In the eyes of his tutor, Qiao Yu is obviously not the kind of student who likes to brag. He can even be said to be very pragmatic.

Most of the older generation of professors are very rigorous.

They evaluate whether a person likes to brag by combining what he says with what he does. Although Qiao Yu's speaking style often sounds unpleasant, there is no doubt that he has done it all.

If you are known to be able to do something, it is not called bragging.

So after listening to Qiao Yu's words, Tian Yan really felt that Qiao Yu probably thought that he could solve the Riemann Hypothesis in half a year. The reason why he said it was one year was to give himself enough buffer time.

Although this sounds a bit unbelievable, but combined with the many scares given by Qiao Yu before, Tian Yanzhen feels that it is actually not bad. As for whether he can succeed, Qiao Yu may have simplified the problem...

Let's see. After all, he doesn't have much energy to understand deeply the things Qiao Yu studies.

I just know it, but I'm definitely not very proficient in it. If you know the specific method of mathematics and can operate it, it won't be that difficult.

So after understanding Qiao Yu's situation, Tian Yanzhen just warned: "Okay, don't talk nonsense before you can do it. It will be fine if you do it in the future, but if you don't do it, it will make people laugh!"

In addition, Academician Liu Zhaoyuan from the Institute of Computing Technology told me that he hopes to invite you to give a lecture. Do you think you can make time to come over and give a casual talk after the year?"

"I don't have time to answer this, right? Director Tian, ​​you know I will definitely be very busy recently. The chief engineer has said that the money for the construction of the data center will be paid after the year. I still have to do research myself.

I’m afraid I won’t be able to spare the time.”

Tian Yan was really annoyed and said: "Don't do this. You can't spare a day? Besides, what does it have to do with you once the money arrives? Mr. Yuan took over this matter and will naturally help you arrange it."

appropriate.

Do you know why you should receive this investment in the name of Huaqing Digital Research Center? On the one hand, it is a qualification issue, and the most important thing is that if you want to build this data center, Huaqing can provide a lot of resources.

Leave professional matters to professional people. The school has experts who specialize in this, and they will give you optimal configuration suggestions. In addition, they can also solve the problem of the location of the computer room.

Do you still plan to do these things yourself? Don't worry about someone hacking your money. Having Mr. Yuan help you keep an eye on it will be more effective than keeping an eye on it yourself. If Mr. Yuan gets angry, it can really affect other people's jobs!

In this case, let me help you set a time. Just go there on Friday the 27th. You don’t need to go into detail, but don’t be too perfunctory to make people feel that it is worth the money.

When you talk about it, think about it, at least half of the 30 million you spent to build the computer room was calculated by others. This amount of money is not a small number for those who do mathematics!"

After scolding Qiao Yu, the other party immediately became honest.

"Oh, okay! I'll go there then."

"That's it. Liu Zhaoyuan and I will confirm the time. I may not be in the capital by then, so I will arrange for someone to take you there."

After hanging up the phone, Tian Yan felt inexplicably relieved.

If Qiao Yu can really solve the Riemann Hypothesis...

As soon as this idea came to his mind, Tian Yanzhen shook his head subconsciously and stopped thinking about this issue.

After all, the greater the hope, the greater the disappointment, so just pretend you don’t know! And really speaking, if Qiao Yu really does what he said, he would have solved the Riemann Hypothesis in one year...

He probably wouldn't force him to give a lecture on mathematics like he did today.

It's not that Qiao Yu is no longer his student after he solved this problem, but he should give such a great mathematician a certain amount of respect.

Yes, as long as Qiao Yu can really solve the Riemann Hypothesis, no matter how old he is or what his status is, he must be a great mathematician! Otherwise, other mathematicians will feel embarrassed.

After all, this problem is the crown jewel of mathematics!

…

United States, Princeton University.

In the small classroom, the words "Introduction to the Generalized Modal Axiom: Fundamentals and Applications" are displayed on the electronic screen next to it.

Zhang Shuwen stood on the podium, carrying the students in the audience, and quickly started writing on the blackboard.

After finishing writing, Zhang Shuwen tapped the blackboard lightly and said: "Today, as entrusted by Professor Dugan, I will discuss with you in depth this very cutting-edge mathematical framework, the generalized modal axiom system.

