​

For my Bachelor of Science in Mathematics at Rhodes College, I conducted three semesters of novel pure mathematics research under the supervision of Dr. Christopher Seaton. Along with the written thesis linked below, I presented this research for Rhodes College Mathematics Faculty multiple times, as well as at the Hendrix-Rhodes-Sewanee Mathematics Conference.

 

As pure mathematics research, the main motivation behind this work is to expand the lexicon of mathematical knowledge, rather than to apply findings to any worldly problems. While some of the mathematical objects which I investigated can be used to model real-world systems, we are not interested in them because of that quality. Rather, this research was intended to broaden the scope of pure mathematical knowledge, if ever so slightly. 

 

I studied directed graphs or digraphs – sets of vertices and directed edges (or points and arrows connecting them). Each of these graphs are associated with an algebraic object called a C*-Algebra, and these objects are further associated with various Morita Equivalence classes. For our purposes, we need not know the specifics of C*-Algebras and Morita Equivalent classes. We only need to recognize that these objects create a definition of “sameness” between digraphs, allowing for the classification of directed graphs into various groups of similar digraphs. Further, there are specific ways we can change a digraph that ensure this defined “sameness” remains – six moves defined in a paper by Eilers et al [2] (see references in my thesis). 

 

Every digraph is further associated with a number of different matrices, and all of these matrices have sets of eigenvalues, also known as spectra. I investigated the Hermitian Adjacency Matrix, which ostensibly describes which vertices in a digraph are connected by edges, and which direction these edges point, and the Hermitian Adjacency Spectrum. My primary question of investigation was: How do digraphs’ Hermitian Adjacency Spectrum change when we perform the moves outlined in [2] on the digraph? Or, in another sense, how much (and what type of) variation is possible in these spectra when maintaining some form of similarity in our digraphs? More colloquially, I looked at classes of digraphs with a particular similarity, and compared some of the important numbers associated with these digraphs. 

 

Often, mathematics is concerned with this notion of sameness or similarity – consider the “equal” sign, perhaps the simplest and most common representation of sameness in mathematics. Once we find a way to define a similar property across objects (such as the Morita Equivalence class of the C*-Algebra) the next question often is: how do the other properties of objects which are similar in this way vary? Think of studying all plants that have purple flowers, and then investigating the geographic diversity of those plants. Can purple plants grow in tropical locations? The essence of that question is similar to one I investigated: Can a set of digraphs with Morita Equivalent C*-Algebras include ones whose spectral radii tend to infinity? 

​

​​

​