I am interested in problems that combine number theory, geometry, combinatorics, and algebra. Though I have worked in graph theory and lattice-based cryptography, the most consistent theme of my work is that of curves over finite fields. These curves are fascinating because they bridge the finite and the infinite, are amenable to both computational exploration and geometric intuition, and present a surprisingly wide variety of intriguing questions.
Here is my 2018 CV.
A table of bounds for S-unit solutions to x+y=1 in certain fields.
Here is my current publications list.
- The de Rham Cohomology of the Suzuki Curves. With R. Pries and C. Weir.
To appear in Contemporary Mathematics: Arithmetic, geometry, cryptography,
and coding theory (Luminy), AMS Publishing, 2018.
- What (quilting) circles can be squared? With K. Haymaker. To appear in Mathematics Magazine.
- Locally recoverable codes with availability t≥2 from fiber products of curves. With K. Haymaker and G. Matthews. Advances in Mathematics of Communication, volume 12 (2), 2018, pp. 317-336.
- “Variations of the McEliece Cryptosystem.” With J. Bolkema, H. Gluesing-Luerssen, C. Kelley, K. Lauter, and J. Rosenthal. Algebraic Geometry for Coding Theory and Cryptography, Springer Publishing, 2017, pp. 129-149.
- “Picard curves over Q with good reduction away from 3.” With C. Rasmussen, LMS Journal of Computation and Mathematics, volume 19, 2016, pp. 382-408.
- “Zeta functions of a class of Artin-Scheier curves with many automorphisms.” With I. Bouw, W. Ho, R. Scheidler, P. Srinivasan, and C. Vincent, Research Directions in Number Theory: Proceedings from the 2014 WIN3 Workshop, Springer Publishing, 2016, pp. 87-124.
- “Local and Global Zeta Functions of Gauss’ Curve.” With J. Muskat. Rocky Mountain Journal of Mathematics, volume 45, 2015, pp. 275-285.
- “The a-numbers of Jacobians of Suzuki Curves.” With H. Frielander, D. Garton, R. Pries, and C. Weir. Proceedings of the American Mathematical Society, vol. 141, 2013, pp. 3019–3028.
- “The Automorphism Groups of a Family of Maximal Curves.” With R. Guralnick and R. Pries. Journal of Algebra, vol. 361, 2012, pp. 92-106.
- “Ramified Covers of Graphs and the Ihara Zeta Functions of Certain Ramified Covers.” With M. Manes. WIN — Women in Numbers, Fields Institute Communications, vol. 60, Amer. Math. Soc., Providence, RI, 2011, pp. 237–247.
- “An efficient digital signature scheme based on the learning with errors problem over polynomial rings.” With K. Lauter, M. Naehrig, and V. Vaikuntanathan. Patent 20120159179
- “Almost Divisibility in the Ihara Zeta Functions of Certain Ramified Covers of q+1-regular graphs.” With M. Manes. Linear Algebra and its Applications, vol. 432, 2010, pp. 2486-2506.
- Discovering networks of phonetic cues for speech sound categorization using graph theory. With Anne Marie Crinnion and Joseph Toscano. Submitted to Cognitive Science, 2017.