Symmetry gaps for geometric structures

For a given type of differential geometric structure, there is often a gap between the maximal and "submaximal" infinitesimal symmetry dimensions. This was first observed in the 19th century for Riemannian metrics and such symmetry gaps were subsequently classified for various other geometric structures on a case-by-case basis. I will describe joint work with Boris Kruglikov that gave a uniform approach to the symmetry gap problem for the class of parabolic geometries. This is a diverse class of geometric structures that include conformal, projective, CR, 2nd order ODE systems, and large classes of generic distributions. A priori, submaximally symmetric structures need not even be homogeneous, but remarkably, in many cases this geometric problem reduces ultimately to Dynkin diagram combinatorics, and some submaximally symmetric models can be "immediately" found (in a sense that I will make precise).