Chapter 16 Hello, Wiles Zhang Lixian

Andrew Wiles giving a lecture at Peking University Photo by He Yanguang On August 29, 2005, I had the wisest dinner of my life. Among the twelve people present, there are five academicians of the Chinese Academy of Sciences, as well as the dean, vice-dean and a professor of the School of Mathematics of Peking University. They were greeted by Andrew Wiles, a foreign academician of the American Academy of Sciences. I joked at the dinner table that even though I dropped a lot of points, this is still the dinner with the highest average IQ ever in Beijing. "Super star!" On August 16, Zhang Jiping, Dean of the School of Mathematics, Peking University, described this guest who was about to visit to me.

Andrew Wiles. The most famous mathematician in the world today, because he solved the most famous mathematical problem in the world - Fermat's last theorem. In June 1993, Andrew Wiles gave three academic lectures at the Newton Institute of Cambridge University. At the end of the last lecture, he completed the proof of Fermat's last theorem.This news quickly appeared on the front pages of the world's major newspapers, and even spread the word in the mathematics world.The news spread to Paris at the first time, and several mathematicians raised their glasses to celebrate, including Titz, the winner of the Wolf Prize, French mathematicians Brouet, Puig, Rouquier, and the French mathematicians at the Ecole Normale Supérieure de Paris. Zhang Jiping, a visiting professor of the school.

Twelve years later, Andrew Wiles, who is already the dean of the Department of Mathematics at Princeton University, came to China to visit the School of Mathematical Sciences of Peking University, where Zhang Jiping was the dean. "Don't do this question again" In the evening of August 28, Zhang Jiping arrived at Beijing International Airport by car. He was accompanied by me, photographer He Yanguang, and Chen Lu, an undergraduate student of the 2002 class of the School of Mathematics, Peking University.The Wuhan girl majoring in basic mathematics is holding a bouquet of lilies and red roses in her arms, ready to dedicate the flowers to her idol: "Many classmates in our college regard him as an idol."

Director Ma Micang of the Department of International Cooperation and Exchange of the Ministry of Education has already led several colleagues to make arrangements here.Thanks to his arrangements, Andrew Wiles was able to set foot on the VIP passage. "The Ministry of Education attaches great importance to and supports such an important guest," he said. At 19:40, Air China flight CA982 landed 100 minutes late, and Dean Zhang Jiping, who was waiting at the exit, said to me, "I haven't seen him for almost ten years, so you can help me keep an eye on him." I said to him and Director Ma: "I just saw a black-and-white photo of him more than ten years ago. The book "Fermat's Last Theorem" says that he is a gentle and elegant person."

Passengers began to pour in one after another, and almost at the same time, several people all recognized the VIP who walked out of the passage with their daughter.There are those who may be called "word types" who seem to be born to interpret certain words—Andrew Wiles is recognizable by the word "gentle" alone. "Welcome to Beijing." Zhang Jiping said, and the heads of mathematics departments of two world-renowned universities held hands together.This is the first time for fifty-two-year-old Andrew Wiles to set foot on Chinese soil, and also his first visit to Asia.

In the VIP room, I watched the British man who had become a legend.He has thin blond hair, a pale face, and is nearly 1.8 meters tall, but he looks a bit thin-it was later learned in an interview that he does not live in a study, but likes golf, swimming and skiing.If you know the American actor James Cromwell, you can imagine him, and Wiles is almost Cromwell shrunk down and softened.But this mathematical genius with special intelligence, it seems that his head is not special, even smaller than the average person's, on the contrary, his fingers are extraordinarily fair and long.I believe his palms, if fully extended, would be longer than the height of his head.

Andrew Wiles was born in England in 1953, graduated from Oxford University in 1974, and then obtained a doctorate from Cambridge University. He came to the United States to teach at Princeton University in 1980.Academician Tian Gang from the School of Mathematics at Peking University is also a professor of the Department of Mathematics at Princeton University. He described Andrew Wiles as a colleague to me: He is very low-key and does not show up often. We respect him very much, and he is equally serious about his work. For example, when admitting students, he will read each student's materials very carefully.

In the next few days in Beijing, everyone who came into contact with Andrew Wiles continued to appreciate his gentleness, including the passers-by who passed him in the bustling tourist attractions.His face is basically an exchange of two expressions: a gentle smile and concentrated thinking; his eyes shoot a kind and shy look behind the lens; , the sound is also constant below a certain number of decibels - he likes the classical music of Bach and Handel.Seeing him with his 15-year-old daughter Helen, Wu Qiaoming, a teacher at Peking University's International Mathematics Research Center, said that it was obvious that he loved his daughter very much.And it seems to me that he often makes claims about his daughter as if she were her friend.

