Chapter 188 The Mathematician Who Conquered the World

Institute for Advanced Study in Princeton.

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

Deligne, who was studying the latest issue of the "Annals of Mathematics", was startled. He calmed down after seeing the person coming.

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

"Did you get it?"

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

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

"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.

Log in to the mailbox, and among the senders of special concern, there is an email from his student.

Clicking it open, the title of the file attached to the email caught his eye.

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

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

The noisy office quickly became quiet. 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 it's been." Putting down the glasses in his hands, Deligne looked up at Witten and said with a sigh.

"Do you think he has proven it?" Witten couldn't help but ask.

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

After all, he was not a pure mathematician in the first place. Even if he had 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 abilities and those of 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 about the proof of such a major conjecture."

After a pause, he continued: "However, after briefly browsing this paper, I did not find any major errors. As for other details, it will take longer to check and confirm them all."

After hearing this, Witten continued to ask: "What's the answer in your heart?"

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

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

In short, he is more willing to believe that the paper in his hand has removed the Hodge conjecture from the seven millennium problems.

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

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

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

This Xu Chuan is not surprised. After all, this is Hodge's conjecture. In terms of difficulty and importance, it is comparable to Fermat's last theorem proved by Professor Wiles, or the Poincaré conjecture proved by Professor Perelman. .

In the office, the two instructors stared at Xu Chuan for a long time. They have many questions in their minds, but they don’t know where to start.

Finally, it was Deligne who spoke: "Witten and I have already 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 knew the difficulty of this paper better than him.

Nodding, he said, "If you have any questions about this paper, I can answer them."

Xu Chuan originally thought that Professor Deligne or Professor Witten would raise some questions about the paper, but unexpectedly, Deligne shook his head.

"Clearing our doubts is secondary. I want to know if you have submitted this paper?"

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

Regarding the submission of the paper proving the Hodge conjecture, he has not yet decided which journal to submit it to.

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

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

"Then don't be in a hurry to submit your article."

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

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

With Deligne and Witten, two top mathematics masters, helping him with his paper, as long as these two masters are sure that there are no problems, the possibility of proving Hodge's conjecture will increase from 90% to 99%. .

Professor Deligne said it again, time is longer than Xu Chuan imagined.

Originally, he expected that he would be able to discuss the problems in the paper in three or four days, but he waited in the dormitory for a whole week 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 this was almost impossible, he endured it again.

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

He thought it was Professor Deligne calling, so he walked over quickly, but the contact person displayed on the screen was another name that he did not expect.

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

The old professor had met him at the Crafford Award 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 the Hodge conjecture on the arxiv preprint website, domestic calls from his mentor Chen Zhengping, Professor Zhou Hai of the School of Mathematics, Dean Rong Zhizhuan, Academician Zhang Weiping of Sun Moon University, and even college friends Fan Hong and Kong Zhong have called him.

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

His finger swiped across the screen, and Xu Chuan answered the phone.

"Did you write the paper on arxiv that solves the Hodge conjecture?"

As soon as the call was connected, Academician Pan's urgent question came over.

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

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

Maybe he was amazed, or maybe he 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 as soon as possible.

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

Maybe it involved too many mathematical fields, or maybe he was old.

But no matter what, when he confirmed this fact again on the phone, he still couldn't help but be surprised and sighed.

As a person born in the 1930s, he can be said to have experienced the establishment, development and rise of the country all the way.

In the mathematical world, he has met many talented and capable mathematicians and has communicated with them, including Hua Luogeng, Chen Jingrun, Qiu Chengtong and others in China, Faltings, Deligne and even Grothendieck in foreign countries.

But there are probably only a few people who can make such great contributions to the mathematical world at the age of 20 throughout history.

For example, Galvaro, this great mathematician who died young, proposed the concept of group theory at the age of 18, and used group theory to rigorously prove that equations above the fifth order have no radical solutions, and equations below the fourth order have radical solutions, solving this problem that has troubled the mathematical world for 250 years.

