Wang Zhaoyu, trainee reporter of China Science Daily
"Money doesn't matter to top mathematicians, many of whom can easily become billionaires with their brilliant minds. In fact, the beauty of mathematics attracts them even more. Ivan B. Fesenko, a professor at Westlake University's Institute of Theoretical Sciences, told China Science Daily.
Recently, Ivan turned down a bonus. The $100,000 prize was awarded to him for a mathematics paper ($20,000 each) co-authored with Professor Shinichi Mochizuki of the Institute of Mathematical Analysis at Kyoto University in Japan. In addition to Ivan, the other four co-authors donated their awards to the Institute of Mathematical Analysis to continue to support related research.
Ivan et al., who received the IUT Innovator Award, made a new and outstanding contribution to Shinichi Mochizuki's "magic theory", the IUT theory. What do you think of this contribution? "This is a seminal achievement in number theory, a paper that revolutionizes a large field of mathematics, which is very rare in the mathematical world," Ivan said. ”
Ivan Fesenko received the IUT Innovator Award trophy Courtesy of the interviewee
The "Theory of Magic" has been around for 12 years, and only 20 people can read it
As the name suggests, the IUT Innovator Award is dedicated to IUT theory. The full name of IUT theory is "Teichmüller theory", which once shocked the mathematical community with its "thinking method rooted in nature" and "magical" way of thinking. However, Ivan told China Science Daily that although the theory has been proposed for nearly 12 years, only 20 people in the world can currently understand it.
On August 30, 2012, Shinichi Mochizuki posted a paper of more than 500 pages on his website, proposing the IUT theory and claiming to be able to prove an extremely important conjecture in mathematics, the "abc conjecture".
This conjecture is very famous in the mathematical community and was proposed in 1985. This conjecture is about the relationship between integer addition and multiplication, described as three coprime positive integers a, b, and c, where c is the sum of a and b. The "ABC conjecture" has a high status in number theory, and as long as it can be proved, many unsolved problems in number theory can be solved.
After Shinichi Mochizuki's paper was released, the mathematical community was boiling. When mathematicians heard that the "ABC conjecture" was expected to be solved, they came to see the true face of the paper, but they were "dissuaded" by its heavenly language.
This is due to the fact that the language used in IUT theory is a completely new mathematical language that "opens up a new paradigm that goes beyond modern mathematics". Even a trained mathematician can find it difficult to accept a series of unfamiliar symbols for a while. Moreover, to fully understand this theory, it is necessary to read many of the research work of Shinichi Mochizuki since 1995, which adds up to more than 2,000 pages, and mathematicians from all over the world have their own work, and few people have the motivation and patience to fully understand it.
To try to summarize the IUT theory in a paragraph, it is as follows: "The θ bond can be constructed using only multiplicative singularity and the local Galova group as an abstract group, and the 'addition' can be restored from this unary system of 'unilateral representation + group'." Rather, we need to calculate the extent to which deviations have been generated in the recovery process. ”
Moreover, it is not feasible to disseminate IUT theory through academic reports. Shinichi Mochizuki has tried no less than 5 times. For example, in December 2015, the University of Oxford in the United Kingdom held an international conference on IUT theory, and Shinichi Mochizuki participated remotely via Skype in Japan. Shinichi Mochizuki's mother is American, and he grew up in the United States and is fluent in English. However, in such seminars, it is often the audience who is either not interested because they "don't understand" or "can't speak clearly" because of limited time.
Shinichi Mochizuki Image source network
Therefore, Shinichi Mochizuki chose a method of communication that he believed was best – a long-term and continuous discussion between two or a few people to enhance understanding of the problem.
The Japanese mathematician Fumito Kato made an analogy: "Imagine that an alien comes to Earth, and he can only speak the language of the alien." If he speaks in front of a large number of earthlings, surely no one will understand what he is saying, and no matter how many times he repeats it, there will be no progress. However, if an earthling stands up at this time, hoping to understand what the alien is saying, then after they have been together long enough, they can deepen their understanding step by step by accumulating the common points of their thoughts. ”
Ivan Fesenko was one of the mathematicians who had a long-time communication with Shinichi Mochizuki and was able to understand him. "Since January 2015, we've been chatting on Zoom or Skype every month." Ivan told China Science Daily.
Ivan Fesenko Photo courtesy of Westlake University
$100,000 for one paper
The paper that recently won the grand prize of US$100,000, entitled "Explicit estimates in inter-universal Teichmüller theory", was published in 2022 and is the latest development and outstanding contribution to IUT theory.
这篇论文共有5位合著者:望月新一教授和他之前的日本学生Yuichiro Hoshi和Arata Minamide,以及Ivan Fesenko教授和他之前的波兰籍学生Wojtek Porowski。
"This paper has greatly improved the IUT theory and produced hundreds of applications. The most important thing is that through new ideas and painstaking calculations, a valid abc inequality was established, which was not present in the original IUT theory. Ivan said when talking about the reason for the award.
The improved IUT theory is powerful and can turn the field of Diophantine Geometry upside down. Diophantine geometry is one of the oldest parts of number theory, with a history of 2,200 years (the founder of Diophantine geometry was the ancient Greek mathematician Apollonius, about 225 BC, and Diophantine made a greater contribution to the field, so it was named after Diophantine).
Excerpts from the paper. The source of the picture is a public paper
Here's a brief introduction to the paper in an interview:
First, the theorem proved in this paper is that for two positive integers a and b coprime and their sum c=a+b, the following basically valid abc inequality holds: log(abc)<max{1.7·10^{30},6log rad(abc)}.
where log is the natural logarithm; rad(abc) is the product of all prime divisors of abc, so we "ignore" the powers of prime numbers in abc's factorization and treat them as 1; 10^{30} represents 10 to the power of 30.
