Great Country Academician

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

"This is also a way to solve the isospectral problem, and the road you conceived is far more perfect than mine."

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

After a pause, Feifer continued: "But I don't intend to give up my own ideas, maybe we can make a big one."

Xu Chuan asked curiously, "What's the big one?"

"Separate research!"

Fefferman smiled gently, and continued: "We start from our own paths and use our own methods to make breakthroughs on the isospectrum problem. If we can complete the solution of this problem within three months, then by It's time to send it out together."

"This can be regarded as a kind of pressure on us. It takes three months to solve a difficult problem, and it is exciting to think about it."

Like Xu Chuan, he is also optimistic about the idea provided by the other party, thinking that it can also solve the problem of isospectrum.

But he is not ready to give up his thinking. In his opinion, the thinking of both of them is based on their own unique mathematical knowledge.

In this case, it is better to study separately.

Given a time limit, make 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 isospectrum problem, even if they have the idea to solve this problem in their minds, it is not so easy to do.

If you agree, it means that the two of them will explode within these three months, and the pressure will be full.

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

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

Facing challenges, how can he shrink back, not to mention that he still has the advantage of age.

He is only twenty years old this year, and Fefferman is already sixty-six. In terms of liver explosion, 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 famously gentle and humble, he treats people kindly and has almost no temper, but this is Princeton after all, and everyone has their own arrogance in their hearts.

Even if he is not as old as Xu Chuan, and he is not as good as the other party in terms of explosive liver, but he is not without advantages.

Being older means that he has more than forty years of experience and knowledge.

From this point of view, he has an absolute advantage.

However, what he didn't know was what kind of monster was standing in front of him.

Princeton Institute for Advanced Study.

In a villa, a figure was prostrate in the study, writing hard.

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

But unlike staying up late, the figure in front of the desk had bright eyes and high spirits, and the ballpoint pen in his hand was constantly scribbling 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 each i∈ N, there is λi =i

".According to theorem [1][6][11], a pair of bounded connected regions with smooth boundary (at least C1 smooth boundary) can be constructed on the plane R2, they are isospectral, but not equal distance isomorphism."

"Thus, it can be proved that the isospectral non-isometric isomorphism conjecture is established in the three-dimensional bounded region!"

After the last point fell, Xu Chuan put down the ballpoint pen in his hand, stared at the manuscript paper on the desk and let out a long sigh of relief, with a smile on his face.

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

Nearly two months have passed since Fefferman challenged him in the office.

In the past two months, with the help of his previous research on the Weyl-Berry conjecture, he used 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 the equilibrium and stable state in physics, such as the electromagnetic field, gravitational field, and reaction-diffusion phenomena in a steady state.

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 made him very distressed.

Fortunately, searching and browsing the literature on isospectral problems and partial differential equations, he found a suitable remedy.

The symplectic geometry algorithm for preserving the symplectic structure of the Hamilton system, and the Lie group method for the differential equation of the Paule Lie group.

These two algorithms proposed by mathematicians in 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.

And these two characteristics can just be applied to his mathematical calculations, which can properly fill the last loophole and allow him to complete the final construction.

Staring at the answer on the manuscript paper, Xu Chuan raised a smile on his face.

He has completed his work here, and I don't know how Fefferman's progress is.

Three months, even including the time the two worked together before, is only more than four months.

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

He was able to solve it, relying on his research on analysis and topology in his previous life, and the fact that the first mathematical problem he solved in this life was in the isospectral direction, so he could do so at such a fast speed.

As for Fefferman, it's not clear.

But presumably, he must have some confidence in raising 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 at Fefferman's previous ideas, he was keenly aware that starting from Dirichlet functions and nonlinear partial differential equations to solve isospectral problems was much better than his It is much easier to start from the proposed Laplacian operator.

This is not only a scientific intuition, but also derived from his research on the equispectral problem during this period.

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

There are also some hazy speculations about other ways to solve this problem.

