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- Globally optimizing small codes in real projective spaces

(with Dustin G. Mixon)

- We classify the optimal packings of 7 points in \(\mathbf{RP}^4\) and 8 lines in \(\mathbf{RP}^5\).

- Small unit-distance graphs in the plane

(with Aidan Globus)

- We classify the unit-distance graphs in \(\mathbf{R}^2\) on up to 9 vertices with 74 minimal forbidden subgraphs.

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

(with Mark Magsino and Dustin G. Mixon)

- We resolve a conjecture of Haikin, Zamir and Gavish on the distribution of singular values of random subensembles of equiangular tight frames.

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

(with Dustin G. Mixon)

to appear in*Experimental Mathematics*- We prove that 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*- We prove that 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- We prove that 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- We prove that 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- We prove that 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- We prove that 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- We classify primes represented by idoneal quadratic forms with 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- We show that a linear programming method frequently beats 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)- We prove the Bukh–Cox bound for line packings with a linear programming method.

- Exact line packings from numerical solutions

(with Dustin G. Mixon)

*13th International Conference on Sampling Theory and Applications*(SampTA 2019)- We apply cylindrical algebraic decomposition to exactify numerical line packings.