Mathematics world abuzz with a proof of the ABC ConjectureSeptember 11, 2012 9:48 PM   Subscribe

Shinichi Mochizuki believes that he has found a connection between prime numbers by developing a 500 page proof of the ABC conjecture

"If the ABC conjecture yields, mathematicians will find themselves staring into a cornucopia of solutions to long-standing problems," says Dorian Goldfeld, a mathematician at Columbia University in New York,
posted by zeoslap (53 comments total) 31 users marked this as a favorite

Like what?

Does this represent a threat to the RSA cipher, for instance?
posted by Chocolate Pickle at 9:53 PM on September 11, 2012

Are there any mathematicians on the plane? Is there a mathematician on this airplane? If you are a mathematician please press the flight attendant call button immediately.
posted by TwelveTwo at 10:02 PM on September 11, 2012 [21 favorites]

I'd like to think anyone can stare at the cornucopia, but only mathematicians can pull out the food and distribute it. Or maybe they just spill it on the table triumphantly.
posted by michaelh at 10:04 PM on September 11, 2012 [3 favorites]

Initial response and discussion on this (from 9/3). I found the Barry Mazur link helpful for understanding the statement of the theorem.
posted by klausman at 10:04 PM on September 11, 2012

RSA is safe. This proof, if valid, proves that some properties of prime numbers we think are true, are actually true. If those properties were useful in attAcking RSA, nobody would wait for a proof; they would just try it.

The long-standing problems are other unanswered questions in mathematics. For example proving this also proves FLT, though that has already been proven in other ways.
posted by jeffamaphone at 10:04 PM on September 11, 2012 [3 favorites]

michaelh: " Or maybe they just spill it on the table triumphantly."

They approach the cornucopia, take a careful look, step back, and proudly proclaim "We could spill this!"
posted by schmod at 10:09 PM on September 11, 2012 [16 favorites]

There's a sense of the amazing strangeness of mathematics in Simon Singh's book "Fermat's Enigma", which is about the discovery of Fermat's Last theorem. I just re-read it, and it's a pretty approachable book which I highly recommend.

I mention it because the article here describes how this proof can be applied to many problems including Fermat's Theorem, not just the ABC conjecture. That would be prodigious.

Fermat's Theorem was solved, finally, but finding a relationship between two seemingly unrelated branches of mathematical research. That discovery is part of an effort to discover more such connections.

It is a wondrous thing. Math would be totally valid in the universe if life and consciousness had never evolved.
posted by bc_fred at 10:09 PM on September 11, 2012 [4 favorites]

I am no jedi. But I know there are jedi in the blue. Hopefully one will speak up because this appears to be quite a revelation. From what I read, I think this has serious implications in predicting (or verifying) prime numbers. And that should have some implication in cryptography, yes?
posted by chemoboy at 10:10 PM on September 11, 2012

And that should have some implication in cryptography

This isn't my strong suit, but my understanding of the role primes play in crypto is that it's factoring non-primes to find two primes that are factors, that is the computationally hard thing that secures RSA and other algorithms. Finding primes themselves doesn't change that.
posted by fatbird at 10:13 PM on September 11, 2012

jeffamaphone For example proving this also proves FLT, though that has already been proven in other ways.

Ah, but is it elegant and small enough to not quite fit within the margin of a textbook?
posted by aeschenkarnos at 10:36 PM on September 11, 2012

Math would be totally valid in the universe if life and consciousness had never evolved.

Sure, but life and consciousness were totally valid long before math got out of bed.
posted by weapons-grade pandemonium at 10:46 PM on September 11, 2012 [4 favorites]

The proof is spread across four long papers, each of which rests on earlier long papers. “It can require a huge investment of time to understand a long and sophisticated proof"

He has developed techniques that very few other mathematicians fully understand and that invoke new mathematical ‘objects’ — abstract entities analogous to more familiar examples such as geometric objects, sets, permutations, topologies and matrices. “At this point, he is probably the only one that knows it all.”

posted by stbalbach at 10:47 PM on September 11, 2012

Shinichi Mochizuki entered Princeton at 16, graduated in three years, and earned his Ph.D by age 22. He's never been married, which seems an odd thing to put on your CV, but then it seems brilliant mathematicians are seldom normal in habit or disposition.

