Chapter 202 NS Equation

When Xu Chuan followed Fefferman to the office to discuss smooth manifolds, his first class at Princeton caused quite a stir in the university network in North America.

Some well-known university forums were discussing this matter.

[Hey, do you know? The genius who proved the Hodge conjecture said in his first class that it only took five months to prove the Hodge conjecture! ]

[Five months? Are you kidding me. ]

[I can swear to God that I didn't tell a single lie. ]

[If this is true, it would be too terrifying, but in fact it is impossible. No one can prove the Hodge conjecture in five months. In fact, he later said that he laid the foundation for more than ten years. ]

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

[This is a fairy magic from the East. ]

As Xu Chuan expected before, almost no one would believe that he really proved the Hodge conjecture in five months. This is too outrageous.

In fact, if this happened to someone else, Xu Chuan himself would not believe it.

After all, it took him only five months to prove the Hodge conjecture, but this was inseparable from his research in topology and mathematical analysis in his previous life, and from the algebraic geometry and differential equations he learned from Deligne in this life.

It is not an exaggeration to spend more than ten years sharpening a sword.

But if a scholar can sharpen such a sword and cut the evil dragon entrenched high above, it will be the greatest achievement in his life.

However, Xu Chuan was not satisfied. After conquering the Hodge conjecture, he joined hands with Fefferman and charged towards the ultimate goal of smooth manifolds, the NS equation.

This proposal was made by Fefferman.

After communicating with Xu Chuan twice about the ideas in the field of smooth manifolds, Fefferman could not hold back his thoughts.

After all, he had made great contributions in the field of multi-complex function theory and smooth manifolds, and had a deep understanding of this field.

In 1974, he proved the world's difficult 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 was a problem that many mathematicians in the 20th century tried to prove but failed.

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

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

Based on this, Feferman had several experiences of charging at the NS equation, but all ended in failure.

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

As for Feferman'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 previous life.

Solving it may be possible to curb the dragon of ultra-high temperature plasma turbulence in controlled nuclear fusion and put a bridle on it to tame it.

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

Unfortunately, due to his limited mathematical ability and Professor Feferman's physical ability, this problem did not get a result in the end.

After reincarnation, he came to Princeton again and started working with Feferman again, and the object to be solved was still the NS equation.

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

Princeton Institute for Advanced Study.

In Xu Chuan's office, Feferman was writing on a blackboard with white chalk.

"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), G× H→ H"

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

Behind him, four students who were doing homework also came over curiously, standing in a place where they would not affect the two and watching curiously.

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

For those things written on the blackboard by Professor Fefferman, it was extremely difficult to understand even for doctoral students.

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

In the office, the slightly younger Shashi Perez poked the older brother beside him: "Dean, did you understand what Professor Fefferman wrote? What are the professors studying?"

Roger Dean did not take his eyes off the blackboard, but he still responded to his junior's question. He shook his head and whispered: "I don't know, but I guess it should be a difficult problem in the direction of manifolds or Lie groups."

"Manifolds? Are there any problems in the direction of manifolds that are worth the two Fields Award bigwigs working together?" Shashi Perez muttered softly.

On the side, Gu Bing, who had also long been unable to keep up with the pace, rubbed his sore eyes and said: "Of course there are."

"For example?"

"N-S equation!"

"Are you saying that the professors are working together to solve the NS equation?"

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

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

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

So far, he is the only one who can keep up with Feferman and understand his ideas.

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

In other words, the unitary representation of the Lie group G is a homomorphic mapping from the group G to the group U(H) composed of all unitary operators on a certain Hilbert space H.

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

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

This is a brand new idea, different from Feferman and his research on NS equations in his previous life. It is an extension of the Lie group direction he had previously suggested, but it is almost completely separated.

He is worthy of being Professor Feferman, the youngest scholar appointed as a professor in a US university.

His knowledge and thinking are inspiring and admirable.

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

And if your idea is right, the door of hope will emit light in the darkness, like a lighthouse, guiding you forward.

