Before Lott Dugan left for China from the small town of Princeton, he had already made an appointment with José for today's meeting.

This is due to the experience of the last time I came to China to participate in the report meeting of Qiao Ze.

It's not that they're sure they'll be able to meet Qiao Ze when they come to cheer.

If he hadn't said it in advance, the little guy would probably have turned away all the old guys like him for all sorts of reasons.

Yes, this guy can do such a thing.

It's infuriating.

Even without mentioning the tradition of respecting the elderly, the world academic community respects an old guy like him. Especially in Asian countries, except for coming to Xilin, no matter where he goes, especially in countries like Dongben and Beixiandao, if he wants to meet a junior in the academic world, he doesn't need to come to the door in person.

He just needs to stay in the most prosperous city, and then make a casual phone call, and people will come over in a hurry.

Especially on Higashimoto Island, that kind of humility is really clearly defined. The younger generation of scholars has even left their mentors and simply does not have the resources to start research independently.

But then again, although José is a little willful, it is understandable to Lotter Dugan.

People who are capable tend to be more tolerant.

Even the old dean, who has never been able to rub sand in his eyes, has learned how to make excuses for the other side after being rejected countless times.

For example, this kid wants to be motivated after all, but he is just a little dull and has higher requirements for communication, maybe he is really busy...... Wait a minute.

Well, to put it bluntly, the almost unlimited increase in his tolerance threshold in Qiao Zeli is still caused by the misalignment of supply and demand after all.

He needed to be able to understand the strange but extremely useful knowledge in Josser's mind, and even dreamed of carrying this young scholar, who was too young but already occupied an important position in the world's academic circles, to Princeton.

But Joze didn't need much of his help.

Even ignoring the amount of resources he could get for this little guy.

That's why you can't see what you can't think of.

Lott Duggan understands this very well, and he doesn't want to waste the hard-won opportunity to meet, so he came to talk to Peter Schultz about his thoughts first, so that he can make full use of the opportunity to meet face-to-face and squeeze out more of what is useful to him in Joze's mind.

What a headache for this little guy.

He doesn't even have the idea of a young man being competitive.

Lott Duggan knows that the Institute for Advanced Study in Princeton has put the progress of research on supercoiled space algebra on its website.

Anyone else, even an old guy like him, will probably be proud when he sees the theory he created being put on the official website of the world's top research institutions to discuss and research with global scholars, and will definitely respond.

Even if the relationship between the two parties is not good, they probably can't help but tell them whether there is a problem with the research direction by not being limited to public shouting, or show off some ideas to solve the problem, so as to prove how impeccable their theory is.

At least Lott Dugan put himself in Jozer's shoes, and thought he would definitely do it. Even if it will help the other party deepen their understanding of this new theory.

What theoretical mathematician wouldn't want to see his theory flourish in his lifetime?

This is the lifelong pursuit of this profession!

But José is really different.

Lott Duggan didn't know how the young man could hold back, and for a whole quarter, he never said a word about the progress of Princeton's research, either publicly or privately, except for the occasional public release of a few related topics.

He silently watched a group of scholars discuss the theoretical framework he had already built.

Cruel and terrible.

What is this something that a normal mathematician can do?

Occasionally, when communicating with José via video, he also asked relevant questions, and even often hinted at José what he thought of the research on Princeton's official website, but Jozer coped with it in a few words.

He could even memorize Joze's phrases.

"Not bad." "Worthy of recognition." "It's good. "The theorem is simple. ”……

This is also the reason why he cherishes this face-to-face communication with Qiao Ze.

He was afraid that the video would be recorded, and he couldn't pull it off completely.

But in the face of communication, he can put down this old face for a while and ask everything he wants to know.

This is very important, because today's defense meeting really inspired him a lot, and he felt that if he could learn from Qiao Ze, it would greatly simplify the amount of calculations he was working on now.

This is an n-body project.

If he succeeds, he may be able to produce a Fields Medal student.

Yes, Lott Dugan may be a stubborn little old man, and even many times his attitude towards his students is not friendly, and he may give extra homework to those poor students because he is in a bad mood.

But from the perspective of pure academic training, Lott Dugan is actually a very good and excellent mentor.

For example, he will neither hide his personal interests during teaching, nor will he snatch the honor that belongs to the students, and he is even willing to take the blame for the harmless little mistakes of those students.

If you ask Qiao Ze to give him an evaluation, he is probably a very pure scholar.

……

At seven o'clock in the evening, Lott Dugan took Peter Schultz to the courtyard of the Xilin Institute of Mathematics on time.

