  The Eulerian transformation is the linear operator on polynomials in one
variable with real coefficients which maps the powers of this variable to the
corresponding Eulerian polynomials. The derangement transformation is defined
similarly. Br\"and\'en and Jochemko have conjectured that the Eulerian
transforms of a class of polynomials with nonnegative coefficients, which
includes those having all their roots in the interval $[-1,0]$, have only real
zeros. This conjecture is proven in this paper. More general transformations
are introduced in the combinatorial-geometric context of uniform triangulations
of simplicial complexes, where Eulerian and derangement transformations arise
in the special case of barycentric subdivision, and are shown to have strong
unimodality and gamma-positivity properties. General real-rootedness
conjectures for these transformations, which unify various results and
conjectures in the literature, are also proposed.