If you look through the Past And Current Research document on his website, it appears he's been developing the tools and ideas for this project since at least the summer of 2000. So this proof has been twelve long years in the works.
posted by dephlogisticated at 10:49 PM on September 11, 2012 [5 favorites]

Something tells me that it's going to take a while before "Inter-universal Teichmuller theory" achieves the same popular usage that "modulo" has.
posted by benito.strauss at 10:52 PM on September 11, 2012 [1 favorite]

His website is awesomely terrible. I mean, it's bad in 1996 terms.
posted by fatbird at 11:20 PM on September 11, 2012 [8 favorites]

RSA is safe.

As is Shinichi Mochizuki.
posted by mr_roboto at 11:32 PM on September 11, 2012 [18 favorites]

How to know someone's really, really good: when mathematicians say that his or her papers feel like they came from aliens.
posted by Malor at 11:33 PM on September 11, 2012 [2 favorites]

Math would be totally valid in the universe if life and consciousness had never evolved.

I think I'm going to dispute this, but not strongly enough to go into a derail over it. I think it'd mostly be me quibbling over terms anyway.
posted by JHarris at 12:10 AM on September 12, 2012 [6 favorites]

I was looking at his "Past and Current research" when I saw just about the only thing I recognize and understand in that PDF -- a Gaussian Integral. But then he said something about his work on Elliptic Curves as a "scheme-theoretic discretization" of said integral. And though I know what "scheme," "theoretic," and "discretization" all mean, I have only the vaguest, foggiest idea what they might mean together in that order.
posted by chimaera at 12:13 AM on September 12, 2012 [1 favorite]

Dammit! He just beat me too it.
posted by From Bklyn at 12:32 AM on September 12, 2012 [7 favorites]

I have learnt that, especially on the internet it is to be skeptical about claimed proofs for longstanding mathematics problems that I have no hope of understanding. So, this is definitely cool but call us again in 2 years?
That said a) everyone should click the first link because this guy has an amazing website.
b) the fact he is a respected and renowned academic makes me think this is much more likely than say that P/NP proof that everyone was excited by a while ago.
posted by Another Fine Product From The Nonsense Factory at 1:03 AM on September 12, 2012 [1 favorite]

Let me report on my progress in writing up the series of papers on
IUTeich. The task of writing up and making a final check on the
contents of the series of papers is proceeding smoothly, and it
appears that the four papers in the series will come to a total of
stated that I hope to finish this project by the summer of 2012;
although it is not clear whether or not I will be able to meet
this summer deadline, I do hope to finish by the latter half of the
year 2012. (Nevertheless, of course, I am not able to guarantee
anything at the present time.)

No wonder this guy's never been married. He's, like, totally noncommittal .
posted by armoir from antproof case at 1:23 AM on September 12, 2012

That the theorem is true isn't useful for attackers (as pointed out, if it was useful, we'd just presume it to be true) but there might be interesting math buried in the proof that could improve the factorization of large two-prime composites (which would break RSA).
posted by effugas at 2:16 AM on September 12, 2012

So, this is definitely cool but call us again in 2 years?

The Riemann hypothesis - which also contains implications about the location of prime numbers - is about 150 years old and (IIRC) still not proven or disproven. I wonder if this ABC conjecture will have any bearing on it.
posted by three blind mice at 3:31 AM on September 12, 2012

The long-standing problems are other unanswered questions in mathematics. For example proving this also proves FLT, though that has already been proven in other ways.

Damn, I thought for a second we had Faster Than Light travel.
posted by wilko at 3:34 AM on September 12, 2012 [4 favorites]

Simon Singh's book "Fermat's Enigma"

Slight tangent here, but he also wrote the awesome The Code Book, which goes into detail not only about things like the Enigma machine and Bletchley Park, but also the deciphering of Egyptian hieroglyphics and Linear B. It's well worth reading.
posted by Mr. Bad Example at 3:52 AM on September 12, 2012

A pretty good post about the ABC problem.
posted by DU at 3:58 AM on September 12, 2012 [3 favorites]

> I'd like to think anyone can stare at the cornucopia, but only mathematicians can pull out the food and distribute it. Or maybe they just spill it on the table triumphantly.

