Nathan Couchet (University of Clermont-Auvergne, France)
The aim of this talk is to present a groupoidal approach to pseudodifferential calculi and how it can be implemented to define pseudodifferential operators on smooth manifolds, circumventing the need for technical theorems of « invariance by diffeomorphims ».
Alain Connes’ tangent groupoid has become essential in non commutative geometry and has been used to some three decades to build pseudo-differential calculi. In 2017, van Erp and Yuncken (vEY), inspired by Debord, Skandalis, Lescure, Manchon and Vassout, built a pseudo-differential calculus on filtered manifolds of which a good example are Heisenberg manifolds (particularly, the Heisenberg group). Many pseudo-differential calculi are built to detect hypo-ellipticity. Whether these calculi are equivalent to the one of vEY is a central question in this talk.
With Robert Yuncken, we proved that for particular Heisenberg manifolds the calculus of vEY and BG are indeed equivalent.
To do that, we first charactherized polyhomogeneous symbols - which govern the realm of pseudo-differential operators . This uses graded dilatations and in a way suggests to adopt a groupoidal approach.
Provided time allows, I may also discuss how we built a groupoidal residue which is equivalent to the one of Wodzicki (1984) in the context of trivially filtered manifolds and the one of Ponge (2007) in the context of Heisenberg manifolds.
Although all this requires various prerequisites, this talk based on my PhD supervised by Robert Yuncken and Dominique Manchon, will be historically oriented and will provide – I hope - enough reminders for non-experts.
Also be accessible on zoom. Those who want to follow it online can contact Lisa Franz (firstname.lastname@example.org) to get the login details.