  We present a novel method for computing slow manifolds and their fast fibre
bundles in geometric singular perturbation problems. This
coordinate-independent method is inspired by the parametrisation method
introduced by Cabr\'e, Fontich and de la Llave. By iteratively solving a
so-called conjugacy equation, our method simultaneously computes
parametrisations of slow manifolds and fast fibre bundles, as well as the
dynamics on these objects, to arbitrarily high degrees of accuracy. We show the
power of this top-down method for the study of systems with multiple (i.e.,
three or more) timescales. In particular, we highlight the emergence of hidden
timescales and show how our method can uncover these surprising multiple
timescale structures. We also apply our parametrisation method to several
reaction network problems.
