On IUT in a few words

Ivan Fesenko

IUT

IUT is Inter-universal Teichmüller theory,also known as arithmetic deformation theory.

Its author is Shinichi Mochizukiwho worked on his theory for 20 yearsat Research Institute for Mathematical Sciences, University of Kyoto.

IUT is not only a new area in mathematics, but it already includes its maintheorem:

the proof of several fundamental conjectures and celebrated problems,

including the abc conjecture:

for every ε > 0 there is a positive number κ such that for every three non-zerocoprime integers a,b,c satisfying a+b = c, the inequality

max(|a|, |b|, |c|)≤ κ C(abc)1+ε holds.

IUT

IUT is Inter-universal Teichmüller theory,also known as arithmetic deformation theory.

Its author is Shinichi Mochizukiwho worked on his theory for 20 yearsat Research Institute for Mathematical Sciences, University of Kyoto.

IUT is not only a new area in mathematics, but it already includes its maintheorem:

the proof of several fundamental conjectures and celebrated problems,

including the abc conjecture:

for every ε > 0 there is a positive number κ such that for every three non-zerocoprime integers a,b,c satisfying a+b = c, the inequality

max(|a|, |b|, |c|)≤ κ C(abc)1+ε holds.

IUT

IUT is Inter-universal Teichmüller theory,also known as arithmetic deformation theory.

Its author is Shinichi Mochizukiwho worked on his theory for 20 yearsat Research Institute for Mathematical Sciences, University of Kyoto.

IUT is not only a new area in mathematics, but it already includes its maintheorem:

the proof of several fundamental conjectures and celebrated problems,

including the abc conjecture:

for every ε > 0 there is a positive number κ such that for every three non-zerocoprime integers a,b,c satisfying a+b = c, the inequality

max(|a|, |b|, |c|)≤ κ C(abc)1+ε holds.

The place of IUT

CFT = class field theoryAAG = adelic analysis and geometry2d = two-dimensional (i.e. for arithmetic surfaces)

Galois theory Kummer theory

CFT

Langlands program 2d CFT anabelian geometry

2d Langlands program 2d AAG IUT

Original texts on IUT

Inter-universal Teichmüller theory (IUT), preprints 2012–2016

I: Constructions of Hodge theaters

II: Hodge–Arakelov-theoretic evaluation

III: Canonical splittings of the log-theta-lattice

IV: Log-volume computations and set-theoretic foundations

The author has invested more than 1000 hours of his time in answeringquestions and giving lectures on his work in Japan.

Original texts on IUT

Inter-universal Teichmüller theory (IUT), preprints 2012–2016

I: Constructions of Hodge theaters

II: Hodge–Arakelov-theoretic evaluation

III: Canonical splittings of the log-theta-lattice

IV: Log-volume computations and set-theoretic foundations

The author has invested more than 1000 hours of his time in answeringquestions and giving lectures on his work in Japan.

Introductory texts and surveys of IUT

The order is chronological,The numbers may or may not correspond to easiness of reading (5 is very easy)

2. A Panoramic Overview of Inter-universal Teichmüller Theory, by ShinichiMochizuki

4. Arithmetic Deformation Theory via Algebraic Fundamental Groups andNonarchimedean Theta-Functions, Notes on the Work of Shinichi Mochizuki,by Ivan Fesenko

1. Introduction to Inter-universal Teichmüller Theory (in Japanese), byYuichiro Hoshi

3. The Mathematics of Mutually Alien Copies: From Gaussian Integrals toInter-universal Teichmüller Theory, by Shinichi Mochizuki

5. Fukugen, by Ivan Fesenko

Materials of two international workshops on IUT coorganized by S&C

1. CMI and S&C Workshop on IUT Theory of Shinichi Mochizuki, Oxford,December 7-11 2015

its online proceedings

2. RIMS and S&C workshop on IUT Summit, Kyoto, July 18-27 2016

Total number of participants of the two workshops: more than 100.

12 mathematicians (from Japan, UK, Poland, USA, France, the Netherlands),and the referees, have studied the theoryor are currently intensively studying it.

