The distribution function of the rightmost particle in a branching Brownian
motion satisfies the Fisher-KPP equation:

∂u/∂t = ∂²u/∂x² + u – u²

Such an equation appears also in biology, chemistry or theoretical physics
to describe a moving interface, or a front, between a stable and an unstable
medium.

Thirty years ago, Bramson gave rigorous sharp estimates on the position of
the front, and, fifteen years ago, Ebert and van Saarloos heuristically
identified universal vanishing corrections.

In this presentation, I will present a novel way to study the position of
such a front, which allows to recover all the known terms and find some new
ones. We start by studying a front equation where the non-linearity is
replaced by a condition at a free boundary, and we show how to extend our
results to the actual Fisher-KPP.