Overdetermined systems, conformal differential geometry, and the BGG complex

Wednesday, July 19, 2006 - 2:00pm - 3:00pm
EE/CS 3-180
Andreas Cap (Universität Wien)
The starting point of my lectures will be a way to rewrite
certain overdetermined systems on Riemannian manifolds in closed
form. The method is based on including the orthogonal group O(n) into
the pseudo-orthogonal group O(n+1,1) and analyzing the standard
representation of O(n+1,1) from the point of view of this subgroup.

Next, I will indicate how, replacing direct observations by tools from
representation theory, this method can be generalized to a large class
of systems.

Then I will explain how the inclusion of O(n) into O(n+1,1) that we
started from is related the passage from Riemannian to conformal
geometry. Refining the methods slightly, one obtains a construction
for a large family of conformally invariant differential operators.

In the end, I want to sketch how the ideas generalize further to a
large class of geometric structures called parabolic geometries.