【 μ((C∩Br(x))\E……】
[|u(y)u(z)|/d(y,z)……]
Li Mu on the stage continued to write the following steps and did not care about what happened in the audience.
However, he could also imagine the surprise of the audience.
For solving any mathematical problem, ideas and directions are the most important. Wrong directions can only bring unnecessary waste.
And luckily he usually finds the right direction.
This can probably be regarded as the effect of mathematical intuition.
In this way, as time passed, the blackboard continued to be filled with writing, and then he continued to erase it.
The cycle repeats over and over again.
Because the audience at the scene all had the original text of his paper, there was no need to drag in a lot of blackboards to record all the processes.
Just let them take their own notes.
Gradually, more than forty minutes passed.
More than forty minutes is neither long nor short, but for most ordinary people, it is difficult to maintain concentration for more than forty minutes.
However, there are many extraordinary people in today's audience. At least the mathematicians sitting in the front rows remained absolutely serious after more than 40 minutes.
And as Li Mu's narration continued to reach a critical point, their eyes would light up from time to time, and they would feel wonderful about a certain step of Li Mu.
Until an hour passed——
"...let us start to consider the situation in the general limit space Mn j → X..."
"In subsection 6.28, by applying the results of the first two subsections, we can immediately conclude that the metric μ satisfies the Ahlfors regularity..."
"We can observe that Nj on all compact subsets approaches C^(1,α)..."
"Then here..."
Li Mu suddenly stopped calculating on the blackboard and turned around to face the audience.
He smiled slightly and said, "Now that I'm here, everyone should probably guess what I'm going to do next."
His words immediately attracted the attention of everyone in the audience.
What’s next?
Those who didn't understand could only say that they didn't know anything and they wanted to ask this question.
For those who understood, they immediately opened the first paper in their hands, which was the tenth page from the bottom of "Self-consistent Properties of Elliptic Curves in K-Modes".
"He is going to demonstrate the connection between elliptic curves and k-theory..."
Faltins whispered from the seat in the first row.
This is the most critical step in the entire proof.
none of them.
In terms of value, among Li Mu's complete proofs, this step is the most critical.
Because it builds a bridge between two originally unrelated theories.
Li Mu, how did you do it?
Wiles on the side did not speak, focusing entirely on Li Mu's proof.
His eyes narrowed slightly under his glasses.
Over the past month, he has also gone through Li Mu's proof process. It can be said that he is very familiar with every process.
However, when he saw this part, he was always very confused. How did Li Mu think?
These great mathematicians were extremely quiet, waiting for Li Mu to give the answer.
Before Li Mu said his next words, the entire venue seemed to have turned on mute mode.
Finally, Li Mu spoke.
"Please let us recall here the Taniyama-Shimura theorem and its proof."
"If p is a prime number and E is an elliptic curve over a field of rational numbers, we can simplify the equation defining E modulo p; in addition to a finite number of p values, we get an elliptic curve over a finite field Fp with np elements .”
"When my teacher Andrew Wiles was proving it, he first considered using Iwasawa's theory to prove it, but after finding that this method didn't work, he tried using the Kolyvagin-Fletcher method, but it failed. Another problem was encountered in a special type of Euler system."
“Until the end, he thought about trying to combine these two methods, and with a single thought, my teacher completed the proof.”
"Now, K-modular theory has made K theory connected with modular form, and all elliptic curves in the rational number field are modular. Therefore, we only need to use the modular form as a bridge to connect K theory and elliptic curves. Realize communication——”
"Successful, it becomes very simple."
"And here, I must say that the combination between Iwasawa theory and the Kolyvarkin-Fletcher method also has a wonderful application."
After speaking, Li Mu turned around and continued writing on the blackboard.
As he demonstrated in just a few steps, the eyes of the world-class mathematicians sitting in the front row immediately lit up.
"I see!"
"Iwasawa theory and the Kolyvarkin-Fletcher method! He can actually think of such an idea! Then using the Pontryagin duality theorem, Γ is dual to the discretization formed by p-th unit roots in all complex fields group……"
Faltings' body, which had been sitting up straight, now relaxed and leaned on the back of the seat, with a smile on his face.
As a very pure mathematician, his interest is nothing but mathematics, so seeing Li Mu's wonderful mathematical interpretation at this moment is no less satisfying to him than watching a super blockbuster with a score of 9.9. happy mood.
Deligne also shook his head and sighed: "Unbelievable, unbelievable."
"Li Mu's knowledge reserve is really bottomless."
"I'm old, I'm old."
At this time, Deligne had a very profound feeling.
As there are more and more branches of mathematics, and the degree of refinement becomes deeper and deeper, these mathematical masters can basically only be said to be mathematical masters who specialize in a certain direction, and no one can do it. Almighty.
Not even his teacher, the Emperor of Mathematics Grothendieck, could do it.
And those mathematical problems are like the enemies they want to challenge. Facing these enemies, they can only use the only mathematical weapon they have in their hands to deal with them.
Therefore, they always fail, because to defeat these enemies, they often need to master more weapons to break through their flaws.
But Li Mu happened to be proficient in many directions and mastered many weapons, so when he faced these enemies, he was often able to discover their weaknesses and defeat them.
Such as the hail conjecture and the twin prime conjecture in the past, and the current Goldbach conjecture.
Maybe……
Li Mu is also able to continuously find the path to success when studying physical problems. Is this the same reason?
Deligne shook his head, filled with sighs.
But suddenly he glanced out of the corner of his eye and saw Wiles next to him, almost bursting into laughter.
