Current Research
At the moment, my research is in the world of arithmetic geometry and number theory, specifically elliptic curves and class groups of number fields. In particular, I am studying ideal classes in class groups of orders in quadratic and cubic fields arising from rational points on elliptic curves.
Past Research
I was accepted to the REU at the University of California-Santa Barbara for the summer of 2020. Under the guidance of Dr. Zhirayr Avetisyan, I studied problems relating to almost Abelian Lie algebras (Lie algebras with a co-dimension 1 Abelian Lie subalgebra) along with their Lie groups, such as their curvature, the Laplace-Beltrami operator on the Lie groups, and the action of automorphisms on orbit space.
In the spring of my sophomore year, I took an independent study in Functional Analysis, studying from Elementary Functional Analysis by Barbara MacCluer. This provided me with the necessary background to begin research with Dr. Robert Allen in operator theory, which I did throughout my junior year. We studied the derivative operator on discrete weighted Banach spaces, and its interactions with the multiplication and composition operators. We have since published at Involve in 2022:
The differentiation operator on discrete function spaces of a tree, by Robert F. Allen and Colin M. Jackson
Involve Vol. 15 (2022), No. 1, 163-184. DOI: https://doi.org/10.2140/involve.2022.15.163
I presented on our work at the Young Mathematicians Conference at The Ohio State University. My slides are available here. I presented at the Joint Mathematics Meetings in January, and here is my poster and a short video on my poster for JMM.
During my freshman year, I met Dr. Sylvester Gates, a National Medal of Science Laureate, after he gave a talk at UWL. That summer, I was accepted to Dr. Gates' Summer Student Theoretical Physics Research Session at Brown University. In this program, Dr. Gates taught a group of twenty undergraduates math and physics that relate to his research.