Shinichi Mochizuki

Shinichi Mochizuki
Born	(1969-03-29) March 29, 1969 Tokyo, Japan
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

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

Mochizuki was an invited speaker at the International Congress of Mathematicians in 1998.

Mochizuki proved Grothendieck conjecture on anabelian geometry in 1996. In 1999, he introduced Hodge–Arakelov theory. In 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, 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.

Mochizuki documented the relevant progress in two reports, the first in December 2013 and the second in December 2014.

Personal life

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. He entered Princeton University as an undergraduate at age 16 and graduated salutatorian in 1988. He then received a Ph.D. under the supervision of Gerd Faltings at age 23. He joined the Research Institute for Mathematical Sciences in Kyoto University in 1992 and was promoted to professor in 2002.

Inter-universal Teichmüller theory

Main article: Inter-universal Teichmüller theory

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, through discussion with Y. Hoshi and G. Yamashita of the Research Institute for Mathematical Sciences in Kyoto University and M. Saidi of University of Exeter, 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. 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". 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."

National workshops on IUT were held at RIMS in March 2015 and in Beijing in July 2015. An international workshop on IUT was held in Oxford in December 2015, talks by its 15 speakers covered many relevant areas, its materials are available online. A further international workshop will be held at RIMS in July 2016.

Publications

Inter-universal Teichmüller theory

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External links

• Shinichi Mochizuki at the Mathematics Genealogy Project
• Personal website
• Papers of Shinichi Mochizuki
• A brief introduction to inter-universal geometry
• Arithmetic deformation theory via arithmetic fundamental groups and nonarchimedean theta functions, notes on the work of Shinichi Mochizuki
• Introduction to inter-universal Teichmüller theory (in Japanese), a survey by Yuichiro Hoshi
• RIMS Joint Research Workshop: On the verification and further development of inter-universal Teichmuller theory, March 2015, Kyoto*
• CMI workshop on IUT theory of Shinichi Mochizuki, December 2015, Oxford*