My research applies combinatorial and computational techniques to solve problems motivated by data science. I frequently use tools from convex optimization, discrete geometry, graph theory, random matrix theory and real algebraic geometry. Much of my past work has leveraged ideas from additive combinatorics and Fourier analysis to solve problems in geometric Ramsey theory.

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 \(1 + 5x - 8x^2 - 80x^3 - 78x^4 + 146x^5 - 80x^6 - 584x^7 + 677x^8 + 1537x^9\).

- 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 \(\mathbf{F}_q\).

- Embedding distance graphs in finite field vector spaces

(with Alex Iosevich)

*Journal of the Korean Mathematical Society*56-2 (2019), 1489–1502- Large subsets of \(\mathbf{F}_q^{2t}\) contain isometric copies of all distance graphs with maximum degree \(t\).

- 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.