Theory: We have investigated this cornucopia and establish a high probability that any hypothetical attempts at spillage would be successful.
Applied: We have parameterized the consequences of a spillage, if attempted, and pending grants approval we will prepare controls and embark on an observed spillage attempt in 2018.
Practical: Cornucopia spillage associates needed immediately for seasonal work beginning immediately. Good pay, convenient hours!
posted by ardgedee at 4:23 AM on September 12, 2012 [1 favorite]

Then, assuming the abc conjecture is true, we have these results:

How many hits have h ≥ 1? Answer: infinitely many.
How many hits have h > 1? Answer: finitely many.
How many hits have h = 1? Answer: zero.

need.. more.. coffee..
posted by charlie don't surf at 4:37 AM on September 12, 2012 [4 favorites]

Initial response and discussion on this (from 9/3).

That's my blog, by the way! I'm a number theorist and I've been thinking about ABC and related questions for a long time, so I'm happy to chat about it here.
posted by escabeche at 4:45 AM on September 12, 2012 [6 favorites]

Does this sort of math actually do anything other than solve a puzzle? I mean, is this proof anything more than a giant riddle? Will it lead to cold fusion or Faster Than Light travel? I ask not to snark but as a mathematical ignoramus.
posted by Hobgoblin at 5:12 AM on September 12, 2012

He's never been married, which seems an odd thing to put on your CV, but then it seems brilliant mathematicians are seldom normal in habit or disposition.

In some places, including apparently Japan, putting your marital or family status on your CV is normal. It doesn't seem like a stretch that specifying you've never been married is common practice, though obviously someone familiar with Japanese CVs could tell us.
posted by hoyland at 5:28 AM on September 12, 2012

Does this sort of math actually do anything other than solve a puzzle?

Prime numbers are pretty useful, yes. Also, "what is the use of a newborn child?"
posted by DU at 5:36 AM on September 12, 2012 [4 favorites]

The Riemann Hypothesis

That always makes me think of a Robert Ludlum novel.
posted by Egg Shen at 5:43 AM on September 12, 2012 [7 favorites]

charlie don't surf, you should note that the reason the passage you link to seems so paradoxical is that it's wrong -- the author has corrected it in the current version of the post.
posted by escabeche at 5:55 AM on September 12, 2012 [1 favorite]

That always makes me think of a Robert Ludlum novel.

The Riemann Legacy didn't even feature Riemann himself though! I mean, I liked the new character David Hilbert but it was still sorta bullshit.
posted by kmz at 6:21 AM on September 12, 2012 [4 favorites]

Does this sort of math actually do anything other than solve a puzzle?

I'm not sure if it's your intent here, but often the "what's it for?" question (directed at math or at other such pursuits) is secretly "can someone get rich from this?". If not, the question is more or less equivalent to "What is music for?" or "Why have non-procreative sex?" -- lots of folks spend lots of energy on things deemed "impractical", but I think asking what it's for is, depending on the intent of the question, basically unfair, or at least gives way too much credence to shared fictions that are fundamentally kind of bizarre; insurance salesfolk, say, or professional athletes, are rarely asked what the objects of their efforts are "for".

If there weren't riddles and poetry and art and games and idle discussion and music -- in short, a whole cornucopia of "impractical" things, including pure math, that folks do -- there'd be very litte point in any of the practical things -- practical as in basic human physical needs -- that, say, "cold fusion" would help to secure.

That said, part of being a mathematician is to grin and bear this question, even though it's less often asked of people whose vocation is equally arcane*. Part of being a good mathematician is to answer the question in such a way that the excitement and narrative tension and emotional investment and joy of math comes across, because these things are the answer to the question, just as they are for numerous other pursuits.

*[How come people who do extremely rarefied physics don't have to put up with it? It's not like "explaining the universe" on a level nobody experiences viscerally is any less arcane than anything in pure math. Plenty of abstract things feel more "real" are more real to me, and any other non-physicist, than abstract descriptions of ostensibly "physical" things.]

In other news about well-known-in-the-right-circles conjectures now having claimed proofs, Justin Moore seems to have shown that Thompson's group F is amenable.
posted by kengraham at 7:41 AM on September 12, 2012 [9 favorites]

Thanks, kengraham. It seems that my question hits you somewhat like "Did you play basketball?" annoys me (I'm tall). I apologize. It's just that I never "got" stuff like this, but I'm glad some folks do.
posted by Hobgoblin at 8:04 AM on September 12, 2012

So inter-universal Teichmuller theory is a mathematical theory which describes the geometric structure of a recursive Level IV multiverse? Did I get that correct or am I just tripping balls over here?
posted by AElfwine Evenstar at 8:18 AM on September 12, 2012

Awesome. Now we can finally leave the Cube.

