Yang–Mills existence and mass gap
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The Clay Mathematics Institute has offered the prize of 1 million dollars for each of 7 great problems in mathematics. One of them is a proof that Yang-Mills theory exists according to the rigorous standards of mathematical physics (i.e. constructive quantum field theory) and it has a mass gap. The latter means that the lightest one-particle state in this theory must have a strictly positive mass.
[edit] Background
Most nontrivial (i.e. interacting) quantum field theories that we know of in 4D are effective field theories with a cutoff scale. Since the beta-function is positive for most models, it appears that most such models have a Landau pole as it is not at all clear whether or not they have nontrivial UV fixed points. This means that if such a QFT is well-defined at all scales, as it has to be to satisfy the axioms of axiomatic quantum field theory, it would have to be trivial (i.e. a free field theory).
However, the quantum Yang-Mills theory (no quarks) with a non-abelian gauge group is an exception. It has a property known as asymptotic freedom, meaning that it has a trivial UV fixed point. Because of this, this is the simplest model to pin our hopes on for a nontrivial constructive QFT model in 4D. (QCD, with its fermionic quarks is obviously more complicated).
It has already been well proven at the standards of theoretical physics, but not mathematical physics, that the quantum Yang–Mills theory for a non-abelian Lie group exhibits a property known as confinement. This is covered in more detail in the relevant QCD articles (QCD, colour confinement, lattice gauge theory, etc.), although not at the level of rigor of mathematical physics. Basically, this means that beyond a certain scale, known as the QCD scale (or since this is a quarkless model, we should say confinement scale), the color charges are connected by chromodynamic flux tubes leading to a linear potential (the tension of the "string" multiplied by its length) between the charges. This means that it is impossible to have free color charges like free gluons. In the absence of such a confinement, we would expect to see massless gluons, but since they are confined, all we see are color-neutral bound states of gluons, called glueballs. All the glueballs are massive, which is why we have a mass gap.
Results from lattice gauge theory have shown beyond any reasonable doubt that this model exhibits confinement (as indicated by an area law for the falloff of the VEV of a Wilson loop), but unfortunately, this isn't mathematically rigorous.
