Seminarium KMMF "Teoria Dwoistości"

I will give an introduction to a class of geometric structures known as parabolic geometries: these are Cartan geometries modelled on homogeneous spaces of the from G/P, where G is a semisimple Lie group and P is a parabolic subgroup. The most prominent example of a parabolic geometry is conformal geometry in dimension >2; the symmetry group G of the flat homogeneous model in this case is the conformal group. A more exotic but still classical example is the geometry of (2,3,5) distributions, which is related to the exceptional simple Lie group G=G_2. In this talk I will review some history, explain how the Lie group G_2 appears in this context, and discuss recent developments in the field.