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

"It's also a solution to the problem of equivalence, and the path you conceived is far more complete than mine."

"I just came up with an idea, but you've expanded on it and even done some calculations."

After a pause, Pfeiffer continued, "But I'm not going to give up on my own ideas, maybe we can make a big one." ”

Xu Chuan asked curiously, "What's big?" ”

"Study separately!"

Feverman smiled mildly, and continued: "Let's start from our own paths, use our own methods to make breakthroughs in the equispectral problem, and if we can complete the solution of this problem within three months, then we will send it out together when the time comes." ”

"It's also a kind of pressure on us, three months to solve a problem, it's exciting to think about."

Like Xu Chuan, he is also optimistic about the ideas provided by the other party, believing that this can also solve the problem of equivalence.

But he is not ready to give up his own ideas, and in his opinion, both of them are based on their own unique mathematical knowledge.

In that case, it is better to study them separately.

Given a time frame, it's good to have a healthy competition with each other to see whose ideas and abilities are better.

Xu Chuan's eyes moved, understood Feverman'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. ”

Three months to solve the problem of equivalence, even if they have the idea of solving the problem in their minds, it is not so easy to do.

If you agree, it means that both of them will burst their livers in these three months, and they are full of pressure.

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

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

In the face of challenges, how can he back down, not to mention that he still has an age advantage.

He is only twenty years old this year, and Feverman is sixty-six, and I believe he will last longer than Feverman in the area of liver bursting.

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

Although he is known for his mildness and humility in Princeton, and his kindness and almost no temper, this is Princeton after all, and everyone has their own pride in their hearts.

Even if he is not as old as Xu Chuan and inferior to the other party in terms of liver explosion, he is not without advantages.

Older age means that he has more than 40 years of experience and knowledge.

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

It's just that what he doesn't know is what kind of monster is standing in front of him.

.......

Institute for Advanced Study in Princeton.

In a villa, a figure crawls in the study.

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

But unlike staying up late, the figure in front of the table has unusually bright eyes and high spirits, 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∈......

“... According to theorem [1][6][11], a pair of bounded connected regions with smooth boundaries (at least C1 smooth boundaries) can be constructed on plane R2, which are isospectral but not equidistant isomorphism. ”

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

.......

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

His eyes fell on the calendar next to him, and before he knew it, the time had reached the beginning of June.

Nearly two months have passed since Peverman and his challenge in the office.

In the past two months, with the help of his previous research on the Weyl-Berry conjecture, he has constructed a set of bounded open fields that do not intersect by using the spectral asymptotic theorem in the xu-weyl-berry theorem.

But when it came to using the Laplace operator to construct a pair of bounded connected regions with smooth boundaries, he ran into some trouble.

The Laplace operator is a second-order differential operator in the 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 and stable states in physics, such as electromagnetic fields, gravitational fields, and reaction diffusion phenomena in a stationary 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 average curvature.

This once made him distressed.

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

Symplectic geometry algorithm for symplectic structure of Hamilton system, Lie group method for Lie group differential equations.

These two algorithms, proposed by mathematicians in the last century, can accurately simulate the changes of differential equations over a long period of time, and can approximate the conservation of momentum and energy of differential equations.

And these two features can just be applied to his mathematical calculations, which can just fill in the last loophole and allow him to complete the final build.

......

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

He's done his job, and he doesn't know what's going on on Feverman's side.

Three months, even if you add the time that the two have worked together before, is only more than four months.

Four months to solve a world-class puzzle is difficult even for a Fields Medal winner.

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

On the other hand, it is not clear.

But presumably he must have some certainty in raising this challenge.

After all, Feverman 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 Feverman's previous ideas, he was keenly aware that it was much easier to solve the isospectral problem from the Dirichlet function and nonlinear partial differential equations than from the Laplace operator he proposed.

This is not only a scientific intuition, but also comes from his research on the problem of equivalence spectrum during this time.

After all, after solving the isospectral nonisomorphism conjecture, he has a deeper understanding of this problem than before.

There is also some vague speculation about other ways to solve this problem.

