Robert Yuncken (Metz), Emmanuel Trélat (Paris)
14:00 Robert Yuncken (Université de Lorraine, Metz): Smooth solutions to PDES — from elliptic operators to the Helffer-Nourrigat conjecture.
14:45 Tea and Coffee Break
15:15 Emmanuel Trélat (Sorbonne Université, Paris): Spectral analysis of sub-Riemannian Laplacians and Weyl measure.
Robert Yuncken (Université de Lorraine, Metz): Smooth solutions to PDES — from elliptic operators to the Helffer-Nourrigat conjecture.
Abstract: A partial differential operator D is called hypoelliptic if all solutions of the partial differential equation Df=g are smooth wherever g is smooth. The name comes from the foundational result that elliptic differential operators are hypoelliptic. But there are many natural hypoelliptic operators which are not elliptic, for instance those discovered by Hörmander and by Rockland. A wide generalisation of these results was conjectured by Helffer and Nourrigat in the 1970s. Starting from Alain Connes’ elegant point of view on elliptic pseudodifferential operators, we will show how the Helffer-Nourrigat conjecture can be proven using ideas from noncommutative geometry.
Emmanuel Trélat (Sorbonne Université, Paris): Spectral analysis of sub-Riemannian Laplacians and Weyl measure.
Abstract: Prototypes of hypoelliptic operators are sub-Riemannian Laplacians, namely selfadjoint hypoelliptic operators satisfying the Hörmander condition. In collaboration with Yves Colin de Verdière and Luc Hillairet, we study their spectral properties.
Thanks to the knowledge of the small-time asymptotics of heat kernels in a neighborhood of the diagonal, we establish the local and microlocal Weyl laws, putting in light the Weyl measure in sub-Riemannian geometry.
When the Lie bracket configuration is regular enough (equiregular case), the Weyl law resembles that of the Riemannian case. But in the singular case (e.g., Baouendi-Grushin, Martinet) the Weyl law reveals much more complexity. In turn, we derive quantum ergodicity properties in some sub-Riemannian cases.
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