**Symmetries and invariant differential pairings**

**Abstract:**

In constructing symmetries of various natural differential operators such as the Laplacian or the Dirac
operator, there arise certain invariant differential pairings. The talk will begin with some examples of this phenomenon.
These pairings seem to persist in the curved setting even if the original construction is in flat space. Moreover, these
curved analogues enjoy the same invariance as does the original operator: in the examples above, this means conformal
invariance. Since a general theory is lacking, most of the talk will be devoted to a discussion of interesting examples.
However, I shall briefly describe some preliminary results obtained by Jens Kroeske in the projective setting.