  We consider the derivative nonlinear Schr\"odinger equation in one space
dimension, posed both on the line and on the circle. This model is known to be
completely integrable and $L^2$-critical with respect to scaling.
  The first question we discuss is whether ensembles of orbits with
$L^2$-equicontinuous initial data remain equicontinuous under evolution. We
prove that this is true under the restriction $M(q)=\int |q|^2 < 4\pi$. We
conjecture that this restriction is unnecessary.
  Further, we prove that the problem is globally well-posed for initial data in
$H^{1/6}$ under the same restriction on $M$. Moreover, we show that this
restriction would be removed by a successful resolution of our equicontinuity
conjecture.
