Difference between revisions of "ABC conjecture"

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# (IUTT-III) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice], Shinichi Mochizuki
 
# (IUTT-III) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20III.pdf Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice], Shinichi Mochizuki
 
# (IUTT-IV) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations], Shinichi Mochizuki
 
# (IUTT-IV) [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV.pdf Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations], Shinichi Mochizuki
 +
 +
Additional papers on IUT
 
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf A Panoramic Overview of Inter-universal Teichmuller Theory], Shinichi Mochizuki
 
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf A Panoramic Overview of Inter-universal Teichmuller Theory], Shinichi Mochizuki
 +
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/Bogomolov%20from%20the%20Point%20of%20View%20of%20Inter-universal%20Teichmuller%20Theory.pdf Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from the Point of View of Inter-universal Teichmuller Theory], Shinichi Mochizuki: "Bogomolov’s proof may be thought of as a sort of useful elementary guide, or blueprint (perhaps even a sort of Rosetta stone!), for understanding substantial portions of inter-universal Teichmüller theory."
 +
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/Alien%20Copies,%20Gaussians,%20and%20Inter-universal%20Teichmuller%20Theory.pdf The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory.]
  
 
Progress reports:
 
Progress reports:
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202013-12.pdf ON THE VERIFICATION OF INTER-UNIVERSAL TEICHM¨ULLER THEORY: A PROGRESS REPORT (AS OF DECEMBER 2013)], Shinichi Mochizuki
+
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202013-12.pdf On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2013)], Shinichi Mochizuki
 +
# [http://www.kurims.kyoto-u.ac.jp/~motizuki/IUTeich%20Verification%20Report%202014-12.pdf On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2014)], Shinichi Mochizuki
  
 
See also [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Brief%20Introduction%20to%20Inter-universal%20Geometry%20(Tokyo%202004-01).pdf these earlier slides] of Mochizuki on inter-universal Teichmuller theory.  The answers to [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture this MathOverflow post] (and in particular [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658 Minhyong Kim's answer]) describe the philosophy behind Mochizuki's proof strategy.  Go Yamashita has a [http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-Universality.pdf short FAQ on inter-universality], which is a concept that appears in Mochizuki's arguments, though it does not appear to be the central ingredient in these papers.
 
See also [http://www.kurims.kyoto-u.ac.jp/~motizuki/A%20Brief%20Introduction%20to%20Inter-universal%20Geometry%20(Tokyo%202004-01).pdf these earlier slides] of Mochizuki on inter-universal Teichmuller theory.  The answers to [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture this MathOverflow post] (and in particular [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/106658#106658 Minhyong Kim's answer]) describe the philosophy behind Mochizuki's proof strategy.  Go Yamashita has a [http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-Universality.pdf short FAQ on inter-universality], which is a concept that appears in Mochizuki's arguments, though it does not appear to be the central ingredient in these papers.
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* The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture).  There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/#comment-10605 this blog comment].  However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.
 
* The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture).  There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/#comment-10605 this blog comment].  However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.
  
* There is some discussion at [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279 this MathOverflow post] as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc.  In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture. Mochizuki has written [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf comments] in October 2012 to say that he hopes to post a revised version of Theorem 1.10 and its proof in the not too distant future.
+
* There is some discussion at [http://mathoverflow.net/questions/106560/philosophy-behind-mochizukis-work-on-the-abc-conjecture/107279#107279 this MathOverflow post] as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc.  In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture. Mochizuki has written [http://www.kurims.kyoto-u.ac.jp/~motizuki/Inter-universal%20Teichmuller%20Theory%20IV%20(comments).pdf comments] in October 2012 to say that he hopes to post a revised version of Theorem 1.10, which were revised in 2013.
  
 
* The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ this blog post].
 
* The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to [http://quomodocumque.wordpress.com/2012/09/03/mochizuki-on-abc/ this blog post].
  
 
* The category and topos theory viewpoint is discussed at the [http://nforum.mathforge.org/discussion/4260/abc-conjecture nForum page for the abc conjecture].
 
