Great Country Academician

When Xu Chuan went to the office with Fefferman to communicate on smooth manifolds, his first class at Princeton caused quite a stir in Beimi’s university network.

Some well-known university forums are discussing this matter one after another.

【Hi, you know? The genius who proved Hodge's conjecture said in his first class that it only takes five months to prove Hodge's conjecture! 】

[Five months? What are you kidding. 】

【I can swear to God, there is not a single lie in what I said. 】

[If this is true, it would be too terrifying, but in fact it is impossible. It took five months to prove Hodge's conjecture. No one can do it. In fact, he said later that he paved the way for this based on decades of experience. 】

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

[This is a fairy art from the East. 】

Just as Xu Chuan had predicted before, few people would believe that he really proved Hodge's conjecture within five months, which is too outrageous.

In fact, Xu Chuan himself wouldn't believe it if it was put on other people.

After all, it took him only five months to complete the proof of Hodge's conjecture, but this is inseparable from his research in the fields of topology and mathematical analysis in his previous life, and also inseparable from the algebraic geometry and differential equation.

It is not an exaggeration to sharpen a sword for more than ten years.

However, if a scholar can sharpen such a sword and slash at the evil dragon entrenched high above, it is already the greatest achievement in this life.

However, Xu Chuan was not satisfied. After conquering Hodge's conjecture, he joined hands with Fefferman and launched a charge towards the smooth and popular ultimate goal "NS equation".

The proposal was sent by Fefferman.

After exchanging ideas on the smooth manifold field with Xu Chuan twice, Fefferman couldn't hold back his thoughts.

After all, he has made great contributions in the field of multi-complex variable function theory and smooth manifolds, and he also has a deep understanding of this area of ​​knowledge.

In 1974, he proved the world problem of "a biholomorphic mapping from a strictly pseudo-convex region with a smooth boundary to another can be smoothly extended to the boundary".

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

Because the region with multiple complex variables is different from the case of single complex variable, two simply connected regions may not necessarily be completely equivalent, so the method of single complex variable cannot be applied.

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

Based on this, Fefferman has made several attempts to attack the NS equation, but all of them ended in failure.

The arrival of Xu Chuan brought him a new dawn. After thinking for a long time, he finally mustered up the courage to propose to Xu Chuan to jointly try to solve the NS equation.

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

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

To solve it, there may be hope to curb the evil dragon of ultra-high temperature plasma turbulence in controlled nuclear fusion, put a rein on it and tame it.

For this reason, he chose to cooperate with Professor Fefferman in his previous life.

But unfortunately, limited by his mathematical ability and Professor Fefferman's physical ability, this kind of problem did not get a result in the end.

In the first reincarnation, he came to Princeton again, and cooperated with Fefferman again, and the object of solution was still the NS equation.

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

Princeton Institute for Advanced Study.

In Xu Chuan's office, Fefferman is writing with white chalk on a blackboard.

"λ1(u)\u003cλ2(u)\u003c···\u003cλn(u) for i = 1,···,n, set ri(u)=(r1i(u),···,rni(u) )T is the right eigenvector corresponding to λi(u): A(u)ri(u)=λi(u)ri(u).”

"π: G→U(H), G×H→H"

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

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

At the beginning, the four of them could more or less understand what Professor Fefferman wrote on the blackboard, but as time went by, some people began to slowly fall behind.

It is extremely difficult for even a doctoral student to understand what Professor Fefferman wrote on the blackboard

And when the chalk on the blackboard was used up and replaced, the faces of the four students squatting behind were filled with confusion, and they began to discuss in a low voice.

In the office, the slightly younger Shashi Perez poked the big brother beside him: "Dean, did you understand what Professor Fefferman wrote? Professor them, what are you 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 a problem in the direction of manifolds or Li groups. .”

"Manifold? Is there any problem with the direction of the manifold? Is it worth the cooperation of the two Filipino bigwigs?" Shashi Perez muttered in a low voice.

On the side, Gu Bing, who also couldn't keep up with the rhythm, rubbed his sore eyes and said, "Of course."

"for example?"

"N-S equation!"

"You mean the professor and they are working together to solve the NS equation?"

"I never said that." Gu Bing shrugged and said in a low voice.

But this still caused waves in the hearts of others. After Hodge's conjecture, is their professor going to attack another seven millennium problems?

Xu Chuan ignored the muttering of the students behind him, and stared intently at the calculations on the blackboard.

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

In general, Fefferman uses a Lie group with a smooth differential manifold structure to carry out smooth mapping, so that the Lie group G unitary table G performs a continuous action on the Hilbert space, and these actions can keep the space inner product unchanged .

That is to say, the unitary representation of a Lie group G is a homomorphic mapping from the group G to the group U(H) formed by all unitary operators on a certain Hilbert space H.

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

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

This is a brand new way of thinking, which is different from Fefferman and his research on NS equations in his previous life. It is an expansion in the direction of Lie groups that he suggested before, but it is almost completely separated from it.

As expected of Professor Fefferman, the youngest scholar to be appointed as a professor in an American university.

