When Xu Chuan followed Fefferman to the office to exchange smooth manifolds, his first class at Princeton caused quite a stir in Beimi's college network.

Some well-known university forums are discussing this matter.

[Hey, you know? The genius who proved Hodge's conjecture stated in his first class that there were only five months left to prove Hodge's conjecture! 】

[Five months? What are you kidding. 】

I can swear to God that I have not spoken a single lie. 】

[If this is true, it is too terrifying, but in fact it is impossible, five months to prove Hodge's conjecture, no one can do it, in fact, he also said later, he laid the foundation for this for more than ten years. 】

[Nine years of education + three years of college entrance examination + five years of simulation? ] (o?v?) ノ】

[This is an immortal magic method from the East. 】

......

As Xu Chuan had predicted, almost no one would believe that he had actually proved the Hodge conjecture in five months, which was outrageous.

In fact, if this matter is put on other people, Xu Chuan himself does not believe it.

After all, it took him only five months to complete the proof of the Hodge conjecture on the surface, but this is inseparable from his research in the field of topology and mathematical analysis in his previous life, as well as the algebraic geometry and differential equations that he studied with Deligne in this life.

More than ten years of sharpening a sword, this is not an exaggeration.

But if a scholar can sharpen such a sword and slash at the evil dragon entrenched in the sky, it will already be the greatest achievement in this life.

However, Xu Chuan was not satisfied, and after conquering Hodge's conjecture, he and Fafmann joined hands and launched a charge towards the ultimate goal of sleek popularity, the 'NS equation'.

The offer was made by Fepferman.

After exchanging ideas in the field of smooth manifolds with Xu Chuan twice, Feverman couldn't hold back his thoughts.

After all, he has made great contributions in the field of multiple complex function theory and smooth manifolds, and he also has in-depth knowledge in this field.

In 1974, he demonstrated that a strict pseudo-convex region with a smooth boundary can be smoothly extended to the boundary by a double holopure mapping of the other.

This is something that many mathematicians in the 20th century have tried to prove unsuccessfully.

Because the multi-complex region is different from the single complex region, the two single-connected regions are not necessarily both pure equivalent, so the single complex method cannot be applied.

And he solved this problem with his own original new method.

Based on this, Fefferman has made several attempts to charge the NS equation, but in the end they have failed.

And Xu Chuan's arrival brought him a new dawn, and after thinking about it for a long time, he finally mustered up the courage to propose to Xu Chuan the ns equation that he tried to solve together.

As for Feffman's proposal, Xu Chuan agreed directly without any hesitation.

The Navier-Stokes equation was one of the problems he wanted to solve most in his last life.

If you solve it, you may be able to curb the evil dragon of ultra-high temperature plasma turbulence in controlled nuclear fusion, and put the reins on it for taming.

For this reason, he chose to cooperate with Professor Feverman in his last life.

Unfortunately, due to his mathematical abilities and Professor Feverman's physics skills, this problem did not come to fruition.

In Samsara I, he came to Princeton again and cooperated with Feverman again, but the object of solving was still the NS equation.

This can't help but make people sigh, fate is indeed wonderful.

.......

Institute for Advanced Study in Princeton.

In Mr. Xu's office, Mr. Feverman was writing in white chalk on a blackboard.

“λ1(u)<λ2(u)<··· <λn(u)........ For i = 1, ···, n, let ri(u)=(r1i(u),···, rni(u))t be the right eigenvector corresponding to λi(u): a(u)ri(u)=λi(u)ri(u). ”

“π: g→ u(h)，gx h→ h.......”

On the side, Xu Chuan stared intently at the blackboard.

Behind him, the four students who were originally doing their homework and studying also approached curiously, standing in a place that did not affect the two of them and watched curiously.

At first, the four of them were able to read some of what Professor Feverman had written on the blackboard, but as time went on, some people began to fall behind.

What Professor Feverman wrote on the blackboard would be extremely difficult for even a PhD student to understand

When the chalk on the blackboard was used up, the faces of the four students crouching behind were full of confusion, and then they began to discuss in a whisper.

