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.

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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. During my last year at Bard, I wrote my senior thesis on APN functions, Sidon sets, and their connections. Some of this work appears in my senior thesis done at Bard College, titled Symmetry and Structures of APN Functions and Sidon Sets and also my most recent paper, Uniform exclue distributions of Sidon sets.

REU 2023-2024. I spent the summers of 2023 and 2024 participating in the CUBE REU (led by Dan Margalit) at Georgia Tech and Vanderbilt University, repsectivley. The mentors of the projects I was involved in for CUBE (along with the year I worked in their project) were Wade Bloomquist (2023), Ryan Dickmann (2024), Dan Margalit (2023-2024), and Abdoul Karim Sane (2024). During my time at CUBE 2023 and 2024, we studied the hyperbolicity constant of the curve graph of a surface, right-angled Artin groups (RAAGs), and graph theoretical properties that are forbidden in any induced subgraph of the curve graph.

REU 2022. During the summer of 2022, I did a summer of undergraduate research at the Bard Summer Research Institute (BSRI), under the mentorship of Lauren Rose. 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.

Papers

A Sidon set \(S\) in \(\mathbb{F}_2^n\) is a set such that the pairwise sums of distinct points are all distinct. The exclude points of a Sidon set \(S\) are the sums of three distinct points in \(S\), and the exclude multiplicity of a point in \(\mathbb{F}_2^n \setminus S\) is the number of such triples in \(S\) it is equal to. We call the function \(d_S \colon \mathbb{F}_2^n \setminus S \to \mathbb{Z}_{\geq 0}\) taking points in \(\mathbb{F}_2^n \setminus S\) to their exclude multiplicity the exclude distribution of \(S\). We say that \(d_S\) is uniform on \(\mathcal{P}\) if \(\mathcal{P}\) is an equally-sized partition \(\mathcal{P}\) of \(\mathbb{F}_2^n \setminus S\) such that \(d_S\) takes the same values an equal number of times on every element of \(\mathcal{P}\). In this paper, we use APN plateaued functions with all component functions unbalanced to construct Sidon sets \(S\) in \((\mathbb{F}_2^n)^2\) whose exclude distributions are uniform on natural partitions of \((\mathbb{F}_2^n)^2 \setminus S\) into \(2^n\) elements. We use this result and a result of Carlet to determine exactly what values the exclude distributions of the graphs of the Gold and Kasami functions take and how often they take these values.

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.

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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