Research

Arithmetic Statistics · Arithmetic Dynamics · Number Theory

My research sits at the intersection of arithmetic statistics and arithmetic dynamics. I compute finite Fourier transforms of invariant functions on spaces of binary forms, extending the Bhargava–Taniguchi–Thorne line of work used to compute similar transforms to higher degree spaces. I also study the arithmetic of critical orbits of the family f(x) = xd + c: the primitive prime divisors that appear along these orbits, the polynomials that arise as their reductions over local and finite fields, and the extent to which such orbits can be captured by simple arithmetic progressions.

Publications

  1. Arithmetic progressions in polynomial orbits
    with M. Sadek and T. Yesin · International Journal of Number Theory, 20(8) (2024), 2009–2025
  2. Construction of polynomials with prescribed divisibility conditions on the critical orbit
    with M. Sadek · preprint
  3. Finite Fourier transform of doubly singular binary quintic forms
    preprint

Research Experience

Doctoral Research, University of South Carolina
2022 – present · Advisor: Frank Thorne

Computing finite Fourier transforms of invariant functions on spaces of binary forms, extending prehomogeneous-vector-space techniques to the non-coregular setting of binary quintic forms. The computation reduces to point counts on auxiliary varieties indexed by splitting type, the most delicate governed by a hyperelliptic curve arising as the Hessian covariant.

MSc Thesis Research, Sabancı University
2020 – 2022 · Advisor: Mohammad Sadek

"Primitive Prime Divisors in the Critical Orbit of Polynomial Dynamical Systems" — established existence results for polynomials with arbitrarily many primitive prime divisors in their critical orbits over global fields; results extended into two subsequent papers.