In the office, the slightly younger Shashi Perez poked the big brother next to him: "Dean, do you understand what Professor Feverman wrote?" Teach them, what exactly are they studying? ”

Roger Dean didn't take his eyes off the blackboard, but he still responded to his junior brother's question, he shook his head, and whispered, "I don't know, but I guess it should be the manifold or Li Qun's direction problem." ”

"Manifolds? Is there any problem with the direction of the manifold that deserves the cooperation of the two Filipino award elders? Shahi Perez muttered quietly.

】

On the side, Gu Bing, who had also been unable to keep up with the rhythm for a long time, rubbed his somewhat sour eyes and said, "Of course there is." ”

"For example?"

"N-S equation!"

"You're saying they're working together to solve the ns equation?"

"I didn't say that." Gu Bing shrugged his shoulders and whispered.

But it still made waves in the minds of the others, and after Hodge's conjecture, did their professor go on the offensive for another of the Seven Millennium Puzzles?

......

Xu Chuan ignored the dripping of the student behind him, and he stared intently at the equation on the blackboard.

So far, he is the only one who can keep up with Feverman and understand his train of thought.

In general, Feverman uses the smooth mapping of Lie groups with smooth differential manifold structures, so that Lie groups g unitary table g make a continuous effect on the Hilbert space, and these actions can keep the spatial product constant.

That is to say, the unitary representation of Lie group g is a homomorphic map from group g to group u(h) consisting of all unitary operators in a certain hilbert space h.....

The calculations and formulas on the blackboard made Xu Chuan's eyes as bright as stars and shining.

From this line of thought, he saw the possibility of advancing the NS equation.

This is a completely new line of thinking, different from the previous life of Feverman and his study of the ns equation, which is an extension of the direction of Li Qun that he had previously suggested, but it is almost completely detached.

He is worthy of Professor Feverman, the youngest scholar to be appointed as a professor at the University of the United States.

His knowledge and thinking are admirable.

Mathematics is like this, once you think wrongly, no matter how hard you try, you are groping forward in chaos and darkness, and you can't see the future.

And if you're thinking right, the door of hope will shine in the darkness and guide you forward like a beacon.

Xu Chuan felt this when he was in junior high and high school.

Sometimes he encounters some multiple-choice questions or fill-in-the-blank questions, and the first answer that comes to his mind intuitively is often the correct answer.

Perhaps, this is what ordinary people call mathematical talent.

........

In the office, in front of the blackboard, after filling both sides of the huge moving blackboard in front of him with mathematical formulas, Feverman turned around and looked at Xu Chuan behind him.

"Xu, from the exchange a few days ago, I got some inspiration, using the smooth properties of Lie groups in differential manifold structures, to generalize the orbital method to reduce Lie groups, which is helpful for studying the overall existence of smooth solutions of three-dimensional incompressible okes equations."

After a pause, he continued, "But I feel that if I continue to push down, there seems to be a problem......

Before Peverman finished speaking, Xu Chuan continued: "How can a pair of bounded connected regions be constructed on the plane R2, and their boundaries are very unsmooth, and even have fractal boundaries, making them isospectral but non-equidistant isomorphic. ”

Hearing this, Feverman nodded abruptly and said, "No wonder I haven't been able to push forward, it's a matter of equivalence." ”

"If we can solve it, maybe we can solve the equation of conservation of momentum in the ns equation even further."

Staring at the equation on the blackboard, Xu Chuan touched his chin and nodded.

He agrees with Feverman's statement.

Both of them are top mathematicians, and they have the same opinion on the same problem, so there is a high probability that behind this opinion is the correct answer.

But the problem now is that there is a mountain that cannot see the height of the mountain.

Neither of them knew how long it would take to turn over or go around.

There is no clear idea of what to do or which path to choose.

......

After staring at the equation on the blackboard and thinking for a full five minutes, Xu Chuan came back from his contemplation, shook his head and said:

"I'm afraid this problem is not so easy to solve, and if I'm not mistaken, it involves a difficult problem in the other direction."

