Equiangular Tight Frames

An equiangular tight frame (ETF) is a collection of n lines in d-dimensional space so that the angles between each of the lines is equal, and a certain symmetry condition (tightness) is satisfied. Traditionally, these objects have fell into the purview of quantum information theory and discrete geometry. However, recent exciting results indicate that there is a deep connection between these frames and Stark units in number fields.

LSU offers special courses called Math 4999, vertically integrated research, which are deigned to involve undergraduates with math research in an academic setting by working with graduate students. As part of a Math 4999 course in the spring of 2026 taught be Gene Kopp and Joe DiCapua, I led a team of three undergraduates on a project related to ETFs. The main focus of this work was to study the relationship between ETFs over the real field, p-adic fields, and finite fields. With Gene's guidance, I was responsible for designing the project and working with the undergraduates. The project continued into the Summer of 2026, and we should have a paper coming soon.

(The Gram matrix for a p-adic ETF. See the upcoming paper for more details!)