Chapter 203: Solving the Equispectral Non-Isometric Isomorphism Conjecture
In the office, Fefferman stared at the calculations on the blackboard for a long time before he said:
"This is also a way to solve the problem of equal spectrum, and the path you have conceived is far more complete than mine."
"I just conceived an idea, but you have expanded on this idea and even done some calculations."
After a pause, Feiffer continued: "But I don't intend to give up my own ideas. Maybe we can make a big one."
Xu Chuan asked curiously: "What big one?"
"Study separately!"
Fefferman smiled gently and continued: "We each started from our own paths and used our own methods to make breakthroughs in the equal spectrum problem. If we can complete the solution to this problem within three months, then by then
We will send them out together when the time comes.”
"This can be considered a kind of pressure on us. It takes three months to solve a difficult problem. It's exciting just thinking about it."
Like Xu Chuan, he is also optimistic about the ideas provided by the other party and believes that this can also solve the problem of equal spectrum.
But he was not prepared to give up his own ideas. In his opinion, both of their ideas were based on their own unique mathematical knowledge.
In this case, it is better to study them separately.
Given a time limit, let's have a healthy competition with each other to see whose ideas and abilities are better.
Xu Chuan's eyes moved, he understood Fefferman's thoughts, and said with a smile: "I have to say, this is indeed a very interesting proposal, and it is also very exciting."
After a pause, he continued: "Of course, it's also very stressful."
It takes three months to solve the equal spectrum problem. Even if they have ideas to solve this problem in their minds, it is not that easy to do.
If you agree, it means that both of you will have liver failure within these three months, which is a lot of pressure.
"So, what do you think?" Fefferman asked with a grin.
"I accept this challenge." Xu Chuan responded decisively.
How can he back down when faced with challenges, not to mention that he has the advantage of age.
He is only twenty years old this year, while Fefferman is already sixty-six. When it comes to liver damage, I believe he will last longer than Fefferman.
"Hahahaha, okay, let's get started!" Fefferman responded with a happy smile.
Although in Princeton, he is famous for being gentle and humble, being kind to others and having almost no temper, but this is Princeton after all, and everyone has his own pride in his heart.
Even though he is not as old as Xu Chuan and is not as good as Xu Chuan in terms of liver damage, he is not without advantages.
Being older means that he has more than 40 years of experience and knowledge.
From this aspect, he has an absolute advantage.
However, what he didn't know was what kind of monster was standing in front of him.
.......
Institute for Advanced Study in Princeton.
In a villa, a figure is crawling in the study and writing furiously.
The messy hair, clumps of stubble, dark circles under the eyes, and bloodshot eyes all showed how long this figure had been suffering.
But what is different from staying up late is that the figure in front of the table has extremely bright eyes and high energy, and the ballpoint pen in his hand is constantly scratching on the manuscript paper.
"...."
"(d1){??u =λu, x∈Ω1,u|?Ω1 = 0;"
"(d2){??v =μv, x∈Ω2,v|?Ω2 = 0;"
"...Then the eigenvalue problems (d1) and (d2) have discrete spectra {λi}i∈n and {μi}i∈n respectively. If for each i∈ n, there is λi =μi.
.....
"...According to the theorem [1][6][11], a pair of bounded connected regions with smooth boundaries (at least the smooth boundaries of c1) can be constructed on the plane r2. They are equispectral, but
Non-isometric isomorphism.”
“Thus, it can be proved that the isometric isomorphism conjecture is established in the three-dimensional bounded region!”
.......
After finishing the last point, Xu Chuan put down the ballpoint pen in his hand, stared at the manuscript paper on the desk and let out a long sigh, with a smile on his face.
My eyes fell on the calendar next to me, and before I knew it, it was early June.
Nearly two months have passed since Fefferman challenged him in the office.
In the past two months, he used his previous research on the Weyl-Berry conjecture and the spectral asymptotic theorem in the Xu-Weyl-Berry theorem to construct a set of pairwise disjoint bounded open domains.
But when using the Laplacian operator for transformation to construct a pair of bounded connected regions with smooth boundaries, he encountered some troubles.