Our goal today is to master the basic concepts of modal space, the construction of modal paths, and how to use modal distances to quantify state changes in complex systems.

Then we start from the modal space. Assume that the modal space M is a high-dimensional geometric space, and each point r represents the state of a system. The coordinates of this point are defined by the key parameters of the system..."

There were not only students but also many professors in the audience. In fact, this was not a formal class, but a Colloquium.

The research on generalized modal axiom systems is really hot now. Many Princeton professors, including Zhang Shuwen, have also begun to try to introduce this method into their respective research fields.

Especially number theory and analytic number theory.

After all, Qiao Yu has made a start. No one can refuse to re-describe and quantify complex mathematical problems through geometry and algebra.

After all, a new perspective means being able to break through traditional methods and adopt new methods to solve those annoying problems.

Zhang Shuwen's advantage in this regard is that he can maintain smooth academic exchanges with the country. Of course, most of them are purely theoretical.

For example, he didn't know much about the calculation work Qiao Yu was doing.

After more than an hour of explanation, this class is coming to an end. Zhang Shuwen prepared the same seminar for about three classes.

It does not take just three lectures for everyone to fully grasp the entire generalized modal axiom system.

Simply because as a field of mathematics that is still developing, the core idea of ​​the generalized modal axiom system has been initially formed, but it still lacks a systematic and completely unified theoretical framework.

There is no corresponding textbook compiled yet. With the current depth and breadth of Zhang Shuwen's research, the content can be completed in about five or six hours.

"...To sum up, the core idea of ​​the generalized modal axiom system is to map number theory problems to high-dimensional geometric space and use the structural relationship between point states and path evolution in modal space to transform the originally abstract number theory into

The problem is geometricized and structured to achieve more intuitive description and quantitative analysis.

This method not only opens up new avenues for number theory research, but is also expected to provide new tools for analyzing classic problems such as number theory. Therefore, understanding the basic construction of modal space is only the first step.

In order to realize the mathematical description and application of more complex systems, we also need to delve into how to modularize the generalized modal axiom system to form a more operational mathematical tool. This will be the main content of our next lesson.

If you have any questions about the content of this class, or have ideas that need to be discussed, you can start asking questions now."

After Zhang Shuwen finished speaking, someone in the audience soon raised their hands.

"Say." Zhang Shuwen pointed to the person raising his hand in the audience. He knew this young man. He was a graduate student of his colleague Peter Sarnak. He was currently mainly studying L functions.

His teacher is also there, but sitting in the back row.

"Professor Zhang, when you were talking about the modal path just now, the picture you used was, well, it was the dynamic picture of the red curve in the three-dimensional space.

When you showed this picture, you mentioned that it seems that under certain conditions, the symmetry of the path is related to the zero point of the Riemann zeta function. I want to know if this judgment is accurate?"

Zhang Shuwen smiled and replied: "If I could make an accurate judgment, I wouldn't use such inaccurate words. I can only say that the relevant research is still at a relatively early stage.

Whether there is a specific connection between the two requires further rigorous proof. But we have observed and deduced some interesting symmetry phenomena.

If you want to completely combine the two, there are still three directions of work to be done. First, you need to define the properties of the modal density function more accurately. There is no doubt that the proposer of the theory was lazy in this area.

Everyone who is here today must have read Qiao Yu's paper, which is the document we mainly quoted today. Qiao Yu did not accurately describe the symmetry and local properties of the modal density function.

Secondly, we also need to prove the relationship between the integral form and the analytic continuation of the ζ function under modal path symmetry. We must know that Pm is not arbitrary, it needs to satisfy specific geometric constraints. This is not clearly stated in Qiao Yu's paper.

.

Of course, the most important thing is to construct a two-way mapping, which is also the most difficult point. From the perspective of number theory, we also need to find a broader equivalence relationship between the modal path and the prime number distribution.

From a geometric point of view, the zero point distribution of the ζ function is reanalyzed through the symmetry of the path or the modal distance. In Qiao Yu's paper, some work was done, but it was not comprehensive.

In other words, if you are interested in this direction, you need to find a geometric structure in the modal space whose symmetry completely matches the deep laws in number theory problems.

