exploring which input bits effect which output bits (1, 2 or 3 at a time) might be interesting
I don't expect SHA2 to be broken by the ways discussed in this thread, but it's a good thing that someone tries it. A good result from such approach would be not analytical solution (seems impossible, but there always is a hope that equations system will collapse 
), but representation of double SHA2 better (by operations count or height) that ones currently used in ASICs. I think I know how to chop 10% (or so) off from ~120000 binary operations, but let's hope that this thread will bring some fresh ideas