FOM: Clay Mathematics Institute $1,000,000 Prize problems

[Joe Shipman]

Last Wednesday, the Clay Mathematics Institute of Cambridge,
Massachusetts, a one-year-old private foundation, announced seven
"Millennium Prize Problems", with a $1,000,000 prize for each.

The seven problems are:

1) P versus NP

2) The Hodge Conjecture

3) The Poincaré Conjecture

4) The Riemann Hypothesis

5) Yang-Mills Existence and Mass Gap

6) Navier-Stokes Existence and Smoothness

7) The Birch and Swinnerton-Dyer Conjecture


See http://www.claymath.org for more details.

The rules are rather interesting. (See
http://www.ams.org/claymath/prize_problems/rules.htm ).  Only a positive
solution to a problem gets the full prize (for the first problem, it is
explicitly stated that settling it in either direction gets the full
prize).  All important decisions are the responsibility of the CMI's
Scientific Advisory Board, which currently consists of Alain Connes,
Arthur Jaffe, Andrew Wiles, and Edward Witten.

As in Hilbert's original list a century ago, some of the problems are
more precisely formulated than others, some of the problems can be
considered applied rather than pure math, and the Riemann Hypothesis is
one of the problems.

The problems chosen seem rather "mainstream", and there is nothing wrong
with that; but in my opinion Hilbert's list was much more wide-ranging.
Since the CMI is backing up the problems with money, it is
understandable that they would not want a very long list.

In my opinion, there can be no serious questioning of the
appropriateness of problems 1, 3, and 4. Problems 5 and 6 are not
precisely expressible as sentences in a formal language, but they seem
to be well-chosen.  Problems 2 and 7, to my mind, lack "g.i.i." and a
different committee might easily have selected other problems instead.
(My personal favorite is the Invariant Subspace Conjecture.)

It is good to see that the problem of the most interest for the
Foundations of Mathematics (P versus NP) was given the top place, just
as Hilbert led with the Continuum Hypothesis.  Another problem of
foundational interest, which ought to be on this list except that it is
easy to imagine much more heated and inconclusive debate over the
awarding of the prize than for the other problems, is "Does Mathematics
Need New Axioms"?

I applaud the Clay Mathematics Institute's proposal of these prize
problems and also their other programs promoting mathematics, and look
forward to comments from FOM subscribers.

-- Joe Shipman