Originally, he had planned to come alone, but after discussing with Peter Schultz, he found it necessary to bring the young mathematician with him, so that he could think more fully about the problem.

The only bad thing is that there may be one more person who sees him as an old guy playing a scoundrel, which will have a big impact on his face.

But thinking that this could be an opportunity to solve the problem he had been studying for many years, after weighing it, he still felt that the face thing was actually not worth a few dollars.

It turned out to be the right decision.

Because when he arrived at the appointment on time, he found out that this conversation was not two.

"Ha, Joze, finally seeing you again! You know, it's so hard to meet you face-to-face. Unfortunately, you refused to go to Princeton. Really, if you're willing to check it out, you'll love it. ”

When greeting each other, it's still a very old-fashioned way of talking.

The same is true for online video.

When Lott Dugan sees Josser, he always opens with the beautiful surroundings of Princeton or the strong academic atmosphere.

Even if he was rejected every time, he never tired of it.

Anyway, it was just a casual remark, and it would be a surprise if Qiao Ze agreed on a whim.

It's a pity that I still got the same reply today: "Xilin Research Institute is very good, you come." ”

"Hehe, well, you know, I can't leave. Oh, by the way, Peter Schultz, we met last time. Well...... Also, who are those people? ”

Lott Dugan looked at the figures standing in the courtyard and asked.

"Bodyguards?" Qiao Ze said uncertainly.

Since the director surnamed Du came to talk to him last time, the group codenamed "Tinder", which had been silently following him before, simply moved into the courtyard of the research institute, and directly cleared out the room where the debris was piled up on the first floor and became their stronghold.

I often wander around the yard on a regular basis, and even the garbage in the research institute has to be checked.

Fortunately, these things did not have any impact on Qiao Ze's work and life, so he went with these people.

In fact, there are benefits to it.

It was as if he had made an appointment to talk to Lott Dugan early tonight, and Qiao Ze also knew that Su Mucheng was definitely not interested in the content of the conversation between them, and then Xiao Nizi went out shopping with the two young women that Director Du promised to arrange last time.

also took Lu Xiuxiu with him.

Today is just his Ph.D. graduation, but the two closest women around him are excited as if they are celebrating a holiday.

Joze's answer also made Lott Dugan stunned, and his eyes flickered on the faces of those people in the small courtyard before he spoke: "Oh, okay...... Forget it, you better take a look at this first, you know, I've been working on the n-body problem.

Your presentation today gave me a lot of inspiration, and I talked a lot with Peter Schultz in the afternoon, and everything that came to my mind was here. You take a look at it first, and then be sure to give me some advice. ”

As he spoke, Lott Dugan handed over the manuscript he had written after the afternoon of discussion with Peter Schultz.

"It's a doctoral graduation defense."

Jozer corrected Lot Dugan's slip of the tongue, took Lot Dugan's manuscript, and glanced at it carefully, feeling a little surprised.

It is clear that the content of the manuscript uses the formula that he has just talked about today, and it touches on the concept of transcendent geometry.

A new mathematical direction and the mathematical tools derived from the ideas often mean that many mathematical problems can be solved. It's as if calculus has become the most basic tool for studying mathematics in this century.

But Lott Dugan was inspired by his Ph.D. defense, and even Joszer did not expect to use transcendental geometry to solve the n-body problem.

And it seems that this idea does have some meaning, and even gave him some inspiration.

Sure enough, what Su Mucheng said was right, these old people are still very capable, it's just that they haven't been exposed to these before.

But if you really want to talk about it, the difficulty of going from 0 to 1 is far more difficult than going from 1 to 100.

After about ten minutes of reading the manuscript, Jozer looked up at Lott Dugan, who looked expectantly, and thought quickly in his head, before he said, "It's better to be bolder." ”

"Bolder? What do you mean? Peter Schultz, who had not spoken, tilted his head and asked.

"Do you have a pen?" José glanced at Peter Schultz and asked.

Without waiting for Peter Schultz to react, Lott Dugan hurriedly pulled a pen from his pocket and handed it over.

Qiao Ze picked up the pen and began to write directly on the manuscript, casually explaining: "Instead of such indirect transformations, it is better to directly describe the position of objects by introducing transcendent geometry." ”

Lott Dugan craned his neck to look at what José had written, while Peter Schultz, who was beside him, said, "Will this work?" ”

"Of course, but there is a need to use a new coordinate system based on transcendental geometry that represents the position of each object as a transcendental number."

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"Wait, I don't understand, doesn't that make the calculations more complicated?"

