Great Country Academician

Princeton Institute for Advanced Study.

After Xu Chuan sent the thesis to Deligne and Witten's two mentors, at noon the next day, Deligne's office was suddenly pushed away in a hurry.

Deligne, who was studying the latest issue of the "Annual Journal of Mathematics", was taken aback, and he was relieved when he saw the person coming.

Casting a puzzled look at the intruder, Deligne put down the journal in his hand.

"Did you get it?"

Edward Witten hurried to the desk and asked the friend in front of him.

"What did you receive?" Deligne asked with some doubts.

"The proof of Hodge's conjecture!" Witten said word by word.

Hearing this, Deligne was stunned for a moment, then reacted and quickly turned on the computer.

Logging into the mailbox, among the senders who are particularly concerned, there is an email from his student lying in it.

Clicking on it, the title of the file attached to the email came into his eyes.

[On nonsingular complex projective algebraic varieties, any Hodge class is a rational linear combination of algebraic closed chain classes. 】

Seeing this email, Deligne stopped breathing for a moment, then quickly took the glasses from the table, wiped the lenses with the lens cloth in his jacket pocket, and put them on.

The noisy office quickly quieted down. After a long time, the old man sitting at the desk put down the mouse in his hand, took off the glasses on the bridge of his nose, and pinched the base of his nose.

"It's really unimaginable, how long it's only been." Putting down the glasses in his hand, Deligne looked up at Witten, and said something like a sigh.

"Do you think he proved it?" Wei Teng couldn't help asking.

He has already read this paper, but to be honest, he has not thoroughly studied the original algebraic variety and group mapping tool, let alone the current paper on the proof of Hodge's conjecture.

After all, he was not a pure mathematician in the first place. Even if he had made some achievements in mathematics, it was based on auxiliary physics. Even if he has won the Fields Medal, there is still a gap between his mathematical ability and those of the top traditional mathematicians.

Deligne shook his head and said, "I don't know. It is impossible for me to make a conclusion in a short time to prove such a major conjecture."

After a pause, he continued: "However, after browsing this paper for a short time, I didn't find any major mistakes. As for other details, it will take longer to check and confirm all of them."

Hearing this, Wei Teng continued to ask: "Then what's the answer in your heart?"

Deligne was silent for a moment, and then slowly said: "If I start from my heart, I would like to believe that he did it."

This may be from mathematical intuition, or it may be from the confidence in the extremely young disciple, or it may be the feeling brought by the perfection of the previous algebraic variety and group mapping tools.

In short, he prefers to believe that the paper in his hand has taken Hodge's conjecture out of the seven millennium problems.

Maybe there are still some problems with the paper in hand, but that is just a matter of revision.

After receiving a call from his mentor Deligne, Xu Chuan, who was sorting out his things, rushed to the Princeton Institute for Advanced Study.

He knocked on the door and walked in. In the office, the two mentors Deligne and Witten were there.

This is not surprising to Xu Chuan. After all, this is Hodge's conjecture. It is comparable to Fermat's last theorem proved by Professor Wiles or the Poincaré conjecture proved by Professor Perleman in terms of difficulty and importance. .

In the office, the two tutors stared at Xu Chuan for a long time. They have a lot of questions in their minds, but they don't know where to ask them.

In the end, it was Deligne who said: "Witten and I have read the paper you sent. However, it may not be easy to understand this paper."

Xu Chuan had no doubts about Professor Deligne's words. The proof of Hodge's conjecture was written by him. No one knows the difficulty of this paper better than him.

Nodding his head, he said, "If you have any doubts about this paper, I can answer it."

Xu Chuan thought that Professor Deligne or Professor Witten would ask some questions about the thesis, but Deligne shook his head instead.

"It's secondary to clarifying our doubts. I want to know if you have submitted this paper yet?"

Xu Chuan shook his head and said, "I haven't submitted a manuscript yet, but I've published it on the arxiv preprint website."

For the submission of the proof paper of Hodge's conjecture, he has not yet decided which journal to submit to.

Moreover, the proof papers thrown on the arxiv preprint website are somewhat different from the serious paper submissions in terms of format and specifications.

He just sorted out the paper yesterday and threw it out. If he wants to submit it officially, he needs to sort it out again.

"Then don't rush to submit."

Deligne pondered for a while, and then said: "For this paper, Witten and I will go through it for you first, and then you will organize it after confirmation, and then submit it after it is completed."

Hearing this, Xu Chuan's eyes lit up, and he said quickly, "Then trouble the two instructors."

Two top mathematicians, Deligne and Witten, have helped him with the thesis. As long as these two masters are sure that there is no problem, the possibility of proving Hodge's conjecture will increase from 90% to 99% from what he thinks. .

After repeating what Professor Deligne said, the time is longer than Xu Chuan imagined.

