Shinichi Mochizuki

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Shinichi Mochizuki
Born (1969-03-29) March 29, 1969 (age 47)[1]
Tokyo, Japan[1]
Nationality Japanese
Fields Mathematics
Institutions Kyoto University
Alma mater Princeton University
Doctoral advisor Gerd Faltings
Known for Proposed proof of abc conjecture,
Proved Grothendieck conjecture on anabelian geometry.
Notable awards JSPS Prize, Japan Academy Medal[1]

Shinichi Mochizuki (望月 新一 Mochizuki Shin'ichi?, born March 29, 1969) is a Japanese mathematician working in number theory and geometry. He is the main contributor to anabelian geometry where he solved the famous Grothendieck conjecture about hyperbolic curves over number fields. He initiated related and new areas such as absolute anabelian geometry and mono-anabelian geometry. He introduced p-adic Teichmüller theory, Hodge–Arakelov theory and two theories which study Frobenioids and anabelioids as a categorical geometry generalization of conventional aspects of arithmetic geometry. He is the author of inter-universal Teichmüller theory (IUT), which is also called arithmetic deformation theory or Mochizuki theory. It goes outside the realm of conventional arithmetic geometry and essentially extends the scope of arithmetic geometry.

Biography[edit]

Early life[edit]

Shinichi Mochizuki's mother was Japanese, and his father was American. When he was five years old, Shinichi Mochizuki and his family left Japan to live in New York City. Mochizuki attended Phillips Exeter Academy and graduated in 1985.[2] He entered Princeton University as an undergraduate at age 16 and graduated salutatorian in 1988.[2] He then received a Ph.D. under the supervision of Gerd Faltings at age 23.[1] He joined the Research Institute for Mathematical Sciences in Kyoto University in 1992 and was promoted to professor in 2002.[1]

Career[edit]

Mochizuki proved Grothendieck conjecture on anabelian geometry in 1996. Mochizuki was an invited speaker at the International Congress of Mathematicians in 1998.[3] In 1999, he introduced Hodge–Arakelov theory. During 2000-2007, he introduced the theory of Frobenioids and mono-anabelian geometry.

In August 2012, Mochizuki released four preprints which develop inter-universal Teichmüller theory and also its applications to proof of several famous conjectures in diophantine geometry,[4] including the abc conjecture over every number field. The theory is very complex and involves many novel concepts and objects. It has already been verified more than 10 times by several mathematicians.[5] Mochizuki documented the relevant progress in two reports, the first in December 2013 and the second in December 2014.

Inter-universal Teichmüller theory[edit]

In the specific situation of a number field and an elliptic curve over it, this theory deals with full Galois and fundamental groups of various hyperbolic curves associated to the elliptic curve and related enhanced categorical structures (systems of frobenioids). It applies deep algorithmic results of mono-anabelian geometry to reconstruct the groups and schemes after applying various links which are not compatible with ring or scheme structure. Resulting synchronizations, rigidities and mild indeterminacies lead to applications to the strong Szpiro conjecture and its equivalent forms.

As of December 2014, Mochizuki wrote "I have yet to hear of even a single problem that relates to the essential thrust or validity of the theory" on the progress report.[6] According to Mochizuki, "At least with regard to the substantive mathematical aspects of such a verification, the verification of Inter-universal Teichmüller theory is, for all practical purposes, complete".[6] He wrote, however, "Nevertheless, as a precautionary measure, in light of the importance of the theory and the novelty of the techniques that underlie the theory, it seems appropriate that a bit more time be allowed to elapse before a final official declaration of the completion of the verification of Inter-universal Teichmüller theory is made."[6]

Shinichi Mochizuki invested a very substantial amount of time into dissemination of his results.[7] Two surveys of IUT were produced by its author,[8][9] one survey by Ivan Fesenko[10] and two surveys by Yuichiro Hoshi.[11] National workshops on IUT were held at RIMS in March 2015 and in Beijing in July 2015.[12] An international workshop on IUT was organized by Ivan Fesenko in Oxford in December 2015, its materials are available online.[13] A further international workshop on IUT Summit, organized by I. Fesenko, S. Mochizuki and Y. Taguchi will be held at RIMS in July 2016.[14] The organizers produced a document[15] which includes, "As of July 2016, the four papers on IUT have been thoroughly studied and verified in their entirety by at least four mathematicians (other than the author), and various substantial portions of these papers have been thoroughly studied by quite a number of mathematicians (such as the speakers at the Oxford workshop in December 2015 and the RIMS workshop in July 2016). These papers are currently being refereed, and, although they have not yet been officially accepted for publication, the refereeing process is proceeding in an orderly, constructive, and positive manner."

Publications[edit]

Inter-universal Teichmüller theory[edit]

References[edit]

  1. ^ a b c d e Mochizuki, Shinichi. "Curriculum Vitae" (PDF). Retrieved 14 September 2012. 
  2. ^ a b "Seniors address commencement crowd". Princeton Weekly Bulletin. 20 June 1988. p. 4. 
  3. ^ "International Congress of Mathematicians 1998". 
  4. ^ Inter-universal Teichmüller theory IV: log-volume computations and set-theoretic foundations, Shinichi Mochizuki, August 2012
  5. ^ Fesenko, Ivan (2015), Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Eur. J. Math., 2015 (PDF) 
  6. ^ a b c Mochizuki, Shinichi (2014), "link On the Verification of Inter-Universal Teichmuller Theory: A Progress Report(as of December 2014)", Research Institute for Mathematical Sciences Kyoto university, p.7-8.
  7. ^ Seminars, meetings, lectures on IUT in Japan (PDF) 
  8. ^ Mochizuki, Shinichi (2014), A panoramic overview of inter-universal Teichmüller theory, In Algebraic number theory and related topics 2012, RIMS Kôkyûroku Bessatsu B51, RIMS, Kyoto (2014), 301–345 (PDF) 
  9. ^ Mochizuki, Shinichi (2016), The mathematics of mutually alien copies: from Gaussian integrals to inter-universal Teichmüller theory (PDF) 
  10. ^ Fesenko, Ivan (2015), Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki, Eur. J. Math., 2015 (PDF) 
  11. ^ "On questions and comments concerning Inter-universal Teichmüller Theory" (PDF). 
  12. ^ Future and past workshops on IUT theory of Shinichi Mochizuki 
  13. ^ Workshop on IUT theory of Shinichi Mochizuki 
  14. ^ Inter-universal Teichmüller Theory Summit 2016 (RIMS workshop, July 18-27 2016) 
  15. ^ "On questions and comments concerning Inter-universal Teichmüller Theory" (PDF). 

External links[edit]