Parabolic geometries: Overdetermined differential equations, holonomy reductions and compactifications

Parabolic geometries provide a uniform framework to describe, treat and analyse a number of differential geometric structures, most prominently projective structures, conformal structures & CR-structures. I will give an introduction to the most important features of parabolic geometries, most importantly in the area of conformal (spin) structures and how this framework can be used to treat interesting geometric differential equations via the BGG-machinery. A major advance in this area was a uniform holonomy reduction theorem, known as 'curved orbit decompositions'. I will explain via some examples how curved orbit decompositions can be used to understand the geometric implications of the existence of solutions to BGG-equations and in particular sheds light on 'singularity sets'. A final topic of this talk will be compactifications of parabolic geometries which are again intimately related with the concept of holonomy reductions and curved orbit decompositions.