  A famous conjecture in gauge theory mathematics, attributed to Witten,
suggests that the polynomial invariants of Donaldson are expressible in terms
of the Seiberg-Witten invariants if the underlying four-manifold is of simple
type. Mathematicians have sought a proof of the conjecture by means of a
`cobordism program' involving $PU(2)$ monopoles. A higher rank version of the
Donaldson invariants was recently introduced by Kronheimer. Before being
defined, the physicists Mari\~no and Moore had already suggested that there
should be a generalisation of Witten's conjecture to this type of invariants.
We adopt a generalisation of the cobordism program to the higher rank situation
by studying $PU(N)$ monopoles. We analyse the differences to the $PU(2)$
situation, yielding evidence that a generalisation of Witten's conjecture
should hold.
