Darrion Thornburgh

I am an incoming graduate student at Vanderbilt University, entering the Ph.D. program in mathematics. I completed my undergraduate studies Bard College majoring in both mathematics and computer science. In 2023, I spent my summer at the CUBE REU program at Georgia Tech, under the mentorship of Dan Margalit and Wade Bloomquist. I also did undergraduate research during the summer of 2022 at the Bard Summer Research Institute (BSRI) under the mentorship of Lauren Rose.

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Research

I have a wide variety of interests, but the work that I have done lies in additive combinatorics, cryptography, affine geometry, and geometric group theory. Over the past year, I have been studying almost perfect nonlinear (APN) functions over finite fields and their connections to Sidon sets in \(\mathbb{F}_2^n\). Some of this work appears in my senior thesis done at Bard College, titled Symmetry and Structures of APN Functions and Sidon Sets.

My research at the BSRI was in finite geometry and combinatorics and can be applied to the card game EvenQuads. During this time, we completely classified the affine equivalence classes of Sidon sets (or 2-caps) in \(\mathbb{F}_2^n\) of size up to 9. At the Georgia Tech REU, my research was in geometric group theory. In collaboration with Sami Aurin, we proved that the curve graph of the 5-punctured sphere is not 1-hyperbolic.

Papers

successor Symmetry and Structures of APN Functions and Sidon Sets
Darrion Thornburgh. Advisors: Bob McGrail and Steven Simon
Undergraduate Thesis, May 2024;

Let \(\mathbb{F}_p^n\) be the \(n\)-dimensional vector space over \(\mathbb{F}_p\). The graph \(\mathcal{G}_F = \{ (x, F(x)) : x \in \mathbb{F}_p^n \}\) of a vectorial function \(F \colon \mathbb{F}_p^n \to \mathbb{F}_p^m\) can have interesting combinatorial properties depending on varying cryptographic conditions on \(F\). A vectorial Boolean function \(F \colon \mathbb{F}_2^n \to \mathbb{F}_2^n\) is almost perfect nonlinear (APN) if there are at most \(2\) solutions to the equation \(F(x+a) + F(x) = b\) for all \(a,b \in \mathbb{F}_2^n\) where \(a \neq 0\). In this paper, we classify APN functions and important subclasses of APN functions in graph theoretical terms using the Kneser graph of all translations of \(\mathcal G_F\). We also study the properties of \(\mathcal G_F\) as a Sidon set. In particular, we introduce the notion of uniform exclude distributions, and we study APN functions whose graphs have uniform exclude distributions.

Topological methods in zero-sum Ramsey theorey
Florian Frick, Jacon Lehmann Duke, Meenakshi McNamara, Hannah Park-Kaufmann, Steven Raanes, Steven Simon, Darrion Thornburgh, Zoe Wellner
Submitted, November 2023;
arXiv

A cornerstone result of Erdős, Ginzburg, and Ziv (EGZ) states that any sequence of \(2n − 1\) elements in \(\mathbb{Z}/n\) contains a zero-sum subsequence of length \(n\). While algebraic techniques have predominated in deriving many deep generalizations of this theorem over the past sixty years, here we introduce topological approaches to zero-sum problems which have proven fruitful in other combinatorial contexts. Our main result (1) is a topological criterion for determining when any \(\mathbb{Z}/n\)- coloring of an n-uniform hypergraph contains a zero-sum hyperedge. In addition to applications for Kneser hypergraphs, for complete hypergraphs our methods recover Olson’s generalization of the EGZ theorem for arbitrary finite groups. Furthermore, we (2) give a fractional generalization of the EGZ theorem with applications to balanced set families and (3) provide a constrained EGZ theorem which imposes combinatorial restrictions on zero-sum sequences in the original result.

successor How Many Cards Should You Lay Out in a Game of EvenQuads: A Detailed Study of Caps in AG(n,2)
Julia Crager, Felicia Flores, Timothy E. Goldberg, Lauren L. Rose, Daniel Rose-Levine, Darrion Thornburgh, Raphael Walker
La Matematica, May 2023;

We define a cap in the affine geometry \(\mathrm{AG}(n,2)\) to be a subset in which any collection of \(4\) points is in general position. In this paper, we classify, up to affine equivalence, all caps in \(\mathrm{AG}(n,2)\) of size \(k \leq 9\). As a result, we obtain a complete characterization of caps in dimension \(n \leq 6\), in particular complete and maximal caps. Since the EvenQuads card deck is a model for \(\mathrm{AG}(6,2)\), as a consequence, we determine the probability that an arbitrary \(k\)-card layout contains a quad.


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