Hypoelliptic geometries

18.11.2016, 11:00 Uhr  –  Haus 9, Raum 2.22
Arbeitsgruppenseminar Analysis

Shantanu Dave (WPI, Wien)

A geometry such as a Riemannian geometry, Lorenzian geometry or CR
geometry allows a local description of a manifold near every point and
hence can be used to construct differential operators which are local
operators. In the case of Riemannian geometry,  most interesting operators
that can be built, such as the Laplace and the Dirac operators, are
elliptic. It is well known that many of the other geometries also
construct operators, but these are not elliptic in general. Only in
contact geometries does one know that the operators constructed are
hypoelliptic due to the existence of a calculus called the Heisenberg
calculus.
The aim of this talk is to show that in many other geometric situations
(like the Cartan geometries) the naturally occuring opperators are in
fact hypoelliptic, which allows  many of the analytic results used in elliptic
theory. This brings these geometries on the same footing as Riemannian
case.

The proof is based on the idea of tangent groupoids, which is a way to
deform operators, and a soft and accessible description of the so called
BGG operators that we introduce.
The talk will be broad and most concepts will be introduced in elementary
way.