Once in every six months there’s a flurry of online excitement about Mochizuki’s alleged proof of the abc-conjecture.
It seems to be that time of the year again.
The twitter-account of the ever optimistic @math_jin is probably the best source for (positive) news about IUT/ABC. He now announces the latest version of Yamashita’s ‘summary’ of Mochizuki’s proof:
山下剛さんのIUTサーベイが更新されました。 Go Yamashita A proof of the abc conjecture after Mochizuki. preprint. last updated on 18/Nov/2017.https://t.co/XtnMEO3zoQ#IUTABC
Big news on Mochizuki's groundbreaking IUT: Over 1000 comments on his 4 papers have been addressed & the final versions sent back to the journal for approval. Hopefully, will be published soon. Here's Ivan Fesenko's interview about IUT on the AMS website.https://t.co/6GLk3Xh0lm
In case you prefer an English translation: The big ABC.
Here’s her opening paragraph:
“In a children’s story written by the Swiss author Peter Bichsel, a lonely man decides to invent his own language. He calls the table “carpet”, the chair “alarm clock”, the bed “picture”. At first he is enthusiastic about his idea and always thinks of new words, his sentences sound original and funny. But after a while, he begins to forget the old words.”
The article is less optimistic than other recent popular accounts of Mochizuki’s story, including:
“Table is called “carpet”, chair is called “alarm clock”, bed is called “picture”. In the story by Peter Bichsel, the lonely man ends up having so much trouble communicating with other people that he speaks only to himself. It is a very sad story.”
Perhaps things will turn out for the better, and we’ll hear about it sometime.
“Sometimes ideas, like men, jump up and say ‘hello’. They introduce themselves, these ideas, with words. Are they words? These ideas speak so strangely.”
“All that we see in this world is based on someone’s ideas. Some ideas are destructive, some are constructive. Some ideas can arrive in the form of a dream. I can say it again: some ideas arrive in the form of a dream.”
Here’s such an idea.
It all started when Norma wanted to compactify her twisted-prime-fruit pies. Norma’s pies are legendary in Twin Peaks, but if you never ate them at Double R Diner, here’s the concept.
Start with a long rectangular strip of pastry and decorate it vertically with ribbons of fruit, one fruit per prime, say cherry for 2, huckleberry for 3, and so on.
For elegance, I argued, the $p$-th ribbon should have width $log(p)$.
“That may very well look natural to you,” she said, “but our Geometer disagrees”. It seems that geometers don’t like logs.
Whatever. I won.
That’s Norma’s basic pie, or the $1$-pie as we call it. Next, she performs $n$ strange twists in one direction and $m$ magical operations in another, to get one of her twisted-pies. In this case we would order it as her $\frac{m}{n}$-pie.
Marketing-wise, these pies are problematic. They are infinite in length, so Norma can serve only a finite portion, limiting the number of fruits you can taste.
That’s why Norma wants to compactify her pies, so that you can hold the entire pastry in your hands, and taste the infinite richness of our local fruits.
“Wait!”, our Geometer warned, “You can never close them up with ordinary scheme-dough, the laws of scheme-pastry prohibit this!” He suggested to use a ribbon of marzipan, instead.
“Fine, then… Margaret, before you start complaining again, how much marzipan should I use?”, Norma asked.
“Well,” I replied, “ideally you’d want it to have zero width, but that wouldn’t close anything. So, I’d go for the next best thing, the log being zero. Take a marzipan-ribbon of width $1$.”
The Geometer took a $1$-pie, closed it with marzipan of width $1$, looked at the pastry from every possible angle, and nodded slowly.
“Yes, that’s a perfectly reasonable trivial bundle, or structure sheaf if you want. I’d sell it as $\mathcal{O}_{\overline{\mathbf{Spec}(\mathbb{Z})}}$ if I were you.”
“In your dreams! I’ll simply call this a $1$-pastry, and an $\frac{m}{n}$-pie closed with a $1$-ribbon of marzipan can be ordered from now on as an $\frac{m}{n}$-pastry.”
“I’m afraid this will not suffice,” our Geometer objected, ” you will have to allow pastries having an arbitrary marzipan-width.”
“Huh? You want me to compactify an $\frac{m}{n}$-pie with marzipan of every imaginable width $r$ and produce a whole collection of … what … $(\frac{m}{n},r)$-pastries? What on earth for??”
“Well, take an $\frac{m}{n}$-pastry and try to unravel it.”
Oh, here we go again, I feared.
Whereas Norma’s pies all looked and tasted quite different to most of us, the Geometer claimed they were all the same, or ‘isomorphic’ as he pompously declared.
“Just reverse the operations Norma performed and you’ll end up with a $1$-pie”, he argued.
