After the closing ceremony of the second phase of training, Qiao Yu and Professor Zhou got on the train back to the capital.

Qiao Yu sat at the window and looked at the scenery outside the window with nostalgia. My hometown is like this. If you stay here for a long time, you will be dissatisfied with many things. But once you leave, the beautiful side will appear.

Distance can always produce some unique filter beauty, just like the beauty software on your mobile phone. This probably shows from the side that the human brain and computers have something in common, and the more advanced part of the human brain is that it also has its own Adaptable software.

"The team will have some activities next, but I have talked with your instructor, Professor Tian, ​​so you don't need to participate. In addition, we have created a group and we have not included you. Don't have any other ideas. The main thing is that you leave It's so close. It's very convenient to notify you.

Zhou Liang was talking next to him, but Qiao Yu just went in and out of one ear and didn't pay much attention. Participating in the IMO competition can probably be understood as an obsession that has a beginning and an end. In fact, Qiao Yu also knows that winning a gold medal is very important to him. Doesn't make much sense.

But Professor Schultz had won a gold medal that year, and Professor Pan across from him had also won a gold medal. If he didn't win one, he always felt like he was sorry for his youth. As for the activities before going to Australia, Qiao Yu had no interest in participating.

It's nothing more than brushing up on questions.

The most important thing is that he is too lazy to expose Professor Zhou's little thoughts.

Why is he so close, so we don’t include him in the group? It’s just because he’s afraid that he’ll cause trouble again and hit a few friends who are competing together. Of course, this can’t be blamed on Zhou Liang for thinking too much, it’s mainly because of his small size. The psychological quality of my partners is so poor.

Faced with a small setback, he is not calm at all. Playing a game can make his mentality collapse once. It will probably be difficult for him to be the leader or coach of the team. Eliminate him as a guy who may bring instability. Outside, it is indeed the simplest way to optimize.

However, the constant nagging still gave Qiao Yu a headache, so he simply responded: "Don't worry, Professor Zhou, you really don't need to explain so much to me. I can understand your arrangement, and it's for everyone's benefit this time." .”

Professor Zhou showed a smile on his face and said with emotion: "Yes, the main reason is that you don't have time. Professor Tian is also worried about you. He knows that you are interested in the geometric Langlands conjecture recently and have collaborated with Huaqing several times. There are a series of related lectures and seminars held today, and many experts are specially invited from abroad, which is enviable.

Qiao Yu finally retracted his gaze from the window, turned to look at Professor Zhou next to him, and asked in surprise: "Is there such a thing?" Zhou Liang also looked at Qiao Yu in surprise and asked: "You don't know?

Qiao Yu suddenly felt the warmth of spring.

Sure enough, his mentor and Grandpa Shi still loved him. Although they didn't tell him explicitly, they had been working silently behind the scenes. Qiao Yu also deeply realized the value of what Lao Xue said to him back then.

Sure enough, if the instructor can handle things, he can do whatever he wants.

How could such a large-scale seminar be held as easily as a tutor who could not handle the task? After all, inviting relevant professors and scholars from abroad requires money, let alone inviting people to eat and stay, which is all money. .In addition, you have to keep up with the appearance.

Of course, we have to thank China for its good development over the years, and its external publicity work has also kept up. Otherwise, just talking about money would be useless. In short, at this moment, Qiao Yu really understood how important having a good tutor is to his studies. How helpful it is.

Lao Xue, sincerely do not deceive him!

On the other side of the ocean, in Madison, the state capital of Wisconsin, is the University of Wisconsin-Madison.

Pan Jingyuan was standing outside the conference room, looking at the tutor who was surrounded by many scholars and still discussing some detailed issues. The lecture just now had ended, and his tutor Dennis was the keynote speaker.

The content of the lecture is naturally about the proof of the geometric Langlands conjecture.

In fact, he could also participate in the discussion, but since chatting with Mr. Yuan that day, Pan Jingyuan has been feeling a little uneasy. This is a very strange feeling.

Although he knew that Qiao Yu was just a student who had just come into contact with the Langlands Conjecture of geometry, some of the questions he asked did not need to be taken too seriously. But the questions Qiao Yu asked that day did make him feel a little uneasy. .

After all, it has been more than ten years since this project was established, and it has been six years since he was invited to participate. He does not want the results of these six years of hard work to become a joke, and it will take several more years to make up for it.

This uneasy feeling even made him re-examine these five papers, trying to make sure that the situation that Qiao Yu judged was impossible. However, this obviously took time, and in fact Pan Jingyuan had already started to do this work.

This is also the reason why he is getting slower and slower in replying to Qiao Yu's messages.

This feeling of uneasiness reached its peak a few days ago when Mr. Yuan forwarded the questions raised by Qiao Yu. He wanted to talk to his tutor about Qiao Yu's thoughts, but he never made up his mind.