* The category and topos theory viewpoint is discussed at the [http://nforum.mathforge.org/discussion/4260/abc-conjecture nForum page for the abc conjecture].
 +
 +
===Workshops===
 +
* '''9.-20. March 2015''': RIMS Joint Research Workshop: On the verification and further development of inter-universal Teichmuller theory: [http://www.kurims.kyoto-u.ac.jp/~motizuki/2015-03%20IUTeich%20Program%20(English).pdf program]
 +
** Lecture Series by [http://www.kurims.kyoto-u.ac.jp/~gokun/myworks.html Go Yamashita] at Kyushu University ([http://www.math.kyushu-u.ac.jp/seminars/view/1373 announcement in japanese])
 +
** Slides by Yuichiro Hoshi on [http://www.kurims.kyoto-u.ac.jp/~yuichiro/talk20150309.pdf Mono-anabelian Reconstruction of Number Fields]
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 +
* '''7.-11. December 2015''': [https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.html Clay Mathematics Institute workshop on the theory of Shinichi Mochizuki], Oxford
 +
 +
* [https://www.maths.nottingham.ac.uk/personal/ibf/files/symcor.iut.cf.html Activities on the study of IUT theory of Shinichi Mochizuki]
 +
 +
===Survey articles===
 +
*Ivan Fesenko, [https://www.maths.nottingham.ac.uk/personal/ibf/notesoniut.pdf Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki]
 +
*Shinichi Mochizuki, [http://www.kurims.kyoto-u.ac.jp/~motizuki/Panoramic%20Overview%20of%20Inter-universal%20Teichmuller%20Theory.pdf Panoramic overview of inter-universal Teichmuller theory]
 +
*Yamashita, [http://www.kurims.kyoto-u.ac.jp/~motizuki/FAQ%20on%20Inter-Universality.pdf FAQ on ‘Inter-Universality’]
 +
*Ivan Fesenko, [http://inference-review.com/article/fukugen Fukugen]
 +
 
===Blogs===
 
===Blogs===
 
*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], Secret Blogging Seminar, 12 June, 2012
 
*[http://sbseminar.wordpress.com/2012/06/12/abc-conjecture-rumor-2/ ABC conjecture rumor], Secret Blogging Seminar, 12 June, 2012
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*[http://www.math.columbia.edu/~woit/wordpress/?p=6514 Latest on abc], Not Even Wrong, 19 Dec 2013
 
*[http://www.math.columbia.edu/~woit/wordpress/?p=6514 Latest on abc], Not Even Wrong, 19 Dec 2013
 
*[https://plus.google.com/+RichardElwes/posts/jMVfRcnRaoV Richard Elwes, Google+], 20 Dec 2013
 
*[https://plus.google.com/+RichardElwes/posts/jMVfRcnRaoV Richard Elwes, Google+], 20 Dec 2013
 +
*[http://www.math.columbia.edu/~woit/wordpress/?p=7451 Peter Woit on Progress-Report 2014], 13 Jan 2015
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*[https://plus.google.com/+AlexanderKruel/posts/dvUQWL7tDg2 Alexander Kruel Google+], 8 Sept 2015
 +
*[https://plus.google.com/+DavidRoberts/posts/UMKqSdvf2WB David Roberts Google+], 8 Sept 2015
 +
*[https://mathbabe.org/2015/12/15/notes-on-the-oxford-iut-workshop-by-brian-conrad/ Notes on the Oxford IUT workshop by Brian Conrad], Mathbabe, 15 Dec 2015
  
 
===2013 study of Geometry of Frobenioids===
 
===2013 study of Geometry of Frobenioids===
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*[http://mathoverflow.net/questions/107379/model-theoretic-content-of-mochizukis-teichmuller-theory-papers Model-theoretic content of Mochizuki’s Teichmüller theory papers], Mathoverflow, 17 Sept 2012
 
*[http://mathoverflow.net/questions/107379/model-theoretic-content-of-mochizukis-teichmuller-theory-papers Model-theoretic content of Mochizuki’s Teichmüller theory papers], Mathoverflow, 17 Sept 2012
 
*[http://math.stackexchange.com/questions/199609/groupification-and-perfection-of-a-commutative-monoid Groupification and perfection of a commutative monoid], Mathematics Stackexchange, 20 Sept 2012
 
*[http://math.stackexchange.com/questions/199609/groupification-and-perfection-of-a-commutative-monoid Groupification and perfection of a commutative monoid], Mathematics Stackexchange, 20 Sept 2012
 +
*[http://www.quora.com/As-of-September-2014-what-is-the-mathematical-communitys-current-understanding-of-Mochizukis-proof-of-the-abc-conjecture As of September 2014, what is the mathematical community's current understanding of Mochizuki's proof of the abc conjecture?] Quora, September 2014.
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*[http://mathoverflow.net/questions/195353/what-is-a-frobenioid What is a Frobenioid?], Mathoverflow, 31 January 2015.
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*[http://mathoverflow.net/questions/195841/what-is-an-%C3%A9tale-theta-function What is an étale theta function?], Mathoverflow, 07 February 2015.
 +
 +
 
Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see [http://mathoverflow.net/faq this FAQ for details].
 
Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see [http://mathoverflow.net/faq this FAQ for details].
  
