  We study the maximum of Branching Brownian motion (BBM) with branching rates
that vary in space, via a periodic function of a particle's location. This
corresponds to a variant of the F-KPP equation in a periodic medium,
extensively studied in the last 15 years, admitting pulsating fronts as
solutions. Recent progress on this PDE due to Hamel, Nolen, Roquejoffre and
Ryzhik ('16) implies tightness for the centered maximum of BBM in a periodic
environment. Here we establish the convergence in distribution of specific
subsequences of this centered maximum, and identify the limiting distribution.
Consequently, we find the asymptotic shift between the solution to the
corresponding F-KPP equation with Heavyside initial data and the pulsating
wave, thereby answering a question of Hamel et al. Analogous results are given
for the cases where the Brownian motion is replaced by an Ito diffusion with
periodic coefficients, as well as for nearest-neighbor branching random walks.
