I enjoy applying explicit combinatorial methods to number theoretic problems. I have worked on projects relating to finite geometry, arithmetic statistics, arithmetic geometry, and computer network design. My projects often involve coding in Sage, Magma, or Mathematica. Common themes in my research include counting rational points on varieties and studying the behavior of functions of the form f(p) as the prime p varies.
See my CV for previous talks/conferences.
Upcoming Talks/Visits:
Visitor at Laboratory for Physical Sciences, Catonsville, MD, February 7 - 22
Keynote speaker at the Villanova Undergraduate Research Symposium, Villanova, PA, April 28
Speaker at the CanaDAM conference, Ottawa, Canada, May 20-23
Speaker at IPDPS 2025, Milan, Italy, June 3 - 7
If the author order is based on significance of contribution, the paper is marked with a *. Otherwise, the author order is alphabetical.
Published Math Papers:
K. Isham, Is the number of subrings of index pe in ℤn a polynomial in p, Communications in Algebra 51 (12) (2023), doi.org/10.1080/00927872.2023.2225600. [arXiv]
K. Isham, An algorithm for counting arcs in higher-dimensional projective space, Finite Fields and Their Applications 80 (2022), https://doi.org/10.1016/j.ffa.2022.102006. View associated Sage code here. [arXiv]
K. Isham, Lower bounds for the number of subrings in ℤn, Journal of Number Theory 234 (2022), 363-390, https://doi.org/10.1016/j.jnt.2021.06.026. [arXiv]
Published Interdisciplinary Papers:
†*K. Isham, L. Monroe, K. Lakhotia, A. Dawkins, D.Hwang, and A. Kubicek. “Edge disjoint spanning trees in star product networks” in Proceedings of the 38th IEEE International Parallel and Distributed Processing Symposium (IPDPS '25), 2025. Accepted
*K. Lakhotia, L. Monroe, K. Isham , M. Besta, N. Blach, T. Hoefler, and F. Petrini. PolarStar: Expanding the Scalability Horizon of Diameter-3 Networks. In Proceedings of the 36th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA ’24), June 17–21, 2024, Nantes, France, pages 345 - 357. https://dl.acm.org/doi/10.1145/3626183.3659975
*K. Lakhotia, K. Isham, L. Monroe, M. Besta, T. Hoefler, and F. Petrini. "In-network Allreduce with Multiple Spanning Trees on PolarFly." In Proceedings of the 35th ACM Symposium on Parallelism in Algorithms and Architectures (SPAA ’23), June 17–19, 2023, Orlando, FL, pages 165 - 176. dl.acm.org/doi/10.1145/3558481.3591073
*K. Lakhotia, M. Besta, L. Monroe, K. Isham, P. Iff, T. Hoefler, F. Petrini. “PolarFly: A Cost-Effective and Flexible Low-Diameter Topology” in 2022 SC22: International Conference for High Performance Computing, Networking, Storage and Analysis (Supercomputing '22), 2022, pages 146-160. https://www.computer.org/csdl/proceedings-article/sc/2022/544400a146/1I0bSNInMqc
† The first three authors contributed equally
Submitted Papers:
K. Isham. Counting Special Sets of Points in P^3(Fq), submitted.
J. Desjardins, Y. Fu, K. Isham, and R. Winter. Generalized Eckardt points on del Pezzo surfaces of degree 1, submitted.
Sophia Child, K. Isham, and Omshi Samal, Exploring the degree and diameter of various graph products. Submitted.
K. Isham and N, Kaplan with an appendix by Gautam Chinta. Most subrings of ℤn have large corank. Submitted . Mathematica Code
Grants:
Colgate Picker Interdisciplinary Science Institute, PI with Kartik Lakhotia and Laura Monroe: April 2024 - August 2026.
Subaward from Los Alamos National Lab LDRD Reserve grant number 20230692ER (PI Laura Monroe): November 2022 - September 2023
Current Projects
(with Gary Gordon, Nathan Kaplan, and Liz McMahon) Counting combinatorial configurations of 12 points in AG(4, 3) containing exactly k lines: An analysis of the Daily SET Puzzle.
(with several authors) The number of 10-arcs in a projective plane is nonquasipolynomial.
Other Research Experiences:
I worked at Los Alamos National Laboratory (LANL) during the summer of 2019 via funding by the NSF Mathematical Sciences Graduate Internship (MSGI).
I am currently a Guest Scientist at LANL and a Visiting Researcher at Laboratory for Physical Sciences.