On the afternoon of August 29, Andrew Wiles, who had seen the Temple of Heaven, Tiananmen, and the Forbidden City, eaten Peking duck and steamed mandarin fish, and taken taxis and buses, sat by the lake in Beihai Park and accepted my interview. ——After the interview, Dean Zhang Jiping asked me: "Have you experienced the speech of a real mathematician?" Yes.His eyes often pass me across the street, looking into the distance, his conversation is full of pauses and contemplation, and what he says is as simple and thoughtful as mathematics. "During the research process of Fermat's last theorem, you relied on paper, pen and mind, and did not use a computer. Have you started using a computer now? Will computers play an increasingly important role in the work of mathematicians in the future? ?"I asked.

"Now I just use the computer to find some examples and verify them, as well as information in some special cases. I never use a computer to do direct proofs. Different mathematicians have different attitudes towards computers, and I rarely use computers myself. , but one of my students used a computer to solve a very important problem. Of course, the characteristics of the problem he solved with a computer were very different from mine. There are very few problems in the field of mathematics that can be solved by a computer. "His answer seemed to be narrating a theorem. A question that must be asked, the impression of China.He just said simply: "It's more lifelike than I imagined." And what is his impression of Chinese students? "I have never taught graduate students from China. In the United States, the vast majority of Chinese students do geometry and analysis, and fewer study number theory. I have very little contact with them."

