Switching Equivalence of Systems of Lines over Finite Fields Algebra and Number Theory Seminar, LSU, Baton Rouge, Louisiana (March 2026). Abstract: In this talk we will discuss important frame theoretic objects such at equiangular tight frames (ETFs) whose existence has important applications in fields as diverse as compressed sensing to quantum state tomography. We will then discuss some new approaches to tackling some open problems, on the existence and structure of these frame theoretic objects, by using tools from geometric algebra, and specially looking at frames over finite field vector spaces with Hermitian forms. We will then show that the switching equivalence classes of systems of lines over finite fields which are frames, often only depend on the double and triple products. This allows us to understand ETFs over finite fields in terms of their double and triple products, with a result similar to saturating the Welch bound over $\mathbb{C}$. We also show that similar to the case over $\mathbb{C}$, collections of vectors are similar to a regular simplex essentially when their triple products satisfy a certain property.
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Conference Talks
Saturating the Welch Bound for Frames over Finite Fields Special Session on Symmetric Subspace Configurations at Constructive Functions 2025, Vanderbilt, Nashville Tennessee (May 2025). Abstract: This talk concerns frames and equiangular lines over finite fields. We find a necessary and sufficient condition for systems of equiangular lines over finite fields to be equiangular tight frames (ETFs). As is the case over subfields of $\mathbb{C}$, it is necessary for the Welch bound to be saturated, but there is an additional condition required involving sums of triple products. We also prove that similar to the case over $\mathbb{C}$, collections of vectors are similar to a regular simplex essentially when the triple products of their scalar products satisfy a certain property. Finally, we investigate switching equivalence classes of frames and systems of lines focusing on systems of equiangular lines in finite orthogonal geometries with maximal incoherent sets, drawing connections to combinatorial design theory.
Presentation: Presentation
Maximal Systems of Equiangular Lines in Finite Unitary Geometry Greenslopes Graduate Seminar (October 2023) Abstract: This talk will explore Unitary Geometries and systems of equiangular lines and how they relate to luxury cars (Mercedes-Benz) as well as quantum computing. We will start by talking about the speaker's limited view of "geometry" which allows them to effortlessly generalize geometry to be a special case of frame theory. We will then discuss some well-known results in frame theory for complex frame and real frames and the more recent generalizations in their finite field analogs. These connect back to the original goal of the talk which is exploring systems of equiangular lines. We will then look at some open problems and connections to quantum computing.
Simple Types Applied Category Theory Seminar (January 2023)
