5 ms·

Shin Mochizuki has released his long-rumored proof of the ABC conjecture

70 points by bdr 14y ago

nicholassmith 14y ago
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.
- aaronbrethorst 14y ago
  Wikipedia has somewhat intelligible (to me) information on the topic: http://en.wikipedia.org/wiki/Abc_conjecture#Some_consequences http://en.wikipedia.org/wiki/Abc_conjecture#Some_consequence...
  - TallboyOne 14y ago
    I still don't particularly understand. So basically this really helps out number theory? Where does that lead to?
toomuchcoffee 14y ago
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: http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html Clearly this man is a force to be reckoned with.
- caseydurfee 14y ago
  "Inter-universal Geometer" is a pretty badass job title.
  - impendia 14y ago
    He presumably translated his own webpage into English. This should not be taken for evidence that he's a crackpot.
    - nu23 14y ago
      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 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.
- impendia 14y ago
  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.
  - aamar 14y ago
    He has a solid command of English. I was in his (very difficult) class several years back. He was maybe a little quirky, but he was not a crackpot. Not surprised to see his name on something--potentially--big like this.
  - toomuchcoffee 14y ago
    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.
    - impendia 14y ago
      > 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.
- geofft 14y ago
  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.
- zvrba 14y ago
  He's Japanese. When I visited Japan a few years ago, it looked like cartoon-land to me.
  - zvrba2 14y ago
    your name seems to imply that you don't know how to spell.
- ktizo 14y ago
  By the same standard you wouldn't pay much attention to Freeman Dyson - http://www.sns.ias.edu/~dyson/ http://www.sns.ias.edu/~dyson/ - or John Nash - http://web.math.princeton.edu/jfnj/ http://web.math.princeton.edu/jfnj/ - either. 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.
  - toomuchcoffee 14y ago
    I actually like the austere minimalism of JFN's page; it practically exudes sprezzatura, we might say. Reminds me of some of his early papers, in that sense.
    - ktizo 14y ago
      Just looked up "sprezzatura". Is a really cool word, thanks for that. :)
- smokey_the_bear 14y ago
  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.
impendia 14y ago
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.
- nthitz 14y ago
  Had him for CS 240 Intro to Discrete Mathematics, I believe. Great prof and a great guy. He knows his stuff so this is pretty cool news!
  - ryanfox 14y ago
    He was my favorite professor and that was my favorite class in college. I can't believe I discovered he has blog on hacker news, of all places. If any undergrads at Madison are reading this, take his class. You definitely won't regret it.
- djao 14y ago
  Also among the blog commenters is NDE (Noam Elkies), who was my Ph.D advisor.
  - djing0 14y ago
    congratulations
tylerneylon 14y ago
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: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5980 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5... (Click on the PDF icon to download the paper for free.)
- prof_despistado 14y ago
  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.
sadfaceunread 14y ago
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.
- ninguem2 14y ago
  If you are referring to comment #15 in http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rum... this was apparently a joke in bad taste. See the next comment.
soseng 14y ago
If anyone's wondering. This is Shin Mochizuki's page: http://www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html http://www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html
deleted 14y ago
[deleted]

70 points by bdr 14y ago

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.

aaronbrethorst 14y ago
Wikipedia has somewhat intelligible (to me) information on the topic: http://en.wikipedia.org/wiki/Abc_conjecture#Some_consequences http://en.wikipedia.org/wiki/Abc_conjecture#Some_consequence...
- TallboyOne 14y ago
  I still don't particularly understand. So basically this really helps out number theory? Where does that lead to?

toomuchcoffee 14y ago

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: http://www.kurims.kyoto-u.ac.jp/~motizuki/top-english.html Clearly this man is a force to be reckoned with.

caseydurfee 14y ago
"Inter-universal Geometer" is a pretty badass job title.
- impendia 14y ago
  He presumably translated his own webpage into English. This should not be taken for evidence that he's a crackpot.
  - nu23 14y ago
    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 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.
impendia 14y ago
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.
- aamar 14y ago
  He has a solid command of English. I was in his (very difficult) class several years back. He was maybe a little quirky, but he was not a crackpot. Not surprised to see his name on something--potentially--big like this.
- toomuchcoffee 14y ago
  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.
  - impendia 14y ago
    > 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.
geofft 14y ago
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.
zvrba 14y ago
He's Japanese. When I visited Japan a few years ago, it looked like cartoon-land to me.
- zvrba2 14y ago
  your name seems to imply that you don't know how to spell.
ktizo 14y ago
By the same standard you wouldn't pay much attention to Freeman Dyson - http://www.sns.ias.edu/~dyson/ http://www.sns.ias.edu/~dyson/ - or John Nash - http://web.math.princeton.edu/jfnj/ http://web.math.princeton.edu/jfnj/ - either. 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.
- toomuchcoffee 14y ago
  I actually like the austere minimalism of JFN's page; it practically exudes sprezzatura, we might say. Reminds me of some of his early papers, in that sense.
  - ktizo 14y ago
    Just looked up "sprezzatura". Is a really cool word, thanks for that. :)
smokey_the_bear 14y ago
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.

impendia 14y ago

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.

nthitz 14y ago
Had him for CS 240 Intro to Discrete Mathematics, I believe. Great prof and a great guy. He knows his stuff so this is pretty cool news!
- ryanfox 14y ago
  He was my favorite professor and that was my favorite class in college. I can't believe I discovered he has blog on hacker news, of all places. If any undergrads at Madison are reading this, take his class. You definitely won't regret it.
djao 14y ago
Also among the blog commenters is NDE (Noam Elkies), who was my Ph.D advisor.
- djing0 14y ago
  congratulations

tylerneylon 14y ago

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: http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5980 http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.24.5... (Click on the PDF icon to download the paper for free.)

prof_despistado 14y ago
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.

sadfaceunread 14y ago

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.

ninguem2 14y ago
If you are referring to comment #15 in http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rum... this was apparently a joke in bad taste. See the next comment.

soseng 14y ago

If anyone's wondering. This is Shin Mochizuki's page: http://www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html http://www.kurims.kyoto-u.ac.jp/~motizuki/news-japanese.html

deleted 14y ago

[deleted]