Hypoellipticit{\'e} de polyn{\^o}mes de champs de vecteurs et conjectures de Helffer etNourrigat

We study here the sub-Riemannian geometry on a manifold $M$ induced by a finite family $F$ of vector fields satisfying the H{\"o}rmander condition, as well as the differential operators obtained as polynomials in the elements of $F$. Such an operator $D$ is hypoelliptic if, for any smooth function $f$, the solutions $u$ of the equation $Du=f$ are also smooth. A more refined notion, that of maximal hypoelliptic operators, extends this property in terms of Sobolev regularity, offering a parallel in sub-Riemannian geometry to elliptic operators. In 1979, Helffer and Nourrigat proposed a conjecture characterizing maximal hypoellipticity, generalizing the main regularity theorem for elliptic operators. This conjecture has recently been confirmed using tools from non-commutative geometry. A central element of this work is a natural generalisation in sub-Riemannian geometry, introduced by Mohsen, of the Connes tangent groupoid, in which appear all the tangent cones, key ingredients in the work of Helffer and Nourrigat. In collaboration with Androulidakis and Yuncken, Mohsen developed a pseudodifferential calculus in this context, introducing in particular the notion of principal symbol. They obtained that the invertibility of this symbol is equivalent to maximal hypoellipticity, thus validating the conjecture. This talk will present the ingredients and broad outlines of these innovative advances.