Chapter 202 NS Equations
When Xu Chuan went to the office with Fefferman 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 are discussing this matter.
[Hey, do you know? The genius who proved the Hodge conjecture said in his first class that he only had five months to prove the Hodge conjecture! 】
[Five months? Are you kidding me? 】
[I can swear to God that nothing I said is false. 】
[If this is true, it would be too scary, but in fact it is impossible. Five months have proved Hodge’s conjecture that no one can do it. In fact, he also said later that he paved the way for this. based on more than ten years. 】
[Nine years of education + three years of college entrance examination + five years of simulation? (ov)ノ】
[This is magic and magic from the East. 】
Just as Xu Chuan had expected, few people would believe that he had actually proved the Hodge conjecture within five months. This was outrageous.
In fact, Xu Chuan himself would not believe it if this matter were brought to others.
After all, it took him only five months to complete the proof of Hodge's conjecture, but this was inseparable from his research in the fields of topology and mathematical analysis in his previous life, and also from the algebraic geometry and algebraic geometry he studied with Deligne in this life. Differential Equations.
It is not an exaggeration to sharpen a sword for more than ten years.
But if a scholar can sharpen such a sword and kill the evil dragon that looms high above him, it would be the greatest achievement in his life.
However, Xu Chuan was not satisfied. After conquering the Hodge conjecture, he and Fefferman joined forces and launched a charge towards the smooth and popular ultimate goal of 'NS equation'.
The proposal came from Fefferman.
After communicating with Xu Chuan twice about the idea of smooth manifolds, Fefferman couldn't hold back his thoughts.
After all, he has made huge contributions in the fields of function theory of multiple complex variables and smooth manifolds, and has an in-depth understanding of this area of knowledge.
In 1974, he proved the world's difficult problem that "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 to prove without success.
Because the region of multiple complex variables is different from the situation of single complex variables, two singly connected regions are not necessarily holomorphically 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, Fefferman has attempted to attack the NS equation several times, but all ended in failure.
The arrival of Xu Chuan brought him a new dawn. After thinking for a long time, he finally plucked up the courage to propose to Xu Chuan that they jointly try to solve the NS equation.
As for Fefferman's proposal, Xu Chuan agreed directly without any hesitation.
The Navier-Stokes equation was one of the problems he most wanted to solve in his previous life.
By solving it, there may be hope to curb the evil dragon of ultra-high temperature plasma turbulence in controllable nuclear fusion and put reins on it to 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 problem did not get the 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 makes people sigh, destiny is indeed wonderful.
Institute for Advanced Study in Princeton.
In Xu Chuan's office, Fefferman was writing with white chalk on a blackboard.
"λ1(u)<λ2(u)<···<λn(u) For i=1,···,n, let 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.
Behind him, the four students who were doing their homework also came over curiously and stood in a place that did not affect the two of them, watching curiously.
At first, the four of them could more or less understand some of what Professor Fefferman wrote on the blackboard, but as time went by, some 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, the four students squatting behind them had confusion written all over their faces, and they began to discuss in a low voice.
In the office, the slightly younger Shashi Perez poked the big brother beside him: "Dean, do you understand what Professor Fefferman wrote? What on earth are they studying?"
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Roger Dean did not look away from the blackboard, but he still responded to his junior brother's question. He shook his head and whispered: "I don't know, but I speculate that it should be a problem in the direction of manifolds or Lie groups. .”
"Manifold? Is there any problem with the direction of the manifold that warrants the two Philippine Prize bosses joining forces?" Shashi Perez muttered in a low voice.
On the side, Gu Bing, who had also been unable to keep up with the rhythm for a long time, rubbed his sour eyes and said, "Of course."
"For example?"
"N-S equation!"
"You mean the professor and the others are working together to solve the NS equation?"
"I never said that." Gu Bing shrugged and said softly.
But this still caused waves in the hearts of others. Following Hodge's conjecture, is their professor going to attack another seven millennium problems?
Xu Chuan ignored the mutterings of the students behind him and stared at the calculations on the blackboard.
Until now, he is the only one who can keep up with Fefferman and understand his thinking.
In general, Fefferman uses the Lie group with a smooth differential manifold structure to perform smooth mapping, allowing the Lie group G unitary table G to perform 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 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 calculations and formulas on the blackboard made Xu Chuan's eyes shine as bright as stars.
From this idea, he saw the possibility of advancing the NS equation.
This is a brand-new idea. It 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 had hinted before, but it is almost completely separated from it.
As expected of Professor Fefferman, the youngest scholar to be appointed a professor in an American university.
His knowledge and thinking are inspiring and admirable.
Mathematics is like this. Once your thinking is wrong, no matter how hard you try, you will still be groping forward in chaos and darkness, unable to see the future.
And if your thinking is right, the door of hope will shine in the darkness, like a beacon, guiding you forward.
Xu Chuan had a lot of experience with this when he was in middle school and high school.