The Laplacian operator is a second-order differential operator in n-dimensional Euclidean space, defined as the divergence div of the gradient grad.
It is suitable for elliptic partial differential equations, and can also be used to describe equilibrium stable states in physics, such as steady-state electromagnetic fields, gravitational fields, reaction diffusion phenomena, etc.
This is the key to solving the isospectral problem, but it cannot construct a stable closed Willmore hypersurface in the calculation of eigenvalues, nor can it calculate the constant mean curvature.
This made him very distressed.
Fortunately, he found a suitable remedy by searching and browsing the literature related to equispectral problems and partial differential equations.
An symplectic geometry algorithm that preserves the symplectic structure of Hamilton systems, and a Lie group method that preserves Lie group differential equations.
These two algorithms, proposed by mathematicians from the country where the sun never sets in the last century, can accurately simulate changes in differential equations for a long time, and can approximately maintain the momentum and energy conservation characteristics of differential equations.
These two characteristics can be applied to his mathematical calculations, filling in the last loopholes and allowing him to complete the final construction.
...
Staring at the answers on the manuscript paper, Xu Chuan had a smile on his face.
This chapter is not finished yet, please click on the next page to continue reading the exciting content! He has completed his work, but I don’t know what the progress is on Fefferman’s side.
Three months, even if the previous cooperation time between the two is included, it is only more than four months.
In four months, it is difficult to solve a world-class problem, even for a Fields Medal winner.
His ability to solve it relied on his previous research on analysis and topology, plus the fact that the first mathematical problem he solved in this life was in the isospectral direction, so he could achieve such a fast speed.
As for Fefferman, I don't know.
But he must be somewhat confident in proposing this challenge.
After all, Fefferman himself is a top expert in the field of partial differential equations, and his research on smooth manifolds is also unique.
On the other hand, after two months of research, when he looked back at Fefferman's previous ideas, he could keenly realize that solving equal spectral problems based on Dirichlet functions and nonlinear partial differential equations was much easier than he had done.
It is much easier to start from the Laplacian operator as proposed.
This is not only scientific research intuition, but also comes from his research on equal spectral problems during this period.
After all, after solving the isometric isomorphism conjecture, he has a deeper understanding of this problem than before.
There are also some vague speculations about other ways to solve this problem.
This also made Xu Chuan sigh. Even though he has solved one of the seven millennium problems, his understanding of the field of mathematics is still not as good as those of these top experts who have been immersed in mathematics research for decades.
There is still a certain gap between him and these top mathematicians who specialize in mathematics research.
However, this gap is gradually narrowing as he grows older and absorbs mathematical knowledge.
...
After coming out of the villa, Xu Chuan did not bother Fefferman. He just took a look at his office and returned to his office after seeing no one in the office.
When they saw him appear at the door of the office, the four students who were studying inside all showed surprised expressions on their faces.
"Professor, are you back?"
"Have you solved that isospectral problem?"
Several students asked in surprise.
Xu Chuan nodded and said, "If nothing else happens, it should be. Has Professor Fefferman been here these days?"
Hearing this, the four students couldn't help but swallow their saliva.
Is their mentor too perverted?
In fact, in addition to studying these days, the four of them have also tried to study the issues studied by Professor Xu Chuan and Professor Fefferman for a period of time.
I want to see if the four of them can solve this problem.
In other words, I want to see how far they are from the two professors.
The proposal was initiated by Shahi Perez and quickly won the approval of the other three.
After all, to be able to come to Princeton, it can be said that the four of them are top academics, so they naturally feel arrogant.
The Three Stooges are still rivaling Zhuge Liang. If they can solve this problem, maybe their mentor and Professor Fefferman will look at them with admiration?
Then the reality is cruel.
Even if the isospectral problem is a small branch problem in the ns equations, and even if Professor Fefferman and their instructors have two different solution ideas in front of them, there is nothing they can do.
Let alone solving this equal spectral problem, even if we follow the ideas of the two professors one step further, even with the combined efforts of the four of them, we cannot achieve it.
