The parabolic geometries are curved deformations of (real) homogeneous
spaces G/P, such that the complexification
is a parabolic subgroup in a semisimple Lie group. The general ideas go back
to Cartan's 'generalized spaces'. Examples involve the conformal,
projective, almost quaternionic, and CR geometries.
The first talk will present an introduction, survey of recent existence
results, and a more detailed explanation of new techniques leading to an
effective calculus for invariant operators. The explicit construction of
curved analogues of the famous Bernstein-Gel'fand-Gel'fand resolutions for
all these geometries (without any use of its representational theoretical
version) will be presented as the first major application of our new
approach.
The second talk will focus on the parabolic geometries with irreducible
tangent bundles. In this case, all the standard geometrical structures known
from conformal Riemannian geometries admit nice general counterparts and,
in particular, there is a more straightforward explicit construction of the
operators from the above mentioned resolutions based on finite dimensional
representation theory. It is remarkable that the closed formulae for all
these operators of a given order do not depend on the choice of the
structure groups.