Sometimes he encounters some multiple-choice questions or fill-in-the-blank questions that he doesn't know how to answer, 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 mobile blackboard with mathematical formulas, Fefferman turned around and looked at Xu Chuan behind him.
"Xu, from the exchanges a few days ago, I got some inspiration. I used the smooth properties of Lie groups on differential manifold structures to extend the orbital method to reduced Lie groups, which is very useful for research on three-dimensional incompressible Navier-Stokes The global existence of smooth solutions to the equations helps ”
After a pause, he continued: "But I feel that there seems to be a problem if we continue to push forward."
Before Fefferman finished speaking, Xu Chuan continued: "How can we construct a pair of bounded connected regions on the plane R2, whose boundaries are very not smooth, and even have fractal boundaries, making them isospectral?" But it is not isomorphic. ”
Hearing this, Fefferman suddenly nodded and said: "No wonder I haven't been able to push forward. This is an equal spectrum problem."
"If we can solve it, maybe we can further solve the momentum conservation equation in the NS equation."
Staring at the calculations on the blackboard, Xu Chuan touched his chin and nodded.
He agrees with Fefferman's statement.
Both of them are top mathematicians, and they have the same opinion on the same problem. So behind this opinion, there is a high probability that it is the correct answer.
But the problem now is that standing in front of this problem is a mountain with an invisible height.
Neither of them knew how long it would take to get over or around.
We don’t even have a clear idea of what to do and which path to choose to start.
After staring at the calculations on the blackboard and thinking about it for five minutes, Xu Chuan came back from his thoughts, shook his head and said:
"I'm afraid this problem is not that easy to solve. If I guess correctly, it involves a difficult problem in another direction."
"What's the problem?" Fefferman asked quickly.
"The conjecture of isospectral isomorphism."
Xu Chuan uttered a few words, and a look of surprise suddenly appeared on Fefferman's face: "So it's this."
The conjecture of isospectral isomorphism is a difficult problem at the intersection of analysis (spectrum of elliptic operators), geometry and topology.
The time from being proposed to today is not that long.
It was a question raised by Gordon Webber Walpert in 1992 when he broke through the field of equal spectrum.
That is: "Is there a pair of bounded connected regions on the plane R2 with smooth boundaries (at least the smooth boundaries of C1), which are isospectral but not isometrically isomorphic?"
This problem is a difficult problem at the intersection of the three major fields of analysts, geometry, and topology, and not many mathematicians are interested in it.
After all, it is too difficult to understand 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 fame and benefits brought by solving it are far outweighed by the effort required.
After asking the question, Xu Chuan pinched the bridge of his nose and continued with a headache: "I'm afraid I don't have many ideas about this issue for the time being."
Although he has already solved the Weyl-Berry conjecture about the isometric direction problem before, the Weyl-Berry conjecture and the isometric isomorphism conjecture are two completely different problems in the same field.
World-class problems can never be solved so easily.
Even if it is just an inspiration, it is not so easy to harvest.
Fefferman was not surprised. He nodded in agreement and said: "This is a step in solving the NS equation. If it were so easy to solve, we would have already achieved results in advancing the NS equation."
After a pause, he continued: "There's no rush, we still have time."
"And we have been able to push it this far these days, and the harvest has been enough. Now it is time to stop and take a break, reflect on and sort out the harvest."
“Perhaps, in the process of reviewing and sorting out, inspiration will come to us on its own?”
Xu Chuan nodded and agreed: "Then let's come here first today."
The exchanges and gains of these days were 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 tell me as soon as possible."
"Of course."
Professor Fefferman left and huddled in the office as the background wall. Four students quickly gathered around him.
"Professor, are you studying the NS equations with Professor Fefferman?"
Amelia asked with her blue eyes open, and the three people next to her also cast expectant glances.
Xu Chuan nodded and said: "Just give it a try. Professor Fefferman is a top expert in this field. Even if you don't succeed, you can still gain a lot."
Hearing this, another student Shashi Perez quickly asked: "Professor, can we also participate in your and Professor Fefferman's project? Even doing odd jobs is fine."
As soon as these words came out, the other three students once again cast expectant glances.
It is undoubtedly exciting to participate in the scientific research project of two top experts.
Xu Chuan smiled and said: "If you also want to participate, please complete the study 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 ourselves."
Hearing this, Shashi Perez cast a resentful look. In the professor's eyes, he was just a tool for making coffee, which was really hurtful.
Moreover, it is not easy to delve into algebraic varieties and group mapping tools.
Over the past month, he has been looking at the paper tool.
But to be honest, even for doctoral students, that stuff is too esoteric. It involves a lot of mathematical fields. It is very difficult to completely understand it in a few months.
For many things he doesn't understand, he has to refer to other textbooks to understand them.
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 instructor.
This also made Shashi Perez's respect for his mentor grow day by day.
(End of chapter)