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My friend is going to attempt to solve the Yang-Mills/Mass Gap hypothesis problem over the summer. Any suggestions?

For those of you who aren't familiar with the problem, it's an attempt to understand theoretically and mathematically why some quantum particles have positive mass even though classical waves travel at the speed of light.

Official problem description:
http://www.claymath.org/millennium/Yang-Mills_Theory/Official_Problem_Description.pdf

I'm sure we can all agree that he's doomed, because he's a college sophomore, but still...

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He's more than doomed. I would be impressed if a sophomore could understand the problem to begin with.
If he has exceptional talent, then he stands a chance at understanding some of the work done on the problem so far.
However he definitely won't solve it, or even make a decent attempt.

This, the Navier Stokes and P = NP are probably the three hardest of the problems, in my opinion.

My reasons for considering them as such are as follows:

Unlike most of the others they don't have a significant amount of work behind them. They haven't the same kind of bounds and refinements and there isn’t as many active researchers in the relevant areas. Universities which list these three are much rarer than universities that list the other four as research areas.

You have Lilley, Atiyah, Simms et al for the Hodge conjecture, not to mention the fact that the Hodge conjecture is in such an active area. In fact probably maths most active area if you average out over the post-Euler era. (Possibly even post-Descartes)

An entire cross-disciplinary group for the Riemann Hypothesis, composed of complex analysists, number theorists, geometers. Even people who work on foundational QM. The Riemann Hypothesis ties into so much that it attracts talent from many different areas.

The Poincaré conjecture is so old a problem that it ties into the very foundations of algebraic topology and even point-set topology, so it has quite a background in the literature of said subject.

I'll grant that the Birch and Swinnerton-Dyer conjecture isn't worked on that much, but it does have solutions in special cases. Something the other problems can't boast.
(Although I admit that point is mute for the Navier-Stokes equations as all they're looking for is a special case.)

Now let's look at the other three.

The Navier-Stokes Equation is counter-intuitively possibly the most difficult in my opinion. Whoever solves it may actually have to extend the current body of knowledge concerning PDEs, before they attempt to solve it.
It also is given much more lee-way than any of the other problems. You only have to prove one of four statements. However each of these are still open in the case of the Euler Equations, so you'd have to be able to show it for the Euler Equations, before you could really think about moving on to the Navier-Stokes.

Now P = NP, is located within a new and relatively active area, complexity theory. As the area is new, there isn't really much written yet with regards to the problem. (Compared with other problems.) Not only that, but complexity theory is still not entirely axiomatic and quite a lot of foundational issues are in question. Some think that Complexity Theory may need to be redefined for P = NP to be probably discussed. Others think it is entirely independent of the current axioms.
To sum up, it's too young a problem in too young an area. Not to mention the area's popularity is confined to certain geographical regions. It isn't even offered as a course in a lot of Universities.

And finally, Yang-Mills and the mass gap. The first difficulty lies in the sheer amount of prerequisite knowledge required to fully understand the problem. First a good knowledge of QFT, which itself can take years to assemble. Then a full appreciation of the work in constructive quantum field theory. That alone isn't a task I'd like to set for myself.
Now moving on, the problem itself has as a prerequisite that you construct the first ever completely axiomatic and well-defined Quantum Field Theory. Nobody has ever done that. Then you have to show that this leads to the mass-gap.
Edward Witten said "It is a problem for 21st century mathematics, but it's not for now". We don't know enough about operator-valued distribution themselves to start talking about the problem.

I hope that explains why I think what I think.


You have to understand, for that to be true he would have to have uncharacteristically extensive knowledge for his age. This isn't about him being to young to get it, I just can't imagine somebody having read what is necessary by that age.

Is it really that likely though? I tend to agree with most complexity theorists in that it will require some major revision of the subject.
Also the problem can be explained to a layman, but that doesn't mean a layman can easily grasp its rigorous formulation.


Yang-Mills hasn't been hugely concentrated on though compared with most of them. The Hodge conjecture is similarly very niche. Algebraic Topology on complex manifolds is a very minor area.
Again I think the Riemannian Hypothesis has had so much work done on it, that it is one of the most complete of the problems. Remember its study didn't really take off until the 90s.

Perhaps.
Again all I can offer is my reasons. You make a good case against mine.
As you said we can't really rank them in order of difficulty. This is merely how I rank them.

With regards to Yang-Mills, the necessary area, Axiomatic Quantum Field Theory, still hasn't really been brought to life fully.
Gauge Theory has been studies to a very extreme degree, but attempts to make QFT axiomatic and rigorous, which is what the problem is about, are few and far between.

The problem is:
Prove that for any compact simple gauge group G, a non-trivial quantum Yang-Mills theory exists on R4...........

Again, what I'm saying is that the Navier-Stokes, Yang-Mills and P = NP, will be the last to be solved because the solution lies in areas that have not yet been created fully.