Chapter 204 Solving the Isospectral Non-Isometry Isomorphism Conjecture

In the office, Fefferman stared at the formula on the blackboard for a long time before he spoke:

"This is also a way to solve the problem of equal spectrum, and the perfection of your idea is far beyond mine."

"I just conceived an idea, but you have expanded on it and even done some calculations."

After a pause, Feffer continued: "But I don't intend to give up my own ideas. Maybe we can come up with a big one."

Xu Chuan asked curiously: "What big?"

"Study separately!"

Fefferman smiled gently and continued: "We will start from our own paths and use our own methods to make breakthroughs in the problem of equal spectrum. If we can solve this problem within three months, then we will send it out together at that time."

"This is also a kind of pressure on us. It's exciting to think about solving a difficult problem in three months."

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 is not ready to give up his ideas. In his opinion, the ideas of both of them are based on their unique mathematical knowledge.

In this case, it is better to study 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, and he understood Fefferman's idea, and smiled: "I have to say, this is indeed a very interesting proposal, and it is also very tempting."

After a pause, he continued: "Of course, it is also very stressful."

It is not so easy to solve the equal spectrum problem in three months, even if they have ideas to solve this problem in their minds.

If they agree, it means that both of them will work hard in these three months, which is very stressful.

"So, what do you think?" Fefferman asked with a grin.

"I accept this challenge." Xu Chuan responded decisively.

How can he retreat in the face of challenges, not to mention that he has the advantage of age.

He is only 20 years old this year, and Fefferman is already 66 years old. In terms of working hard, I believe he will be more persistent than Fefferman.

"Hahahaha, okay, let's get started!" Fefferman responded with a happy smile.

Although he is well-known for his gentleness and modesty in Princeton, and he is kind to others and has almost no temper, this is Princeton after all, and everyone has their own pride in their hearts.

Even though he is not as old as Xu Chuan and is not as hardworking as him, 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 doesn't know is what kind of monster is standing in front of him.

Princeton Institute for Advanced Study.

In a villa, a figure is crawling in the study and writing furiously.

Messy hair, stubble, dark eye circles, and bloodshot eyes all show how long this figure has been working.

But unlike staying up late, the figure in front of the table has unusually bright eyes and is excited, and the ballpoint pen in his hand is constantly moving on the manuscript paper.

"."

"(D1){u =λu， x∈Ω1，u|Ω1 = 0；"

"(D2){v =v， x∈Ω2，v|Ω2 = 0；"

".Then the eigenvalue problems (D1) and (D2) have discrete spectra {λi}i∈N and {i}i∈N respectively. If for every i∈N, λi =i

".According to Theorem [1][6][11], a pair of bounded connected regions with smooth boundaries (at least C1 smooth boundaries) can be constructed on the plane R2. They are isospectral but not isometry. "

"Therefore, it can be proved that the isospectral non-isometry conjecture holds in three-dimensional bounded regions! ”

After the last point was dropped, Xu Chuan put down the ballpoint pen in his hand, stared at the manuscript on the desk, breathed a sigh of relief, and smiled.

His eyes fell on the calendar next to him, and before he knew it, it was already early June.

It has been nearly two months 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 bounded open domains that do not intersect with each other.

But when he used the Laplace operator to transform and construct a pair of bounded connected regions with smooth boundaries, he encountered some trouble.

The Laplace operator is an n-dimensional A second-order differential operator in Euclidean space, defined as the divergence div of the gradient grad.

It is applicable to elliptic partial differential equations and can also be used to describe the equilibrium stable state in physics, such as the steady state electromagnetic field, gravitational field and reaction diffusion phenomenon.

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 once troubled him.

Fortunately, through searching and browsing the literature related to the isospectral problem and partial differential equations, he found a suitable remedy.

Symplectic geometry algorithm for preserving the symplectic structure of Hamiltonian systems and Lie group method for preserving Lie group differential equations.

These two algorithms proposed by mathematicians from the British Empire in the last century can accurately simulate the changes of differential equations for a long time, and can approximately maintain the conservation characteristics of momentum and energy of differential equations.

These two characteristics can be applied to his mathematical calculations, which can just fill the last gap and allow him to complete the final construction.

Staring at the answer on the manuscript, Xu Chuan smiled.

He has completed his work here, and he doesn't know how Feferman is progressing.

Three months, even if the time they worked together before is added, it is only more than four months.

It is not easy to solve a world-class problem in four months, even for a Fields Medal winner.

He can solve it because of his research on analysis and topology in his previous life, and the first mathematical problem he solved in this life is in the direction of equal spectrum, so he can solve it so quickly.

As for Feferman, it is not clear.

But I guess he must have some confidence in proposing this challenge.

After all, Feferman 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 at Fefferman's previous ideas again, he could keenly perceive that solving the isospectral problem from the perspective of Dirichlet functions and nonlinear partial differential equations was much easier than starting from the Laplace operator he proposed.

This was not just a scientific intuition, but also came from his research on the isospectral problem during this period.

After all, after solving the isospectral non-isometry conjecture, he had a deeper understanding of this problem than before.

And there were some vague speculations about other methods that could solve this problem.

