  This is an announcement of some new computational methods in stable homotopy
theory, in particular, methods for using the cohomology of small-height Morava
stabilizer groups to compute the cohomology of large-height Morava stabilizer
groups. As an application, the cohomology of the height four Morava stabilizer
group is computed at large primes (its rank turns out to be $3440$).
Consequently we are able to formulate a plausible conjecture on the rank of the
large-primary cohomology of the Morava stabilizer groups at all heights.
