  We study the real-valued modified KdV equation on the real line and the
circle, in both the focusing and the defocusing case. By employing the method
of commuting flows introduced by Killip and Vi\c{s}an (2019), we prove global
well-posedness in $H^{s}$ for $0\leq s<\tfrac{1}{2}$. On the line, we show how
the arguments in the recent paper by Harrop-Griffiths, Killip, and Vi\c{s}an
(2020) may be simplified in the higher regularity regime $s\geq 0$. On the
circle, we provide an alternative proof of the sharp global well-posedness in
$L^2$ due to Kappeler and Topalov (2005), and also extend this to the
large-data focusing case.