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*[http://www.dailyprincetonian.com/2012/09/20/31183/ U.-educated mathematician offers proof of pivotal number theory conjecture], The Daily Princetonian, 20 Sept 2012
 
*[http://www.dailyprincetonian.com/2012/09/20/31183/ U.-educated mathematician offers proof of pivotal number theory conjecture], The Daily Princetonian, 20 Sept 2012
 
*[http://bostonglobe.com/ideas/2012/11/03/abc-proof-too-tough-even-for-mathematicians/o9bja4kwPuXhDeDb2Ana2K/story.html An ABC proof too tough even for mathematicians], Kevin Hartnett, 3 Nov 2012.
 
*[http://bostonglobe.com/ideas/2012/11/03/abc-proof-too-tough-even-for-mathematicians/o9bja4kwPuXhDeDb2Ana2K/story.html An ABC proof too tough even for mathematicians], Kevin Hartnett, 3 Nov 2012.
*[http://projectwordsworth.com/the-paradox-of-the-proof/ The Paradox of the Proof], Caroline Chen, May 10, 2013.
+
*[http://projectwordsworth.com/the-paradox-of-the-proof/ The Paradox of the Proof], Caroline Chen, 10 May 2013.
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*[http://www.newscientist.com/article/dn26753-mathematicians-anger-over-his-unread-500page-proof.html  Mathematician's anger over his unread 500-page proof], Jacob Aron, 02 Jan 2015.
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*[https://www.newscientist.com/article/dn28065-our-numbers-up-machines-will-do-maths-well-never-understand/ Our number’s up: Machines will do maths we’ll never understand], New Scientist, 26 August 2015
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*[http://www.nature.com/news/the-biggest-mystery-in-mathematics-shinichi-mochizuki-and-the-impenetrable-proof-1.18509 The biggest mystery in mathematics: Shinichi Mochizuki and the impenetrable proof], Davide Castelvecchi, 07 October 2015, Nature News
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*[http://www.nature.com/news/biggest-mystery-in-mathematics-in-limbo-after-cryptic-meeting-1.19035 Biggest mystery in mathematics in limbo after cryptic meeting], Davide Castelvecchi, 16 December 2015, Nature News
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*[https://www.newscientist.com/article/dn28682-mathematicians-left-baffled-after-three-year-struggle-over-proof/ Mathematicians left baffled after three-year struggle over proof], New Scientist, 16 December 2015
 +
*[https://www.quantamagazine.org/20151221-hope-rekindled-for-abc-proof/ Hope Rekindled for Perplexing Proof], Quanta Magazine, Kevin Hartnett, December 21, 2015
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*[http://www.nature.com/news/monumental-proof-to-torment-mathematicians-for-years-to-come-1.20342 Monumental proof to torment mathematicians for years to come], 28 July 2016, Nature News
 +
*[https://www.newscientist.com/article/2099534-mathematicians-finally-starting-to-understand-epic-abc-proof/ Mathematicians finally starting to understand epic ABC proof], New Scientist, 2 August 2016
  
 
===Crowd News===
 
===Crowd News===
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*[http://theconversation.edu.au/the-abc-conjecture-as-easy-as-1-2-3-or-not-10836 The abc conjecture, as easy as 1, 2, 3 ... or not ], The Conversation, Alexandru Ghitza, 26 Nov 2012.
 
*[http://theconversation.edu.au/the-abc-conjecture-as-easy-as-1-2-3-or-not-10836 The abc conjecture, as easy as 1, 2, 3 ... or not ], The Conversation, Alexandru Ghitza, 26 Nov 2012.
 
*[http://www.sciencenews.org/view/generic/id/349199/description/A_theorem_in_limbo_shows_that_QED_is_not_the_last_word_in_a_mathematical_proof A theorem in limbo shows that QED is not the last word in a mathematical proof], March 25, 2013.
 
*[http://www.sciencenews.org/view/generic/id/349199/description/A_theorem_in_limbo_shows_that_QED_is_not_the_last_word_in_a_mathematical_proof A theorem in limbo shows that QED is not the last word in a mathematical proof], March 25, 2013.
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===See also===
 +
*[http://ncatlab.org/nlab/show/inter-universal+Teichm%C3%BCller+theory inter-universal Teichmüller theory] at the nLab
 +
*[https://en.wikipedia.org/wiki/Inter-universal_Teichm%C3%BCller_theory Inter-universal Teichmüller theory] at Wikipedia
 +
*[https://en.wikipedia.org/wiki/Frobenioid Frobenioid] at Wikipedia

Latest revision as of 06:44, 5 November 2016

The abc conjecture asserts, roughly speaking, that if a+b=c and a,b,c are coprime, then a,b,c cannot all be too smooth; in particular, the product of all the primes dividing a, b, or c has to exceed [math]c^{1-\varepsilon}[/math] for any fixed [math]\varepsilon \gt 0[/math] (if a,b,c are smooth).