After all, he has only been involved for six years and is already very nervous. For a mentor, he has spent fifteen years in it. He is not sure what Dennis's attitude will be when he hears such remarks, and whether it will have the opposite effect. .

The relationship between students and tutors is never necessarily a good one. Because some conflicts turn into enemies, whether in China or abroad, it is just that in most cases, everyone has a tacit understanding not to expose it.

However, Huaqing is planning to organize a series of seminars on the geometric Langlands conjecture with Yanbei University, and has invited many foreign professors. There is a high probability that the news will reach the ears of the instructors.

Of course Pan Jingyuan knew who this series of seminars was for.

To be honest, Pan Jingyuan really had no objections to this decision. Of course he knew what Qiao Yuzhen's growth meant to the Chinese mathematics community.

Over the years, although China has developed well, its voice in the academic world is not very big, at least compared to the size of the country, it is very unreasonable. But there is no way, the academic world is often like the Olympics. If you want to speak and be listened to, it ultimately depends on your strength.

Why is it generally recognized that English journals are more authoritative? Because a series of tools used in mathematical research are basically summarized by Westerners. The main theorems encountered in college classes are basically named after Westerners.

Abel's theorem in algebra, Gauss' theorem, Fermat's theorem in number theory, Euler's theorem, Biggras' theorem in geometry, Heron's formula, De Morgan's law in analytic geometry, L'Hourbid's law in calculus...

In addition to Markov's random process, Yebes' theorem for calculating probability...

The efforts of our predecessors have allowed the modern Western world to firmly grasp the right to speak in academic circles. Even if you don’t know English, especially English for mathematics majors, you cannot accurately understand the mathematical ideas of those predecessors.

Huaxia is a latecomer, so if it wants to gain more say, someone must be able to take the lead in the world of mathematics in the future. For example, in the future, there will be a mathematician in Huaxia whose achievements are comparable to those of Newton and Euler. Even if he speaks in a low voice, he can Deafening.

There has never been a pure land in the real world, or places like the Peach Blossom Spring only exist in people's imaginations. People's intelligence and social nature have long been determined. Anyone who has achieved success will definitely stick together and raise the entry threshold for related fields.

School valves have always existed. The more developed various operations are, the more common this phenomenon is. The consideration of higher-ranking people is never to ban this kind of thing, but how to use this inevitable phenomenon to Promote the development of science.

For example, elite governance, through the most basic education, directly physically separates the elites from the children of ordinary people, and achieves the strategic goal of starting children in the third, sixth, and ninth grades.

Pan Jingyuan felt that Qiao Yu had hope of becoming such a scholar. But the current situation made him feel confused.

But Pan Jingyuan made up his mind today. Instead of waiting for Dennis to hear Qiao Yu's "fallacies" from others, he should take the initiative to report them. And he had already thought of a way to deal with it.

Finally, Dennis ended the discussion with several professors and walked out of the conference room in high spirits. When he saw Pan Jingyuan waiting for him, he couldn't help but made a joke: "What's wrong? Pan, can't this be the case today?" Does it get you excited? Don’t worry, there will be more important meetings in the future.”

Pan Jingyuan shook his head, then smiled and said: "No, Dennis, actually I benefited a lot from this meeting. I just want to tell you something... By the way, do you still remember what you planned to let me review before? A paper?"

Dennis replied: "Of course, that paper left a deep impression on me. I also exchanged opinions with Schulz specifically about this paper. In fact, he also spoke highly of this paper.

But I haven't issued a final review opinion yet. You know, I want to issue a more professional review opinion. The author is still young, and he may not know that publishing a paper of this level requires some patience."

Pan Jingyuan said: "No, you misunderstood. It's not that the author of the paper knows that you are one of the reviewers and wants me to urge you to speed up the review. It's that he has recently become interested in the geometric Langlands conjecture, and is targeting our The work raises some questions.

Phew...I have actually been thinking about how to systematically answer his questions for the past two days, so that I can give him a very definite answer so that the child will not waste his efforts in this regard."

"Oh?" Dennis was obviously interested and said, "I heard that the author of that paper, Qiao Yu, was still very young. This is what Schulz told me. Obviously he knew something that I didn't know.

Information, so can you tell me how old that Qiao Yu is this year?"

"Sixteen years old." Pan Jingyuan replied.

"Sixteen years old?" Dennis frowned and looked at Pan Jingyuan again, with some disbelief: "You mean a sixteen-year-old student not only independently completed a high-quality and significant paper, but also

Did you understand our paper and ask some questions that you personally find difficult to answer?"

"I know this sounds unbelievable, but it is true." Pan Jingyuan nodded, and then added: "We have had some exchanges on academic aspects. He is a young man who is very easy to learn and studies well."

Things are really fast.