(although, let's face it, only Mochizuki will actually get to leave the Cube. The rest of us will be undone by our various human failings)
posted by Afroblanco at 8:19 AM on September 12, 2012 [3 favorites]

posted by fearfulsymmetry at 8:23 AM on September 12, 2012

There's a great discussion of this over at mathoverflow. As a non-math person, the comment I found most helpful was explaining why we're not going to get much useful discussion of this for a while:
That being said, although the new ABC developments are potentially very exciting, and it is understandable to want to "share in the excitement", for reasons specific to this situation it seems to be much too premature to ask for a sketch on MO or in a blog of Mochizuki's vision/proof with an expectation of insight into the new work. Let me try to indicate why this is the case.

As has been explained clearly by JSE elsewhere, there are plenty of top experts in arithmetic geometry who are presently struggling to get even a small handle on what is really going on in Mochizuki's papers (due entirely to the experts' lack of prior study of these ideas; Mochizuki's writing is extremely precise, detailed, thorough, and full of intuitive asides!). So the situation seems to be rather different from that of other tremendous advances in recent decades (by Perelman, Faltings, Wiles, etc.), for which the deep new work took place within a context that was already somewhat familiar to a good-sized community of experts in the field (who could then use their experience and expertise to quickly disseminate a "bird's eye view" to others of some of the key new ideas).

Because of the rather unique circumstances of this case, as just indicated, I believe that quid's initial urging of patience (if one isn't going to be directly engaged with the struggle to read the actual papers and the prior work upon which they depend) is appropriate.
From a purely linguistic perspective, though, I recommend reading the comment by Professor Minhyong Kim, one of the aforementioned experts in arithmetic geometry who seems to have written important papers on the subject. His first paragraph says "as with many of my answers, there's the danger I'm just regurgitating common knowlege in a long-winded fashion, in which case, I apologize." The game is to count the number of technical words he uses once in his 2,600-word comment and then never mentions again:
The beginning of the investigation is indeed the function field case (over C, for simplicity), where one is given a family
[math]
of elliptic curves over a compact base, best assumed to be semi-stable and non-isotrivial. There is an exact sequence
[math]
which is moved by the logarithmic Gauss-Manin connection of the family. (I hope I will be forgiven for using standard and non-optimal notation without explanation in this note.) That is to say, if S⊂B is the finite set of images of the bad fibers, there is a log connection
[math]
which does not preserve ωE. This fact is crucial, since it leads to an OB-linear Kodaira-Spencer map
[math]
and thence to a non-trivial map
[math]
From this, one easily deduces Szpiro's inequality:
[math]
And then it goes on like that for another eight pages with barely any repeats! I'm not saying this is beyond the reach of mortals, I'm just saying the sheer density of information is amazing. It's like a Martian who learns basic English, lands on planet Earth and asks for a quick summary of the significance of Das Kapital. Where do you start?
posted by jhc at 9:05 AM on September 12, 2012 [7 favorites]

Can someone please explain this to me using animated gifs?
posted by the painkiller at 9:08 AM on September 12, 2012 [5 favorites]

the painkiller: Can someone please explain this to me using animated gifs?
Count all the cat videos on Youtube. Now count all the snarky comments on Youtube. Multiply them together, creating a gigantic Youtube channel of cat videos with nothing but snarky comments.

Wait, I forgot what I was trying to do.
posted by IAmBroom at 9:28 AM on September 12, 2012 [3 favorites]

So inter-universal Teichmuller theory is a mathematical theory which describes the geometric structure of a recursive Level IV multiverse? Did I get that correct or am I just tripping balls over here?

And what does this have to do with primes???? I'm more confused than ever.
posted by AElfwine Evenstar at 10:47 AM on September 12, 2012

Does this sort of math actually do anything other than solve a puzzle?

Even if it doesn't right now, as it said at the top "Mathematicians will find themselves staring into a cornucopia of solutions to long-standing problems." Some of these problems may be immediately applicable to problems in applied mathematics.

And even if that weren't true, then you can think of it as a tool. Pure math develops the tools, and applied math uses them. Many applications of applied math used tools from pure math that were originally developed outside of the context of a real-world application. A good example is the case of complex/imaginary numbers. The idea of a construct for the square root of -1 was originally an exercise in pure math, but without this tool you'd never have the math for aerodynamics, for example.
posted by Hubajube at 12:51 PM on September 12, 2012 [3 favorites]

As has been explained clearly by JSE elsewhere,

That's me, too, actually!
posted by escabeche at 2:07 PM on September 12, 2012 [1 favorite]

Speaking as a total lay person here, but to chime in on the "but what is it good for" question, math is full of theoretical and recreational topics that turned out to be important later on.