Humor bursts through his simple answers. "Through the proof of Fermat's Last Theorem, the whole world is paying attention to you. So, has the public's attention influenced and changed your life?" "For me personally, the biggest change is that I don't have to do Fermat's last theorem anymore." "In the Hollywood movie "A Beautiful Mind", after the protagonist John Nash won the Nobel Prize, a group of mathematicians presented pens to him in the conference hall as a way of paying tribute. Is this the tradition of Princeton University? Have you received Have you ever had a pen?" "Fictional, of course." He let out a burst of laughter, "But it's a good idea and should be done from now on." The simplest and most thorough answer appears here.I asked, "Would you mind telling me how you and your wife fell in love and got married?" "We met in Princeton, we got married in Princeton." "I have solved Fermat's last theorem" It's time to go back to Andrew Wiles' tortuous and brilliant moment. After the death of Fermat, a French mathematician in the 17th century, his notes on a mathematics book were discovered, resulting in the most difficult problem for future generations to answer: "It is impossible to write a cubic number as two the sum of cubes; or write a power of 4 as the sum of two powers of 4; or, in general, the impossibility of writing a power higher than 2 as the sum of two powers of the same power." It's a mystery articulated in words familiar to every middle school student, yet it's baffled the brightest minds in the world.Even more excruciatingly, Fermat also left a note suggesting that he already had a solution, although he did not write the proof. As a result, generations of mathematical geniuses continued to challenge this conjecture.More than three hundred years have passed, and this theorem still lacks a complete and rigorous proof.No problem in any discipline, expressed so simply and clearly, has been able to stand up to the onslaught of advanced knowledge for so long. Andrew Wiles was fascinated by Fermat's last theorem when he was ten years old, and has since chosen mathematics as his lifelong career.But after I went to college, “I kept thinking that many people in history have thought of all conceivable solutions, but they didn’t solve Fermat’s Last Theorem in the end, so I felt that I had to learn more advanced mathematics. From the graduate stage, I put I put more energy into broadening my horizons." In the interview, he said, "It seems that I have temporarily left the last theorem." In 1986, Andrew Wiles decided to challenge Fermat's Last Theorem.He first spent eighteen months collecting the necessary mathematical tools for this impact, and his comprehensive estimate is that what he will do next will be a dedicated effort that may last as long as ten years. I asked: "At that time, many mathematicians felt that this problem was difficult, or that the hope of solving this problem was very slim, so they gave up, but you persisted for seven years. When you started your research, were you confident? Did you know it? Do it if you are not sure?" "Historically, not many serious mathematicians decided to study Fermat's Last Theorem, because they first had to consider whether the development of mathematics provided them with enough tools under their historical conditions. Reach the level of solving this problem. By 1986, I decided to study the last theorem. At that time, most people thought that the tools in their hands were not enough, and I thought there was hope." "So I'm not romantic, but very realistic." A common theory is that Wiles conducted his research in complete secrecy, keeping no one from knowing what he was doing and communicating with no one.During those seven years, only his wife knew what he was doing. In this interview, Andrew Wiles clarified this statement to me face to face: "In fact, at the beginning, I still told some colleagues. But after they knew, as long as they saw me, they kept asking about the progress. situation, which caused me a lot of stress and distraction. So I thought it would be better not to speak out. And I realized that it would take a long, long time to solve this problem. During this process, people kept asking And, I will bear a lot of pressure, just like a child in the process of growing up, if he is always asked what's wrong and where he is, then the child will be embarrassed." In this way, he gradually turned into a kind of fighting in a secret state.Seven years later, Andrew Wiles said to his wife: "I have solved Fermat's last theorem." After the Newton Institute report in June 1993, while mathematicians around the world toasted him, Andrew Wiles' paper submitted to the journal Mathematical Inventions was undergoing rigorous review.The reviewers ran into a problem in Chapter 3 that prevented Wiles from being able to guarantee that a certain method would work, as he had originally thought.He has to strengthen his proof. Just two weeks before her birthday, Andrew Wiles' wife told her husband that the only birthday present she wanted was proof that he was right. Andrew Wiles didn't make this birthday present on time. As time passed, the people who had just cheered hung their hearts again.For more than three hundred years, among the many attempted proofs of Fermat's last theorem, no one has been able to remedy the loopholes that have occurred.The most recent failure of time was on March 8, 1988, when the Washington Post and the New York Times claimed that Yoichi Miyaoka of the University of Tokyo had discovered the solution to Fermat's Last Theorem, but had to withdraw it a month later.Could it be that Wiles could not escape this fate? John Lynch, editor of BBC Television, said: "It's hard for me to imagine that Andrew wouldn't be another tombstone in that mathematics graveyard." This time the proof work was carried out almost under the attention of the whole world. It is said that there were only two things discussed by colleagues at Princeton University at that time: the Simpson case and Wiles' proof.Andrew Wiles recalls this time now, still vividly: "I was very happy in the first stage of secret research, and I was enjoying the process. In the second stage, I felt like I was under the public , At the conferences of the mathematics community, many people keep asking me, I don’t like this state.” In his most desperate moments, he was even ready to publicly admit that his proof was flawed.When Professor Tian Gang accepted my interview, he once commented on Andrew Wiles as a "brave man" because he was under the pressure of being ruined at this time.Referring to this passage, I