Prolongation on contact manifolds

written jointly with Rod Gover and also why one might care. Rather amazingly, one of the main motivations I presented in the first talk was a mysterious phenomenon in parabolic geometry that was totally explained in

On a new normalization for tractor covariant derivatives

by Matthias Hammerl, Petr Somberg, Vladimír Souček, and Josef Šilhan released as a preprint in the early hours of the following morning! Nevertheless, the related phenomenon of directly constructing Cartan connections from invariant differential operators remains mysterious and still provides a strong motivation.