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

There is still a certain gap between him and these top mathematicians who are dedicated to mathematical 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 Peverman, but just went to his office to take a look, and returned to his office after seeing that there was no one in the office.

When he appeared at the door of the office, the four students who were studying inside had surprised expressions on their faces.

"Professor, you're back?"

"Have you solved that equivalence problem?"

Several students asked in surprise.

Xu Chuan nodded and said, "If nothing else, it should be." Has Professor Feverman been here these days? ”

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

Isn't this mentor of theirs too perverted?

In fact, these days, in addition to studying, the four of them have also tried to study the problems that Professor Xu Chuan and Professor Feverman have been studying for a while.

I want to see if the four of them can solve this puzzle with the help of the four of them.

Or rather, to see how far they were from the two professors.

The proposal was initiated by Sahi Perez and quickly won the consent of the other three.

After all, to come to Princeton, it can be said that the four of them are all scholars, and they naturally have arrogance in their hearts.

If they can solve this problem, maybe the mentor and Professor Feverman will be impressed by them?

And then the reality is brutal.

Even if the isospectral problem is a small branch of the NS equation, even if Professor Feverman and their supervisor have two different solutions in front of them, they can't do anything about it.

Not to mention solving this equation problem, it is to take a step forward along the lines of thought of the two professors, and the four of them can't do it together.

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

.....

Recovering from his shock, Roger Dean replied respectfully, "Professor Feverman was here the other day, but you weren't there. ”

Hearing this big disciple's words, Xu Chuan was a little surprised, and asked in surprise, "Has Professor Feverman solved the problem of equivalence?" ”

Roger Dean shook his head and said, "I don't know, Professor Feverman didn't say this, but when he came over the other day, he told me that if you come back, please give him a call." ”

Xu Chuan nodded and said, "I know." ”

Pulling out his phone from his pocket, Xu Chuan walked out of the office and dialed Peverman.

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

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

Peverman: "There's a place that's stuck and you're trying to figure it out, what about you?" ”

Before Xu Chuan could answer, Feverman remembered something, and then asked hesitantly: "Shouldn't you have solved it?" ”

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

And every time he comes out, it proves that the problem he has studied has a result.

Isn't it, he's solved the isospectral nonisomorphism conjecture, right?

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

As soon as the words fell, there was a sound of 'poof' on the other side, as if the mobile phone had fallen on the wooden table, but it was grabbed for a long time, and the sound of fire came from inside.

"You're in the office now, I'll be right over."

......

Peverman came over faster than Xu Chuan thought.

And he wasn't the only one who came, he was also Professor Deligne.

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

"Have you solved the isospectral nonisomorphism conjecture? What about the proof process? Did you bring it here? Peverman asked urgently.

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

"Together, of course." Feverman said without hesitation.

.......

In the quiet villa, Xu Chuan handed the proof process to Feverman, 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, "There is no copy, but I still have the original manuscript paper." ”

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

Although the cleanliness of the original manuscript is not as good as the proofing process after collation, it is not a problem to look at the prover's thinking from the inside.

In a secluded villa, time passes little by little.

Professor Feverman was the first to put down the proof process in his hand and looked at Xu Chuan opposite, his eyes a little complicated.

Before he came, 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 himself.

He didn't find any loopholes in the paper, especially the step of using the Laplace operator transformation to build a stable closed willmore hypersurface, which made him can't help but clap in his heart.

It's subtle.

As if math was born that way.

But he didn't dwell on this, after all, the young man in front of him, regardless of his age, was not inferior to him in mathematical ability at all, and could even be said to be better.

He didn't solve the Seven Millennium Problems and didn't become a tenured fellow at the Institute for Advanced Study in Princeton at the age of 20.

"Very good proof process, at least I didn't find a loophole." Feverman put down the manuscript paper in his hand, looked at Xu Chuan and said, "Is your mathematical talent really exaggerated to this extent?" It took a month and a half to solve this problem. ”

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

"I've been sleeping no more than six hours a day for all this time, and even if I add other times like eating and running, it's not more than ten hours, and the remaining fourteen hours I've been working on it."

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

.......