I've just been sent a fun paper by Harvard mathematician Lakshminarayanan Mahadevan on the physics of flying carpets. It's published in a recent edition of the physics journal Physical Review Letters.
It addresses this question: "From a physical perspective ... [is] a flying carpet possible and, if so, under which conditions might it operate?" The answer turns out to be a tentative yes, carpets can fly, if vibrated appropriately.
Read the whole thing.