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

Sometimes he encountered some multiple-choice questions or fill-in-the-blank questions that he couldn't answer, and the first answer that emerged intuitively in his mind was often the correct answer.

Perhaps this is what ordinary people call mathematical talent.

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

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

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

Before Feferman finished speaking, Xu Chuan continued: "How to construct a pair of bounded connected regions on plane R2, whose boundaries are very non-smooth, or even have fractal boundaries, so that they are isospectral but not isomorphic."

Hearing this, Feferman nodded suddenly and said: "No wonder I have not been able to move forward. This is an isospectral problem."

"If it can be solved, perhaps we can make a further solution to the momentum conservation equation in the NS equation."

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

He agreed with Feferman's statement.

Both of them are top mathematicians. If they have the same view on the same problem, then the correct answer is most likely behind this view.

But the problem now is that there is a mountain whose height is invisible in front of this problem.

Neither of them knows how long it will take to climb over or bypass it.

Even what to do and which road to choose to start, there is no clear idea.

After staring at the formula on the blackboard for 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'm not mistaken, it involves a problem in another direction."

"What problem?" Feferman asked quickly.

"Isospectral non-isometric isomorphism conjecture."

Xu Chuan uttered a few words, and Feferman's face suddenly showed a look of realization: "So that's it."

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

It has not been long since it was proposed.

This is a question raised by Gordon Weber Wolpert in 1992 when he broke through the field of isospectral.

That is: "Is there a pair of bounded connected regions with smooth boundaries (at least C1 smooth boundaries) on plane R2, which are isospectral but non-isometric?"

This problem is a difficult problem that crosses the three major fields of analysis, geometry, and topology, and 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 is Terence Tao. It is difficult for most mathematicians to study a mathematical problem across multiple fields.

Moreover, this problem is not very famous, and the reputation and benefits brought by solving it are far less than the effort required.

After stating the problem, Xu Chuan pinched his nose and continued with a headache: "I'm afraid I don't have too many ideas about this problem for the time being."

Although he had previously solved a Weyl-Berry conjecture in the isospectral direction, the Weyl-Berry conjecture and the isospectral non-isometric conjecture are two completely different problems in the same field.

World-class problems are not so easy to solve.

Even just an inspiration is not so easy to obtain.

Feferman was not surprised, nodded in agreement, and said: "This is a step in solving the NS equation. If it is so easy to solve, we would have achieved results in the advancement of the NS equation long ago."

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

"And we have been able to advance it here in these days, and we have gained enough. Now it's time to stop and rest, and savor and sort out the harvest."

"Maybe, in the process of savoring and sorting, inspiration will come to us by itself?"

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

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

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

"Of course."

Professor Feferman left, and the four students who were huddled in the office as a background wall quickly surrounded him.

"Professor, are you studying the NS equation with Professor Feferman?"

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

Xu Chuan nodded and said, "Just give it a try. Professor Feferman is a top expert in this field. Even if we don't succeed, we can still gain a lot."

Hearing this, another student Shashi Perez quickly asked, "Professor, can we also participate in your and Professor Feferman's project? Even if we just do some odd jobs, it's fine."

As soon as this was said, the other three students cast their expectant eyes 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, finish learning the algebraic cluster and group mapping tools as soon as possible."

"If you pass my assessment before August, you may have a chance to participate."

"As for now."

After a pause, Xu Chuan shrugged and continued, "Professor Feferman and I don't lack four tools for making coffee and holding blackboards. We can do those things ourselves."

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

Moreover, it is not so easy to thoroughly study the algebraic variety and group mapping tools.

He has been reading the paper tool for the past month.

But to be honest, even for a doctoral student, that thing is too profound. It involves quite a lot of mathematical fields. It is very difficult to thoroughly understand it within a few months.

He has to refer to other textbooks to understand many things he doesn't understand.

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

Every time he encounters a problem, he can get a perfect answer from this young mentor.

This also makes Shashi Perez respect his mentor more and more.