The top student must be diligent.

Chapter 72 Faltings' Advice

"Professor Faltings, here I come."

At the Max Planck Institute of Mathematics, Xiao Yi knocked on the door of an office.

After walking in, he spoke respectfully to the mathematics master sitting at the desk.

Faltings smiled and nodded.

According to the previous agreement, Xiao Yi came to him to seek help in academic research.

Although it was a bit of a pity that Xiao Yi did not ask to become his student, he soon felt relieved. Perhaps the young man had his own ideas.

Of course, he would not lose face and force the other party to become his student.

Of course he couldn't do such a thing.
Well, today let’s listen carefully to the questions this young man wants to know and try to give him some good advice.

"Come on, sit down first."

Faltings left his seat, walked to the coffee machine and asked, "Would you like coffee or tea?"

Xiao Yi, who had just sat down, stood up and walked over before he even had time to sit down, saying, "I'll do it, just have some tea."

Faltings nodded with a smile and handed the cup to Xiao Yi.

"You've been all over the news these past two days. What do you think?"

Xiao Yi shook his head helplessly: "I don't have any ideas, um... it's too noisy."

"Too noisy?" Faltings was stunned, then burst into laughter. "Well said, it is indeed too noisy, just like a group of monkeys."

"They don't even understand mathematics, and they can't even explain what tele-Abelian geometry is, but the reporters are talking about this as a major event in the mathematics world. It's really interesting."

Xiao Yi felt ashamed. Perhaps only a big shot like the one in front of him could say such a thing?
After expressing his disdain for the media, Faltings said again: "Okay, let's not talk about those things. Tell me what question you want to ask me."

"Of course I need to say in advance that if it is something too simple, I will not answer you. Simple things need you to figure it out yourself. According to your ability, the questions you want to ask should be the kind that are at least worthy of publishing an IHES article."

The Faltings remained strict.

Xiao Yi thought about the question he wanted to ask, well, it should be similar. Not to mention publishing in IHES, it should be no problem to publish in the mathematics yearbook.

After bringing the tea to the table, he took out two sheets of A4 paper from his bag and handed them to Faltings.

"Professor, this is the question I'm thinking about now."

"Hmm... let me take a look." Faltings nodded, took the A4 paper and started reading.

Soon, his brows raised.

"Is this... the twin prime conjecture?"

Xiao Yi was amazed in his heart, he really was worthy of being a big shot, he could figure out so quickly that his goal was the twin prime conjecture.

He nodded and said, "Yes, Professor."

"The work on etale cohomology in tele-Abelian geometry reminds me of Bombieri's theorem. I think from a tele-Abelian perspective, perhaps we can continue to improve Bombieri's theorem and further strengthen the results of the three GPY mathematicians."

"Coupled with the improvements made by mathematicians Terence Tao and Maynard, perhaps we can actually break through the current upper limit of 246 and reduce this number. It is even possible to successfully prove the twin prime conjecture."

Listening to Xiao Yi's words, Faltings couldn't help but look at the young man again.

Prove the twin prime conjecture?

He wouldn't have dared to say such words when he was 17 years old.

This young man not only has talent, but also courage.

No wonder I was able to write an IHES paper right away.

He lowered his head again and continued to look at the two A4 papers from Xiao Yi.

"Bombieri's theorem...if you ask old man Bombieri to come over and talk about this problem, maybe he can tell you a lot." Enrico Bombieri, who proposed this theorem, is probably next door now.

"But, regarding your idea..."

Faltings continued to look at it for a while, then asked, "Have you considered the Siegel-Walfisz hypothesis? If this premise is not included, etale will not apply in the first place."

Xiao Yi replied: "I have considered it."

"How to solve?"

Xiao Yi looked around.

Faltings understood what he meant and dragged a blackboard behind him.

"use this."

Xiao Yi's eyes lit up, and then he picked up the blackboard pen next to the blackboard and started writing on it.

[Siegel-Walfisz hypothesis: ∑n≡a(r)(n, q)=1β(n)1φ(r)∑(n, qr)=1β(n)……]

"Since the modulus d is at most slightly larger than x^1/2 on a logarithmic scale, we only need to save a small power of x from the trivial estimate, after reducing the problem to estimating some incomplete Kloosterman sums."

"In that case, Siegel-Walfisz wouldn't be a problem."

Looking at what Xiao Yi wrote, Faltings thought for a moment and then nodded.

"Well, you solved that problem."

Without saying anything more, he continued to look down and soon asked again.
"So, what about the boundary problem of S1? Have you ever thought about this? Don't forget the requirements of the Chinese Remainder Theorem."

"This question directly involves the application of far Abelian geometry in number theory."

Faltings believes that this problem is a key one and is more important than the above problem. Moreover, in terms of the difficulty of solving it, Faltings estimates that it is also much more difficult.

If I can solve this problem, I will probably be able to publish a letter in the first area at any time.

However, Xiao Yi's expression remained unchanged, and he said directly: "I have already thought about this issue."

Then he picked up the pen and started writing on the small blackboard next to him.

After a while, Faltings looked at the new derivations appearing on the blackboard with a little surprise.

"Not bad, very good."

"It seems that you are indeed well prepared for this idea."

Just like that, he continued to look down, and from time to time he would ask questions about some of the issues, and when faced with these questions, Xiao Yi was always able to give answers that he had already prepared.

Even if Xiao Yi was not prepared, he only needed to think for a short while before he could give an answer.

Perhaps, these questions that he had not prepared for were questions that he had already thought about before, but they were not difficult for him at all.

In this way, in more than half an hour, Faltings asked more than a dozen questions, and each question was successfully answered by Xiao Yi.

Finally, Faltings put the papers down.

"Your preparations are more than I could have imagined."

"So, if you come to me for help, I may not be able to give you any useful help."

"Otherwise, I would have probably proved this problem long ago."

Faltings smiled slightly: "But, I have a suggestion that might be good."

“Why not think about it in terms of the Elliott-Halberstam conjecture?”

(End of this chapter)