Research

Malmskog's Lemma!

Malmskog’s Lemma!

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 2020 CV.

A table of bounds for S-unit solutions to x+y=1 in certain fields.

Colin is having this much fun collaborating with me!!

Colin is having this much fun collaborating with me!!

Publications

Submitted:

Hermitian-Lifted Codes. With Hiram Lopez, Gretchen Matthews, Fernando Pinero-Gonzalez, and Mary Wootters, 2020. Submitted to Codes, Designs, and Cryptography.

Colorado in Context: Congressional Redistricting and Competing Fairness Criteria in Colorado [August 9 CO Ensemble Analysis]. With Jeanne Clelland, Haley Colgate, Daryl Deford, and Flavia Sancier-Barbosa, 2020

A robust implementation for solving the S-unit equation and several applications. With Alejandra Alvarado, Angelos Koutsianas, Christopher Rasmussen, Christelle Vincent, and Mckenzie West, 2020. Under revision for Simons Symposia series.

Published:

A graph-theoretic approach to identifying acoustic cues 2020 for speech sound categorization. With Anne Marie Crinnion and Joseph Toscano. Psychonomic Bulletin and Review, published online July 15, 2020, print to follow.

The de Rham Cohomology of the Suzuki Curves. With R. Pries and C. Weir. In Arithmetic geometry: Computation and Applications. Contemporary Mathematics Series, AMS Publishing, 2019, pp.105-120. 

What (quilting) circles can be squared? With K. Haymaker. Mathematics Magazine, volume 92 (3), 2019, pp. 173-186.

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.

Educational:

Quilt-Doku!  Parts 1, 2, and 3, in Girls Angle Bulletin, vol. 10 nos. 2 and 3, 2018.

Non-mathematical:

Ordovician-Silurian boundary strata of the Indian Himalaya: Record of the latest Ordovician Boda event. With Paul Myrow, David A. Fike, Stephen A. Leslie, Tianran Zhang, Birendra P. Singh, Ravi S. Chaubey, and Subhay K. Prasad. GSA Bulletin 131, no. 5-6 (2019): 881-898.

Math with friends at the Arizona Winter School–Go Team F_p!