Climate scientist here. I'm not sure about next but I'd really like to see an efficient means of removing CO2 from the atmosphere. For carbon capture and storage tech I don't think there will be a 'breakthrough' on the storage side so much as a progression of finding better and better locations and injection methods. But on the capture side I think there is a lot of room for breakthroughs in C binding catalysts. This is a bit out of my specialty but inspired by an article in the Jan 15th issue of Science about a group that has made some progress into doing just this, synthesizing a Cu based catalyst that binds CO2 into oxalate.