3 ms·

Ask HN: Understanding inter-universal Teichmüller theory?

13 points by valera_rozuvan 11y ago

Hi! My name is Valera Rozuvan, and I want to understand inter-universal Teichmüller theory. For those of you who never heard of this theory, please take a look at the Wikipedia's entry https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory . Why is this important? This theory might have a proof for the abc conjecture. My background is in physics, optics. I finished the Taras Shevchenko National University of Kyiv. I am a hobbyist mathematician. What should be my path of learning that will lead to understanding this theory? Any advice on specific texts, courses, online resources, etc. will be appreciated!

valera_rozuvan 11y ago
A very-very good YouTube video explaining what the abc conjecture is: https://www.youtube.com/watch?v=RkBl7WKzzRw https://www.youtube.com/watch?v=RkBl7WKzzRw .
asdftemp 11y ago
hi there, you might start with this mathoverflow thread: http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture http://mathoverflow.net/questions/106560/philosophy-behind-m...
impendia 11y ago
It would take an extraordinary effort. I doubt there is anyone who could learn this material given anything less than a year of solid study -- and I am thinking of the leading researchers in the world, who are already experts in everything related. I am a professional mathematician myself, and I doubt I could manage it even within a year. This is dictated by the difficulty of the subject, and it is even more so since Mochizuki's papers are notoriously difficult to read. What I have heard from other research mathematicians is that Mochizuki doesn't make much of an effort to make himself comprehensible, to answer questions, or in general to explain his results to the community. Also please be warned that the consensus is generally that Mochizuki's theory is probably incorrect. That said, if you want to learn the stuff, there is no reference other than Mochizuki's papers themselves. If you would like to learn some general background theory I would recommend learning some algebraic geometry, for which see Ravi Vakil's book: http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pd... and maybe Milne on etale cohmology: http://www.jmilne.org/math/CourseNotes/LEC210.pdf http://www.jmilne.org/math/CourseNotes/LEC210.pdf If you find that incomprehensible, start with Dummit and Foote's Abstract Algebra (read it cover to cover), Neukirch's Algebraic Number Theory, Atiyah and Macdonald's Commutative Algebra, and Fulton and Harris's Representation Theory. If you indeed want to read all that, you may as well enroll in a Ph.D. program in math....
- valera_rozuvan 11y ago
  Thanks for your answer! I will think about enrolling in a Ph.D = ) On a side note, apparently I missed an earlier Hacker News post about the approach to understanding this theory: https://news.ycombinator.com/item?id=8866144 https://news.ycombinator.com/item?id=8866144 .
- upwg24 11y ago
  the mathematical advice contained in this comment is incorrect. For correct information see e.g. http://www.claymath.org/events/iut-theory-shinichi-mochizuki http://www.claymath.org/events/iut-theory-shinichi-mochizuki "the consensus is generally that Mochizuki's theory is probably incorrect", "I am a professional mathematician" - these two claims contradict each other: no professional mathematician would make such false claims
  - smeyer 11y ago
    >no professional mathematician would make such false claims Well, that's obviously incorrect, because the previous commenter did in fact make those claims and is in fact a professional mathematician. It's not like you can make some sort of no true scotsman argument about his profession: http://people.math.sc.edu/thornef/ http://people.math.sc.edu/thornef/ .
  - impendia 11y ago
    The consensus is that Mochizuki's theory, at least insofar as it claims to prove ABC, is probably incorrect. (Otherwise this workshop would be a lot more popular.) But not definitely incorrect. Mochizuki is certainly not an obvious crank. His theory may be right, perhaps only partially -- in which case it will be enormously consequential. So it's still worth paying attention to. (In Minhyong Kim's judgment, which carries a hell of a lot more weight than mine.) I do stand corrected a little bit: I see that a couple other reputable mathematicians have already invested some time digesting his work, and further that Mochizuki has decided to be more open to answering questions.
    - valera_rozuvan 11y ago
upwg24 11y ago
see http://www.claymath.org/events/iut-theory-shinichi-mochizuki http://www.claymath.org/events/iut-theory-shinichi-mochizuki and https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor... for more information, including the 1st letter to its participants.
- valera_rozuvan 11y ago
  There are several open-ended questions for Shinichi Mochizuki: 1.) What is a Frobenioid? http://mathoverflow.net/questions/195353/what-is-a-frobenioid http://mathoverflow.net/questions/195353/what-is-a-frobenioi... 2.) What is an étale theta function? http://mathoverflow.net/questions/195841/what-is-an-%C3%A9tale-theta-function http://mathoverflow.net/questions/195841/what-is-an-%C3%A9ta... I hope the workshops planned will shed some light on them!
deleted 11y ago
[deleted]

valera_rozuvan 11y ago

A very-very good YouTube video explaining what the abc conjecture is: https://www.youtube.com/watch?v=RkBl7WKzzRw https://www.youtube.com/watch?v=RkBl7WKzzRw .

