My understanding is that Mars' gravity is much too weak to maintain an atmosphere -- any gas that's produced will literally escape into space. There's also the

My understanding is that Mars' gravity is much too weak to maintain an atmosphere -- any gas that's produced will literally escape into space. There's also the problem of radiation due to Mars' lack of a strong magnetic field. You could mitigate these problems by having humans live permanently underground. But that's not exactly terraforming. How can we overcome these significant hurdles to permanent human colonization of Mars?