Sums of squares of complex vector fields

Hypoellipticity is far from being understood, especially that of degenerate operators of the kind sums of squares of complex vector fields. In recent times, J. J. Kohn discovered complex vector fields that, despite their finite-type generation of the complexified tangent space at any given point, nevertheless are losing many derivatives, a phenomenon also discovered by C. Parenti and myself at about the same time for operators with multiple transversal characteristics. In this talk, I will survey Kohn’s results and some extensions of it, that I have given, and through examples give some speculations in order to deepen the understanding of this phenomenon.