Institute for Advanced Study in Princeton.

After Xu Chuan sent the paper to his two supervisors, 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 his 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 came into his eyes.

[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 proved 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 is not a pure mathematician. Even if he has achievements in mathematics, it is based on auxiliary physics. Even if he has won the Fields Medal, his mathematical ability is not the same as that of the top traditional mathematicians.

There is still a difference.

Deligne shook his head and said: "I don't know. It is impossible for me to make a conclusion in a short time on 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 about 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.

I 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 Poincare conjecture proved by Professor Perreman.

.

In the office, the two instructors stared at Xu Chuan for a long time. They had many questions in their minds, but they didn'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 the proof papers posted on the arxiv preprint website are still somewhat different from those of serious paper submissions.

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

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

So much so that he thought the two instructors 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 when his cell phone on the table suddenly vibrated.

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

This chapter is not finished yet, please click on the next page to continue reading the exciting content! The person who called him was Professor Pan Deming from Peking University in China.

The old professor had met him once before at the Crafford Award party, and the two exchanged contact information.

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

However, since he posted the Hodge conjecture paper on the arxiv preprint website, domestic phone calls have started from his mentor Chen Zhengping, to Professor Zhou Hai from the School of Mathematics, Dean Rong Zhizhuan, and Academician Zhang Weiping from Sun Moon University.

Even his college friends, Fan Hong and Kong Zhong, called him.

From this point of view, it was nothing for Professor Pan to call him.

His fingers crossed the screen and Xu Chuan answered the phone.

"The paper on arxiv that solved the Hodge conjecture was written by you?"

As soon as the call was connected, Academician Pan's hurried inquiry came over.

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

After hearing this, the other end of the phone was silent for a long time.

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

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

But after studying for several days, he still had many things he didn't understand about the paper in his hand.

Maybe it involves too much mathematics, or maybe he is old.

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

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

In the world of mathematics, he has met many talented and capable mathematicians and communicated with them, whether they are Hua Luogeng, Chen Jingrun, Qiu Chengtong and others in China, or Faltings, Deli and others abroad.

He had seen even Grothendieck.

But looking at the entire history, there are probably only a few people who can make such a huge contribution to the world of mathematics at the age of twenty.

For example, Gavarro, this 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 of the fifth degree or above have no radical solutions, and equations of the fourth degree or below have radical solutions.

It solves this problem that has plagued the mathematical community for 250 years.

After being distracted for a long time, Pan Deming finally calmed down his thoughts, took a deep breath and asked calmly: "How confident are you that you can solve 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: "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 are no major problems, there should be no problems.

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 embark on a great path in mathematics, but I didn't expect that you would make such a huge breakthrough just one year later."

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

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

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

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

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, "We don't know yet. It depends on the situation."

He really hasn't thought about the time to return to China yet.
To be continued...