Here...well, I personally guess that this may involve some kind of high-dimensional symmetry group, or special constraints on a custom modal space.

As far as I know, a team from Huaxia Yanbei University is already involved in this problem, including the introduction of the modal space of the group structure. I guess that after this problem is solved, we will have more sufficient tools to pry into the truth of the ζ function.

"

After answering this question, Zhang Shuwen briefly answered a few more questions and then announced the end of get out of class.

After all, today is just a very basic content and not in-depth. The real difficulty level starts with modularizing the modal system and converting it into usable mathematical tools.

When Zhang Shuwen was packing up course materials on the podium, his colleague Peter Sarnak, the supervisor of the graduate student who had just asked the question, came to the podium and asked: "Are you in a hurry to go home? If you are not in a hurry, why not go for a drink?"

Zhang Shuwen looked at Peter and said with a smile: "Forget the wine, coffee is fine."

"Haha, Zhang, drinking coffee at night is not a good habit, it will cause insomnia." Peter Sarnak said with a smile.

"No, I will sleep better if I drink coffee at night." Zhang Shuwen shook his head and said.

"Oh my god, didn't your doctor tell you that you might be allergic to caffeine?" Peter Sarnak said exaggeratedly.

Zhang Shuwen smiled, then sped up his movements. After finishing the courseware, he said, "Let's go."

The issue of allergies was not discussed.

When Zhang Shuwen first went abroad, he only thought it was hypocritical for people around him to regard allergies as a scourge. After all, in Zhang Shuwen's original understanding, in addition to the dangers of drug allergies, the things he encountered in life were really different.

I think allergies are a big deal.

But after I actually saw people around me who were admitted to the ICU due to anaphylactic shock due to allergy to some kind of foreign pollen, I realized that these people may not be pretentious, but have physical reasons.

Who would have thought that I usually don’t have symptoms of pollen allergy, but just because I encountered a flower I had never seen before, I thought it was very beautiful, so I picked it and smelled it a few times, and I went into shock within a few minutes...

From that time on, Zhang Shuwen expressed respect and blessing for the word allergy, but never discussed it.

Even if he is really allergic to coffee, Zhang Shuwen feels that it is at least much better than alcohol. There is no other way, he will feel a headache after drinking alcohol.

So the two of them didn't go out to find a small bar, and simply started chatting in the professor's lounge. After all, there was a ready-made coffee machine here.

After all, in this country, professors are different from professors. Unless they are the kind of big bosses, the salary of professors is generally not high, so they naturally save as much as they can.

"Zhang, you know, my tutor Paul has been doing research on the Selberg conjecture. Well, actually I am also interested in this problem, but there is no good way yet. So do you think using that... um,

Can Qiao Yu’s method solve this problem?”

After making coffee, Peter Sarnak took the initiative to say.

The Selberg Hypothesis can be regarded as a special case of the generalized Riemann Hypothesis. Its mathematical expression is that the non-trivial zero points of all automorphic L-functions on the key strip 0
(s)=1/2.

Because this conjecture covers all automorphic L-functions, including the Riemann ζ function and the Dirichlet L-function, it is also a subset of the generalized Riemann conjecture.

In the history of mathematics, there have always been people who like to raise some strange but significant questions.

The Selberg Hypothesis is undoubtedly one of them. Although it is not as well-known as the Riemann Hypothesis, its significance is also far-reaching.

For example, it inspired research on modular forms, automorphic forms and spectral theory, and was even crucial to the Langlands Program.

Of course, because of this, the Selberg Hypothesis relies on higher-level tools and the research threshold is very high, so there are relatively few researchers. In addition, the history is not as long as the Riemann Hypothesis, so the popularity is naturally not that high.

Zhang Shuwen shrugged, and then replied: "How should I answer your question? Well, Peter, theoretically I can answer you for sure, yes!

But at present, I still have no idea how to operate it. I believe you should have read the relevant literature before asking this question! Although there are many related papers, most of them are still very shallow.

In fact, we are the same. Qiao Yu's theory is still at a stage that needs to be fully developed. As I said in class, there are still many articles to be done on this axiom system."

Peter Sarnak smiled and said: "See, this is exactly why I came to talk to you. I went to see Professor Dugan this morning, hoping that he could help me introduce him.