This attempt allowed them to see clearly that there are differences, no, gaps between people.
.....
After recovering from the shock, Roger Dean respectfully replied: "Professor Fefferman came here a few days ago, but you were not here."
Hearing what this senior disciple said, Xu Chuan was a little surprised and asked in surprise: "Has Professor Fefferman solved the equal spectrum problem?"
Roger Dean shook his head and said: "I don't know, Professor Fefferman didn't say this, but when he came over a few days ago, he told me that if you come back, please give him a call.
.”
Xu Chuan nodded and said, "I understand."
Taking out his cell phone from his pocket, Xu Chuan walked out of the office and dialed Fefferman.
The phone rang for a while before it was answered, and Fefferman's voice quickly came through: "Oh, Xu, are you finally willing to leave your villa?"
Xu Chuan smiled and asked, "How is your research progress?"
Fefferman: "I'm stuck somewhere and I'm trying to figure it out. What about you?"
After Xu Chuan finished speaking, before Xu Chuan could answer, Fefferman remembered something, and then asked hesitantly: "Have you already solved it?"
He recalled his previous communication with Deligne, who said that Xu Chuan habitually locked himself in the room alone when solving problems.
And every time he comes out, it proves that the problem he studied has already produced results.
Could it be that he has already solved the isometric isomorphism conjecture?
Xu Chuan nodded and said: "Well, it has been constructed using my method."
As soon as he finished speaking, a sound of "Pudong" came from the opposite side, as if the mobile phone had been dropped on the wooden table, but he grabbed it after a long time, and a furious voice came from inside.
"You are in the office now, I will be there right away."
...
Fefferman came over faster than Xu Chuan thought.
And he was not the only one who came over, Professor Deligne also came with him.
In fact, Fefferman was in Deligne's office just now, and the two were discussing how to solve the problems they encountered.
"Have you solved the isometric isomorphism conjecture? Where is the proof process? Have you brought it over?" Fefferman asked urgently.
"It's still in my villa. Do you want to go there together or let me get it?" Xu Chuan replied.
"Of course we'll go together." Fefferman said without hesitation.
.......
In the quiet villa, Xu Chuan handed the compiled proof process to Fefferman, who immediately took it and started reading it.
On the side, Deligne asked with interest: "Do you still have a copy?"
Xu Chuan shook his head and said, "I don't have a copy, but I still keep the original manuscript paper."
As he spoke, he dug out the original manuscript from the bookcase on the side and handed it to Deligne.
Although the neatness of the original manuscript is not as good as the organized proof process, there is still no problem in taking a look at the prover's thinking from inside.
In the quiet villa, time passed little by little.
Professor Fefferman was the first to put down the proof process in his hands and looked at Xu Chuan opposite, with a somewhat complicated look in his eyes.
Before coming, he was still thinking about whether he could find some flaws in the proof process to comfort himself.
But now it seems that he lost.
He lost to the boy in front of him who was forty-six years younger than him.
He did not find any loopholes in the paper, especially the step of constructing a stable closed Willmore hypersurface using Laplacian operator transformation, which made him couldn't help but admire it in his heart.
So subtle.
It’s as if mathematics was born this way.
However, he did not dwell on this. After all, the young man in front of him, regardless of his age, was not weaker than him in terms of mathematical ability, and could even be said to be better.
He did not solve the Seven Millennium Problems, nor did he become a lifelong researcher at the Institute for Advanced Study in Princeton at the age of twenty.
"A very excellent proof process, at least I didn't find any loopholes." Fefferman put down the manuscript paper in his hand, looked at Xu Chuan and continued: "Is your mathematical talent really exaggerated to this extent? It was solved in one and a half months.
this problem."
Xu Chuan grinned, pointed at the dark circles under his eyes, and said, "A month and a half? Maybe."
"Since this period of time, I have not slept more than six hours a day. Even if I include other time such as eating and running, it has not been more than ten hours. And for the remaining fourteen hours, I have been studying this
question."
Hearing this, Fefferman opened his mouth but stopped talking.