Originally, he expected that in three or four days, he should be able to start discussing the problems in the paper. As a result, he waited in the dormitory for a full week, but he did not receive a call from the old professor.

So much so that he thought that the two mentors had forgotten about it.

But thinking that this is almost impossible, he endured it again.

In the dormitory, Xu Chuan was sorting out his manuscripts when the phone on the table vibrated suddenly.

He thought it was Professor Deligne calling, and walked over quickly, but the contact person displayed on the screen was another name that he hadn't thought of.

It was Professor Pan Deming from Peking University who called him.

The old professor had met him at the Crawford Awards party before, and the two exchanged contact information.

Xu Chuan was still a little surprised that the old professor called him.

However, since he posted the paper on Hodge's conjecture on the arxiv preprint website, domestic calls, from his tutor Chen Zhengping, to Professor Zhou Hai of the School of Mathematics, Dean Rong Zhizhuan, Academician Zhang Weiping of Sun Moon University, Even college friends like Fan Hong and Kong Zhong called him.

From this point of view, it's no big deal for Old Professor Pan to call him.

Swiping his finger across the screen, Xu Chuan answered the phone.

"You wrote the paper on arxiv that solved Hodge's conjecture?"

As soon as the phone was connected, Old Academician Pan's hasty questioning student passed it on.

Xu Chuan nodded in response: "If there is no second person named Xu Chuan who uploaded a paper on Hodge's conjecture on arxiv at the same time, it should be me."

Hearing this, there was a long silence on the other end of the phone.

Maybe I was amazed, maybe I didn't know what to say.

In fact, Pan Deming knew about this a few days ago, and he downloaded and printed out the paper immediately.

But after studying for several days, he still has a lot of things he can't understand about the paper in his hand.

It may be that it involves too many fields of mathematics, or it may be that he is old.

But no matter what, when he reconfirmed this fact on the phone, he still couldn't help being surprised and moved.

As a person born in the 30th century of the last century, he can be said to have experienced the establishment, development and rise of the country completely along the way.

In the field of mathematics, he has seen many talented and capable mathematicians, and has communicated with them, whether it is Hua Luogeng, Chen Jingrun, Qiu Chengtong and others in China, or Faltings and Deli in foreign countries. Nirvana, even Grothendieck he has seen.

But at the age of twenty, there are probably very few people who can make such a great contribution in the field of mathematics.

For example, Galvalo, a great mathematician who died young, proposed the concept of group theory at the age of eighteen, and used group theory to rigorously prove that there are no radical solutions for equations of degree 5 or higher, and equations of degree 4 or less have radical solutions Solved the problem of solving this problem that has plagued the mathematics community for 250 years.

After being distracted for a long time, Pan Deming finally calmed down, took a deep breath and asked calmly: "How sure are you of solving Hodge's conjecture?"

Xu Chuan shook his head, and was about to say 'I don't know', but he couldn't help it when the words came to his lips.

After thinking about it, he said again, "I'm more than 90% sure."

After a pause, he continued: "Now my two supervisors, Professor Deligne and Professor Witten, are checking the proof paper, if they confirm that there is no big problem, there should be no problem.

Hearing this, there was another silence on the other end of the phone, but this time it wasn't long before the old man sighed, "It's really unexpected."

"After seeing you in Stockholm last year, I have always believed that you can make a great breakthrough in mathematics, but I didn't expect you to make such a huge breakthrough in just one year."

"If this Hodge conjecture is finally verified to be correct, you will be the first person in the younger generation of mathematics. The future will be written by you."

Xu Chuan smiled, thanked him, and calmly accepted the compliment from the old man on the other side of the phone.

This is Hodge's conjecture. If it is finally confirmed, even if it is the sum of the two lifetimes, this achievement will be among the best, and it is extremely important.

After a pause, the voice came back from the phone.

Pan Deming asked with a smile: "How long are you going to stay there? When are you going to come back?"

Xu Chuan thought for a while and said, "I don't know yet, let's see the situation."

As for the time to return to China, he really hasn't made up his mind yet.

He wanted to stay in Princeton as long as possible, if the situation allowed.

Even if he proves Hodge's conjecture, both Deligne and Witten still have a lot to teach him.

And in the Institute for Advanced Study in Princeton, there are many other top mathematicians and physicists.

For example, Professor Fefferman, a winner of the Fields Medal, although he is mainly engaged in the research of classical analysis, has in-depth research in the theory of multiple complex variables, smooth planes, and fluids.

The latter, related to fluid mechanics, is an area he has always wanted to study.

For these top mathematicians and physicists, often a communication may bring you huge gains.

So when he was studying at Princeton, if there were public classes from these big guys, he never missed them.

Even at the important moment of proving Hodge's conjecture, he took time out to listen to some professors' public lectures.

So he cherishes his time in Princeton very much, and he can learn more if he can learn more.