On the ordinary scale, the addition and multiplication of positive integers are very different. This theorem, however, tells us that on the huge scale of exp(1.7·10^{30}), addition and multiplication are almost "integrated". Here exp is an exponential function with e as the base.
This theorem can be applied to hundreds of integer equations to find their integer solutions.
Second, using some computational number theory results obtained by Preda Mihailescu, a mathematician at the University of Göttingen in Germany, and the latest classical results on the lower bound of the possible integer solutions of Fermat's equation, our theorem is applied to Fermat's equation and a new proof of Fermat's grand theorem is obtained.
We didn't use anything from Andrew Wiles' proof of Fermat's Great Theorem in 1995 at Princeton University in the United States, because our approach was based on completely different ideas. Wiles's method can't be used to study other integer equations to find their integer solutions, but our method can.
In the future, collaborative work involving three types of number theorists (far-abelian geometers, classical number theorists, and computational number theorists) will become popular. It's good for people from different directions to work together.
Back in 2019, before the paper was published, authors such as Ivan and Shinichi Mochizuki were invited to announce the results of the paper at the British Prime Minister's Office, but they did not accept the invitation.
Ivan Fesenko and the heads of the mathematics departments of the Universities of Cambridge, Oxford, Warwick, Edinburgh and Imperial College London in front of the Prime Minister's residence
"Money doesn't matter to top mathematicians"
Talking about the first time he met Shinichi Mochizuki, Ivan told China Science Daily that it was at a conference in Tokyo in 1998.
In 2014, Ivan decided to start working on IUT theory, and in December 2014, he visited Kyoto University in Japan, where he had several conversations with Shinichi Mochizuki and young mathematicians from the Institute of Mathematical Analysis at Kyoto University.
之后，Ivan撰写了IUT理论的第一篇国际评论文章《通过算术基本群和非阿基米德θ函数的算术形变理论（Arithmetic deformation theory via arithmetic fundamental groups and non-archimedean theta-functions）》。
In March and December 2015, July 2016 and September 2021, Shinichi Mochizuki and Ivan co-organized four international IUT workshops.
"Mathematics is very international, mathematicians can travel from one country to another, to be with their local math peers, just like the locals, and to learn about the customs and traditions of many different countries." Ivan said.
"Seeing the Universe in the Language of Mathematics" is an easy-to-understand introduction to IUT theory
In order to encourage more people to study IUT theory, IUGC has established the "IUT Innovator Award".
IUGC is an international research institution, similar to the Clay Institute for Mathematics and the Simmons Foundation, both funded by math magnaires. The IUGC is funded by Nobuo Kawakami, the founder of Japan's DWANGO company, which has close ties to Studio Ghibli, which produced famous animated films such as Spirited Away.
The IUT Innovator Award will be awarded annually to the best paper that has made significant progress in IUT theory and related fields from 2024 to $20,000 to $100,000 for a period of 10 years.
Ivan's paper, co-authored with Shinichi Mochizuki and others, won the top prize of $100,000.
After receiving "the highest award ever given for a single mathematical paper," Ivan declined to accept the prize.
He believes that the work of mathematicians is not for money, and the source of motivation for engaging in mathematical research is to be curious about revealing new and beautiful mathematical theorems and laws, "I follow the tradition of mathematicians in my city. In my hometown of St. Petersburg, there is a mathematician Grisha Perelman who refused to accept the Fields Medal (less than $25,000) and the Cray Prize ($1 million). ”
According to Ivan, the biggest significance of the Mathematics Paper Award is to attract attention and motivate young researchers to devote years to solving difficult mathematical problems, especially "difficult households" like IUT theory, which few people can fully understand.
Many groundbreaking theories, such as Albert Einstein's theory of relativity, are so advanced that others need years to learn and further develop such theories. The same is true for IUT theory. To understand it, one needs to study the Far Abelian geometry that Japan has developed over the past 30 years.
Ivan estimates that it takes more than 3,000 hours to learn far-abelian geometry and IUT theory. And many senior mathematicians can't spare so much time because of their duties. He hopes that the establishment of this award will encourage more young mathematicians to learn IUT theory.
"Since we worked hard to help a new generation of mathematicians learn IUT theory, there are now about 20 IUT theory experts around the world, from Japan, Russia, the United Kingdom, France, Poland, and more recently, mathematicians from the United States. I hope that there will also be IUT theorists in China as soon as possible. Ivan said he had already given an academic presentation on the study at Zhejiang University, and that one of his Chinese PhD students would begin studying IUT theory in September.
Poster of Ivan Fesenko's presentation at Zhejiang University Courtesy of the interviewee
"In the history of mathematics, groundbreaking theories were first understood by a very small number of contemporary mathematicians, usually no more than 10 people." Ivan said.
While IUT theory is so difficult that it is "not even necessary to understand" for most people, who can guarantee that pure mathematical theory will not someday burst into unpredictable power? After all, few people know that the IC cards we use every day use the elliptic curve theory in the field of pure mathematics in ancient times; Modern computers, smartphones, and even the field of "deep learning" are assembled with mathematical theories.
Perhaps, one day in the future, through the tireless efforts of mathematicians, the IUT theory will truly demonstrate its secular value.
Resources:
1.论文doi:10.2996/kmj45201
2. [J] Fumito Kato, "Looking at the Universe in the Language of Mathematics"