Publications

1. Investigation of the Qualitative Behavior of the Equilibrium Points for a Modified Lotka-Volterra Model (with Jemal Mohammed-Awel and Andreas Lazari), Georgia Journal of Science, Volume 67 Number 3 (2009), pages 46-59. Links: Journal | Local | Google Drive
   Abstract

   We are interested in a modified Lotka-Volterra model to analyze population dynamics of two competing species which are ecologically identical (that is, they use the same resource). The model incorporates a non-linear relationship to represent the interaction between the species. We study the stability of the equilibrium points of the system and compare the qualitative behavior of the equilibrium points in our model with qualitative behavior of the classical Lotka-Volterra equations. Our result suggests that in some cases the modified model may have more than one equilibrium points in the interior of the first quadrant, which biologically means that the two species may co-exist at multiple positive population sizes.

Other Works

1. Sutured 3-Manifolds, Finite-Depth Foliations, and Related Topics (Dec 2014). DOI: 10.13140/2.1.4212.8965. Links: Local
   Description

   This was the literature component of my advanced topics / candidacy exam (i.e., my prospectus). The main component consists of a review/reproduction/elaboration of some foliation-theoretic results from Gabai's '83 article concerning the existence of Reebless foliations in a large class of 3-manifolds admitting sutured manifold hierarchies. The document details the basics of foliation theory as well as defining sutured manifolds (including hierarchies & decompositions), foliation depth, and a considerable number of lemmas and theorems of Gabai and others. It also contains a somewhat-significant number of scalable (vector) images and is largely self-contained.
2. Topics in Complex and Hypercomplex Geometry (May 2014). DOI: 10.13140/2.1.4406.4324. Links: Local
   Description

   This was the write-up for a Spring 2014 class in Dirac Analysis in which various aspects of complex and hypercomplex geometry are documented. One interesting property of hypercomplex geometry is that the classically-discovered classification of (compact) hypercomplex 4-manifolds doesn't carry over to $4n$-manifolds for $n>1$ to a degree that's almost pathological, so one of the guiding principles of this paper was to attempt a survey of the then-present state of information related thereto. A fairly large collection of literature is surveyed and the document is mostly self-contained with some assumed knowledge of geometry.
3. A Survey of Quatenrionic Analysis (Jan 2014). DOI: 10.13140/2.1.3357.8560. Links: Local
   Abstract

   The group $\mathbb{H}$ of (Hamilton) quaternions can be described, algebraically, as a four-dimensional associative-but-non-commutative normed division algebra over the ring $\mathbb{R}$ of real numbers. This paper is meant to be a survey on these numbers. In particular, the paper is broken into four main sections. The quaternions are introduced in section one, and in section two, a bit of preliminary information needed to understand the study of functions of a quaternionic variable is introduced. Section three is meant to give an overview of some of the properties of quaternion-valued functions including (but not limited to) properties that parallel main ideas in complex-valued theory. The fourth and final section of the paper describes in varying degrees of detail several different applications of quaternionic analysis. An appendix of notational standards is included for convenience.