"No, quite the opposite, it allows us to describe the interaction between objects more precisely, especially if the objects are very close together, and the interaction force can be approximated by expanding it in a series. Moreover, it is not only the classical n-body problem, but also the relativistic n-question problem. ”

Hearing Qiao Ze's words, Peter Schultz couldn't help but stand up, leaned over to Qiao Ze and began to look at his calculations on the manuscript paper.

"This is an elliptical model?"

"Yes, first presuppose a three-body problem, and represent the position of each object in the three-body system as the solution of the elliptic function."

After saying that, Qiao Ze wrote three more formulas on the manuscript.

[ x_i(t)= a_i \cos(\omega_i t +\phi_i)，]

[ y_i(t)= b_i \sin(\omega_i t +\phi_i)，]

[ z_i(t)= c_i \cos(\omega_i t +\phi_i)，]

Then he explained: "where (a_i, b_i, c_i) are the semi-major, semi-minor, and half-height axes of the ellipse, (\omega_i) is the angular frequency of the ellipse, and (\phi_i) is the initial phase. ”

Lott Dugan first showed a stunned look, and then asked with a frown: "But how does this affect the calculation of the interaction force?" ”

"Approximating the interaction force by means of a series expansion, e.g. if we consider the gravitational force between objects (i) and (j), then define the interaction force as ......"

As he spoke, Qiao Ze wrote a series of formulas on the manuscript.

[\mathbf{f}_{ij}=-g \frac{m_i m_j}{|\mathbf{r}_i -\mathbf{r}j|^2}\hat{\mathbf{r}}{ij}，]。

Then he says: "(g) is the gravitational constant that needs no explanation, (m_i, m_j) are the masses of objects (i) and (j) respectively, (\hat{\mathbf{r}_{ij}=(\mathbf{r}_j -\mathbf{r}_i)/|\mathbf{r}_j -\mathbf{r}_i|) is a unit vector. ”

"How does this progression expand? Beyond geometry also involves the calculation of mechanics? ”

"Beyond geometry allows the use of item-by-item approximation techniques, which can be used to do series expansions, and the specific ...... can be obtained"

[\frac{1}{|\mathbf{r}_i -\mathbf{r}j|} =\sum{k=0}^{\infty}\frac{\psi_k}{r^{k+1}}，]

"By the way, (\psi_k) is the coefficient."

Looking at the formula on the manuscript that began to gradually become fuller, Peter Schultz suddenly felt that something was wrong, and he frowned and said: "No, there will be a truncation error of the series expansion, and this error is uncontrollable, right?" ”

"There is a way, have you forgotten how I solved it when I explained the paper today? Set a truncation parameter n to consider only the first n terms of the series expansion. As long as the value of n is large enough, the model mathematically approaches the exact solution.

Of course, when it comes to calculating the position of celestial bodies, we don't need such high precision, and we can directly consider the performance of the supercomputer and the required accuracy to set the truncation parameters, at least the amount of computation is much less than that of solving higher-order differential equations. ”

Lott Dugan subconsciously raised his hand and gestured twice, and couldn't help asking: "In the process of iterative solving, the problem of numerical instability will be encountered, how to solve it?" ”

"Huh?"

Qiao Ze wrote down the last stroke, carefully looked at his derivation process again, and then handed the pen back to Lott Dugan, and said: "I remember that there was a numerical integration algorithm with adaptive step size, and the superiority of transcendence geometry in solving such complex problems, which should ensure that the numerical solution is still stable even when the distance between objects is small.

Of course, you can also use numerical stability analysis to adjust the parameters of the algorithm. In short, there should be many more methods, but if I were to solve this problem, I would definitely use this idea. In addition, if you want to calculate the relativistic n-body problem, you can use Einstein's field equation instead of Newton's gravitational law, and the idea will probably not change. ”

With that, José handed the manuscript and pen back to Lott Dugan.

Lotter Dugan stupidly took the manuscript handed by Qiao Ze, and looked at the contents on it, the expression on his face changed rapidly.

Really, after studying the n-body problem for so many years, it was the first time he felt that his thinking was so clear.

Most importantly, he realized how bold José was in solving mathematical problems.

Or how flexible his thinking is, of course, it may also be because he has a deeper understanding of transcendental geometry.

In short, the ideas he gives are completely subversive.

Of course, the effect still needs to be verified by supercomputing, and the specific algorithm also needs to be designed by his team.

Although Qiao Ze didn't say it explicitly, Lott Dugan knew very well that Qiao Ze could talk about his thoughts to this point, which was already very face-saving. If the problem is solved in a few minutes, the n-body problem is inevitably too simple......

More importantly, if there is a paper in the future, whose name will be written?

……

On the other side, the three people in the small courtyard also got together and chatted casually.

