In 1941, J.H.C. Whitehead posed the question of whether asphericity is a hereditary property for 2-dimensional CW complexes. This question remains unanswered, but has led to the development of several algebraic and topological properties that are sufficient (but not necessary) for the asphericity of presentation 2-complexes. While many of the...
Institutions of higher education have experienced a rapid increase in international student enrollment within a short period of time, especially in mathematics classrooms. It is therefore important for instructors to have the knowledge and skills to support international students in the learning of mathematics. There are few research studies of...
The topic of statistical mechanics has been studied for over a century, and it is one of the pillars of modern physics. This theory can be applied to the study of the thermodynamic behavior of large systems of interacting particles, in which case it is referred to as equilibrium statistical...
There is an underrepresentation of minority groups and women in the field of mathematics. With growing interest in equity and inclusion within the mathematics education community, it is important to look at how such concepts can be implemented in the classroom. Equitable and inclusive teaching practices have been shown to...
Enumerative combinatorics is an area of mathematics that is both highly accessible for students and widely applicable to other sciences and areas of mathematics (Kapur, 1970; Lockwood, Wasserman, & Tillema, 2020). One important class of problems in combinatorics is combinatorial proofs of binomial identities, which is a type of proof...
Mapper is a tool designed by Gurjeet Singh, Facundo Mémoli, and Gunnar Carlsson for Topological Data Analysis (TDA) that constructs a simplicial complex from a finite subset of a metric space, which has been met with great success. Due to the many moving parts of Mapper and the potential it...
In this dissertation, we use Fourier-analytic and spectral theory methods to analyze the behavior of solutions of the incompressible Navier-Stokes equations in 2D and 3D (with an eye towards better understanding turbulence). In particular, we investigate the possible existence of so-called ghost solutions to the Navier-Stokes Equations. Such solutions, if...
Let H be a cyclically-presented group on n generators with a single defining relator. Attempts have been made to classify such groups by their order, their status as a 3-manifold group, and the asphericity status of their presentations. For groups with a defining relator of length 3 these classifications are...
This is a program that searches for reduced spherical pictures over 1-generator, 1-relator relative group presentations with finite cyclic coefficient group. The search is depth-first and builds pictures face-by-face using a prescribed order. This program was written as part of Matthias Merzenich's PhD research. More details are available in the...
The theory behind magnetohydrodynamics (MHD) is utilized to present a 3-D solution to how the induced magnetic field changes with respect to time. Several MHD-based assump- tions are made to simplify the coupling of Maxwell’s equations with two constitutive laws and Ohm’s law. The velocity field is assumed to be...