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Topic: Super Imaginary Numbers - What are they and how do they help?
Replies: 9   Last Post: Oct 26, 2012 9:53 AM

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Paul A. Tanner III

Posts: 5,920
Registered: 12/6/04
Re: Super Imaginary Numbers - What are they and how do they help?
Posted: Sep 20, 2012 12:17 PM
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> ============================
> WHOLE numbers, addition and multiplication are among
> the first things schoolchildren learn, but a new
> mathematical proof shows that even the world's best
> minds have plenty more to learn about these seemingly
> simple concepts.
> Shinichi Mochizuki of Kyoto University in Japan has
> torn up these most fundamental of mathematical ideas
> and reconstructed them as never before. The result is
> a fiendishly complicated proof for the 27-year-old
> "ABC conjecture" - and an alternative mathematical
> universe that should prise open many other
> outstanding enigmas.
> Not only that, Mochizuki's work also offers an
> alternative way to prove Fermat's last theorem, a
> long-standing problem that became one of the most
> famous results in the history of mathematics when it
> was proved in 1993
> =============
> The above extract is from a 15 September 2012 New
> Scientist cover story titled, Super Imaginary
> Numbers.
> While I read the full article, I neither understand
> Shinichi Mochizuki's work nor the magnitude of his
> breakthroughs.
> Would someone please explain this all to me in a way
> that makes some sense to a naïve mathematician?
> Thank you
> Jonathan Crabtree
> Here are the original papers currently being
> studied...
> sal%20Teichmuller%20Theory%20I.pdf
> sal%20Teichmuller%20Theory%20II.pdf
> sal%20Teichmuller%20Theory%20III.pdf
> sal%20Teichmuller%20Theory%20IV.pdf
> You can register free and read the full article at
> ndish-abc-proof-heralds-new-mathematical-universe.html
> ------- End of Forwarded Message

This claimed proof of the abc conjecture at this time has not been verified to be a proof. There was a claimed proof 5 years ago that was shown to be not a proof. Here is a good article on all this:

"Not Even Wrong
Proof of the abc Conjecture?"


- -------------
Jordan Ellenberg at Quomodocumque reports here on a potential breakthrough in number theory, a claimed proof of the abc conjecture by Shin Mochizuki. More than five years ago I wrote a posting with the same title, reporting on a talk by Lucien Szpiro claiming a proof of this conjecture (the proof soon was found to have a flaw). One change over the last five years is that now there are excellent Wikipedia articles about mathematically important questions like this conjecture, so you should consult the Wikipedia article for more details on the mathematics of the conjecture. To get some idea of the significance of this, that article quotes my colleague and next-door office neighbor Dorian Goldfeld describing the conjecture as "the most important unsolved problem in Diophantine analysis", i.e. for a very significant part of number theory.

Jordan is an expert of this kind of thing, and he has some of the best mathematicians in the world (Terry Tao, Brian Conrad and Noam Elkies) commenting, so his blog is the place to get the best possible idea of what is going on here. After consulting a couple experts, it looks like this is a very interesting and possibly earth-shattering moment for this field of mathematics. In the case of the Szpiro proof, the techniques he was using were relatively straightforward and well-understood, so experts very quickly could read through his proof and identify places there might be a problem. This is a very different situation. What Mochizuki is claiming is that he has a new set of techniques, which he calls "inter-universal geometry", generalizing the foundations of algebraic geometry in terms of schemes first envisioned by Grothendieck. In essence, he has created a new world of mathematical objects, and now claims that he understands them well enough to work with them consistently !
and show that their properties imply the abc conjecture.

What experts tell me is that, very much unlike the case of Szpiro's proof, here it may take a very long time to see if this is really a proof. They can't just rely on their familiarity with the usual scheme-theoretic world, but need to invest some serious time and effort into becoming familiar with Mochizuki's new world. Only then can they hope to see how his proof is supposed to work, and be able to check carefully that a proof is really there, not just a mirage. It's important to realize that this is being taken seriously because such experts have a high opinion of Mochizuki and his past work. If someone unknown were to write a similar paper, claiming to have solved one of the major open questions in mathematics, with an invention of a strange-sounding new world of mathematical objects, few if any experts would think it worth their time to figure out exactly what was going on, figuring instead this had to be a fantasy. Even with Mochizuki's high reputation, few were willin!
g in the past to try and understand what he was doing, but the abc conjecture proof will now provide a major motivation.

Mochizuki has been at this for quite a while. See this page for some notes from him about how he has been pursuing this project in recent years. This page has notes from lectures he has given on the topic, starting in 2004 with A Brief Introduction to Inter-universal Geometry. For the proof itself, see here, but this is the fourth in a sequence of papers, so one probably needs to understand parts of the other three too.

Update: Barry Mazur has recently made available his 1995 expository article on the abc conjecture, entitled Questions about Number.
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