where the ground breaks

There are four sources of groundbreaking ideas.

1. Following an interesting track of arguments and discovering where it leads
2. Revisiting the fundamental definitions and recasting them in a new light
3. Making new conjectures based on brand new experiment findings
4. Using an entirely different set of building blocks to make hard problems easier to attack

The first and second are possible in all fields of academia, the third is possibly only in experimental fields and the fourth only in mathematics.