'Algebraic preperiodic points of entire transcendental functions' - 

A complex number \(z\) is called algebraic if it satisfies a nontrivial polynomial equation \(p(z)=0\) with integer coefficients. If \(z\) is not algebraic, it is called transcendental. In a similar way one can distinguish between algebraic and transcendental functions of a complex variable. The study of transcendental numbers that developed following the work of Charles Hermite in the 1870's leads naturally to the question of existence of transcendental functions \(f:\mathbb{C}\to\mathbb{C}\) mapping every algebraic number to an algebraic number. The existence of such functions was proved by Marques and Moreira only recently, in 2016. In this talk we will discuss some of the the history leading up to this recent theorem, and we will explore some dynamical properties of this type of function \(f\), in particular the structure of its directed graph of algebraic preperiodic points.