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.

### Published Math Papers:

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.

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.

### 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 (to appear)

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, 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.

K. Isham, Is the number of subrings of index p^e in Z^n a polynomial in p?, submitted.

M. I. de Frutos-Fernández, S. Garai, K. Isham, T. Murayama, and G. Smith, Rational Linear Subspaces of Hypersurfaces over Finite Fields, submitted.

K. Isham and L. Monroe, Arithmetic of idempotents in Z/mZ, submitted.

K. Isham and L. Monroe, On the structure induced by power sequences of (Z/mZ, *), submitted.

### Grants:

Subaward from Los Alamos National Lab LDRD Reserve grant number 20230692ER (PI Laura Monroe)

### Current Projects:

(with several authors) The number of 10-arcs in a projective plane is nonquasipolynomial.

(with Nathan Kaplan) On the proportion of subrings in Z^n with corank at most k

Understanding the behavior of 8-arcs in P^3(F_q)

(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.

### 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.