  We establish pointwise almost everywhere convergence for ergodic averages
along polynomial sequences in nilpotent groups of step two of
measure-preserving transformations on $\sigma$-finite measure spaces. We also
establish corresponding maximal inequalities on $L^p$ for $1<p\leq \infty$ and
$\rho$-variational inequalities on $L^2$ for $2<\rho<\infty$. This gives an
affirmative answer to the Furstenberg-Bergelson-Leibman conjecture in the
linear case for all polynomial ergodic averages in discrete nilpotent groups of
step two.
  Our proof is based on almost-orthogonality techniques that go far beyond
Fourier transform tools, which are not available in the non-commutative,
nilpotent setting. In particular, we develop what we call a nilpotent circle
method that allows us to adapt some of the ideas of the classical circle method
to the setting of nilpotent groups.