This also made Xu Chuan sigh a little. Even though he has solved one of the seven millennium problems, his understanding of the field of mathematics is still not as good as 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.

Coming out of the villa, Xu Chuan didn't bother Fefferman, but 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 appearing at the door of the office, the four students who were studying inside showed surprise expressions on their faces.

"Professor, are you back?"

"Have you solved that isospectral problem?"

Several students asked in amazement.

Xu Chuan nodded and said, "If there are no accidents, it should be. Has Professor Fefferman been here these days?"

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

Is their mentor too perverted?

In fact, in addition to studying these days, the four of them have also tried to conduct research on the issues studied by 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, being able to come to Princeton, it can be said that the four of them are all academic masters, so they are naturally proud.

The three cobblers are still as good as Zhuge Liang. If they can solve this problem, maybe the tutor and Professor Fefferman will look at them with admiration?

Then reality is cruel.

Even if the isospectral problem is a small branch problem in the NS equation, even if the two different solutions of Professor Fefferman and their supervisors are placed in front of them, they are powerless.

Not to mention solving the problem of isospectrum, it is just a step forward along the line of thought of the two professors, which cannot be achieved with the efforts of the four of them.

This attempt made them see 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 here."

Xu Chuan was a little surprised when he heard the big disciple's words, and asked in amazement: "Has Professor Fefferman solved the isospectrum 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 call him .”

Xu Chuan nodded and said, "I see."

Taking 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, Fefferman's voice quickly passed over: "Oh, Xu, are you finally willing to leave your villa?"

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

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

After the words fell, before Xu Chuan could answer, Fefferman remembered something, and then asked hesitantly: "Could it be that you have already solved it?"

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

And every time he walks out, it proves that the problem he has studied has achieved results.

Shouldn't he have already solved the isomorphism conjecture of isospectrum and non-isometric isomorphism?

Xu Chuan nodded and said, "Well, with my method, it has been constructed."

As soon as the words fell, there was a sound of 'puff' from the opposite side, as if the mobile phone had dropped on the wooden table, but it took a long time to grab it, and the sound of urgency came from inside.

"You are in the office now, I'll be there right away."

Fefferman came over faster than Xu Chuan thought.

And it wasn't just him who came here, Professor Deligne came with him.

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

"Have you solved the isospectral non-isometric isomorphism conjecture? What about the proof process? Have you brought it over?" Fefferman asked urgently.

"It's still in my villa, should we go there together, or I'll fetch it." Xu Chuan replied.

"Of course we went together." Fefferman said without hesitation.

In the quiet villa, Xu Chuan handed the sorted out proof process to Fefferman, who immediately took it and read it.

On the side, Deligne asked with interest: "Is there any copy?"

Xu Chuan shook his head and said, "I don't have a copy, but I still keep the original paper."

As he spoke, he dug out the original manuscript from the bookcase beside him and handed it to Deligne.

Although the cleanliness of the original manuscript is not as good as the proof process after sorting out, it is still no problem to see the proofer's thinking from the inside.

In the secluded villa, time passed bit by bit.

Professor Fefferman was the first to put down the proof process in his hands, and looked at Xu Chuan opposite him with complicated 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.

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 by using the Laplacian operator transformation, which made him uncontrollably applaud in his heart.

So subtle.

As if mathematics was born to be like this.

However, he didn't get entangled in this. After all, the young man in front of him, regardless of his age, was not weaker than him in mathematics at all, and it could even be said to be better.

He did not solve the seven millennium problems, nor did he become a lifelong fellow of the Institute for Advanced Study in Princeton at the age of twenty.

"It's an 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 a month and a half this problem."

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

"During this period of time, I have slept no more than six hours a day, even if I add other time such as eating and running, it will not exceed ten hours, and for the remaining fourteen hours, I have been studying this question."

Hearing this, Fefferman opened his mouth, but stopped talking.