I think it's clear that the first step we would need to take would be to establish a powerful magnetosphere. Without it, not only would any forms of
life be subjected to intolerable amounts of radiation, but any atmosphere that we could built up would slowly be blown out of gravitational pull, as
is currently happening with Mars' remnants.
How best to restart the core then?
Well, initially I thought this could be a simple process of creating a
nuclear
fusion chain reaction within the core, in a manner being experimented with in some places
on Earth. Basically, massive laser energy gets 'light' elements to fuse, giving off
masses of energy.
But unfortunately, as it turns out, this does not really happen to heavy elements as iron, which I'm sure we'd find at the center of mars. Perhaps
there is no 'ignition' cure for this problem, and may take persistent heating from lasers, rather than fusion.
If we could drill shafts to the core, which would be easier than on Earth as it is geologically dead, we could build a massive array of nuclear
powered laser beam arrays geometrically positioned around the surface of the globe aimed inward, melting the core. This of course would take decades,
perhaps even hundreds of years and require a great deal of manned maintenance. Not too easy.
My other idea is a little wild, perhaps pretty dangerous for our own planet, but we could strategically drag Mars toward the Sun, just close enough to
melt the thing over, then place it back out behind Earth. If we throw it in just right it right we might be able to sling shot it back into position
again.
An atmosphere (and pleasant weather)
As for establishing an atmosphere once we've got a magnetosphere, I would suggest doing something wild like dragging in a small moon from Saturn,
such as Phobos, or some other ice moon, and crashing it into Mars. Not only would we deliver a massive quantity of water, but the explosion would
certainly kick up a lot of dust, perhaps just enough to clog up the atmosphere, and warm it up enough to get liquid water and rain going. From then on
out we just throw down some algae and mycelium and boom! we have planet awesome.
edit: Aaah, I forgot to add, that the mass of the small moon might also ammend our gravity problem.
[edit on 24-2-2010 by The_Modulus]