"What's the problem?" Feverman asked quickly.

"Isospectral non-isomorphic conjecture."

Xu Chuan spit out a few words from his mouth, and a sudden look appeared on Feverman's face: "It turned out to be this." ”

The isospectral non-isometric isomorphism conjecture is a difficult problem at the intersection of analytics (the spectrum of elliptic operators), geometry, and topology.

It is not a long time since it was raised.

It was a question posed by Gordon Webb Wahlbert in 1992 when he broke through the field of isospectroscopy.

That is: "Is there a pair of bounded connected regions with smooth boundaries (at least C1 smooth boundaries) on plane R2 that are isospectral but not equidistant isomorphic?" ”

This problem is a difficult problem at the intersection of the three fields of analysts, geometry, and topology, and there are not many mathematicians interested in it.

After all, it is too difficult to understand the three major fields at the same time, not everyone is Tao Zhexuan, and it is a difficult thing for most mathematicians to study a mathematical problem across multiple fields.

And this problem is not very well-known, and the fame and benefits that come with solving it are far less than the effort that goes into it.

After saying the problem, Xu Chuan pinched the bridge of his nose and continued with a headache: "I'm afraid I don't have too many thoughts about this question for the time being." ”

Although he had previously solved a Weyl-Berry conjecture for the problem of isospectral directions, the Weyl-Berry conjecture and the isospectral non-isomorphic conjecture were two completely different directions in the same field.

World-class problems can not be solved so easily.

Even if it's just an inspiration, it's not so easy to get.

Feverman was not surprised, nodded in agreement, and said: "This is a step in solving the ns equation, if it is really so easy to solve, we would have already had results for the advancement of the ns equation." ”

After a pause, he continued: "Don't worry, we still have time." ”

"And we've been able to push it here these days, and we've gained enough. Now it's time to stop and take a break, have a good reminiscence and tidy up the harvest. ”

"Perhaps, in the process of reminiscing and organizing, the inspiration will come to the door on its own?"

Xu Chuan nodded and agreed: "Then let's get here first today." ”

These days of communication and harvest are indeed enough for the two of them to spend some time sorting out.

"Hopefully we can solve this problem, and if you have any new ideas, please be sure to let me know as soon as possible," said Feverman with a smile. ”

"Of course."

......

Professor Feverman left, huddled in his office as the background wall was quickly surrounded by four students.

"Professor, are you working on the NS equation with Professor Feverman?"

Amelia asked with a pair of blue eyes, and the three people next to her also cast expectant glances.

Xu Chuan nodded and said, "It's just a try, Professor Feverman is a top expert in this area, even if he doesn't succeed, he can gain a lot of things." ”

Hearing this, another student, Shashi Perez, quickly asked, "Professor, can we also participate in your and Professor Ffeiferman's project?" Even if it's a chore, it's good. ”

As soon as these words came out, the other three students once again cast their expectant gazes.

Participating in the scientific research projects of two top experts is undoubtedly exciting.

Xu Chuan smiled and said, "If you also want to participate, finish learning algebraic clusters and group mapping tools as soon as possible." ”

"If you pass my test by August, you may have a chance to participate."

"As for now."

After a pause, Xu Chuan shrugged his shoulders and continued: "Professor Peverman and I are not short of four tools for making coffee and serving the blackboard, and we can do those things by ourselves. ”

Hearing this, Shashi Perez cast a resentful look, in the eyes of the professor, he is now just a tool for making coffee, which is really hurtful.

Moreover, it is not so easy to delve into the tools of algebraic cluster and group mapping.

He's been looking at the paper tool for a month.

But to be honest, even for a PhD student, it's too esoteric, and it involves quite a lot of mathematical fields, and it's quite difficult to thoroughly understand it in a few months.

Many things that he can't understand have to refer to other textbooks to understand.

But compared to others who are also studying this tool, he is undoubtedly lucky, because his mentor is the creator of this tool.

Every time I have a question, I can get a perfect answer from this young mentor.

This also makes Shashi Perez's respect for his mentor higher and higher day by day.

......