My research is motivated by problems in combinatorics, particularly when they have a geometric or number theoretic flavor. I have recently found myself applying combinatorial and computational techniques to study optimal Grassmannian packings.

Publications- The optimal packing of eight points in the real projective plane
(arXiv preprint)

(with Dustin G. Mixon)

*submitted*- The minimum coherence of eight points in \(\mathbf{RP}^2\) is roughly 0.6475889787343.

- A Delsarte-style proof of the Bukh–Cox bound
(arXiv preprint)

(with Mark Magsino and Dustin G. Mixon)

*submitted*- The Bukh–Cox bound for line packings is obtained through linear programming.

- Exact line packings from numerical solutions
(arXiv preprint)

(with Dustin G. Mixon)

*submitted*- Numerical line packings are made exact by cylindrical algebraic decomposition.

- Embedding distance graphs in finite field vector spaces
(arXiv preprint)

(with Alex Iosevich)

*to appear in J. Korean Math. Soc.*- Large subsets of \(\mathbf{F}_q^{2t}\) contain isometric copies of all distance graphs of maximum degree \(t\).

- On the quotient set of the distance set
(arXiv preprint)

(with Alex Iosevich and Doowon Koh)

*to appear in Mosc. J. Comb. Number Theory*- The quotient set of the distance set of every \(E \subseteq \mathbf{F}_q^2\) with \(|E| > 9q\) is equal to \(\mathbf{F}_q\).

- Spherical configurations over finite fields
(preprint)

(with Neil Lyall and Ákos Magyar)

*to appear in Amer. J. Math.*- Large subsets of \(\mathbf{F}_q^{10}\) contain isometric copies of all spherical quadrilaterals.

- Simplices over finite fields
(arXiv preprint)

*Proc. Amer. Math. Soc. 145 (2017), 2323-2334*- Large subsets of \(\mathbf{F}_q^{k + 1}\) contain isometric copies of every \(k\)-simplex from \(\mathbf{F}_q^k\).

- Small gaps between configurations of prime polynomials
(arXiv preprint)

*J. Number Theory 162 (2016), 35–53*- There are arbitrarily large affine subspaces in \(\mathbf{F}_q[t]\) consisting of twin prime polynomials.

- Primes represented by binary quadratic forms
(PDF)

(with Pete L. Clark, Jacob Hicks, and Katherine Thompson)

*Integers 13 (2013), A37*- Primes represented by idoneal quadratic forms are determined by explicit congruence conditions.

- Small unit-distance graphs in the plane

(with Aidan Globus)- The unit-distance graphs on up to 9 vertices in \(\mathbf{R}^2\) are classified by 74 forbidden subgraphs.

I also have a few old expository notes on topics in arithmetic combinatorics.