So Norma took an arbitrary $\frac{m}{n}$-pastry and did perform the reverse operations, which was a lot more difficult that with pies as now the marzipan-bit produced friction. The end-result was a $1$-pie held together with a marzipan-ribbon of width strictly larger or strictly smaller than $1$, but never gave back the $1$-pastry. Strange!
“Besides”, the Geometer added, “if you take two of your pastries, which I prefer to call $\mathcal{L}$ and $\mathcal{M}$, rather than use your numerical system, then their product $\mathcal{L} \otimes \mathcal{M}$ is again a pastry, though with variable marzipan-width.
In the promotional stage it might be nice to give the product for free to anyone ordering two pastries.”
“And how should I produce such a product-pastry?”
“Well, I’m too lazy to compute such things, it must follow trivially from elementary results in Picard-pastry. Surely, our log lady will work out the details in your notation. No doubt it will involve lots of logs…”
And so I did the calculations in my dreams, and I wrote down all formulas in the Double R Diner log-book, for Norma to consult whenever a customer ordered a product, or power of pastries.
A few years ago we had a Japanese tourist visiting Twin Peaks. He set up office in the Double R Diner, consulted my formulas, observed Norma’s pastry production and had endless conversations with our Geometer.
I’m told he categorified Norma’s pastry-bizness, probably to clone the concept to the Japanese market, replacing pastries by sushi-rolls.
When he left, he thanked me for working out the most trivial of examples, that of the Frobenioid of $\mathbb{Z}$…
Added december 2015:
I wrote this little story some time ago.
The last couple of days this blog gets some renewed interest in the aftermath of the IUTT-Mochizuki-Fest in Oxford last week.
I thought it might be fun to include it, if only in order to decrease the bounce rate.
If you are at all interested in the maths, you may want to start with this google+ post, and work your way back using the links curated by David Roberts here.
I had hoped the G+ post below of end december would have been the last I had to say on this (non)issue: (btw. embedded G+-post below, not visible in feeds)
A quick recap :
– in august 2012, Shinichi Mochizuki finishes the fourth of his papers on ‘inter-universal Teichmuller theory’ (IUTeich for the aficianados), claiming to contain a proof of the ABC-conjecture.
– in may 2013, Caroline Chen publishes The Paradox of the Proof, summing up the initial reactions of the mathematical world:
“The problem, as many mathematicians were discovering when they flocked to Mochizuki’s website, was that the proof was impossible to read. The first paper, entitled “Inter-universal Teichmuller Theory I: Construction of Hodge Theaters,” starts out by stating that the goal is “to establish an arithmetic version of Teichmuller theory for number fields equipped with an elliptic curve…by applying the theory of semi-graphs of anabelioids, Frobenioids, the etale theta function, and log-shells.”
[quote name=”Caroline Chen”]
This is not just gibberish to the average layman. It was gibberish to the math community as well.
[/quote]
“Looking at it, you feel a bit like you might be reading a paper from the future, or from outer space,” wrote Ellenberg on his blog.
“It’s very, very weird,” says Columbia University professor Johan de Jong, who works in a related field of mathematics.”
– at the time i found these reactions premature. It often happens that the first version of a proof is not the most elegant or shortest, and i was hoping that Mochizuki would soon come up with a streamlined version, more accessible to people working in arithmetic geometry. I spend a couple of weeks going through “The geometry of Frobenioids 1” and recorded my stumbling progress (being a non-expert) on Google+.
– i was even silly enough to feed almost each and every one of Mochizuki papers to Wordle and paste the resulting Word-clouds into a “Je suis Mochizuki”-support clip. However, in the process I noticed a subtle shift from word-clouds containing established mathematical terms to clouds containing mostly self-defined terms:
.
the situation, early 2015
In recent (comments to) Google+ posts, there seems to be a growing polarisation between believers and non-believers.
If you are a professional mathematician, you know all too well that the verification of a proof is a shared responsability of the author and the mathematical community. We all received a referee report once complaining that a certain proof was ‘unclear’ or even ‘opaque’?
The usual response to this is to rewrite the proof, make it crystal-clear, and resubmit it.
Few people would suggest the referee to spend a couple of years reading up on all their previous papers, and at the same time, complain to the editor that the referee is unqualified to deliver a verdict before (s)he has done so.
Mochizuki is one of these people.
His latest Progress Report reads more like a sectarian newsletter.
There’s no shortage of extremely clever people working in arithmetic geometry. Mochizuki should reach out to them and provide explanations in a language they are used to.
Let me give an example.
As far as i understand it, ‘Frobenioids 1’ is all about a categorification of Arakelov line bundles, not just over one particular number ring, but also over all its extensions, and the corresponding reconstruction result recovering the number ring from this category.
Such a one-line synopsis may help experts to either believe the result on the spot or to construct a counter-example. They do not have to wade through all of the 178 new definitions given in that paper.
