  Eulerian polynomials are fundamental in combinatorics and algebra. In this
paper we study the linear transformation $\mathcal{A} : \mathbb{R}[t] \to
\mathbb{R}[t]$ defined by $\mathcal{A}(t^n) = A_n(t)$, where $A_n(t)$ denotes
the $n$-th Eulerian polynomial. We give combinatorial, topological and Ehrhart
theoretic interpretations of the operator $\mathcal{A}$, and investigate
questions of unimodality and real-rootedness. In particular, we disprove a
conjecture by Brenti (1989) concerning the preservation of real zeros, and
generalize and strengthen recent results of Haglund and Zhang (2019) on
binomial Eulerian polynomials.
