Dennis The
“Homogeneous integrable Legendrian contact structures”
While all contact manifolds are locally equivalent, endowing the contact
distribution with additional data often yields a rich geometric structure with
non-trivial local invariants. For Legendrian contact structures, this means that
the contact distribution is split into two complementary subspaces that are
maximally isotropic with respect to a natural (conformal) symplectic form.
In my first talk, I'll give an overview of this
geometry: motivations for its study coming from complex (CR) geometry,
explaining how a large subclass of it can be equivalently viewed as a complete
2nd order PDE system, describing the fundamental curvature quantities, and the
notion of duality. This will be a general talk, requiring only basic
notions from differential geometry.
In a sequel talk, I'll focus on the problem of
classifying homogeneous such structures in dimension five. In particular,
I'll outline a systematic "top-down" method called "Cartan reduction" for
carrying out such classifications. (While this very general method dates
back to Elie Cartan's work in the early 1900's, it is not well-known.)