Polynomial Statistics, Necklace Polynomials, and the Arithmetic Dynamical Mordell-Lang Conjecture

Abstract

This thesis consists of six chapters representing two directions of the author’s graduate research under the advisement of Jeffrey Lagarias and Michael Zieve. The first direction studies new connections between the arithmetic statistics of polynomials over a finite field and the symmetric group representations carried by the cohomology of configuration space. The second direction, joint with Michael Zieve, studies unlikely intersections of orbits on curves, culminating in a non-commutative arithmetic dynamical Mordell-Lang theorem.