Institute for Advanced Study in Princeton.

After Xu Chuan sent the paper to the two supervisors, Deligne and Witten, at noon the next day, Deligne's office was suddenly pushed open.

Deligne, who was studying the latest issue of the Annals of Mathematics, was startled and relieved when he saw someone coming.

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

"Did you get it?"

Edward Witten hurriedly walked to his desk and asked his friend in front of him.

"What did you get?" Deligne asked, a little confused.

"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 of special attention, an email from his student lies in it.

Click on it, and the title of the file attached to the email catches his eye.

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

Seeing the email, Deligne's breathing froze for a moment, and then he quickly took his glasses from the table, wiped them with the cloth in his jacket pocket, and put them on.

The noisy office quickly quieted down, and 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 root of his nose.

"It's hard to imagine how long this has been." Putting down the glasses in his hand, Deligne looked up at Witten and sighed.

"Do you think he proved it?" On the side, Witten couldn't help but ask.

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

After all, he was not a pure mathematician in the first place, and even if he made 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 top traditional mathematicians.

Deligne shook his head and said, "I don't know. The proof of such a great conjecture cannot be concluded in a short time. ”

After a pause, he continued: "But after a brief glance at this paper, I didn't find any major errors. As for the other details, it will take longer to check them all. ”

Hearing this, Witten then continued to ask: "What about the answer in your heart?" ”

Deligne was silent for a moment, then spoke slowly, "If I had to start from my heart, I would believe that he did it." ”

It may have come from mathematical intuition, or it could have come from confidence in the extremely young disciple, or perhaps it was the feeling of perfection with the previous algebraic cluster and group mapping tools.

In short, he prefers to believe that the paper in his hand plucks the Hodge conjecture from the Seven Millennium Problems.

There may still be some problems with the paper in hand, but it is just a matter of revision.

.......

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

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

Xu Chuan is not surprised by this, after all, this is the Hodge conjecture, both in difficulty and importance, comparable to Professor Wiles's Fermat's theorem, or Professor Pell Lehman's Poincarian conjecture.

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

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

Xu Chuan did not doubt Professor Deligne's words, the proof of Hodge's conjecture was written by him, and no one knew the difficulty of this paper better than him.

Nodding, he spoke, "If you have any doubts about what is in this paper, I can answer them." ”

Xu Chuan thought that Professor Deligne or Professor Witten would ask some questions about the paper, but he didn't expect Deligne to shake his head instead.

"It's secondary to solve our doubts, I want to know if you have already submitted this paper?"

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

He had not yet decided which journal to submit to the proof of Hodge's conjecture.

In addition, the certification papers that are thrown to the ARXIV preprint website are still somewhat different from the formal paper submission in terms of format and specifications.

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

"Then don't rush to submit the manuscript yet."

Deligne pondered for a moment, and then said: "For this paper, Witten and I will go through it for you first, and then you will sort it out after confirmation, and then you will submit it when it is finished." ”

Hearing this, Xu Chuan's eyes lit up, and he quickly said, "Then it's troublesome for the two mentors." ”

With Deligne and Witten, two top mathematicians, helping him with his papers, as long as the two elders are sure that there is no problem, the probability of proving Hodge's conjecture will increase from his own belief that 90% to 90%.

......

Professor Deligne said it again, and the time is even older than Xu Chuan imagined.

Originally, he expected that he should be able to start discussing the problems in the paper in three or four days, but he waited for a full week in the dormitory, and 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 it was almost impossible, he put up with it again.

In the dormitory, Xu Chuan was sorting out his manuscript when the mobile phone on the table suddenly vibrate.

Thinking it was Professor Deligne, he walked briskly over to find a contact on the screen with another name he hadn't expected.

The person who called him was Professor Pan Deming of Peking University in China.

The old professor had previously met him at the Clafford Prize party, and the two exchanged contact information.

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

However, since he threw the paper of Hodge's conjecture on the Arxiv preprint website, he has been called by domestic phone calls, from his supervisor Chen Zhengping, to Professor Zhou Hai of the Academy of Mathematics, Dean Rong Zhizhuan, Academician Zhang Weiping of Riyue University, and even friends from Fan Hong and Kong Zhong, who were in college.

In this way, it is not a big deal for Professor Pan to call him.

Fingers swiped across the screen, and Xu Chuan answered the phone.

"You wrote the paper on Arxiv that solved the Hodge conjecture?"

As soon as the phone was connected, the old academician Pan hurriedly asked the student to pass it over.

Xu Chuan nodded in response: "If there was no second person named Xu Chuan who uploaded a paper on the Hodge conjecture on arxiv at the same time, it would have been me." ”

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

Maybe it's an amazement, maybe it's not knowing what to say.

In fact, Pan Deming knew about this as early as a few days ago, and downloaded and printed the paper as soon as possible.

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

It may be that there are too many areas of mathematics involved, or it may be that he is old.

But anyway, when he reconfirmed this fact on the phone, he still couldn't help but be surprised and emotional.

As a person born in the 30th century of the last century, he has gone all the way to fully experience the establishment, development and rise of the country.

In the field of mathematics, he has seen many talented and capable mathematicians, and has also communicated with them, whether it is Hua Luogeng, Chen Jingrun, Qiu Chengtong and others in China, or Faltins, Deligne, and even Grothendieck abroad.

But at the age of twenty, there are probably only a few people who can make such a huge contribution to the mathematical community.

