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Research in arithmetic combinatorics (and closely related areas)

• The Furstenberg-Sárközy theorem for polynomials in one or more prime variables (with John R. Doyle), preprint. (arXiv)
• Multivariate polynomial values in difference sets (with John R. Doyle), Discrete Analysis, 2021:11, 46pp. (arXiv)
• Sets in \(\mathbb{R}^d\) with slow-decaying density and unbounded missing distances, Proceedings of the American Mathematical Society 148 (2020), 523-526. (arXiv)
• Binary quadratic forms in differences sets, Combinatorial and Additive Number Theory III, Springer Proceedings of Mathematics and Statistics vol. 297 (2020), 175-196. (arXiv)
• A maximal extension of the best-known bounds for the Furstenberg-Sárközy theorem, Acta Arith. 187 (2019), 1-41. (arXiv)
• Polynomials and primes in generalized arithmetic progressions (with Ernie Croot and Neil Lyall), International Mathematics Research Notices (2015), no. 15, 6021-6043. (arXiv)
• A purely combinatorial approach to simultaneous polynomial recurrence modulo 1 (with Ernie Croot and Neil Lyall), Proceedings of the American Mathematical Society 143 (2015), 3231-3238. (arXiv)
• A quantitative result on Diophantine approximation for intersective polynomials (with Neil Lyall), INTEGERS volume 15A (2015), Proceedings of Integers 2013: The Erdős Centennial Conference. (arXiv)
• Sárközy’s theorem for \(\mathcal{P}\)-intersective polynomials, Acta Arithmetica 157 (2013), no. 1, 69-89. (arXiv)
• Improved Bounds on Sárközy’s theorem for quadratic polynomials (with Mariah Hamel and Neil Lyall), International Mathematics Research Notices (2013), no. 8, 1761-1782. (arXiv)
• Polynomial differences in the primes (with Neil Lyall), Combinatorial and Additive Number Theory: CANT 2011 and 2012, Springer Proceedings of Mathematics and Statistics vol. 101 (2014), 129-146. (arXiv)
• Improvements and extensions of two theorems of Sárközy (Ph. D. thesis)

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Research with undergraduates

Work from Kinnaird Institute Research Experience, Millsaps College, Summer 2023:

• The sum-product problem for small sets (with Ginny Ray Clevenger, Haley Havard, Patch Heard, Andrew Lott, Brittany Wilson), to appear in Involve. (arXiv)

Work from Kinnaird Institute Research Experience, Millsaps College, Summer 2022:

• Computations and observations on congruence covering systems (with Raj Agrawal, Prarthana Bhatia, Kratik Gupta, Powers Lamb, Andrew Lott, Christine Rose Ward), INTEGERS Volume 24A (2024): Proceedings of the Integers 2023 Conference, Paper A1. (arXiv)

Work from Kinnaird Institute Research Experience, Millsaps College, Summer 2021:

• The pigeonhole principle and multicolor Ramsey numbers (with Vishal Balaji, Powers Lamb, Andrew Lott, Dhruv Patel, Sakshi Singh, Christine Rose Ward), Involve vol. 15 (2022), no. 5, 857-884. (arXiv)
• Schur’s theorem in integer lattices (with Vishal Balaji, Andrew Lott), INTEGERS vol. 22 (2022), Paper A62. (arXiv)

Work from Kinnaird Institute Research Experience, Millsaps College, Summer 2019:

• Sets in \(\mathbb{R}^d\) determining \(k\) taxicab distances (with Vajresh Balaji, Olivia Edwards, Anne Marie Loftin, Solomon Mcharo, Lo Phillips, Bineyam Tsegaye), Involve vol. 13 (2020), no. 3, 487-509. (arXiv)
• Lattice configurations determining few distances (with Vajresh Balaji, Olivia Edwards, Anne Marie Loftin, Solomon Mcharo, Lo Phillips, Bineyam Tsegaye), INTEGERS vol. 20 (2020), Paper A86. (arXiv)

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Research Experience for Undergraduate Faculty (REUF) group on Seidel tournament matrices

• Determinants of Seidel tournament matrices (with Sarah Klanderman, MurphyKate Montee, Andrzej Piotrowski, and Bryan Shader), to appear in Linear Algebra and Applications. (arXiv)

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UGA VIGRE group on elliptic curves with complex multiplication

• Computations on elliptic curves with complex multiplication (with Pete Clark, Patrick Corn, and James Stankewicz), London Mathematical Society Journal of Computational Mathematics 17 (2014), no. 1, 509-535. (arXiv)
• Torsion points on elliptic curves with complex multiplication (author of appendix, paper by Pete Clark, Brian Cook, and James Stankewicz), International Journal of Number Theory 9 (2013), 447-479.

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Miscellaneous/Expository

• Reciprocal sums and counting functions, Amer. Math. Monthly, Vol. 129 (2022), Issue 10, 903-912.
  • Recipient of a 2023 Paul R. Halmos-Lester R. Ford Award.
• A precise probability related to Simpson’s paradox, The College Mathematics Journal, Vol. 55 (2024), Issue 5, 400-405.
• Two theorems of Sárközy (expository note written with Neil Lyall)
• Waring’s problem (undergraduate-targeted expository notes, still in progress)
• Goldbach’s problems (notes from graduate seminar)
• Density and substance (undergraduate project)