If an atmosphere were created one way or another on Mars, would it not be stripped away again by solar winds? I thought the lack of a magnetic field is the reason why Mars has so little atmosphere left.
However, the "habitable" space under the aerogel may mainly be used for agriculture or as a temporary working area for people in vacuum-proof suits. One thin sheet of aerogel also sounds like a flimsy way of keeping atmospheric pressure.
Or do you really need a hot iron core?
I'd further guess that it would be easier to do the same with some sort of electromagnetic field. To enclose the whole planet would also be a giganteous undertaking. Much easier would be local field generators on the surface, maybe powered by a fusion generator.
At best this would be centuries in the future, at worst this doesn't make sense at all because it would require less energy/resources to construct huge greenhouses/domes that achieve the same thing and more.
The more immediate obstacles to colonizing Mars seem to be getting there in the first place, getting enough stuff there on a permanent/continuing basis for a few dozen people to live there for a few years, including radiation shielding.
It would be a really long time until there is a financially self-sustaining "colony", let alone one that can produce all of its own foods and much of its own technology and other resources. I figure it will be centuries until people even think about anything terraforming-related seriously.
Such a project would be achievable in the near future. I think we don't quite have sufficiently strong magnets yet, and it wouldn't be cheap.
The mass required is prodigious, but the energy requirements would be surprisingly small. The magnetic field of Earth above the surface stores about 100 megatons of magnetic energy; on Mars, and if the field were a bit weak, the energy needed would be much less.
Mars is smaller, and so the volume of the magnetic field would be smaller, and a lower field is probably enough. Since stored energy goes as B^2, maybe this energy could be reduced by two orders of magnitude.