For those who don't know, Jordan Ellenberg (the author of this blog) is a relatively famous and extremely well-regarded math professor at the University of Wisconsin. I was fortunate enough to have him on my Ph.D. thesis committee.
Terence Tao (see the blog comments), as many HN readers know, is a Fields medalist and quite brilliant. Brian Conrad (again, see the blog comments) is a professor at Stanford who is incredible, he knows algebraic number theory as well as anyone, and he has a great nose for bullshit (as well as no patience for it).
This paper is outside my immediate area, so I can't comment scientifically. But I can speak for the culture of mathematics. Evidently, this paper is not a bunch of obvious BS. The paper could be correct, perhaps not. It could be like Wiles's first proof of Fermat's Last Theorem, which contained a fairly serious gap, which was nevertheless later fixed.
In short, outstanding people are taking this seriously.
I don't understand the new paper, but here is the ABC conjecture:
Among triples A + B = C, with no common factors between the three, for any given epsilon > 0, there only finitely many with C > R^(1+epsilon); here, R is the "radical" of A,B,C, which is the product of the union of all primes that divide A, B, or C.
The difference between R and the product ABC is that we take out any higher powers of the primes. My intuitive interpretation is something like "in almost every irreducible A+B=C, we almost have C < product(primes(A,B,C))."
Fermat's last theorem (FLT) is a major consequence. Actually, the only proof I've seen connecting the two shows that FLT can have at most finitely many solutions. Here's a useful survey paper:
My intuitive interpretation is something like "in almost every irreducible A+B=C, we almost have C < product(primes(A,B,C))."
^^Another intuitive version the conjecture is: "if A+B=C (with no common prime factors), then it is very difficult for A, B, and C to be divisible by a prime raised to a high power." For instance, if A was divisible by 2^1000, B was divisible by 3^1000, and C by 5^1000, then these prime factors together would contribute only 235 = 30 to the "radical" R, which could allow C to perhaps be much bigger than R. This can't happen "too often" (well, maybe some finite number of times).
It was proved by Tijdeman in 1976 that the equation A + 1 = C has only finitely many solutions where A and C are both perfect powers. Think about this for a minute: they could be perfect squares, cubes, 4th powers... and perhaps A = something^1000 and C = somethingelse^1001. Probably the only obvious example of this is 2^3 + 1 = 3^2.
The ABC conjecture, if true, implies that for any positive integer k, the equation A + k = C has finitely many solutions where both A and C are perfect powers.
This paper is 4th in a sequence. There is a mistake on page 59 of the third paper according to some other math blogs. Not sure how this may impact other results subsequently.
I've just looked at this and my face melted, so could someone say why this is important? It certainly looks impressive but it's gone over my head some what.
Wow. Not only has he just demolished the ABC conjecture as a mere afterthought to his pathbreaking research into inter-universal Teichmüller theory... he's also an awesome Geocities-style web designer:
This is unfair. Yes, his website looks ridiculous, but it is not part of the culture of academia to care much (and I think such websites are especially popular in East Asia). Very few academics spend much time or effort on the design of their website (or are impressed with the same from others). Also keep in mind he is not a native speaker of English.
Laugh at his website if you want, but please do not infer that he's some eccentric who has no idea what he's talking about.
Clearly he isn't a crank, and I didn't mean for a minute to imply that he was one. I just found the design aesthetics of his web page to be an... interesting coincidence.
But that said -- just trust your gut instincts here. There's a long history in mathematics of brilliant people going off into the dark for frighteningly long periods of time, and coming back with announcements of major results that turn out to be... interesting, but ultimately to contain irreparable gaps. And the fact that nobody -- not even people like Tao or Conrad -- seems to know what exactly he's talking about (even though they obviously respect the guy) doesn't sound too encouraging, either.
Tao doesn't know exactly what Mochizuki is talking about because it is not in any of the (many) areas that Tao is an expert in. Even Fields medalists have their limits.
The people that do work in his area don't know exactly what he's talking about because he's been going in a very deep and difficult direction. No one has been following, because they've been waiting to see if he finds anything in there. There was a good description at "Not Even Wrong":
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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 willing in the past to try and understand what he was doing, but the abc conjecture proof will now provide a major motivation.
> I didn't mean for a minute to imply that he was one
Ok, fair enough.
But I am trusting my gut instincts here. Not so much having read the paper, but reading the way Ellenberg, Conrad, and Tao discuss the paper. To argue by analogy, if PG and Mark Zuckerberg both came out and said they were both intrigued by a new startup, that certainly wouldn't mean it was a shoo-in for success, but I imagine that VC's would be lining up to take that gamble.
Please also keep in mind that Perelman's proof of the Poincare conjecture confused the hell out of everyone, and I think it was a couple of years before experts came to a consensus that it was correct.
Of course, your pessimism could well be accurate. In any case, the consensus view in the mathematical community will be that ABC "may have been" proved (unless a mistake is found immediately). For better or worse, we are pretty damn cautious.
The title seems apt actually. I think 'universe' here refers to the concept of a set so large that it is closed under all set theoretic operations and since mathematical operations reduce to set theoretic operations (in the usual foundation scheme) one can start with some elements in this universe set and perform any arbitrary manipulation and the output will still be inside this set. http://en.wikipedia.org/wiki/Grothendieck_universe
Inter-universal presumably refers to geometrical statements which hold
across different such universes. At least, that's my understanding. The paper is well beyond my knowledge.
It's generally the case and expectation that really hard problems need to be solved by inventing new theories and new ways of doing math, and they're worth encouraging people to solve because of the discoveries along the way, not because of the truth or falsehood of the conjecture itself, which is mostly a curiosity.
My father is also a math professor, but not a web design expert. At age 13 I made him a now somewhat embarrassing website for one of his trips, that looks pretty similar to that one. My dad still has it posted (17 years later). It's not because he likes the aesthetics of a 13 year old girl, he likes that the information is accessible and is a path of least resistance kind of guy.
I have noticed that many of the best academics seem to like having horrendously appalling websites that look like they were built for NCSA Mosaic. I suspect that there is an in-joke I am missing. They probably have a contest to see who can construct the worst one.
Terence Tao (see the blog comments), as many HN readers know, is a Fields medalist and quite brilliant. Brian Conrad (again, see the blog comments) is a professor at Stanford who is incredible, he knows algebraic number theory as well as anyone, and he has a great nose for bullshit (as well as no patience for it).
This paper is outside my immediate area, so I can't comment scientifically. But I can speak for the culture of mathematics. Evidently, this paper is not a bunch of obvious BS. The paper could be correct, perhaps not. It could be like Wiles's first proof of Fermat's Last Theorem, which contained a fairly serious gap, which was nevertheless later fixed.
In short, outstanding people are taking this seriously.