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So what happened to the abc conjecture and Navier-Stokes?

Has there been any progress on verifying the proof of the abc conjecture or the solution to the Navier-Stokes equations? It’s been eerily quiet. I’m never going to come close to understanding any of those things, so I’ve been eagerly awaiting news that they’ve been either accepted or rejected.

$abc$ first: in August of 2012, Shin Mochizuki posted a series of papers containing a proof of the $abc$ conjecture to his website. Since then, people have been trying to work out whether it holds up, and Mochizuki has been modifying his papers in response to requests for clarification. If you need to get up to speed, we’ve got a few posts covering the story so far.

Clearly something‘s still happening, because Mochizuki is still updating his papers – his website says the last update was to paper II in May, but as far as I can tell nobody’s said anything in public about which way opinion is leaning.

And then there’s Kazakh mathematician Mukhtarbay Otelbaev’s proposed solution to the Navier-Stokes equations (Warning: it’s in Russian). We never got round to posting about it here (sorry!), but Otelbaev is an eminent mathematician and his paper was taken seriously. There was a lot of hullabaloo in the popular media, including of course New Scientist and Kazakhstan’s international news agency, but it also garnered some quick coverage on BBC Radio 4 (where it got the usual expertise-free handwavey treatment).

Otelbaev set up the problem a bit differently to the way it’s normally stated, so there was some confusion about whether his solution would qualify for the Clay Millennium Mathematics prize. According to this math.stackexchange question the two problems are equivalent, so we just needed to find out if it works or not.

To add some more doubt into the mix, Terry Tao published a paper titled Finite time blowup for an averaged three-dimensional Navier-Stokes equation to the arXiv, which states some limits on what a solution to Navier-Stokes can look like, but comments on his blog post about the paper say that it doesn’t rule out Otelbaev’s solution.

But in the end, with no fanfare, and no mention on Radio 4, Otelbaev sent this message to Stephen Montgomery, which he reproduced on another math.stackexchange thread about the solution:

Dear Prof. Montgomery-Smith,

To my shame, on the page 56 the inequality (6.34) is incorrect therefore the proposition 6.3 (p. 54) isn’t proved. I am so sorry.

Thanks for goodwill.

Oh well! Maybe next time.

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