Shin'ichi Mochizuki

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Shin'ichi Mochizuki ( Japanese 望月 新 一 , Mochizuki Shin'ichi ; born March 29, 1969 in Tokyo ) is a Japanese mathematician . He is Professor at the Research Institute for Mathematical Sciences (RIMS) at Kyoto University . He deals with algebraic geometry and arithmetic geometry .

Life

Mochizuki lived from the age of five years with his parents in New York City who attended Phillips Exeter Academy with the degree in 1985 and then studied at the Princeton University first theoretical physics at Arthur Wightman and Edward Witten , before settling at Gerd Faltings mathematics (Algebraic Geometry) with a Masters degree in 1988, he received the George B. Wood Prize as a student with top grades. He also received the George B. Covington Prize for Mathematics. In 1992 he received his doctorate under Gerd Faltings ( The Geometry of the Compactification of the Hurwitz Scheme ). From 1992 to 1994 he was Benjamin Peirce Instructor at Harvard University and at the same time at RIMS, where he became assistant professor in 1996 and professor in 2002.

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In the early 1990s he developed a p-adic Teichmüller theory , that is, the p-adic analog of the uniformization of hyperbolic curves and their moduli (in the classical complex case by Paul Koebe , Lipman Bers ). He further developed a p-adic Anabelian geometry in the mid-1990s and a Hodge-Arakelov theory of elliptic curves (analogue of the Hodge theory for elliptic curves in the context of Arakelov geometry) in the late 1990s .

In 2012, he announced a proof of the abc conjecture (over the equivalent presumption of Lucien Szpiro on elliptic curves) as part of which he developed in the 2000s theory that the usual used in the arithmetic and algebraic geometry scheme exceeds frames and he calls inter-universal geometry , and here specifically inter-universal Teichmüller theory (IUT, developed by him from 2006). According to his own words, this is analogous to his construction of the p-adic Teichmüller theory of hyperbolic curves, with p-adic bodies being replaced by number fields with associated elliptic curves. Since the proof is over 500 pages long and includes additional references to Mochizuki's previous work, all of which had not yet appeared (except as preprints) and which use entirely novel mathematical concepts and techniques, it is currently still being reviewed by mathematicians. In the last part of his preprints on the abc conjecture, presented in 2015, Mochizuki himself establishes a close relationship between the mathematical apparatus he developed for this and the famous Riemann conjecture .

A gap in evidence found by Akshay Venkatesh and Vesselin Dimitrov in 2012 (part 3, 4 of his preprint series) was admitted by Mochizuki, but declared to be rectifiable - he subsequently corrected his preprints.

At a conference at the Clay Mathematics Institute in Oxford in December 2015, it was revealed at least what kind of mathematical objects Mochizuki's proof could be about. Mochizuki starts from Szpiro's equivalent formulation of the abc conjecture about the theory of elliptic curves and in the course of the conference it became clear that the algebraic concepts of the frobenioids that he introduced play an essential role in this (lecture by Kiran Kedlaya ). However, the proponents of Mochizuki's evidence (Chung Pang Mok, Yuichiro Hoshi, and Go Yamashita; Mochizuki himself was absent but answered questions on Skype) failed to present a convincing account of the next steps in the evidence.

Mochizuki himself was present at a conference in Kyoto in July 2016, and the number of predominantly younger mathematicians who were intensively involved in understanding the work of Mochizuki increased from three at the 2015 Oxford meeting to ten. According to Jeffrey Lagarias preprints included some revolutionary new ideas in number theory.

While in 2017, after five years of confidential peer review, a publication in the Publications of the RIMS (whose editor is Mochizuki) became apparent, not much has changed on the question of acceptance. The core of the proof lies in Corollary 3.12 to Theorem 3.11 (the main theorem of the IUT) and there are also significant problems of understanding. Another point is that one would actually expect that the proof would lead to new insights beyond the proof of the abc conjecture. Jakob Stix and Peter Scholze spent a week with Mochizuki in March 2018 to discuss his evidence with him. Mochizuki stuck to his position that his evidence was correct; on the part of Scholze and Stix, there are misunderstandings and oversimplifications. In September 2018, Stix and Scholze published a ten-page article. In it they write that the aforementioned Corollary 3.12 is a fundamental gap in the evidence that, in their opinion, cannot be remedied.

In 2020, it was announced that Mochizuki's work on the abc issue was accepted for publication in the Publications of the RIMS, of which Mochizuki is editor-in-chief, on February 5th. After that, Mochizuki himself was not involved in the assessment. Despite the criticism from Scholze and Stix, which, according to Nature, Scholze maintains unchanged and which led to considerable doubts among a large number of mathematicians about the validity of the proof, there should be no significant changes in the publication compared to the preprints and only on the criticism to be included in comments. According to number theorist Kiran Kedlaya , the predominantly negative assessment among mathematicians of Mochizuki's attempt at proof has not changed even with the announcement of the publication. Scholze also repeated his criticism.

Honors

In 1997 he received the Fall Prize of the Japanese Mathematical Society and in 2005 the medal of the Japan Academy.

In 1998 he was invited speaker at the International Congress of Mathematicians in Berlin ( The intrinsic Hodge theory of p-adic hyperbolic curves ).

