It seems that physicists at Harvard, no less, have found an aerodynamics solution in which one can make a "carpet" fly.
The researchers have studied1 the aerodynamics of a flexible, rippling sheet moving through a fluid, and find that it should be possible to make one that will stay aloft in air.
No such carpet is going to ferry people around, though. The researchers say that to stay afloat in air, a sheet measuring about 10 centimetres long and 0.1 millimetres thick would need to vibrate at about 10 hertz with an amplitude of about 0.25 millimetres. Making a heavier carpet 'fly' is not forbidden by the laws of physics. But the researchers say that their "computations and scaling laws suggest it will remain in the magical, mystical and virtual realm", as the engine driving the necessary vibrations would need to be so powerful.
I will stick to commercial airlines, though. Just think how difficult it is to eat powdered doughnuts during an open-air magic carpet ride? :)