Darrion Thornburgh

I am a graduating senior from Bard College majoring in both mathematics and computer science. I have the goal of earning a Ph.D. in mathematics and doing research in mathematics as a career. I participated in the 2023 CUBE REU program at Georgia Tech, under the mentorship of Dan Margalit and Wade Bloomquist. During the summer of 2022, I researched at the Bard Summer Research Institute (BSRI) under the mentorship of Lauren Rose. Additionally, I have received the Distinguished Scientist Scholarship at Bard College.

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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 \(\mathbb{F}_2^n\). This work will appear in my senior thesis 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

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 \(a \neq 0\). Equivalently, \(F\) is APN if and only if \(\mathcal{G}_F\) is a Sidon set, that is, a set in \(\mathbb{F}_2^n\) where no four distinct points sum to zero. 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, and we show that most known APN functions exhibit uniformity in their exclude distributions, with explicit permutations given in the case of the Gold function.

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.

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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