Materials of two international workshops on IUT coorganized by S&C

1. CMI and S&C Workshop on IUT Theory of Shinichi Mochizuki, Oxford,December 7-11 2015

its online proceedings

2. RIMS and S&C workshop on IUT Summit, Kyoto, July 18-27 2016

Total number of participants of the two workshops: more than 100.

12 mathematicians (from Japan, UK, Poland, USA, France, the Netherlands),and the referees, have studied the theoryor are currently intensively studying it.

Materials of two international workshops on IUT coorganized by S&C

1. CMI and S&C Workshop on IUT Theory of Shinichi Mochizuki, Oxford,December 7-11 2015

its online proceedings

2. RIMS and S&C workshop on IUT Summit, Kyoto, July 18-27 2016

Total number of participants of the two workshops: more than 100.

12 mathematicians (from Japan, UK, Poland, USA, France, the Netherlands),and the referees, have studied the theoryor are currently intensively studying it.

Materials of two international workshops on IUT coorganized by S&C

1. CMI and S&C Workshop on IUT Theory of Shinichi Mochizuki, Oxford,December 7-11 2015

its online proceedings

2. RIMS and S&C workshop on IUT Summit, Kyoto, July 18-27 2016

Total number of participants of the two workshops: more than 100.

12 mathematicians (from Japan, UK, Poland, USA, France, the Netherlands),and the referees, have studied the theoryor are currently intensively studying it.

On IUT in several sentences

IUT works with elliptic curves over number fieldsand implements deformation of multiplication of various associated rings.

These deformations are not compatible with ring structure.

Deformations are coded in theta-links between theatres,theatres are certain systems of categories, involving two types of symmetry,additive geometric and multiplicative arithmetic.

Ring structures do not pass through theta-links.

Galois and fundamental groups (groups of symmetries of rings) do pass.

To restore rings from such groups (which pass through a theta-link) one usesanabelian geometry results about number fields and hyperbolic curves overthem.

Measuring the result of the deformation produces bounds which eventually leadto solutions of several famous problems in number theory.

IUT addresses such fundamental aspects as to which extent the multiplicationand addition cannot be separated from one another.

On IUT in several sentences

IUT works with elliptic curves over number fieldsand implements deformation of multiplication of various associated rings.

These deformations are not compatible with ring structure.

Deformations are coded in theta-links between theatres,theatres are certain systems of categories, involving two types of symmetry,additive geometric and multiplicative arithmetic.

Ring structures do not pass through theta-links.

Galois and fundamental groups (groups of symmetries of rings) do pass.

To restore rings from such groups (which pass through a theta-link) one usesanabelian geometry results about number fields and hyperbolic curves overthem.

Measuring the result of the deformation produces bounds which eventually leadto solutions of several famous problems in number theory.

IUT addresses such fundamental aspects as to which extent the multiplicationand addition cannot be separated from one another.

On IUT in several sentences

IUT works with elliptic curves over number fieldsand implements deformation of multiplication of various associated rings.

These deformations are not compatible with ring structure.

Deformations are coded in theta-links between theatres,theatres are certain systems of categories, involving two types of symmetry,additive geometric and multiplicative arithmetic.

Ring structures do not pass through theta-links.

Galois and fundamental groups (groups of symmetries of rings) do pass.

To restore rings from such groups (which pass through a theta-link) one usesanabelian geometry results about number fields and hyperbolic curves overthem.

Measuring the result of the deformation produces bounds which eventually leadto solutions of several famous problems in number theory.

IUT addresses such fundamental aspects as to which extent the multiplicationand addition cannot be separated from one another.

On IUT in several sentences

IUT works with elliptic curves over number fieldsand implements deformation of multiplication of various associated rings.

These deformations are not compatible with ring structure.

Deformations are coded in theta-links between theatres,theatres are certain systems of categories, involving two types of symmetry,additive geometric and multiplicative arithmetic.

Ring structures do not pass through theta-links.

Galois and fundamental groups (groups of symmetries of rings) do pass.

To restore rings from such groups (which pass through a theta-link) one usesanabelian geometry results about number fields and hyperbolic curves overthem.

Measuring the result of the deformation produces bounds which eventually leadto solutions of several famous problems in number theory.