Take imaginary numbers for example: take a thing known to be impossible, finding the square root of -1. Pretend it is possible, and assign a constant to be equal to the answer. It turns out that answer is useful, as a symbol, in a lot of ways. Both the theory of relativity and quantum mechanics rely on imaginary numbers, and they are quite real. It's silly to ask what the point is, because we have no way of knowing for sure what math will be important to later generations.

And there's so much novel math being used to arrive at this solution that it may well be the case that the solution to this problem turns out to be less important than the means Mochizuki used to arrive at it.
posted by JHarris at 3:21 PM on September 12, 2012 [1 favorite]

charlie don't surf, you should note that the reason the passage you link to seems so paradoxical is that it's wrong -- the author has corrected it in the current version of the post.

Well that's a huge relief. I'm working with a woman who has an MS in Math, you know the type who can recite 30 digits of Pi from memory. I wrote down those inequalities for her and briefly explained the background. First I got the Spock Eyebrow, then she got a faraway look in her eyes as she contemplated them, then a look of scorn and the query "what what could that possibly mean?" We are working on logic papers that are so badly written, our logical faculties are degrading, and we were both prepared to believe this was true.
posted by charlie don't surf at 3:54 PM on September 12, 2012

it may well be the case that the solution to this problem turns out to be less important than the means Mochizuki used to arrive at it.

Emphatically this, in general, even if it doesn't turn out to be the case in this situation. I'm not a number theorist and am far from being able to speculate intelligently about other applications of machinery introduced by Mochizuki, but aiming at specific goals, and developing machinery in order to progress toward those goals, drives a lot of mathematical development, even when the goals are not reached. When they are, the accomplishment is a cause for celebration for everyone, but for those in the trenches in that particular subfield, a major benefit is retracing the road to the solution and looking for unexplored forks where the alternative route has a reasonable chance of being well-paved, as it were. This stands to reason, especially for long-standing conjectures, because the statement of the conjecture is not a surprise, but the proof necessarily is: everyone's had time to think about the implications of the conjecture, but not necessarily about the implications of the (hitherto unknown) ingredients of the proof.

Of course it all gets very complicated, because theorems can't readily be slotted into "road" or "terminus" pigeonholes. However, JHarris, you've drawn attention to the important fact that the goal is to understand "stuff" and how the aggregate of all the "stuff" relates to itself. Proving theorems is a way of making very precise what is understood, and articulating the understanding, but often (though not always) the understanding is concentrated in the proof -- which almost always illuminates some of the inner workings of the "stuff" -- more than in the bare statement of the theorem.

(This is calling to mind this essay by the late and extremely creative William Thurston, which I recommend highly to folks interested in the "What's it for?" question. A little formal math training helps to understand some of his examples, but it's widely appreciate-able.)

Maybe a more experienced mathematician would take issue with some of this, or have something to add. I'm pretty new to the game, just starting to stray from a small neighborhood of my thesis, so I don't even have evidence that particular philosophical or social statements about one field of math hold in another (and I even have evidence that they in general don't). If it's not too deraily, maybe there is some mathematical eminence grise around here who can tell us their response to the "what's it for?" question.
posted by kengraham at 4:22 PM on September 12, 2012

I was thinking about this while I fell asleep last night, and I started imagining a pineapple with sequences of numbers spiralling down it, with new spirals being added gradually as the pineapple broadened. The first spiral consisted of multiples of 2, the next had multiples of 3, and so on and on. The head of every spiral was a prime number, of course, and just as I fell asleep I was sure that I had worked out a method to calculate prime numbers - just look at the pineapple! The idea still seems vast and portentous as I type this but I don't think it actually makes sense. But still - look at the pineapple!
posted by Joe in Australia at 7:03 PM on September 12, 2012 [3 favorites]

i think you've accurately captured the psychedelic experience, Joe In Australia.
posted by RTQP at 8:05 PM on September 12, 2012

In some places, including apparently Japan, putting your marital or family status on your CV is normal. It doesn't seem like a stretch that specifying you've never been married is common practice, though obviously someone familiar with Japanese CVs could tell us.

Yup, this is a cultural thing.
posted by smorange at 11:41 AM on September 13, 2012

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