Mathematics
Computational Complexity and the Classes P and NP
Quick fact
No one knows whether P equals NP, but if P = NP, then every problem whose solution can be quickly verified could also be quickly solved—potentially breaking modern encryption and solving countless optimization puzzles instantly.
Why this is interesting
Some problems are easy for a computer to solve, and others seem impossible—even with the fastest supercomputers. What makes a problem 'easy' or 'hard'?