# Hans Parshall

My research is motivated by problems in combinatorics, particularly when they have a geometric or number theoretic flavor. I have recently found myself applying combinatorial and computational techniques to study optimal Grassmannian packings.

Publications
• The optimal packing of eight points in the real projective plane (arXiv preprint)
(with Dustin G. Mixon)
submitted
• The minimum coherence of eight points in $$\mathbf{RP}^2$$ is roughly 0.6475889787343.

• A Delsarte-style proof of the Bukh–Cox bound (arXiv preprint)
(with Mark Magsino and Dustin G. Mixon)
submitted
• The Bukh–Cox bound for line packings is obtained through linear programming.

• Exact line packings from numerical solutions (arXiv preprint)
(with Dustin G. Mixon)
submitted
• Numerical line packings are made exact by cylindrical algebraic decomposition.

• Embedding distance graphs in finite field vector spaces (arXiv preprint)
(with Alex Iosevich)
to appear in J. Korean Math. Soc.
• 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
• 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.

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