asked Andrew Wiles: "Do you agree that you are a 'brave man'?" His answer was: "I just know that this problem can be solved, and I hope it can be solved. Even if I admit that my proof is flawed, there will be hundreds of people who see hope and see that we have enough Good tools, they will further solve this problem. Maybe they will use it for some time, eight years, ten years, but the tools are already there, and the direction is already there.” Andrew Wiles was not wrong in his judgment.Fourteen months later, he submitted his second paper to the Annals of Mathematics, consisting of two papers, "Modular Elliptic Curves and Fermat's Last Theorem" and "The Ring Theoretic Properties of Certain Hecke Algebras", this time to prove that There is no longer any doubt. Mrs. Wiles finally got what she wanted for her birthday. "How does your wife react to this year-overdue birthday present?" I asked. He smiled: "She is happier than she got this gift a year ago." "Fermat could not have solved this problem" Jiang Boju, a professor at the School of Mathematics at Peking University and an academician of the Chinese Academy of Sciences, commented that Andrew Wiles' proof of Fermat's Last Theorem was "the most brilliant mathematical achievement of the twentieth century". The accolades followed, and in 1996 Wiles and Robert Langlands shared the $100,000 Wolf Prize.The Langlands program proposed by Langlands is a conjecture that unifies the proofs in various fields of mathematics, and Wiles unifies the elliptic curve and the modular form through the proof of the Taniyama-Shimura conjecture. This program is infused with life—a problem in one domain can be solved by a corresponding problem in a parallel domain—a breakthrough work that could usher mathematics into yet another golden age of hard problem solving. In 1998, the International Congress of Mathematicians was held in Berlin, and the Fields Medal, the "Nobel Prize" in mathematics, awarded Andrew Wiles a special award.Zhang Gongqing, a professor at the School of Mathematics of Peking University and an academician of the Chinese Academy of Sciences, attended the conference, witnessed Andrew Wiles' award and listened to his report. The Fields Medal is named after the Canadian mathematician John Fields. The motivation for establishing the award is to encourage young mathematicians, so it is specially used to reward outstanding achievements of young mathematicians under the age of forty.Andrew was just over forty years old when he successfully proved Fermat's Last Theorem.Professor Zhang Gongqing explained to me that the solution of this three-hundred-year-old problem is a milestone in the field of mathematics, so the International Congress of Mathematicians awarded him a special award. This special case made Andrew Wiles the history of the Fields Medal. The only special award winner in the world. "If Andrew Wiles is not given this award, it will be a loss of the Fields Medal." Zhang Jiping said. The question that hangs over the minds of ordinary people is that Fermat, who raised this question, once wrote "I have a very beautiful proof of this proposition, and the space here is too small to write", and more than three hundred years ago Fermat In his time, he did not have the tools used by Andrew Wiles to prove the last theorem: elliptic curves, modular forms, Taniyama-Shimura conjecture, Galois group theory, Iwasawa theory and Kolivakin-Fletcher method.So, what method did Fermat himself use to prove his conjecture? Andrew Wiles had 200 pages for his first submission to the journal Mathematical Inventions, and 130 pages for his second submission to the Annals of Mathematics.Dean Zhang Jiping said that this is the only impeccable, most rigorous and most economical proof of Fermat's last theorem. So is there a simpler proof? Zhang Jiping said with a smile that there is a saying about Fermat's last theorem in the mathematics circle: any proof that can be understood is wrong. "Have you ever wondered what Fermat's method of proof was and how many pages it would have been if he had written a paper?" I asked Andrew Wiles. "Fermat never wrote a paper." He replied flatly. "Many ordinary readers will have this question, can Fermat himself really prove Fermat's last theorem?" I continued to ask. Professor Zong Chuanming, who acted as a translator, told me directly: It is generally believed in the mathematical circles that the solution Fermat mentioned would not exist. After listening to Zong's translation, Andrew Wiles replied affirmatively: "No, Fermat can't solve this problem." "Do you think there is another solution?" "Although anything is possible, I still don't think there will be a simpler proof than mine. Maybe my proof can be simplified, but the basic idea and complexity of the proof of Fermat's Last Theorem are different. It will change." After Fermat's last theorem was proved, another ten years have passed. As a mathematician, Andrew Wiles' life has not changed. It is still the same as before, getting up in the morning, going to the office, and studying new mathematical problems. "What do you think is the most interesting problem in mathematics today?" I asked. "Of course it is the Riemann hypothesis." The Riemann Hypothesis is one of the seven "Century Problems" known to everyone in mathematics, and was proposed by German mathematician Riemann in the 19th century.A brief description of the hypothesis I found on the Internet is: the distribution frequency of prime numbers is closely related to the behavior of a carefully constructed so-called Riemann Zeita function z(s). The famous Riemann hypothesis asserts that the equation z(s )=0 all meaningful solutions lie on a straight line. "Are you working on the Riemann Hypothesis now?" "Sometimes I also think about this problem, but it takes very little time. When I started to prove Fermat's Last Theorem in 1986, the method discovered by others happened to be my field of expertise and I was able to solve it. But so far the world No one on the Internet can propose any direction for the Riemann Hypothesis, or what field it belongs to. No one knows whether the Riemann Hypothesis should be proved by a number theorist or a function theorist. If the cracking tool is in the field of number theory, I Of course, more time will be spent on research.” He described his situation after becoming famous all over the world: "Fermat's last theorem allowed me to have a lot of contacts with people outside the mathematics community, and to feel how others feel about mathematics. I received invitations from all over the world—including this time Peking University—I