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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), submitted. (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), Involve, vol. 18 (2025), no. 1, 165-180. (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), Linear Algebra and Applications vol. 707 (2025), 126-151. (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)