Prove that for any compact simple gauge group G, a non-trivial quantum Yang–Mills theory exists on R4 and has a mass gap Delta > 0.
This is told in detail here. You also must prove this to the standards of axiomatic quantum field theory. But this is not the only hurdle.
- You have to prove this for ANY (non-commutative) compact simple gauge group. Unfortunately, in nature, only ONE is known: SU(3) (not SU(2), as it is broken), for the strong nuclear interaction. Even if string theory were correct, there would only be a FINITE number of gauge groups in nature. But the problem asks to prove it for all, INFINITELY many gauge groups. So you have to prove something that is contrary to experiment.
- Then you have to prove it on R4, i.e., on classical spacetime. Now, if string theory is correct, spacetime is not R4, but has 10 or 11 dimensions. And if a smallest distance exists, as many string theorists and most quantum gravity experts state, it does not even make sense to speak about 4 dimensions and space-time continuity. It is not a secret: there is only some sort of space-time foam in nature. So you have to prove something that is not really related to the real world.
- At least, a mass gap is indeed observed for the strong interaction; the precise size is unknown. The observed mass gap is larger than 0.5 GeV/c^2, but it could be as large as the Planck mass, depending on whether glueballs are discovered or not.
So you get a million dollar if you prove for ALL groups that a mass gap exists on continuous, four-dimensional space-time, while to our best knowledge it only exists for SU(3) on space-time foam. So, an institute offers a million dollars for showing that mathematics can contradict physics! That is funny, but it is not illegal. It is also funny that all these issues where known already when the problem was posed. But again, it is not illegal. Now, who posed the problem? It was co-authored by Witten.
Wait a moment. Witten, the smartest physicist on the planet? Yes. Is that the same Witten who says all the time that string theory is needed to solve the mass gap problem? Yes. And despite this, he formulated a math problem that makes no sense physically and contradicts string theory? Yes. And he was encouraged and helped by the smartest mathematicians in the world? Yes. And they all claim that showing that math contradicts physics is a 'millennium problem', as important as proving the Riemann hypothesis? Yes. And a serious institute offers a million dollars for solving a problem that makes no sense? Yes - but wait, who says that they are serious?
Indeed, there is a final joke. The rules of the institute state that they will "consider" giving out the prize also for a counterexample. I have one: the real world! I want my million dollars!
There is nothing funnier than real life.