Why is the Yang–Mills existence and mass gap problem so fundamental? Ref: Yang–Mills existence and mass gap
Can anyone please explain why the Yang–Mills Existence and Mass Gap problem is so important / fundamental to contemporary mathematics (and, presumably, theoretical / mathematical physics)?
If possible, please include the following in your answer / response:
(1) Why the problem is so fundamental / important - for both mathematics and theoretical / mathematical physics;
(2) How the solution to the problem would impact research in the above two fields;
(3) (If applicable) What progress has been made so far in resolving it; (references to papers, books, other resources would be helpful).
I'm an upper-level undergraduate in theoretical physics, and so am seeking an answer / response at such level.
 A: This question deserves more than one good answer, so let me offer some comments addressing more directly what's hard about this particular problem.  The mass gap is an important challenge because solving it should force mathematical physicists to confront directly the messy question of exactly what the observables of QCD are. 
It's fairly easy to describe what QCD looks like at short distances.   The theory is asymptotically free, so you can describe quarks and gluons and their dynamics pretty accurately.  This can even be done rigorously; Balaban and Magnen, Seneor, & Rivasseau did it about 30 years ago.
But QCD becomes more complicated at long distances.  Quarks and gluons are still the basic building blocks of the infrared observables, but they are not themselves observables.  Instead the IR observables are things like hadrons and glueballs, which are complicated and difficult to describe in terms of the short distance observables.  A proton is 3 quarks and a cloud of gluons binding them, but it's not easy to say exactly what this cloud of gluons looks like.  
It's pretty likely that a solution to the Millenium Prize problem would involve coming up with a fairly precise description of exactly how hadrons and mesons are formed (or at least with mathematical technology which could be used to address this question).  This would be a real step forward in mathematicians' ability to deal with QFTs, and would probably be rather useful for things like lattice gauge theory.  (In lattice QFT, you can get away with imperfectly describing your observables, but you pay for it in accuracy.)
A: Dear Anonymous, gauge theories such as QCD are among the easiest theories to be formulated yet richest theories when it comes to the phenomena they cover, many of which are important in the real world.
The existence of the mass gap - the absence of arbitrarily small positive values in the mass spectrum - is a simple property of QCD that holds but that hasn't been rigorously demonstrated.
To demonstrate it and win the $1 million from the Clay Institute, she has to define the quantum field theory at a rigorous mathematical level and master much of its physics in an equally rigorous way. So it's a good, simple enough to be formulated, mathematical problem whose solution would bring mathematicians' mastery to a higher level.
At the same moment, the paper that would win the $1 million award would almost certainly not be very important for physicists. Physicists have found lots of complementary ways and insights that made them sure that the mass gap exists. Harboring doubts about the mass gap or trying to "totally" eliminate these doubts is simply not what theoretical physicists in this discipline spend most of their man-hours.
The evidence that the mass gap is real comes from renormalization group calculations of the strength of various interactions; simulations in lattice QCD; and, among other approaches, the most modern tools are based on the holographic AdS/CFT correspondence. These physical insights are arguably much more important and valuable than whatever could be included in the hypothetical math paper that proves the existence of the mass gap.
So I would summarize this answer by saying that despite the positive hype I started with, and despite the prize that has been offered, a fully mathematical proof of the mass gap is not one of the most important problems in physics - and maths. When it comes to maths, I personally view it as much less profound than e.g. the Riemann Hypothesis. When it comes to physics, I could enumerate hundreds of problems that are more important than a rigorous proof of the mass gap.