Wiles also noticed that Deligne looked over, and immediately said: "Did you hear that? Li Mu said it. He used the Iwasawa theory and the Kolyvarkin-Fletcher method, which I used back then. The way he passed it, you still question me as a teacher for not helping him."
"Don't stop spreading rumors like this in the future, otherwise I will sue you for defamation."
Deligne immediately said angrily: "The Iwasawa theory and Kolyvarkin-Fletcher method used by Li Mu are completely different from what you used back then. He made improvements to your original method. A lot of modifications will make it more perfect than your original combination.”
Wiles spread his hands and said, "So this is my student! What? Are you not convinced?"
Deligne didn't want to deal with this guy anymore.
Like a little kid, an old naughty boy?
This was not the case when this guy was still teaching at the Institute for Advanced Study in Princeton.
Of course, although he despised Wiles in his heart, Deligne was also very regretful at this time.
Once upon a time, he also had an opportunity to accept Li Mu as his student, but he did not cherish it. He regrets it until today. If God gives him another chance -
He must give Li Mu a precious gift before Wiles does.
He had watched with his own eyes as Wiles gave the pen to Li Mu.
And he didn't say anything, and even gave Wiles an assist.
If I had known that a situation like this would happen today...
I regret it!
…
Of course, Li Mu's step also made other scholars realize what genius thinking is.
When they see this, they can't help but put themselves into Li Mu's perspective, and then think about whether they can think of using the Iwasawa theory and the Kolyvarkin-Fletcher method to solve this problem, and then use Pang's theory to solve this problem. Triagin's idea of dealing with the duality theorem finally completely realized the unification between K-module theory and elliptic curves.
In the end, 90% of people can only shake their heads, thinking that they must not have thought of such an idea.
Then there are 9% of people who decisively do not think about this kind of thing. They can't even do this step, let alone think about the next solution.
Of course, there are still 1% of people who are more stubborn and feel that they should be able to think of it. However, these people are insignificant.
On the podium, after Li Mu completed this step, the next steps became very clear.
After a few simple steps, Li Mu finally turned his head and said with a smile: "So, from here, we can easily get -"
"All elliptic equations on Q are K-modular."
"That's it."
"We successfully integrated elliptic curves, k-theory and modular forms to achieve final unification."
He opened his hands and said in a declarative tone: "Let's not discuss the proof of Goldbach's conjecture later. At this point, I can say with great confidence that the connection between algebraic geometry and number theory has become more It’s getting closer.”
"The program proposed by Mr. Langlands is now one step closer to its final realization."
As soon as he finished speaking, applause suddenly broke out. Starting from the first row to the end, everyone in the audience applauded.
Realizing the Langlands Program is the common goal of all mathematicians, and Li Mu has achieved this step, which is worthy of their warm applause for this.
Listening to the applause, Li Mu also smiled slightly and listened to the warm applause.
Until the applause gradually subsided, he continued: "In addition, I will also make a prediction here. The elliptic curve based on K-module theory plays a very important role in solving Artin's conjecture."
"If you are interested in solving Artin's conjecture, you might as well try it using elliptic curves under K-module theory."
After hearing Li Mu's words, everyone present was stunned again.
Artin's guess?
Atting's conjecture is also a very important issue in the Langlands program, because it directly corresponds to the functority conjecture, one of the two parts of the Langlands program. That is to say, proving Ating's conjecture will help to prove the functor. The conjecture of functority, and proving the conjecture of functority is equivalent to realizing half of the Langlands program.
For a time, many people started thinking, and finally their eyes lit up.
really!
Elliptic curves under K-module theory are indeed of great help in solving Artin's conjecture.
Artin's conjecture speculates that a given integer a, which is neither a square number nor -1, is the original radical module of infinitely many prime numbers p, and there is also an extended discussion on elliptic curves. Think about it...
Many people present immediately made a decision to try to study Artin's conjecture after returning home.
Even if you can't prove it and achieve some results, you can still publish a first-class paper with less words.
After all, this is Artin’s conjecture!
Li Mu on the stage took in the audience's reactions and smiled. This is the meaning of solving a mathematical problem.
Because the theories and methods born in the process of solving one problem will help solve more problems.
Mathematics has also developed from 1, 2, 3, and 4 thousands of years ago to what it is today.
Then, he turned his head again and continued with the next steps.
"Then, the next step is to completely solve Goldbach's conjecture - in fact, the next steps are very clear at this point."
"So, I'll stop talking nonsense."
Li Mu wiped the filled blackboard clean, and then started the next steps like a broken bamboo.
The audience in the audience also flipped through the second paper, followed Li Mu's proof, and continued to take notes.
Indeed, as Li Mu said, the next steps are very clear. He used the elliptic curve under the K-mode to easily substitute the circle method, and then combined the sieve method.
till the end--
"So, at this point, we can easily see that for all even numbers N greater than or equal to 6, the loop integral D(N) on the unit circle is greater than 0."
"We substitute it into the original sieve function and can easily verify that when λ = 2, the sieve function is greater than zero."
"So far——"
Li Mu put down the blackboard pen in his hand, looked at the audience again, and announced crisply: "Obviously, we have successfully proved the Goldbach conjecture about even numbers."
"The letter sent by Goldbach could not be completely opened in Euler's hands, so Euler sent the letter to the future."
"It spanned the long river of time and successfully reached the end today, 280 years later."
"I am honored to be its unsealer."
"Thank you everyone!"
(End of chapter)
