We consider the derivative nonlinear Schrodinger 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 $M(q)=int |q|^2 < 4pi$. 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.
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