Research Interests

- combinatorics, discrete geometry, graph theory, optimization, and their applications to data science

Preprints (submitted)

- Uniquely optimal codes of low complexity are symmetric

(with Christopher Cox, Emily J. King and Dustin G. Mixon)

- We formulate and test explicit predictions concerning the symmetry of optimal codes in compact metric spaces.

- Lie PCA: Density estimation for symmetric manifolds

(with Jameson Cahill and Dustin G. Mixon)

- We extend local principal component analysis for learning symmetric manifolds.

- 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 points in \(\mathbf{RP}^5\).

Journal Articles

- Small unit-distance graphs in the plane

(with Aidan Globus)

to appear,*Bulletin of the Institute for Combinatorics and its Applications*- We classify the unit-distance graphs in \(\mathbf{R}^2\) on up to 9 vertices via 74 minimal forbidden subgraphs.

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

(with Mark Magsino and Dustin G. Mixon)

to appear,*Constructive Approximation*- We resolve the first case of a conjecture on the 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,*Experimental Mathematics*- We prove that the minimum coherence of 8 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)

*American Journal of Mathematics*142-2 (2020), 373–404- 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.

Conference Proceedings

- Linear programming bounds for cliques in Paley graphs

(with Mark Magsino and Dustin G. Mixon)

*SPIE Proceedings 11138, Wavelets and Sparsity XVIII*(2019), 111381H - 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) - 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.