Sunday, 14 July 2013

the individual v the commons

The twin prime conjecture, a famous open problem in mathematics, states that there are an infinite number of pairs of primes of the form \(p, p+2\).  Earlier this year, Yitang Zhang made a great breakthrough, publishing a proof that there are an infinite number of pairs of primes of the form \(p, p+H\), where \(H \leq 70,000,000\).

I've been watching what happened next with interest. Some people quickly seized on the result, and found better bounds. The encouraged Terrence Tao to propose a Polymath project, Polymath8,  to help coordinate efforts to understand Zhang's proof, and to reduce the bound on \(H\). Subsequent progress has reduced \(H\) considerably, with it currently standing at \(12,006\), and with as yet unconfirmed results of \(5,414\).

That's an amazing 4 orders of magnitude reduction, in just a couple of months.  To appreciate the progress, it's useful to look at a couple of graphs (based on that PolyMath wiki data). Here's how the best bound for \(H\) has fallen over the eight weeks since Zhang's paper was accepted:

best known value for \(H\), linear scale (diamond, confirmed result; +,  unconfirmed result)

best known value for \(H\), log scale (diamond, confirmed result; +,  unconfirmed result)
Here we see a fascinating synergy between individual and group efforts.  Zhang came up with the first, qualitative, breakthrough: a technique for providing a bound, and got a first (and now we see, rough) estimate.  Then the community gathered round, and through a process of cooperation (and presumably, a degree of competition, too), chipping away at the various definitions and terms, have quantitatively improved the technique.

So, to all those administrators trying to force us to work individually, or in groups, depending on the current fashion at headquarters -- the answer is clear: diversity works!  Some of the time progress is made by individuals, sometimes by groups, even on the same problem.  Don't assume one size fits all.

Progress here appears to have tapered off recently, with no new results reported in the last few days.  Is this due to the improvements having been pushed as far as possible (not likely, as there are several unconfirmed results yet), to enthusiasm having flagged (also unlikely, as the results are getting ever closer to the ultimate value of \(2\)), or due to it being vacation time? I'll continue watching with interest.

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