Mostly these bundle gerbes are used in geometry, and in mathematical physics.
So the most interesting example that shows up naturally involves bundles of Fock spaces
These are representations of a certain algebra on separable Hilbert spaces
In that instance, X_0 turns out to be the disjoint union of an open cover of a space of operators where a single open cover is given by what is known as a spectral cut
(spectrum here in the sense of spectrum of an operator)
Perhaps they are not SO interesting on their own, but we can put connections on them and so forth.
And so a bundle gerbe with connection (and curving don't worry what that means) represents a class in Deligne cohomology, much as a line bundle represents a cohomology class in H^2(-,Z)
Does anyone know where I can find a proof of H^*(ko; Z/2)=A//A(1)? I hear this was originally proved by Stong, but I'm having some trouble finding the reference.
try the beginning of chapter 3 of ravenel's green book, where he does some ASS computations with bo. if he doesn't give a proof, he ought to provide a reference. or, check the end of part 3 of adams' generalized homology and stable homotopy, where he does a bunch of bu computations and takes care to ensure his methods also apply to bo