I enjoy applying explicit combinatorial methods to number theoretic problems. I have worked on projects relating to finite geometry, arithmetic statistics, arithmetic geometry, and 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.
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 p^e in Z^n 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 Z^n, Journal of Number Theory 234 (2022), 363-390, https://doi.org/10.1016/j.jnt.2021.06.026. [arXiv]
Published Interdisciplinary Papers:
*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
Submitted Papers:
*K. Lahotia, L. Monroe, K. Isham , M. Besta, N. Blach, T. Hoefler, and F. Petrini. PolarStar: Expanding the Scalability Horizon of Diameter-3 Networks. Submitted.
K. Isham. An algorithm to count the number of caps in P^3(F_q), submitted.
J. Desjardins, Y. Fu, K. Isham, and R. Winter. Generalized Eckardt points on del Pezzo surfaces of degree 1, submitted.
A. Dawkins, K. Isham , K. Lakhotia, A. Kubicek, and L. Monroe, Edge-Disjoint Spanning Trees on Star-Product Networks. Submitted.
Conferences:
Co-organizer of a special session on Topics in Recreational Math and Finite Geometry at the AMS Fall Eastern Sectional in Albany, October 19-20.
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:
The * indicates an undergraduate coauthor.
(with Nathan Kaplan) On the proportion of subrings in Z^n with corank at most k
(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 Sophia Child* and Omshi Samal*) Exploring the degree and diameter of various graph products.
(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.