His knowledge and thinking are inspiring to people.

Mathematics is like this. Once the thinking is wrong, no matter how hard you try, you will still be groping forward in chaos and darkness, and you will not see the future.

And if your thinking is right, the door of hope will shine in the dark, like a lighthouse, guiding you forward.

This point, Xu Chuan felt quite a lot when he was in junior high school.

Sometimes when he encounters some questions that he can't do multiple choice or fill in the blanks, the first answer that emerges in his mind intuitively is often the correct answer.

Perhaps, this is the mathematical talent that ordinary people call it.

In the office, in front of the blackboard, after writing mathematical formulas on both sides of the huge mobile blackboard in front of him, Fefferman turned around and looked at Xu Chuan behind him.

"Xu, from the communication a few days ago, I got some inspiration. Using the smooth property of Lie groups on differential manifold structures, I extended the orbital method to reduced Lie groups. This is useful for the study of three-dimensional incompressible Navier-Stokes The overall existence of smooth solutions to the equations helps somewhat."

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

Before Fefferman finished speaking, Xu Chuan continued: "How can a pair of bounded connected regions be constructed on the plane R2, whose boundaries are very rough, even fractal boundaries, so that they are isospectral But not isometric."

Hearing this, Fefferman nodded suddenly, and said, "No wonder I haven't been able to push forward. This is a problem of equal spectrum."

"If it can be solved, maybe we can further solve the momentum conservation equation in the NS equation."

Staring at the formula on the blackboard, Xu Chuan stroked his chin and nodded.

For Fefferman's statement, he agrees.

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

But the problem now is that in front of this problem, there is still a mountain whose height cannot be seen.

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

Even what should be done, which way to choose to start, there is no clear idea.

After staring at the formula on the blackboard and pondering 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. If I guess correctly, it involves a problem in another direction."

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

"Isospectral non-isometric isomorphism conjecture."

Xu Chuan spat out a few words, and Fefferman suddenly showed a look of surprise on his face: "It turned out to be this."

The isospectral non-isometric isomorphism conjecture is a difficult problem in the cross-disciplines of analysis (spectrum of elliptic operators), geometry and topology.

It has not been a long time since it was proposed.

It is a question posed by Gordon Webb Wolpert in 1992 when he broke through the isospectral field.

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

This problem is a problem that intersects the three major fields of analysts, geometry, and topology. Not many mathematicians are interested in it.

After all, it is too difficult to understand the three major fields at the same time. Not everyone belongs to Terence Tao. It is very difficult for most mathematicians to study a mathematical problem across multiple fields. .

And the problem isn't very well known, and the fame and benefits of solving it outweigh the effort.

After asking the question, Xu Chuan pinched the bridge of his nose and continued with some headaches: "I'm afraid I don't have too many thoughts on this question for the time being."

Although he has solved a Weyl-Berry conjecture for the isospectral direction before, the Weyl-Berry conjecture and the isospectral non-isomorphic isomorphism conjecture are two completely different directions in the same field.

World-class problems are not so easily solved.

Even if it is just an inspiration, it is not so easy to harvest.

Fefferman was not surprised either. He nodded in agreement and said, "This is a step in solving the NS equation. If it is really that easy to solve, we have already achieved results in advancing the NS equation."

After a pause, he continued, "No rush, we still have time."

"And we've been able to push it here these days, enough has been gained. It's time to stop and take a break and reflect and sort out the gains."

"Maybe, in the process of reminiscing and sorting out, the inspiration came to us by itself?"

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

The exchanges and gains these days are indeed enough for the two of them to spend some time sorting out.

Fefferman smiled and said: "I hope we can solve this problem. If you have any new ideas, please be sure to tell me as soon as possible."

"certainly."

Professor Fefferman left and shrank in the office as the background wall. Four students quickly surrounded him.

"Professor, are you working on NS equations with Professor Fefferman?"

Amelia asked with her blue eyes open, and the three people beside her also cast expectant gazes.

Xu Chuan nodded and said, "It's just a try. Professor Fefferman is a top expert in this field. Even if he doesn't succeed, he can still gain a lot."

Hearing this, another student, Shashi Perez, quickly asked: "Professor, can we also participate in your project with Professor Fefferman? Even doing odd jobs is fine."

As soon as these words came out, the other three students looked expectantly again.

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

Xu Chuan smiled and said, "If you also want to participate, you should complete the learning of algebraic varieties and group mapping tools as soon as possible."

"If you pass my assessment before August, you may have the opportunity to participate."

"As for now."

After a pause, Xu Chuan shrugged and continued: "Professor Fefferman and I do not lack four tools for making coffee and serving the blackboard. 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 algebraic varieties and group mapping tools.

For a month, he has been looking at the paper tools.

But to be honest, even for doctoral students, that stuff is too esoteric, it involves quite a lot of mathematical fields, and it is quite difficult to understand it thoroughly within a few months.

For many things that he can't understand, he has to refer to other textbooks to understand.

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

Every time I encounter a problem, I can get a perfect answer from this young mentor.

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