Treves' conjecture on analytic hypoellipticity of sums of squares

Let $P$ be a sum of squares of real-analytic vector fields as in the paper, with characteristic variety stratified into real-analytic manifolds on which the symplectic form has constant rank and the iterated Poisson brackets of the vector-field symbols have constant depth. A stratum is symplectic when the restriction of the fundamental symplectic form to it is nondegenerate.

Treves' conjecture. The operator $P$ is analytic hypoelliptic if and only if every stratum in the above-described Poisson–Treves stratification is symplectic.

The conjecture proposes a geometric characterization of analytic hypoellipticity for sums of squares. Symplectic strata yield analytic hypoellipticity in several important cases, while the necessity and sufficiency in full generality remain open in the supplied text.