Shinichi Mochizuki

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 leader of and main contributor to anabelian geometry; his contributions include a solution of the Grothendieck conjecture in anabelian geometry about hyperbolic curves over number fields. He initiated and developed related several novel areas such as absolute anabelian geometry, mono-anabelian geometry, combinatorial anabelian geometry. Mochizuki introduced and developed p-adic Teichmüller theory and Hodge–Arakelov theory. His more recent theories include the theory of frobenioids, anabelioids and the etale theta-function theory.
Shinichi Mochizuki is the author of the famous inter-universal Teichmüller theory (IUT), also referred to as the arithmetic deformation theory or Mochizuki theory. This theory provides a new conceptual view of numbers by using groups of symmetries: the absolute Galois groups and arithmetic fundamental groups. Applications of IUT in Mochizuki's papers solve several celebrated problems in number theory such as the Szpiro conjecture, the hyperbolic Vojta conjecture, and the abc conjecture and its generalization over arbitrary number fields. The IUT theory may open a fundamentally new development in number theory.

Early life

Shinichi Mochizuki was born to parents Kiichi and Anne Mochizuki. 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.

Career

Mochizuki proved Grothendieck conjecture on anabelian geometry in 1996. Mochizuki was an invited speaker at the International Congress of Mathematicians in 1998. In 1999, he introduced Hodge–Arakelov theory. In 2000-2008 he introduced several new theories including the theory of frobenioids, mono-anabelian geometry and the etale theta-function theory for line bundles over tempered covers of the Tate curve.

At the end of August 2012 Shinichi Mochizuki released four preprints which develop inter-universal Teichmüller theory and apply it to prove several holy grail problems in mathematics, and more specifically, Diophantine geometry, including the strong Szpiro conjecture, the hyperbolic Vojta conjecture and the abc conjecture over every number field. The impact of this work might be the greatest in number theory.

In the specific situation of a number field and an elliptic curve with semi-stable reduction 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 (so called theaters which are certain systems of frobenioids). IUT studies deformations of multiplication on arithmetic structures and how much multiplication can be separated from addition. To achieve that, IUT uses absolute Galois and fundamental groups. It applies deep algorithmic results of mono-anabelian geometry to reconstruct the rings and schemes from their automorphism groups after applying theta- and log- links, which are not compatible with the ring or scheme structure. The study of mild indeterminacies introduced for multiradiality purposes leads to applications to the strong Szpiro conjecture and its equivalent forms. IUT goes outside the realm of conventional arithmetic geometry and it essentially extends the scope of arithmetic geometry. Rarely for mathematics, the IUT theory is not only a program in number theory but also its full realization with applications to the proofs of fundamental problems in number theory.

The IUT theory involves a relatively large number of new concepts. The theory is quite complex, and some of its complexity may be related to the absence of appropriate language to describe it. Shinichi Mochizuki documented the relevant progress of the study of the theory by other mathematicians, as well as minor changes in his preprints, in the first two years since 2012, in his two reports December 2013 December 2014. The study of the theory has proved to be a certain challenge for contemporary mathematicians. To assist mathematicians, various surveys and reviews of the theory were produced by several mathematicians and two international workshops in Oxford and in Kyoto were organized in 2015 and 2016.

To study and understand IUT, one has to invest some appropriate time and effort. Currently, very few number theorists have done that. At the same time, the author of the theory has invested a very substantial amount of his time into dissemination of his work. Surveys of IUT were produced by its author, by Ivan Fesenko, , and by Yuichiro Hoshi (currently available in Japanese only). National workshops on IUT were held at RIMS in March 2015 and in Beijing in July 2015. International workshops on IUT were held in Oxford in December 2015and at RIMS in July 2016. These workshops attracted more than 100 participants. Their outputs can be useful to future learners of the theory.

Files of the RIMS workshop include a document which mentions, "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."

• Mochizuki, Shinichi (1997), "A Version of the Grothendieck Conjecture for p-adic Local Fields" (PDF), The International Journal of Mathematics, singapore: World Scientific Pub. Co., 8 (3): 499–506, ISSN 0129-167X 
• Mochizuki, Shinichi (1998), "Proceedings of the International Congress of Mathematicians, Vol. II (Berlin, 1998)", Documenta Mathematica: 187–196, ISSN 1431-0635, MR 1648069 
• Mochizuki, Shinichi (1999), Foundations of p-adic Teichmüller theory, AMS/IP Studies in Advanced Mathematics, 11, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-1190-0, MR 1700772 

Inter-universal Teichmüller theory

• Mochizuki, Shinichi (2011), "Inter-universal Teichmüller Theory: A Progress Report", Development of Galois–Teichmüller Theory and Anabelian Geometry (PDF), The 3rd Mathematical Society of Japan, Seasonal Institute .
• Mochizuki, Shinichi (2016a), Inter-universal Teichmuller Theory I: Construction of Hodge Theaters (PDF) .
• Mochizuki, Shinichi (2016b), Inter-universal Teichmuller Theory II: Hodge–Arakelov-theoretic Evaluation (PDF) .
• Mochizuki, Shinichi (2016c), Inter-universal Teichmuller Theory III: Canonical Splittings of the Log-theta-lattice (PDF) .
• Mochizuki, Shinichi (2016d), Inter-universal Teichmuller Theory IV: Log-volume Computations and Set-theoretic Foundations (PDF) .