For example, Gavarro, 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 equations more than five times have no radical solution, and equations less than four times have a fundamental solution, solving this problem that has puzzled the mathematical community for 250 years.

After wandering for a long time, Pan Deming finally eased his thoughts, took a deep breath and asked calmly, "How sure are you about solving Hodge's conjecture?" ”

Xu Chuan shook his head, just about to say 'I don't know', but when the words came to his mouth, he somehow held back.

After thinking about it, he spoke again: "More than ninety percent sure." ”

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

Hearing this, there was silence on the other end of the phone again, but this time it didn't take long for the old man to sigh: "It's really unexpected." ”

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

"If the Hodge conjecture turns out to be correct, you'll be the first person in the younger generation of math. The future will be written by you. ”

Xu Chuan smiled, said thank you, and calmly accepted the praise from the old man on the other side of the phone.

This is Hodge's conjecture, and if it is finally confirmed, even if it is combined for two lifetimes, this achievement can be ranked among the best and extremely important.

After a moment's pause, the voice on the other end of the line resumed.

Pan Deming smiled and asked, "How long are you going to stay over there?" When are you going to come back? ”

Xu Chuan thought for a while and said, "I don't know for the time being, it depends on the situation." ”

As for the time to return to China, he really hasn't thought about it yet.

If the situation allows, he wants to spend as much time as he can.

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

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

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

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

For these top mathematicians and physicists, often a single exchange can bring you great gains.

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

Even at important moments to prove Hodge's conjecture, he took the time to attend some professors' public lectures.

So he cherishes his time at Princeton, and he learns a little more if he can learn more.

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

may have something to do with the fact that the field of his research is basically pure theory and openness, and no one in the United States has come to him at present, asking him to join the United States and stay.

But when the proof of Hodge's conjecture is announced, the other party will notice him no matter what.

Xu Chuan estimated that the other party might come to the door in the near future.

He is very familiar with this set of processes.

If you have made a great contribution in a certain field, then at first it may be a familiar friend around you, or a classmate, or a mentor who inadvertently makes a temptation, and if there is no result or is politely rejected or hesitated, then a top university, such as Princeton or Harvard, invites him to become a tenured professor and stay.

If you are still rejected and you have enough value, the next step is for someone from the National Academy of Sciences to come forward and invite you to join the National Academy of Sciences to become an academician.

If you still refuse at this point, the other party may take certain measures.

Like forcibly leaving you behind, or being killed or something.

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

Maybe a month or two, or three or four months, or at the latest at the beginning of next year's International Mathematical Congress.

By that time, he would have to leave Princeton.

Until then, it should be safe overall, or rather, there will be no problem until the Hodge conjecture is fully accepted by the mathematical community.

Although he has now published his paper on the Arxiv website, he is still far from proving that the paper has been verified correctly by the mathematical community.

It took three years for the Poincaré conjecture to conclude that Perleman's proof was correct.

The importance of the Hodge conjecture is no less than that of the Pontler conjecture, and no one knows how long it will take for the mathematical community to accept and complete the verification.

Until the proof paper of Hodge's conjecture is formally accepted by the mathematical community, no matter how quintessential and important the paper is, it is only a manuscript paper.

Although this manuscript paper can attract the attention of the world, as long as it is not sublimated into a theorem, it is far less valuable.

Of course, at this point, he had to be prepared to leave early.

......

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

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

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

There were no surprises, and in addition to Deligne, his other mentor, Edward Witten, was also here.

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

Of course, he is a young man, but in fact he is in his thirties, and this young scholar named Gregory Olier is a doctoral student at the same time as Xu Chuan.

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

However, unlike usual, 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.

"Awesome! Xu Chuan. ”

Seeing Xu Chuan coming in, Gregory Olier walked over quickly, and hit him with a fist in his right hand on the shoulder: "It's incredible to actually solve the Hodge conjecture, are you really only twenty years old?" Isn't it an undercover agent arranged by an alien civilization to infiltrate humans? ”

Xu Chuan rubbed his shoulders and said with a smile: "It's just a little idea, whether it is really solved or not, it still needs to be verified by the mathematical community, and before that, it can't be said that the Hodge conjecture has been solved." ”

"You are too modest, maybe this is the habit of Chinese people? I know a few Chinese friends who are just like you, always saying that they can only do a little bit and then everything. ”

Gregory Olier grinned, and continued, "The professor has already verified your paper, no problem, if I were you, it would have opened the champagne." ”

Xu Chuan smiled and 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 Witten: "Witten and I have reviewed your proof paper these days, and the core proof idea is very perfect, and from the eyes of the two of us, we have not found any major defects so far. Of course, it is possible that other mathematicians have made different discoveries. ”

Xu Chuan nodded and said, "Before a conjecture is not promoted to a theorem, no one can guarantee that its proof paper must be correct, what I can do is to try my best to improve it." ”

Hearing this, Deligne smiled and said, "There are no problems found in the core proof idea at present, but some other places, such as your use of metric space, do have some flaws and deficiencies. ”

"It may be easy to change places, but you should face them squarely. As for where the specific problem is, we have already marked it for you. ”

As he spoke, Deligne handed over the stack of papers on his desk.

Xu Chuan took the paper from Deligne's hand and said solemnly: "I will do my best to revise it." ”

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

"After all, this is the Hodge conjecture, this kind of major mathematical conjecture, the opinions of me and Witten alone cannot determine whether it is correct or not, you need to conquer the mathematicians of the world at the report meeting."

.......