My current projects are focused on applying ideas from arithmetic combinatorics to obtain new density results in geometric Ramsey theory.

- Spherical configurations over finite fields

(with Neil Lyall and Ákos Magyar)

*submitted, preprint available upon request*- 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 all full rank \(k\)-simplices.

- 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 Kate Thompson)

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

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