England, home of Andrew Wiles.
The phone rang suddenly, and Andrew Wiles was awakened. Of course, his wife was also awakened.
It was midnight in the UK at this time, probably around three o'clock, and it was rest time.
Andrew Wiles picked up the phone with a confused look on his face.
Who was calling him at such a late hour?
It turned out to be Simon Donaldson.
This old guy stayed up so late and made harassing phone calls to him?
Don’t you know that sleep is very important for these elderly people?
After answering the phone, he said angrily: "Simon, if you don't give me a satisfactory reason, I will definitely remember the fact that you woke me up tonight."
"Hey, Andrew, don't worry, I was woken up by someone else too." Simon Donaldson said: "Guess, what big thing is your student who hasn't come over yet doing?"
Wiles was stunned for a moment.
This is talking about... Li Mu?
Calculating the time, it seems that it is indeed daytime over there in China.
"What happened again?" he asked doubtfully?
"Just now, my friend in the United States suddenly called me and told me that there was news from China that Li Mu was proving the Polignac conjecture and the Hardy-Littlewood conjecture."
Wiles: "???"
Question marks were written all over his face: "Are you sure? Is it true or false? "
"Why are you lying to me?" Simon Donaldson said: "I am watching the live broadcast of his proof now. He has solved the Polignac conjecture, and now it is Hardy Littlewood's turn to conjecture. "
In an instant, Wiles was no longer sleepy.
Just kidding, how could he miss such a major event in mathematics?
"Give me the live broadcast URL and I'll watch it now."
"I sent it to you. You have to translate it for me later and translate what Li Mu is saying. After all, you have studied Chinese for so long."
"I can't guarantee that I can completely translate it. You also know that Chinese is difficult to learn."
Wiles said as he turned over to get up, and at the same time gave his wife a look, telling her to continue sleeping.
His wife had a look of helplessness on her face.
Sometimes being a mathematician's wife seems to be quite normal when faced with such things.
Perhaps for most mathematicians, mathematics is more important than love.
Wiles came to the study, brought a cup of coffee, and turned on the computer.
Entered the live broadcast room sent by Donaldson.
This is a Chinese website, so the text on the web page is also in Chinese.
Although the web page translation function could be used, Wiles still relied on his Chinese ability to identify it carefully.
"Li Mu... proved... Polignac conjecture and Hardy-Littlewood conjecture!"
"It's actually true!"
He quickly clicked on the live broadcast and saw Li Mu writing mathematical formulas on the blackboard.
"This proves... the Hardy-Littlewood conjecture!"
Just by looking at it, Wiles could tell what Li Mu was writing.
"This is a divergence theory... It actually uses non-Archimedean absolute values... Damn, is this true?"
Simon Donaldson's voice came on the phone again.
"Of course this is true, please translate for me and practice your simultaneous interpretation skills."
Wiles: "Don't worry, I haven't understood it myself yet..."
But Simon Donaldson still urged: "Translate for me quickly! Oh my God, this step of Li Mu is so crucial. Looking at what he wrote on the blackboard, it seems that he has successfully found the twin prime number pair and the prime number theorem. The superposition relationship between? Oh, what is he talking about? "
In this way, a Fields Medal winner, a Fields Special Medal winner, got up in the middle of the night and connected to the line, listening to the report of a young Chinese man.
In fact, there are many like them.
A lot of well-known mathematicians in the world have entered this live broadcast room.
It doesn’t matter what time zone these mathematicians are in, whether it’s time for a break or not.
Even mathematicians from some countries such as Japan, Han, New Apo, etc., because their time zones are not much different from China, they are basically still working hours, but they were attracted by this report.
As a result, students in the mathematics department of universities in these countries temporarily received news of the teacher asking for leave before class. What's more, they were in class, and then the teacher opened the live broadcast room and took them to watch it together. Li Mu’s proof.
Euphemistically called: witnessing history.
In fact, it's just their teacher who wants to see it.
Of course, the result of this situation is that both teachers and students are confused.
Who makes the content too esoteric.
Not only did they not understand Chinese, they also did not understand what was being said.
For them, the only good thing is that the mathematical formula written by Li Mu on the blackboard still allows them to make associations and figure out how to prove it.
However, even if you want to do this, it is very difficult. Only those top mathematicians, and mathematicians with an in-depth understanding of relevant knowledge, can easily understand it. As for other people with slightly less capable abilities, or those with research skills, Mathematicians in other fields can only be half-understood, or confused from beginning to end.
I can't understand it, I can't understand it at all.
…
Beijing Auditorium.
Li Mu on the stage had no idea that his report had attracted so many internationally renowned mathematicians.
He continued to prove it on stage wholeheartedly.
Whether it is divergence theory or non-Archimedean absolute value, these are the methods he has thought of recently.
The source of these inspirations lies in Professor Lin Yao’s report that day.