But Xu Chuan knew that he might not be able to stay in Princeton for too long.

It may have something to do with the fact that his research fields are basically purely theoretical and open. At present, no one in the United States has come to him and asked him to join the United States and stay.

But when the proof of Hodge's conjecture was released, the other party would notice him no matter what.

Xu Chuan reckoned that the other party might come looking for him in the near future.

He is very familiar with this set of procedures.

If you have made a huge contribution in a certain field, then at first it may be a familiar friend or classmate around you, or a mentor who inadvertently tried it out. If there is no result or if you are declined or hesitated, then it is Top universities came forward, such as Princeton and Harvard, invited him to become a tenured professor and stayed.

If you are still rejected and you are of sufficient value, then someone from the National Academy of Sciences of the United States will come forward and invite you to join the National Academy of Sciences of the United States as an academician.

If it is still rejected up to this point, the other party may take certain measures.

For example, you are forced to stay, or you are committed suicide or something.

With the experience of his previous life, Xu Chuan estimated that he should be close to the first stage now.

It may be one or two months, three or four months, or at the latest when the International Mathematical Congress next year begins.

By then, he would almost have to leave Princeton.

Before that, it should be safe on the whole, or in other words, there is no problem before Hodge's conjecture is fully accepted by the mathematical community.

Although he has published the paper on the arxiv website, it is still far away from this proof paper being verified by the mathematical community.

It took three full years for the Poincaré conjecture before the mathematical community concluded that Perleman's proof was correct.

The Hodge conjecture is no less important than the Poincaré conjecture. No one knows how long it will take for the mathematical community to accept and complete the verification.

Before the proof paper of Hodge's conjecture is formally accepted by the mathematical community, no matter how essential and important the paper is, it is nothing more than a manuscript.

Although this manuscript paper can attract the attention of the world, as long as it is not sublimated into a theorem, its value is far from that great.

Of course, now that things have come to this, he has to be prepared to leave early.

After chatting with Academician Pan Deming for a few words, Xu Chuan ended the call.

After waiting for a few more days, he finally received a call from his mentor Deligne, asking him to go to the office.

When I came to Deligne's office again, ten days had passed.

Not surprisingly, besides Deligne, another of his mentors, Edward Witten, is also here.

Not only that, but there was another young man in the office.

Of course, he is said to be a young man, but he is actually in his thirties. This young scholar named Gregory Olier is also a doctoral student of Deligne like Xu Chuan.

Previously, the two often studied and exchanged ideas together in the office.

However, what is different from usual is that at this time Gregory Olier, who can be regarded as a "senior brother", looked at Xu Chuan with admiration, admiration, excitement, surprise and other emotions in his eyes.

"Amazing! Xu Chuan."

Seeing Xu Chuan coming in, Gregory Olier walked over quickly, clenched his right hand and hit him heavily on the shoulder: "It's incredible that you actually solved Hodge's conjecture. You really only have Twenty years old? Don’t you be an undercover agent arranged by some alien civilization to lurk among human beings?”

Xu Chuan rubbed his shoulders and said with a smile: "It's just a thought. Whether it is really solved or not needs to be verified by the mathematics community. Before that, it cannot be said that Hodge's conjecture has been solved."

"You're too modest, maybe this is the habit of Huaguo people? The few Huaguo friends I know are like you. They always say that they can do a little bit, and then they can do everything."

Gregory Olier grinned, and continued: "The professor has already verified your thesis, and there is no problem. If I were you, the champagne would have already been opened."

Xu Chuan smiled, greeted the two mentors in the office, and then his eyes fell on the log desk, where a neat stack of manuscript paper was placed.

Seeing Xu Chuan's arrival, Professor Deligne stopped talking with Wei Teng: "Wi Teng and I have reviewed your proof paper these days, and the core proof idea is very complete. From the perspective of the two of us, there is currently no Discover any major flaws. Of course, it is possible that other mathematicians may have found something different."

Xu Chuan nodded, and said: "Before a conjecture is promoted to a theorem, no one can guarantee that its proof paper must be correct. All I can do is to try my best to perfect it."

Hearing this, Deligne smiled and said: "There are no problems with the core proof idea, but there are some flaws and deficiencies in some other places, such as your use of metric space."

"This may be an easy place to modify, but you should face them squarely. As for where the specific problems are, we have already marked them for you."

With that said, Deligne handed over the stack of papers on the desk.

Xu Chuan took the paper from Deligne and said solemnly, "I will do my best to revise it."

Deligne nodded, and said: "Go ahead, remember to notify me after the revision is completed. If there is no problem in the verification, Princeton will arrange a report meeting for you. You must prepare the report materials in advance."

"After all, this is Hodge's conjecture. For such an important mathematical conjecture, the opinions of me and Witten alone cannot determine whether it is correct. You need to conquer mathematicians all over the world at the report meeting."