This also made Xu Chuan sigh a little. Even if he had solved one of the seven millennium problems, his understanding of the field of mathematics was still not as good as these top experts who had been immersed in mathematical research for decades.

There was still a certain gap between him and these top mathematicians who specialized in mathematical research.

However, this gap was gradually narrowing as he grew older and absorbed more mathematical knowledge.

After coming out of the villa, Xu Chuan did not disturb Fefferman. He just took a look in his office and returned to his office after seeing that there was no one in the office.

When they saw him appear at the door of the office, the four students who were studying inside showed surprised expressions on their faces.

"Professor, you are back?"

"Have you solved the equal spectrum problem?"

Several students asked in surprise.

Xu Chuan nodded and said, "If nothing unexpected happens, it should be. Has Professor Fefferman come these days?"

Hearing this, the four students couldn't help swallowing their saliva.

Is their mentor too abnormal?

In fact, in addition to studying these days, the four of them also tried to study the problems studied by Professor Xu Chuan and Professor Fefferman for a period of time.

Want to see if the four of them can solve this problem together.

Or, want to see how far they are from the two professors.

This proposal was initiated by Shashi Perez, and soon won the consent of the other three.

After all, being able to come to Princeton, it can be said that the four of them are all academic masters, and naturally they are proud.

Three cobblers are better than one Zhuge Liang. If they can solve this problem, maybe their mentor and Professor Feferman will look at them with new eyes?

But the reality is cruel.

Even if the equal spectrum problem is a small branch problem in the NS equation, even if Professor Feferman and their mentor's two different solutions are in front of them, they are powerless.

Let alone solving this equal spectrum problem, even if they follow the ideas of the two professors to move forward, they can't do it with the efforts of the four of them.

This attempt made them see clearly that there is a gap between people, no, there is a gap.

After recovering from the shock, Roger Dean replied respectfully: "Professor Fefferman came here a few days ago, but you were not there."

Hearing the words of this senior disciple, 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 here a few days ago, he told me that if you come back, please call him."

Xu Chuan nodded and said: "I know."

Take out the mobile phone from his pocket, Xu Chuan walked out of the office and called Fefferman.

The phone rang for a while before it was connected, and Feferman's voice came quickly: "Oh, Xu, are you finally willing to leave your villa?"

Xu Chuan smiled and asked: "How is your research progress?"

Feferman: "There is a place stuck, I am trying to solve it, what about you?"

After Xu Chuan finished speaking, Feferman remembered something before he answered, and then hesitantly asked: "You haven't solved it yet?"

He remembered that when he communicated with Deligne before, Deligne said that Xu Chuan habitually locked himself in the room when solving problems.

And every time he came out, it proved that the problem he was studying had a result.

Could it be that he has solved the equal spectrum non-isometry isomorphism conjecture?

Xu Chuan nodded and said: "Well, I have constructed it using my method."

After the words fell, a "plop" sound came from the other side, like a mobile phone fell on the wooden table, but it took a long time to pick it up, and an anxious voice came from inside.

"You are in the office now, I will go there immediately."

Feferman came faster than Xu Chuan thought.

And it was not just him who came, but also Professor Deligne.

In fact, Feferman was in Deligne's office just now, and the two were discussing how to solve the problems they encountered.

"Have you solved the equal-spectral non-isometric isomorphism conjecture? Where is the proof process? Have you brought it?" Feferman asked anxiously.

"It's still in my villa. Do you want to go together or I will get it." Xu Chuan replied.

"Of course we will go together." Feferman said without hesitation.

In the quiet villa, Xu Chuan handed the proof process to Feferman, who immediately took it and looked at it.

On the side, Deligne asked with interest: "Do you have any copies?"

Xu Chuan shook his head and said: "No copies, but I still have the original manuscript."

As he said that, he took out the original manuscript from the bookcase next to him and handed it to Deligne.

Although the neatness of the original manuscript is not as good as the proof process after the arrangement, it is still okay to look at the prover's ideas from it.

In the quiet villa, time passed bit by bit.

Professor Fefferman was the first to put down the proof process in his hand and looked at Xu Chuan opposite with a complicated look.

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 46 years younger than him.

He did not find any loopholes in the paper, especially the step of using Laplace operator transformation to construct a stable closed Willmore hypersurface, which made him involuntarily applaud in his heart.

It's too exquisite.

It seems that mathematics is born like this.

But he didn't dwell on it. After all, the boy in front of him, apart from his age, was not inferior to him in mathematical ability, and could even be said to be better.

He had not solved the seven millennium problems, nor did he become a tenured researcher at the Institute for Advanced Study at the age of 20.

"It's an excellent proof process. At least I didn't find any loopholes." Feferman put down the manuscript in his hand, looked at Xu Chuan and continued, "Is your mathematical talent really so exaggerated? You solved this problem in one and a half months."

Xu Chuan grinned, pointed to his dark circles, and said, "One and a half months? Maybe."

"I have been sleeping for no more than six hours a day for some time. Even if you add other time such as eating and running, it will not exceed ten hours. I have been studying this problem for the remaining fourteen hours."

Hearing this, Feferman opened his mouth, wanting to say something but stopped.