asdftemp 11y ago

hi there, you might start with this mathoverflow thread: http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture http://mathoverflow.net/questions/106560/philosophy-behind-m...

impendia 11y ago

It would take an extraordinary effort. I doubt there is anyone who could learn this material given anything less than a year of solid study -- and I am thinking of the leading researchers in the world, who are already experts in everything related. I am a professional mathematician myself, and I doubt I could manage it even within a year. This is dictated by the difficulty of the subject, and it is even more so since Mochizuki's papers are notoriously difficult to read. What I have heard from other research mathematicians is that Mochizuki doesn't make much of an effort to make himself comprehensible, to answer questions, or in general to explain his results to the community. Also please be warned that the consensus is generally that Mochizuki's theory is probably incorrect. That said, if you want to learn the stuff, there is no reference other than Mochizuki's papers themselves. If you would like to learn some general background theory I would recommend learning some algebraic geometry, for which see Ravi Vakil's book: http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pdf http://math.stanford.edu/~vakil/216blog/FOAGapr2915public.pd... and maybe Milne on etale cohmology: http://www.jmilne.org/math/CourseNotes/LEC210.pdf http://www.jmilne.org/math/CourseNotes/LEC210.pdf If you find that incomprehensible, start with Dummit and Foote's Abstract Algebra (read it cover to cover), Neukirch's Algebraic Number Theory, Atiyah and Macdonald's Commutative Algebra, and Fulton and Harris's Representation Theory. If you indeed want to read all that, you may as well enroll in a Ph.D. program in math....

valera_rozuvan 11y ago
Thanks for your answer! I will think about enrolling in a Ph.D = ) On a side note, apparently I missed an earlier Hacker News post about the approach to understanding this theory: https://news.ycombinator.com/item?id=8866144 https://news.ycombinator.com/item?id=8866144 .
upwg24 11y ago
the mathematical advice contained in this comment is incorrect. For correct information see e.g. http://www.claymath.org/events/iut-theory-shinichi-mochizuki http://www.claymath.org/events/iut-theory-shinichi-mochizuki "the consensus is generally that Mochizuki's theory is probably incorrect", "I am a professional mathematician" - these two claims contradict each other: no professional mathematician would make such false claims
- smeyer 11y ago
  >no professional mathematician would make such false claims Well, that's obviously incorrect, because the previous commenter did in fact make those claims and is in fact a professional mathematician. It's not like you can make some sort of no true scotsman argument about his profession: http://people.math.sc.edu/thornef/ http://people.math.sc.edu/thornef/ .
- impendia 11y ago
  The consensus is that Mochizuki's theory, at least insofar as it claims to prove ABC, is probably incorrect. (Otherwise this workshop would be a lot more popular.) But not definitely incorrect. Mochizuki is certainly not an obvious crank. His theory may be right, perhaps only partially -- in which case it will be enormously consequential. So it's still worth paying attention to. (In Minhyong Kim's judgment, which carries a hell of a lot more weight than mine.) I do stand corrected a little bit: I see that a couple other reputable mathematicians have already invested some time digesting his work, and further that Mochizuki has decided to be more open to answering questions.
  - valera_rozuvan 11y ago

upwg24 11y ago

see http://www.claymath.org/events/iut-theory-shinichi-mochizuki http://www.claymath.org/events/iut-theory-shinichi-mochizuki and https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor... for more information, including the 1st letter to its participants.

valera_rozuvan 11y ago
There are several open-ended questions for Shinichi Mochizuki: 1.) What is a Frobenioid? http://mathoverflow.net/questions/195353/what-is-a-frobenioid http://mathoverflow.net/questions/195353/what-is-a-frobenioi... 2.) What is an étale theta function? http://mathoverflow.net/questions/195841/what-is-an-%C3%A9tale-theta-function http://mathoverflow.net/questions/195841/what-is-an-%C3%A9ta... I hope the workshops planned will shed some light on them!

deleted 11y ago

[deleted]