 
A constrained optimization model for partitioning students into cooperative learning groups
The problem of the constrained partitioning of a set using quantitative relationships amongst the elements is considered. An approach based on constrained integer programming is proposed that permits a group objective ... 
Computational advancements in the Dbar reconstruction method for 2D electrical impedance tomography
We study the problem of reconstructing 2D conductivities from boundary voltage and current density measurements, also known as the electrical impedance tomography (EIT) problem, using the Dbar inversion method, based on ... 
Counting ArtinSchreier curves over finite fields
Several authors have considered the weighted sum of various types of curves of a certain genus g over a finite field k := Fq of characteristic p where p is a prime and q = pm for some positive integer m. These include ... 
Algorithms for Feature Selection and Pattern Recognition on Grassmann Manifolds
This dissertation presents three distinct applicationdriven research projects united by ideas and topics from geometric data analysis, optimization, computational topology, and machine learning. We first consider ... 
Electromechanical and curvaturedriven molecular flows for lipid membranes
Lipid membranes play a crucial role in sustaining life, appearing ubiquitously in biology. Gaining a quantitative understanding of the flows of lipid membranes is critical to understanding how living systems operate. ... 
Algorithms in Numerical Algebraic Geometry and Applications
The topics in this dissertation, while independent, are unified under the field of numerical algebraic geometry. With ties to some of the oldest areas in mathematics, numerical algebraic geometry is relatively young as a ... 
Preconditioning Polynomial Systems Using Macaulay Dual Spaces
Polynomial systems arise in many applications across a diverse landscape of subjects. Solving these systems has been an area of intense research for many years. Methods for solving these systems numerically fit into the ... 
Mathematical Models for HIV1 Viral Capsid Structure and Assembly
HIV1 (human immunodeﬁciency virus type 1) is a retrovirus that causes the acquired immunodeﬁciency syndrome (AIDS). This infectious disease has high mortality rates, encouraging HIV1 to receive extensive research interest ...