I think we can work with Qiao Yu to solve this problem. You know the significance of this topic. But Professor Dugan said maybe I can talk to you."

Zhang Shuwen was stunned, and then looked at Peter Sarnak doubtfully and asked: "You can send an email directly to Qiao Yu. I think if he is interested, he should reply to you, right? I have met that kid, and he is still very worried.

Polite."

Peter Sarnak spread his hands and laughed at himself: "Maybe it's because I'm not well-known. I wrote him three emails, all of which were automatic replies. I doubt he has the habit of checking his mailbox at all."

Zhang Shuwen really didn't know about this situation. He hadn't contacted Qiao Yu recently.

He originally planned to wait until March to visit Huaqing, and then discuss with Qiao Yu how the generalized modal system could be better integrated into the number theory system.

"Well, let me ask you. Of course, I'm not sure that he is interested in this proposition. I heard that he has been quite busy recently and has at least two collaborative research projects on hand."

Zhang Shuwen thought for a while and agreed.

"Thank you, Zhang! By the way, help me tell Qiao Yu that I am very interested in his theory. If he agrees, I can share all the current research data with him.

You know, although we are still some way away from the final solution to the Selberg Conjecture, we have still done some breakthrough work.

For example, we have verified several special cases of zero-point distribution on some specific types of automorphic L-functions, and studied the analogy of zero-point distribution in random matrix theory.

Especially the Berry-Keating model, in short, I believe that this preliminary work should provide some help for the mapping of modal space." Peter Sarnak said confidently.

Zhang Shuwen nodded.

"By the way, tell him that if his method can really help us solve this problem, we can work together."

Peter Sarnak explained again.

There is no way.

Just like many unsolved mathematical problems, the reason why they cannot be effectively solved is mostly because the existing mathematical tools cannot explore the truth at all.

I have been thinking about a certain mathematical problem for more than ten or even decades, and basically I have tried all the tools that can be used.

Zhang Shuwen expressed his understanding, but he would definitely not take over everything. There was no way, Qiao Yu was not his student. It was even difficult to reconcile the relationship between the two.

After all, when the two met for the first time, he severely punished Qiao Yu and even criticized him for not being too ambitious.

Although he did have good intentions at the time, there is a deep gap between today's young people and his generation. Zhang Shuwen didn't know what Qiao Yu was thinking.

Maybe I have been slandering him as an old man who is good at teaching others - this is very possible. Especially since this guy has grown into a genius so quickly that he is no longer easy to evaluate...

This is even more embarrassing. But even if we go back to the first time the two met, Zhang Shuwen felt that what should be said at that time still had to be said. He could not let the trajectory of fate change...

If he had not said that, Qiao Yu would not have constructed the generalized modal axiom system, which would definitely be a great loss to the mathematical community.

Recalling the scene of his first meeting with Qiao Yu, Zhang Shuwen nodded absently and promised: "Don't worry, I will convey the original words to him."

"Thank you, Zhang! I'm waiting for your news."

After speaking, Peter Sarnak couldn't help but look at his cup of coffee that had not been touched much, and commented seriously: "You should protest to Professor Dugan and change the coffee.

This kind of coffee is sour and astringent, I really don’t know how you can tolerate it. Next time you come to the institute, trust me, the coffee there is much better than your garbage coffee.”

Zhang Shuwen smiled.

This guy is not the first person to complain about the bad free coffee at Mathematics College, and he certainly won't be the last.

After sending Peter Sarnak away, Zhang Shuwen sat in the tea room and thought for a while before returning to his office and dialing Yuan Zhengxin's phone number from the office phone.

"Hey, is this Shuwen?"

"Yes, hello Mr. Yuan, how are you doing lately?"

"It's not bad. I can still eat. I should be fine for another ten years."

Listening to Yuan Zhengzhong's angry voice on the phone, Zhang Shuwen subconsciously smiled.

It's not surprising to hear that the old man is in a good mood. Although there are still many disputes between the two parties, with a genius like Qiao Yu acting as lubricant in the middle, it is reasonable to be happy.

After a person has passed the age of knowing his destiny, he actually doesn’t want much. With Mr. Yuan’s status, it is obviously the most important thing to inherit his knowledge.