The topic of Lin Yao’s report that day [Non-Archimedean subshape mapping of hypersurfaces in projective varieties], although it is not closely related to the Hardy-Littlewood conjecture, there are commonalities between mathematics and methods. It is also common.
As long as you find the relationship, you can use it.
"...by Theorem 2.1 and Theorem 2.2, the algebraic extension of finite fields of characteristic p whose non-Archimedean absolute group is isomorphic to ^Zp is..."
[A(F)→ H1(GF, T (A))……]
Another formula was written on the blackboard.
Li Mu stood up straight with a slight smile on his face.
At the same time, the mathematicians who understood the situation were once again surprised.
"Wait... did he transform the entire problem into an algebraic geometry problem?"
Qiu Chengtong narrowed his eyes.
Li Mu surprisingly transformed the original Hardy-Littlewood conjecture into a problem in algebraic geometry through a series of transformations.
"Does he want to use algebraic geometry to solve it?"
Qiu Chengtong already had this idea in his mind.
Use algebraic geometry to solve problems in number theory!
What a crazy thing.
One mathematician who had done this in the past was Gerd Faltings.
He is one of the top mathematicians in the world today. He used algebraic geometry methods to prove the Model's conjecture in number theory, and eventually won the Fields Medal for this.
And now Li Mu also wants to use this method to complete the proof?
Zhang Yitang next to him also had the same expression.
He had seen many geniuses, and he himself was considered a genius, but he never expected that Li Mu would plan to do this.
At the same time, many mathematicians in the live broadcast room who realized Li Mu's intentions took a breath of air.
"Can this really be done?"
In the UK, Andrew Wiles and Simon Donaldson have been on the phone, communicating on the contents of Li Mu's report.
They are all top mathematicians, so they can understand the meaning of what Li Mu wrote.
Regarding this issue, they could not help but remain silent for a long time.
till the end.
"Hopefully he can."
Even if Li Mu had used other methods to prove this conjecture, it probably wouldn't have made them so excited.
But if this problem is really solved using algebraic geometry methods, then this will have far-reaching implications for the mathematical community.
Because this will once again inspire mathematicians' confidence in the realization of the Langlands program and the unification of algebraic geometry and number theory.
This report will also become a classic report in the mathematics community.
…
On the rostrum.
Li Mu turned his head slightly and said with a smile: "I believe some friends have already seen what I am thinking."
"Then here, we will officially enter the field of algebraic geometry——"
"And here, please let me briefly introduce a new theory to you."
"I call it k-mode theory."
"For the time being, you can simply understand it as a combination of k-theory and modulus space."
His words once again shocked the mathematicians present.
Combine K theory and modulus space?
K theory is closely related to algebraic geometry, algebraic number theory and other fields, and modulus space is a key research object of algebraic geometry.
It is not that there has been no precedent for the two to be used together in the past, but it is very rare because there has never been a systematic method that can perfectly combine the two methods.
And now what Li Mu means...is to realize this?
Li Mu didn't explain much, turned his head and started writing on the blackboard.
Everyone in the audience held their breath, even those who couldn't understand, knew that Li Mu was doing something big.
As the formulas were listed on a blackboard, Qiu Chengtong showed a stunned expression.
"So it turns out that each point in the modulus space is calculated according to the K0 functor, thereby generating a semigroup of the projective modular isomorphism class... By the way, coupled with the incompleteness of the modulus space, after that, He will probably use this to estimate the distribution of twin prime pairs..."
As a top mathematician, Qiu Chengtong certainly has strong mathematical intuition.
Almost immediately, he saw Li Mu's purpose.
But even though he could see it, if he was asked to do it, he could only choose to give up.
It is technically too difficult to do this.
Especially the calculation steps that need to be performed later will further test the control of the entire method.
He might have given it a try when he was young, but now, he has to accept his old age.
After that, Li Mu did indeed start a lot of calculations as he expected.
His calculations gave everyone else at the scene the feeling of walking a tightrope. One step of error would lead to an absolute error.
However, Li Mu is like a humanoid computer, handling the entire complex calculation process extremely perfectly, and his intuition in mathematics is even more vivid.
And such calculations also require enough blackboards.
So people saw the staff nearby dragging up a small blackboard from time to time, until all 20 small blackboards were dragged up——
[To sum up, π2(N) is approximately equal to ∫dt/(lnt)^2≈2Ct(N/(ln)^2N]
[Where Ct is the twin prime constant. 】
[Certificate completed. 】
Li Mu wrote the last three lines on the last blackboard, the last blank space.
"At this point, I think the Hardy-Littlewood conjecture is officially history."
"That's all for this report."
"Please allow me to finally introduce to you the Polignac-Lee Theorem and the Hardy-Littlewood-Lee Theorem with great honor and pride."
Li Mu smiled slightly and then bowed to the audience.
Thunderous applause.
He really did it, using algebraic geometry to solve a number theory problem!
(End of chapter)
