My research concentration has been in applying explicit combinatorial methods to number theoretic problems. I have also worked on projects relating to finite geometry, algebraic statistics, and arithmetic algebraic geometry. 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.
Papers:
- K. Isham, Lower bounds for the number of subrings in Z^n, 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.
- K. Isham, An algorithm for counting arcs in higher-dimensional projective space, working paper (to be submitted in June 2021).
- The number of 10-arcs in a projective plane is not a quasipolynomial, working paper.
Upcoming Talks:
- UCSD Number Theory Seminar on June 3, 2021.
Talks:
- Charles University Number Theory Seminar on May 4, 2021.
- UC Berkeley Combinatorics Seminar on April 19, 2021.
- Lower bounds for the number of subrings in Z^n – PAJAMAS (virtual), September 2020.
- A lower bound for the number of subrings in Z^n – Chicago Number Theory Day (virtual), June 2020.
- The number of subrings of prime power index in Z^n – WiMSoCal 2020 at UCI, March 2020.
- Counting n-arcs in projective planes – Invited speaker at JMM 2020 in the AMS Special Session on Algorithms, Experimentation, and Applications in Number Theory
- Counting n-arcs in projective planes – AMS Sectional Meeting in Riverside in the special session on Research in Mathematics by Early Career Graduate Students, November, 2019.
- Counting n-arcs in projective planes – Modular Forms, Arithmetic, and Women in Mathematics, November 2019.
- Power sequences in (Z/mZ, .) – Number Theory Series in Los Angeles I at Occidental College, October 2019.
- Counting n-Arcs in Projective Planes – UCI Math Graduate Student Colloquium, May 2019.
Workshops Attended:
- In June 2019, I participated in the MRC program on Explicit Methods in Arithmetic Geometry in Characteristic p.
- In May 2018, I attended the Connecticut Number Theory Summer School.
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).