"What are the two foreigners talking to Qiao Ze? Zhou Shun, you use English best, translate with us. ”

"I'm sure I'm fine with English, but the English they speak is useless. It seems that you are discussing some kind of celestial calculations, right? It's like José gave them a solution to the problem. ”

"Nonsense, if you want to say that, I can see the expressions of those two foreigners and know that they have benefited a lot. Isn't it very slippery for you to speak English when you are on assignments? Why did the chain fall off at a critical time? ”

"No, Brother Lu, their mouths are full of words that I haven't heard of, there is a kind of English called academic language, which ordinary people can't touch at all, okay?"

"Hey, I said let's stop talking about this. I think, Qiao Ze doesn't seem to be so patient with the professors in China, shouldn't he really be moved by the old man's words and plan to go abroad? That's a big trouble, if this young master makes a fuss, I always feel unbearable. ”

"No, I understood what they said when they first met. The old man wanted José to go to Princeton, and José said that Cillin was good enough to let him come over. In short, the two of them are always pulling back and forth like this in the previous video. ”

"Hey...... I'm tired! If you want me to say, this kind of unilateral exchange should be abolished altogether. ”

"Ahem, someone reported it. Although we don't understand it, didn't we say it at the meeting, the results of theoretical mathematicians are for the benefit of all mankind......"

"Okay, don't talk nonsense, just get your things in order and report it on time. Just the two of you talk a lot of nonsense. ”

"No, Brother Lu, it seems that you were the first to talk ......"

……

After the three of them discussed the questions raised by Lott Duggan, Peter Schultz asked Joser a few more questions about supercoiled space algebra.

Perhaps the biggest inspiration for him in today's report is that he can combine supercoiled space algebra with transcendental geometry and start thinking about similar problems.

It's a pity that the question he prepared didn't seem to arouse much interest in Qiao Ze.

Or the communication on the three-body problem just now made Qiao Ze finish all his words today, and returned to his previous taciturn appearance, just answering a few words.

Well, at least there's still a catch.

A question that had been thought about for a long time and had no answer, under Qiao Ze's suggestion, I figured it out.

But it also made Peter Schultz even more depressed.

After struggling for a moment, he simply said directly: "Dr. Qiao, I think I should do what I said last time about researching and promoting supercoiled algebra, in fact, we are already doing it, but the progress is a little slow." ”

"Well, it's slow." José nodded, agreeing with Peter Schultz's statement.

"So, actually, you can join in. Even if you are in Xilin, we can keep in touch by email. ”

Qiao Ze thought for a while and said, "I promised that others would set up a research team to do research and promotion, and if you want to join, you can wait until the 28th to take the exam." ”

"Exams?"

"Yes, exams."

"You're saying you're going to form a team at the institute that will specialize in supercoiled algebra and transcendental geometry? Is the recruitment of team members in the form of exams? ”

"Yes!"

"Can foreigners join as well?"

"Yes, it has to be in Xilin."

"Must be in Xilin? Can't be remote? ”

"Yes."

Listening to the conversation between the two, Lott Duggan came back to his senses from thinking about the n-body problem and proposed: "Is it okay to cooperate with your Xilin Institute of Mathematics in the name of the School of Mathematics and the Institute for Advanced Study at Princeton University to jointly promote this project?" ”

José looked at Lott Duggan and said, "The resident professors you have appointed can join. ”

"Well, we've already done a lot of research."

"yes, it's just too slow." Qiao Ze nodded approvingly.

I can't refute it, so there's no need to talk.

Sure enough, some opportunities are missed, and it's really hard to get them back.

Lott Duggan shook his head, then suddenly remembered what he was going to say, and hurriedly asked: "By the way, will your doctoral dissertation still be submitted to our Annals of Mathematics?" ”

"Do you accept Chinese papers?" Qiao Ze asked rhetorically.

"Huh?" Lott Dugan was stunned.

It seems that the Chinese paper has not been included.

"Our institute plans to run a journal, which is accepted in both Chinese and English." José explained.

"This ...... Isn't your doctoral dissertation going to be published in English? Peter Schultz asked in an incredulous tone.

He thought that the Chinese used for Qiao Ze's defense today was too bright, and the time was limited, so he couldn't wait to announce it......

"Well, it's tiring to translate with too many new terms and explanations." Qiao Ze nodded.

"You can leave this to me, and I'll find someone to translate and compile it." Lott Dugan hurriedly said.

"Oh, thank you."

"Uh...... That ......"

"I'm tired, let's talk about this today." Qiao Ze spoke.

It's not an excuse, Qiao Ze does feel a little tired.

Especially after talking too much in today's doctoral defense, my throat feels a little painful.

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