Edward Green wrote:

Oh. Thank God. After thinking for five years "I ought to try
looking
at that", and then "damn! It's too late!", I can now procrastinate
for
another five years, or until senility sets in, which ever comes
first.
I think the solution to two Clay problems will probably come out at
the
same time -- one a consequence of the other, in either order (the
other
is the NP complete stuff).

Senility must be like a black hole horizon: you can never tell when
you've crossed it, everything looks the same to you locally.

You better hurry ... on both counts. I've finally decided to actually
take on the problems. All of them.

If their intractibility is not intrinsic, I will find a resolution --
unless someone else gets there first, sooner. Poincare', I think, is
already down.

Navier-Stokes is under discussion in sci.math.research.
At the outset I'm banking on being able to pull off a Bohm-Nelson in
reverse and convert it to a Schroedinger equation or something
similar.
Another approach relates to what I call the "Lagrangian method".
Section 30 of Lecture Notes in Physics 107 covers the symplectic
approach to fluid mechanics. (The only problem is that I think the
problem formulated by the Clay Foundation may not always admit a
Lagrangian). The "Lagrangian method" is an approximation method I
developed a while back to directly resolving Lagrangian problems by
optimizing the action integral. Its chief characteristic is that it
avoids the explicit use of calculus and of solving partial
differential
equations; instead, directly finding the update equations for a
simulated evolution. The task at hand, here, would then be to prove a
multi-variant version of Picard's theorem, showing that the
"Lagrangian
method" approximations uniformly converge to a regular solution over
some interval containing the boundary of the problem in its interior.
(The Navier-Stokes problem, at least as formulated in the Millennium
Prize challenge, is posed as one of two boundary-value problems).

There are two approaches to Yang-Mills; though unfortunately they may
deviate too far from what Clay wants for a solution. One poses
Yang-Mills as a Cauchy problem over a Big Bang universe (or an
inflationary universe), using the compactness of the past light cone,
or a similar spacelike surface, to establish the desired gaps in the
"frequency space". Another formulates quantum field theory as a theory
in a thermal vacuum, and attempts to link the mass gap to the 3rd law
of thermodynamics. The latter approach has been in the back of my mind
for quite a while, but I've never gone too far with it, because it may
deviate too far from what Clay would count as a legitimate solution.

A third approach works directly off of the Connes algebraic
characterization of Yang-Mills theory. Connes found a nice global
algebraic formulation of the Yang-Mills equations that completely
abstracts out the "calculus" and differential equations. There may be
a
way to exploit this to find a quantized theory that admits the
discretization of the frequency spectrum around 0.

In addition, there are a couple other approaches I've entertained,
that
it would take me too far afield to mention here.