My research has brought together techniques from arithmetic combinatorics and Fourier analysis to study problems in geometric Ramsey theory. Recently, I have focused on problems from discrete geometry and frame theory that are motivated by applications in signal processing.

Submitted- 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.

- Kesten–McKay law for random subensembles of Paley equiangular tight frames

(with Mark Magsino and Dustin G. Mixon)

- The spectra of random subensembles of symmetric conference matrices follow a Kesten–McKay distribution.

- The optimal packing of eight points in the real projective plane

(with Dustin G. Mixon)

to appear in*Experimental Mathematics*- The minimum coherence of eight points in \(\mathbf{RP}^2\) is the largest root of \(1 + 5x - 8x^2 - 80x^3 - 78x^4 + 146x^5 - 80x^6 - 584x^7 + 677x^8 + 1537x^9\).

- Embedding distance graphs in finite field vector spaces

(with Alex Iosevich)

to appear in*Journal of the Korean Mathematical Society*- Large subsets of \(\mathbf{F}_q^{2t}\) contain isometric copies of all distance graphs with maximum degree \(t\).

- Spherical configurations over finite fields

(with Neil Lyall and Ákos Magyar)

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

- On the quotient set of the distance set

(with Alex Iosevich and Doowon Koh)

*Moscow Journal of Combinatorics and Number Theory*8-2 (2019), 103–115.- The quotient set of the distance set of every \(E \subseteq \mathbf{F}_q^2\) with \(|E| \gt 9q\) is equal to \(\mathbf{F}_q\).

- Simplices over finite fields

*Proceedings of the American Mathematical Society*145 (2017), 2323–2334- Large subsets of \(\mathbf{F}_q^{3}\) contain isometric copies of all triangles.

- Small gaps between configurations of prime polynomials

*Journal of 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

(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.

- Linear programming bounds for cliques in Paley graphs

(with Mark Magsino and Dustin G. Mixon)

*SPIE Proceedings 11138, Wavelets and Sparsity XVIII*(2019), 111381H- Linear programming frequently improves the Hanson–Petridis bound.

- A Delsarte-style proof of the Bukh–Cox bound

(with Mark Magsino and Dustin G. Mixon)

*13th International Conference on Sampling Theory and Applications*(SampTA 2019)- The Bukh–Cox bound for line packings is obtained through linear programming.

- Exact line packings from numerical solutions

(with Dustin G. Mixon)

*13th International Conference on Sampling Theory and Applications*(SampTA 2019)- Numerical line packings are made exact by cylindrical algebraic decomposition.