  We look at several problems in even dimensional conformal geometry based
around the de Rham complex. A leading and motivating problem is to find a
conformally invariant replacement for the usual de Rham harmonics. An obviously
related problem is to find, for each order of differential form bundle, a
``gauge'' operator which completes the exterior derivative to a system which is
both elliptically coercive and conformally invariant. Treating these issues
involves constructing a family of new operators which, on the one hand,
generalise Branson's celebrated Q-curvature and, on the other hand, compose
with the exterior derivative and its formal adjoint to give operators on
differential forms which generalise the critical conformal power of the
Laplacian of Graham-Jenne-Mason-Sparling. We prove here that, like the critical
conformal Laplacians, these conformally invariant operators are not strongly
invariant. The construction draws heavily on the ambient metric of
Fefferman-Graham and its relationship to the conformal tractor connection and
exploring this relationship will be a central theme of the lectures.
