# Hans Parshall

My research is motivated by problems in combinatorics, particularly when they have a geometric or number theoretic flavor. Much of my work can be described as density Ramsey theory, where the typical problem concerns what "structure" must appear in all "large" sets. Several of my results have applied Fourier analysis to find geometric structure in large subsets of vector spaces over finite fields. I have recently found myself applying combinatorial and computational techniques to study optimal Grassmannian packings.

Publications
• Embedding distance graphs in finite field vector spaces (arXiv preprint)
(with Alex Iosevich)
submitted
• Large subsets of $$\mathbf{F}_q^{2t}$$ contain isometric copies of all distance graphs of maximum degree $$t$$.

• On the quotient set of the distance set (arXiv preprint)
(with Alex Iosevich and Doowon Koh)
to appear in Mosc. J. Comb. Number Theory
• The quotient set of the distance set of every $$E \subseteq \mathbf{F}_q^2$$ with $$|E| > 9q$$ is equal to $$\mathbf{F}_q$$.

• Spherical configurations over finite fields (preprint)
(with Neil Lyall and Ákos Magyar)
to appear in Amer. J. Math.
• 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 every $$k$$-simplex from $$\mathbf{F}_q^k$$.

• 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 Katherine Thompson)
Integers 13 (2013), A37
• Primes represented by idoneal quadratic forms are determined by explicit congruence conditions.
In Preparation
• The optimal packing of eight points in the real projective plane
(with Dustin Mixon)
• The minimum coherence of eight points in $$\mathbf{RP}^2$$ is roughly 0.6475889787343.

• 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 73 forbidden subgraphs.

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