In 2004 he was one of the first recipients of the JSPS Prize (Prize of the Japan Society for the Promotion of Science) for research into the arithmetic geometry of hyperbolic curves, including the solution of the Grothendieck conjecture for Anabel's geometry using p-adic methods .

Fonts

  • A Version of the Grothendieck Conjecture for p-adic Local Fields , The International Journal of Mathematics, Volume 8, 1997, pp. 499-506.
  • Foundations of p-adic Teichmüller theory , AMS / IP Studies in Advanced Mathematics, Volume 11, Providence, RI: American Mathematical Society, International Press 1999
  • Inter-universal Teichmüller Theory. (Preprints)
  • A panoramic overview of Inter-Universal Teichmüller Theory , in: Algebraic number theory and related topics 2012, RIMS Kôkyûroku Bessatsu B51, RIMS, Kyoto (2014), pp. 301–345, pdf (review article)
  • The mathematics of mutually alien copies: from Gaussian integrals to Inter-Universal Teichmüller Theory , 2016, pdf (review article)
  • The étale theta function and its Frobenioid-theoretic manifestations, Publ. RIMS, Volume 45, 2009, pp. 227-349.
  • Arithmetic elliptic curves in general position, Math. J. Okayama Univ., Volume 52, 2010, pp. 1-28.

literature

  • Gerd Faltings: Curves and their fundamental groups, following Grothendieck, Tamagawa and Mochizuki. Seminaire Bourbaki, no.840, March 1998
  • Iwan Borissowitsch Fessenko : Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki. In: Europ. J. Math. 2015, 1, pp. 405-440, online
  • Vesselin Dimitrov: Effectivity in Mochizuki's work on the abc conjecture, Preprint 2016, Arxiv
  • Davide Castelvecchi: The impenetrable proof , Nature, Volume 526, 2015, pp. 179–181, reprinted with additions in: Mircea Pitici (Ed.), The best writings in mathematics 2016, Princeton UP 2017
  • Go Yamashita: A proof of the abc conjecture after Mochizuki , Preprint 2017, Online

Web links

Individual evidence

  1. ^ Princeton Weekly Bulletin. Volume 77, June 20, 1988, Online, Senior Address Commencement Crowd  ( page no longer available , search in web archives )  Info: The link was automatically marked as defective. Please check the link according to the instructions and then remove this notice. , with Mochizuki as the Salutatorian@1@ 2Template: dead link / libserv23.princeton.edu  
  2. ^ Mathematics Genealogy Project
  3. ^ Preprint Inter-universal Teichmüller Theory IV: Log-volume computations and set-theoretic foundations , RIMS, August 2012. Parts 1-3 were only available as preprints until August 2012.
  4. ↑ Holger Dambeck: Japanese presents solution for prime number puzzles. on Spiegel-Online , September 26, 2012
  5. ^ The Paradox of the Proof
  6. ↑ Inter-universal Teichmüller Theory IV: Log-volume Computations and Set-theoretic Foundations. September 2015, pp. 47–53
  7. ↑ Kevin Hartnett, An abc proof too tough even for mathematicians, in: Mircea Pitici (Ed.), The best writings in mathematics 2013, Princeton UP 2014, p. 228, originally Boston Globe November 4, 2012
  8. ↑ Frobenioid , nLab, with web links. See also the article by Fessenko cited in the literature.
  9. ↑ Report by Kevin Hartnett 2015, see web links
  10. ↑ Davide Castelvecchi, Monumental proof to torment mathematicians for years to come, Nature, July 28, 2016
  11. ↑ Latest on abc , Blog Not even wrong by Peter Woit, December 16, 2017
  12. ↑ Frank Calegari, The abc conjecture has (still) not been proved , blog, December 17, 2017. With critical comments from Peter Scholze (PS), Brian Conrad and Terence Tao, among others .
  13. ↑ Mochizuki website with the report by Scholze and Stix and answers from Mochizuki
  14. ↑ Erica Klarreich: Titans of Mathematics Clash Over Epic Proof of ABC Conjecture , Quanta Magazine, September 20, 2018
  15. ↑ see also Manfred Dworschak: Schwurbel aus dem All . In: Der Spiegel 41/2018, p. 110.
  16. ↑ Davide Castelvecchi, Mathematical proof that rocked number theory will be published , Nature, Volume 580, April 3, 2020, p. 177
  17. ↑ Davide Castelvecchi, Mathematical proof that rocked number theory will be published , Nature, Volume 580, April 3, 2020, p. 177
  18. ↑ Peter Scholze, comment on the blog entry Latest on abc in Peter Woit's blog, April 6, 2020 at 9:28 am
  19. ^ Official website JSPS
  20. ^ Research on the Arithmetic Geometry of Hyperbolic Curves, including Solution via p-adic Methods of the Grothendieck Conjecture on Anabelian Geometry
  21. ^ S. Mochizuki: The profinite Grothendieck conjecture for hyperbolic curves over number fields. In: J. Math. Sci. Univ. Tokyo. Volume 3, 1996, pp. 571-627
personal data
SURNAME Mochizuki, Shin'ichi
ALTERNATIVE NAMES 望月 新 一 (Japanese)
BRIEF DESCRIPTION Japanese mathematician
DATE OF BIRTH March 29, 1969
PLACE OF BIRTH Tokyo