A remark on the well-posedness of the modified KdV equation in L ²

We study the real-valued modified KdV equation on the real line and the circle in both the focusing and the defocusing cases. By employing the method of commuting flows introduced by Killip and Vişan (2019), we prove global well-posedness in H^s for. On the line, we show how the arguments in the recent article by Harrop-Griffiths, Killip, and Vişan (2020) may be simplified in the higher regularity regime. On the circle, we provide an alternative proof of the sharp global well-posedness in L² due to Kappeler and Topalov (2005) and also extend this to the large-data focusing case.