After about a week or so, I already felt that many of the questions he raised were a bit difficult to deal with. Do you still remember some of the issues we discussed these days? In fact, most of them were questions raised by him. "

Dennis opened his mouth, wanting to say something, but no sound came out. Finally he asked with a surprised expression: "Well, you have made me more and more interested in this matter. Tell me, what did he propose?"

Are the questions difficult for you to answer?"

Finally waiting for this sentence, Pan Jingyuan took out a printed copy from the information he carried with him and handed it to Dennis. This was specially prepared by him.

In fact, the report that Yuan Zhengxin forwarded to him was translated into English, and some content was added in the middle. The main reason was that Qiao Yu had some doubts about the rigor of the paper during his thinking process when the two of them communicated on WeChat.

Dennis glanced at Pan Jingyuan, then took the printed copy, and instead of moving forward, he stood there and read it.

After all, Qiao Yu's question had no specific examples, so the article was not long. However, Dennis looked at it very carefully. The two of them stood here for about five minutes before Dennis looked away from the printed document.

After returning the printed copy to Pan Jingyuan, Dennis slowly said: "If these are really the questions that a sixteen-year-old child raised after reading the paper after reading the paper, then I can only say that he is extremely talented."

Pan Jingyuan nodded and replied: "In fact, he is indeed gifted. In fact, it can be seen from his paper. It is said that since he decided to become independent, he used the theory of near-complete space to analyze the problem of the upper bound of the rational point of the curve.

It only took about twenty days to write that paper.

What’s even more surprising is that he only came into contact with Professor Schulz’s mathematical ideas and a series of tools for only a month. It is said that he studied Schulz’s paper because of the recommendation of another teacher.

"

Dennis nodded. If the genius Pan Jingyuan mentioned changed his question, he might be interested in talking more about Qiao Yu with Pan Jingyuan. But the question was exactly about their subject.

Dennis shook his head and said: "The question he raised is indeed very interesting. We all know how important the consistency of the structure is to the conclusions we give. This is indeed a difficult question to refute, but it is not based on the proof itself. Rigor.

The interaction of multiple papers does increase the complexity of verification. It is very important to explain that during the entire proof process, the systematic analysis tools we used can ensure that the designed geometric objects, algebraic structures and related properties maintain strict consistency. Complex and massive work.

But to be honest, I can probably understand your feelings. The questions he raised made me feel a little nervous. But you see, he has not been able to find a counterexample. This at least shows that our argument is rigorous, at least relatively Rigorous, counterexamples are not that easy to find.”

Pan Jingyuan nodded with a wry smile, and then asked softly: "What if he finds it? For example, at a certain time point in the future."

Dennis took a long breath, shrugged and said: "Then we can only re-explore this issue and re-examine the proof process of the paper. That will mean that our work will continue.

Otherwise? At least I won't give up on this proposition. It's just to solve the special case proposed by the other party. I believe it will never be more difficult than starting from scratch! Are you right, Pan?"

Pan Jingyuan liked Professor Dennis's open-mindedness, which was probably one of the reasons why he chose Dennis as his tutor, so he nodded and said, "I understand, I will continue.

By the way, the two best universities in China have held a series of seminars specifically on the geometric Langlands conjecture. After I return to China in two days, I will also attend this meeting. I guess Qiao Yu will also talk about it at the meeting. raised the question. Is there anything you want me to bring to him?"

Dennis thought for a while and replied: "I originally wanted you to give him my email address so that we can communicate directly, which might be very interesting. However, considering that I am still reviewing his paper, this time point is not suitable. Well, let’s wait until the final review opinion on his paper comes out.

But if you have any fresh ideas at this meeting, you can send me an email at any time. Of course, if Qiao Yu really does what he thinks and finds a counterexample, you can call me directly. Although I think this possibility is unlikely. .By the way, of course I’m not just talking about the seminar period.”

Pan Jingyuan nodded and agreed.

"Do you know why I want to add the last sentence? Because guys like Schulz often have terrible mathematical intuitions. Especially when they have not studied mathematics in depth. How can I put it, young people always He is good at creating mathematical miracles, which is why I like to teach students.

Dennis explained, then raised his hand and patted Pan Jingyuan on the shoulder, jokingly:

"Okay, let's go eat now. Don't look sad. I think you should be happy that another mathematical genius has appeared in China. Even if you feel uncomfortable, I think it should be Schultz because of his unique genius. The position is being challenged."

Hearing these words, Pan Jingyuan also laughed, and then there was a hint of bitterness in his heart.

Yes, his tutor is a rich second generation, and he doesn’t need to live on the salary of a professor. He studies mathematics purely out of interest, so he takes fame and fortune less seriously than others.

But to be honest, Pan Jingyuan still hopes to win a grand prize in the international mathematics community.

In a sense, guys like Schultz and Qiao Yu who have their own aura of talent are the public enemies of mathematicians around the world. Half a lifetime of their efforts may not be worth their few months of research.
To be continued...