The physicists' Yang-Mills theory is a description of behavior. By caclulating based on the Y-M behavior they can accurately predict things.
But the physicists have never shown that there is something that behaves that way. To mathematicians, these "existence questions" are important. The physicsts can just say suppose there's a system that works like THIS; then we can do thus and so, ain't it great! But the mathematicians worry about rigor.
So what the Clay prize wants is a proof the Y-M theory exists, and that it has an important property that physicists just assume, a mass gap. That means that the theory won't produce a chain of particles with smaller and smaller masses going to a limit of zero mass. If Y-M didn't have a mass gap, you couldn't rely on it to give physical answers*. It has to produce either genuine zero mass particles (like the gluons in QCD) or particles of mass greater than some constant. The standard model does this, but there isn't any proof that a Y-M theory does it automatically.
*Each particle could and would decay into littler particles, and the littler ones to littler ones, and so on ad infinitum. It doesn't rhyme but it shows that the stability of our world depends on having a mass gap. You can decay as far as the top and bottom quarks, but no farther because there ain't no lighter fermions in the theory. So we have protons and all.
The exact statement of this problem that will make you rich is you can solve it is: "Prove that for any compact simple gauge group G, quantum Yang-Mills theory on R4 exists and has a mass gap superior to zero"
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