This shows for instance that [math](1-\varepsilon) \log N / 3[/math]-smooth a,b,c of size N which are coprime cannot sum to form a+b=c. This unfortunately seems to be too weak to be of much use for the finding primes project.

A probabilistic heuristic justification for the ABC conjecture can be found at this blog post.

Mochizuki's proof

Papers

Mochizuki's claimed proof of the abc conjecture is conducted primarily through the following series of papers:

  1. (IUTT-I) Inter-universal Teichmuller Theory I: Construction of Hodge Theaters, Shinichi Mochizuki
  2. (IUTT-II) Inter-universal Teichmuller Theory II: Hodge-Arakelov-theoretic Evaluation, Shinichi Mochizuki
  3. (IUTT-III) Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice, Shinichi Mochizuki
  4. (IUTT-IV) Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations, Shinichi Mochizuki

Additional papers on IUT

  1. A Panoramic Overview of Inter-universal Teichmuller Theory, Shinichi Mochizuki
  2. Bogomolov's Proof of the Geometric Version of the Szpiro Conjecture from the Point of View of Inter-universal Teichmuller Theory, Shinichi Mochizuki: "Bogomolov’s proof may be thought of as a sort of useful elementary guide, or blueprint (perhaps even a sort of Rosetta stone!), for understanding substantial portions of inter-universal Teichmüller theory."
  3. The Mathematics of Mutually Alien Copies: from Gaussian Integrals to Inter-universal Teichmuller Theory.

Progress reports:

  1. On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2013), Shinichi Mochizuki
  2. On the Verification of Inter-Universal Teichmüller theory: A process report (as of december 2014), Shinichi Mochizuki

See also these earlier slides of Mochizuki on inter-universal Teichmuller theory. The answers to this MathOverflow post (and in particular Minhyong Kim's answer) describe the philosophy behind Mochizuki's proof strategy. Go Yamashita has a short FAQ on inter-universality, which is a concept that appears in Mochizuki's arguments, though it does not appear to be the central ingredient in these papers.

The argument also relies heavily on Mochizuki's previous work on the Hodge-Arakelov theory of elliptic curves, including the following references:

Anyone seeking to get a thorough "bottom-up" understanding of Mochizuki's argument will probably be best advised to start with these latter papers first. The papers (AbsTopIII), (EtTh) are directly cited heavily by the IUTT series of papers; the earlier papers (HAT), (GTKS) cover thematically related material but serve more as inspiration than as a source of mathematical results in the IUTT series.

The theory of (IUTT I-IV) is used to establish a Szpiro-type inequality, which is similar to Szpiro's conjecture but with an additional genericity hypothesis on a certain parameter [math]\ell[/math]. In order to then deduce the true Szpiro's conjecture (which is essentially equivalent to the abc conjecture), the results from the paper

are used. (Note that the published version of this paper requires some small corrections, listed here.) See this MathOverflow post of Vesselin Dimitrov for more discussion.

Here are the remainder of Shinichi Mochizuki's papers, and here is the Wikipedia page for Shinichi Mochizuki.

Specific topics

  • The last part of (IUTT-IV) explores the use of different models of ZFC set theory in order to more fully develop inter-universal Teichmuller theory (this part is not needed for the applications to the abc conjecture). There appears to be an inaccuracy in a remark in Section 3, page 43 of that paper regarding the conservative nature of the extension of ZFC by the addition of the Grothendieck universe axiom; see this blog comment. However, this remark was purely for motivational purposes and does not impact the proof of the abc conjecture.
  • There is some discussion at this MathOverflow post as to whether the explicit bounds for the abc conjecture are too strong to be consistent with known or conjectured lower bounds on abc. In particular, there appears to be a serious issue with the main Diophantine inequality (Theorem 1.10 of IUTT-IV), in that it appears to be inconsistent with commonly accepted conjectures, namely the abc conjecture and the uniform Serre open image conjecture. Mochizuki has written comments in October 2012 to say that he hopes to post a revised version of Theorem 1.10, which were revised in 2013.
  • The question of whether the results in this paper can be made completely effective (which would be of importance for several applications) is discussed in some of the comments to this blog post.

Workshops

Survey articles

Blogs

2013 study of Geometry of Frobenioids

Q & A


Note that Mathoverflow has a number of policies and guidelines regarding appropriate questions and answers to post on that site; see this FAQ for details.

News Media

Crowd News

See also