He had heard about it at Princeton. Not only was Qiao Yu very talented in mathematics, but Qiao Yu's mother seemed to be very talented as well. She was just a little older. It was a pity, but it was said that there was not much of a problem in inheriting the mantle.

"Take care of yourself." Zhang Shuwen said sincerely.

"What's the matter with Shuwen calling at this time? It's already nine o'clock in the evening at Princeton, right? It's still in the office."

Zhang Shuwen said quickly: "Professor Sarnak just talked to me specifically. He also spent a lot of thought on Selberg's conjecture, and it has been stuck in a bottleneck for a long time.

During this period, I came across Qiao Yu’s method and was very interested. I wanted to invite Qiao Yu to collaborate on this topic, but I sent Qiao Yu three emails but never received a reply.

So you came to me here and thought about asking me to help ask if Qiao Yu didn't read the email. If possible, I still hope to promote this cooperation."

After saying this, Zhang Shuwen said nothing. To his surprise, Mr. Yuan on the opposite side didn't speak immediately either, and the phone fell into silence.

Zhang Shuwen did not urge him, he just waited quietly, but he began to have doubts in his heart. To be honest, this was not a very excessive request. Whether he agreed or refused, given Mr. Yuan's temperament, he should be very happy.

There was silence for nearly thirty seconds. Just as Zhang Shuwen was about to speak to break the silence, Yuan Zhengxin spoke up: "Shuwen, have you thought about coming back recently?"

Such a heavy topic, Zhang Shuwen thought about it for a few seconds and then asked: "You mean resigning from your teaching position at Princeton?"

"Yes, come back and teach in Huaqing. I can't work for a few more years. Come and help me look after the Mathematics Research Center and look at Qiao Yu. This kid is too young after all. I'm not worried about mathematics, but there are other issues.

It's still not clear enough.

There has to be someone who keeps an eye on him and doesn't let him go astray. I don't trust Tian Yanzhen. Really, without a child like Qiao Yu, I wouldn't advise you like this. But you think, now even your colleagues

They all look forward to cooperating with him. What does that mean?

Do you dare to imagine? In the near future, China will develop from a big country in mathematics to a powerful country in mathematics? Or even become the center of the world's academic community? Perhaps China's national strength could not support such a grand vision before, but what about now?

Think again about the current relationship between China and the other side. Shuwen, think about it carefully! Maybe the vision of our generation can be realized in Qiao Yu’s generation!

History is a cycle of reincarnations, and I have a hunch that another era of technological explosion may be about to begin. In this round, our national strength is no worse than any other country, and we even have many advantages that other countries do not have!"

Zhang Shuwen could feel that Yuan Zhengxin's heart was surging when he said these words. After thinking seriously for a moment, his heart began to waver.

He just put down everything he had here and went back directly, but hesitantly.

After thinking for a moment, he said: "Yuan Laorong, I will think about it."

"Well, this is a big deal, and you need to think about it. As for the matter you mentioned, I will talk to Qiao Yu. But don't have high hopes. There is a reason why he doesn't reply to emails.

Recently, many teams and organizations have invited him to cooperate. The little guy is a little unbearable and even complained to me some time ago. And he does have a lot of things going on recently. You have to understand."

Zhang Shuwen replied: "Understood. I'm just asking questions for Peter. Naturally, Qiao Yu still has to put his own plans first."

"Okay, think carefully about what I just mentioned. Once you make a decision, let me know as soon as possible."

"Okay, Mr. Yuan."

After hanging up the phone, Zhang Shuwen sat there in a daze for a while. In front of Mr. Yuan, he was a junior, but really speaking, he was sixty-one this year.

Whether or not to go back at this age is indeed a question. After thinking for a while, Zhang Shuwen's eyes focused on the courseware on the table.

There was a paper underneath the courseware. Zhang Shuwen subconsciously took out the paper and flipped through a few pages at random.

This is exactly the paper Qiao Yu published in the Annals of Mathematics.

The blank spaces at the top and sides of almost every page are densely filled with various annotations, ideas, and additional derivation processes.

It is only a vision for China to become a powerful country in mathematics, but it would be true to be able to come into contact with the most cutting-edge mathematical ideas if we go back.

A huge system that converts all numbers into elements, it’s a shame that guy can think of it and can do it.
Chapter completed!