am very happy to meet very friendly people from all over the world, but I don’t use many opportunities like this.” It is said that a joke circulated among the teachers of Princeton University: Andrew Wiles was suggested to do commercials, including men's underwear.So I asked him for confirmation: "Have you really received an invitation to shoot an advertisement?" His answer was beyond my expectation: "There was indeed, but the name of the clothing company was 'GAP' (gap means 'flaw'), so I declined." On July 1 this year, Andrew Wiles became the dean of the Department of Mathematics at Princeton University.Princeton's mathematics department is known as "the place that defines what good mathematics is." When I asked him if he liked this position, he replied in a calm tone: "Only when I finish this position, I can conclude that I like it." dislike." Back in 1963, ten-year-old Andrew Wiles saw a book called "The Big Question" in the library on Milton Road in Cambridge, from which he learned about the unrivaled problem, "From that moment on, I knew I will never let it go, I have to work it out."I asked him if he hadn't read that book... "I'll still study math," he said. That night, he briefly talked about his impression of Beijing to his colleagues at the School of Mathematics of Peking University: the Forbidden City where the emperor lived was much grander than he had imagined before, but, "I don't want to be an emperor, I would rather be a mathematician." "This question chose me" "The Terminator of Fermat's Last Theorem and the only winner of the Fields Medal in the world will visit Peking University," Professor Chen Dayue from the School of Mathematics of Peking University posted the news of Andrew Wiles' visit to Peking University on the Peking University campus network.Before Wiles arrived in Beijing, there were already more than 5,000 visitors in two days. "Wiles' visit is the honor of Peking University." Zhang Gongqing said, "He will help Peking University improve its academic level, and his spirit of concentrating on academic research for seven years will also inspire the young people of Peking University." At 1:30 p.m. on August 30, Qin Jin, an undergraduate student of grade 02 of the Department of Mathematics of Beijing Institute of Technology, came to occupy a seat in the Sunshine Hall of Yingjie Exchange Center of Peking University. Halfway through, more than 300 seats were packed. At four o'clock, Andrew Wiles began his public speech to applause.On the podium, he no longer groaned like he did when he was being interviewed, but was as calm and at ease as if he had returned to his own kingdom, and his fluent English had a musical rhythm.In one hour, he reviewed the history of Fermat's last theorem, and the brilliant course of over three hundred years in the mathematics world to overcome Fermat's last theorem.Then he raised some unsolved problems in the field of mathematics, and ended with the abc conjecture.The slide finally froze on a set of huge numbers, and there was a burst of knowing laughter in the Sunshine Hall. During the exchange time that followed, Liu Qi, a 2003 direct PhD student from the School of Mathematics, Peking University, asked Professor Wiles why he chose this research topic that took seven or eight years.Wiles replied, "I didn't choose the question, the question chose me." Yuan Xili, who graduated from the Department of Mathematics of Peking University in 1995, happened to come to his alma mater today and encountered this speech.After listening to the speech excitedly, he stood next to his seat and said with deep emotion: "There are too few mathematicians who devote themselves to theoretical research without seeking practicality." The academic experience of Andrew Wiles made his colleagues in Peking University very impressed.Academician Ding Weiyue, director of the Institute of Mathematics of Peking University, said: "Professor Wiles spent seven years dedicated to solving a difficult problem. Nowadays, few people can endure this kind of loneliness. Many people are eager for quick success and quick success. Everyone should ask Andrew Wiles to study.” Liu Huarong, deputy dean of the School of Mathematics, put it more bluntly: “His devotion to science is worth learning from.” Some mathematicians question our scientific research system.Jiang Boju said: "In China, even if someone has the wisdom to decipher Fermat's last theorem, I am afraid that they may not succeed. Nowadays, everyone is busy with evaluation and must produce some short-term results. A lot of energy and wisdom are wasted." "Three hundred years of difficult problems and seven years of investment, for us, evaluation alone takes up time." Academician Wen Lan of the School of Mathematics of Peking University said. "Like Andrew Wiles, who has devoted himself to researching a problem for seven years. During these seven years, he has not produced any results or published papers. If he wants to be in China, he will have no salary, no allowance, and no funds." Zhang Gongqing added . "In the final analysis, the scientific and technological system needs to be reformed." Zhang Jiping summed it up in one word. On the morning of August 31st, Andrew Wiles gave another special academic report at the School of Mathematics, Peking University.In a classroom with less than 100 people, Andrew Wiles communicated with his colleagues in the Chinese mathematics field in a language that only a very small number of people in the world can understand. Mathematicians are working on topics that are getting further and further away from people's ability to understand, but Andrew Wiles doesn't think he's alone.On the contrary, he felt that Fermat's last theorem he studied was a rare well-known problem in the field of mathematics, including many laymen who were interested.You must know that there are many mathematical problems, and it is not easy to explain clearly to colleagues and interest them, but Fermat's last theorem is much better. Other mathematicians did not feel alone either, and although they worked on subjects less well known than Fermat's last theorem, they felt fulfilled and enjoyed working on them as well. "It's like a coal digger, does he feel lonely when he's working?" Zhang Jiping asked me. On September 1, Andrew Wiles traveled to Hong Kong to receive the 2005 Shaw Prize in Mathematical Sciences and a prize of US$1 million.The award is also in recognition of his great contribution to the final solution of Fermat's Last Theorem. "This is a major event in the history of Chinese mathematics." Zhang Jiping commented on Andrew Wiles' trip to China.