After being distracted for a long time, Pan Deming finally recovered his thoughts, took a deep breath and asked calmly: "How confident are you about solving the Hodge conjecture?"

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

After thinking for a while, he spoke again: "More than 90% sure."

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

After hearing this, the other end of the phone was silent again, but this time it didn't take long before the old man sighed: "It's really unexpected."

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

"If the Hodge conjecture is finally verified to be correct this time, you will be the first of the young generation in the mathematics world. 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 end of the phone.

This is the Hodge conjecture. If it is finally confirmed, even if it is added up in two lives, this achievement can be ranked among the best and is extremely important.

After a pause, the voice came from the other end of the phone again.

Pan Deming smiled and asked: "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."

He really hasn't decided when to return to China.

If the situation permits, he wants to stay in Princeton for as long as possible.

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

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

For example, Professor Fefferman, a Fields Medal winner, although he mainly engages in classical analysis, has in-depth research in multi-complex function theory, smooth planes, fluids, etc.

The latter, related to fluid mechanics, is a field he has always wanted to study.

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

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

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

Therefore, he cherished his time at Princeton and learned as much as he could.

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

It may be related to the fact that the fields he studies are basically purely theoretical and open to the public. No one in the United States has come to him yet and asked him to join the United States and stay.

But if 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 visit him in the near future.

He is very familiar with this process.

If you have made a great contribution in a certain field, then it may be that a familiar friend, classmate, or mentor around you initially tries it out unintentionally. If there is no result or you are declined or hesitant, then Top universities, such as Princeton and Harvard, came forward to invite him to become a tenured professor and stay.

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

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

Such as forcing you to stay, or committing suicide.

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

Maybe in one or two months or three or four months, or at the latest at the beginning of next year's International Mathematics Congress, we will be in contact with the other party.

By that time, he would almost have to leave Princeton.

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

Although he has now published the paper on the arxiv website, it is still far from being verified by the mathematical community as correct.

It took three years for the Poincaré Conjecture to be concluded by the mathematical community that Perelman's proof was correct.

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

Before the mathematical community officially accepts the proof paper of Hodge's conjecture, no matter how essential and important the paper is, it is just a piece of paper.

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

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, a full ten days had passed.

There is no surprise. In addition to Deligne, his other mentor Edward Witten is also here.

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

Of course, although he is said to be a young man, 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.

But what is different from usual is that at this time, this is Gregory Olier, who can be regarded as a 'senior brother', looking at Xu Chuan with eyes full of admiration, adoration, excitement, surprise and other emotions.

"Awesome! Xu Chuan."

Seeing Xu Chuan come in, Gregory Olier quickly walked over, clenched his fist with his right hand and hit him hard on the shoulder: "It's incredible that you actually solved the Hodge conjecture. You really only have Are you twenty years old? Are you an undercover agent arranged by some alien civilization to lurk among humans?"

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

"You are too modest. Maybe this is the habit of Chinese people? Several Chinese friends I know are like you. They always say that they only know 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, I would have opened the champagne by now."

Xu Chuan smiled and said hello to the two instructors in the office, and then his eyes fell on the wooden desk, where a neat stack of manuscript papers was placed.

Seeing Xu Chuan's arrival, Professor Deligne stopped talking to Witten: "Witten and I have reviewed your proof paper these days. The core proof ideas are very complete. From the perspective of the two of us, there is currently no What major flaws have been discovered? Of course, it is possible that other mathematicians have discovered something different.”

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

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

"This may be easy to modify, but you should face them. As for the specific problems, we have marked them for you."

With that said, Deligne handed over the stack of manuscript papers on his 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, and let me know when you are done. If there is no problem with the verification, Princeton will arrange a report meeting for you. You need to prepare the report materials in advance."

"After all, this is the Hodge conjecture. This kind of important mathematical conjecture cannot be determined by the opinions of Witten and I alone. You need to conquer mathematicians all over the world at the report meeting."