IUT addresses such fundamental aspects as to which extent the multiplicationand addition cannot be separated from one another.

On IUT in several sentences

IUT works with elliptic curves over number fieldsand implements deformation of multiplication of various associated rings.

These deformations are not compatible with ring structure.

Deformations are coded in theta-links between theatres,theatres are certain systems of categories, involving two types of symmetry,additive geometric and multiplicative arithmetic.

Ring structures do not pass through theta-links.

Galois and fundamental groups (groups of symmetries of rings) do pass.

To restore rings from such groups (which pass through a theta-link) one usesanabelian geometry results about number fields and hyperbolic curves overthem.

Measuring the result of the deformation produces bounds which eventually leadto solutions of several famous problems in number theory.

IUT addresses such fundamental aspects as to which extent the multiplicationand addition cannot be separated from one another.

On IUT in several sentences

IUT works with elliptic curves over number fieldsand implements deformation of multiplication of various associated rings.

These deformations are not compatible with ring structure.

Deformations are coded in theta-links between theatres,theatres are certain systems of categories, involving two types of symmetry,additive geometric and multiplicative arithmetic.

Ring structures do not pass through theta-links.

Galois and fundamental groups (groups of symmetries of rings) do pass.

To restore rings from such groups (which pass through a theta-link) one usesanabelian geometry results about number fields and hyperbolic curves overthem.

Measuring the result of the deformation produces bounds which eventually leadto solutions of several famous problems in number theory.

IUT addresses such fundamental aspects as to which extent the multiplicationand addition cannot be separated from one another.

Flow of reconstruction

Video animation

Fukugen no nagare = Flow of reconstruction

produced by Etienne Farcot, Shinichi Mochizuki and the speaker

The impact of IUT

It is crucial that one works in IUT with full absolute Galois and algebraicfundamental groups.The use of familiar quotients of these groups such as the maximal abelianquotient or quotients related to the study of representations of these groups arenot sufficient for IUT.

It is important to explore venues of using full absolute Galois and algebraicfundamental groups for investigations in other areas of number theory.

The group-theoretical aspects of pro-finite groups in IUT may lead to newdevelopments in group theory.

Together with strategic developments, there are various option to try to deducemore results from the current or stronger versions of IUT.

One of them is the second proof of Fermat’s Last Theorem from the maintheorems of IUT.

The impact of IUT

It is crucial that one works in IUT with full absolute Galois and algebraicfundamental groups.The use of familiar quotients of these groups such as the maximal abelianquotient or quotients related to the study of representations of these groups arenot sufficient for IUT.

It is important to explore venues of using full absolute Galois and algebraicfundamental groups for investigations in other areas of number theory.

The group-theoretical aspects of pro-finite groups in IUT may lead to newdevelopments in group theory.

Together with strategic developments, there are various option to try to deducemore results from the current or stronger versions of IUT.

One of them is the second proof of Fermat’s Last Theorem from the maintheorems of IUT.

The impact of IUT

It is crucial that one works in IUT with full absolute Galois and algebraicfundamental groups.The use of familiar quotients of these groups such as the maximal abelianquotient or quotients related to the study of representations of these groups arenot sufficient for IUT.

It is important to explore venues of using full absolute Galois and algebraicfundamental groups for investigations in other areas of number theory.

The group-theoretical aspects of pro-finite groups in IUT may lead to newdevelopments in group theory.

Together with strategic developments, there are various option to try to deducemore results from the current or stronger versions of IUT.

One of them is the second proof of Fermat’s Last Theorem from the maintheorems of IUT.

The impact of IUT

It is crucial that one works in IUT with full absolute Galois and algebraicfundamental groups.The use of familiar quotients of these groups such as the maximal abelianquotient or quotients related to the study of representations of these groups arenot sufficient for IUT.

It is important to explore venues of using full absolute Galois and algebraicfundamental groups for investigations in other areas of number theory.

The group-theoretical aspects of pro-finite groups in IUT may lead to newdevelopments in group theory.

Together with strategic developments, there are various option to try to deducemore results from the current or stronger versions of IUT.

One of them is the second proof of Fermat’s Last Theorem from the maintheorems of IUT.