【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 offstage.

However, he could also imagine the surprise of the audience.

Ideas and direction are the most important thing to solve any mathematical problem, and the wrong direction can only lead to unwarranted waste.

Luckily, he often finds the right direction.

This can be regarded as a function of mathematical intuition.

In this way, as time passed, the blackboard kept being filled with writing, and then he kept erasing it.

The cycle repeats itself over and over again.

Because the audience had the original paper in their hands, there was no need to drag a lot of blackboards to record all the process.

Let them take notes on their own.

Gradually, more than forty minutes passed.

More than forty minutes is neither long nor short, but for the vast majority of ordinary people, it is difficult to maintain more than forty minutes of concentration all the time.

However, there are many ordinary people in today's audience, at least the mathematicians sitting in the front rows, after more than 40 minutes, they still maintain absolute seriousness.

And as Li Mu's narration continues to enter a critical point, they will also light up from time to time, and feel wonderful for a certain step of Li Mu.

Until an hour has passed—

“…… Let's start with the case of the general limit space Mn j→ X......"

"In measure 6.28, by applying the results of the first two bars, we can immediately conclude that the measure μ satisfies the Ahlfors regularity ......"

"We can observe that Nj on all compact subsets is close to C^(1,α)......"

"So here ......"

Li Mu's calculation on the blackboard suddenly stopped, and he turned to face the audience.

He smiled slightly and said, "When you come here, you should probably guess what I'm going to do next." ”

His words immediately drew the attention of all the audience.

What's next?

Those who don't understand can only say that they don't know anything, and they want to ask this question as well.

And for those who understood, they immediately opened the first paper in their hands, that is, the penultimate page of "The Self-Consistent Properties of Elliptic Curves under K-Mode".

"He's going to demonstrate the connection between elliptic curves and k-theory......"

From the seats in the first row, Faltins whispered.

This is the most critical step in the entire proof.

Not one of them.

In terms of value, in Li Mu's complete proof, it is also the most critical value of this step.

Because it builds a bridge between two theories that are otherwise unrelated.

Li Mu, how did you do it?

Wiles on the side did not speak, and focused on Li Mu's proof.

His gaze narrowed slightly under his glasses.

In the past month, he has also gone through Li Mu's proof process, and it can be said that he is very familiar with each of them.

However, when he saw this part, he was always very puzzled, how did Li Mu think?

These great mathematicians were all very quiet, waiting for Li Mu to give an answer.

Before Li Mu's next sentence was spoken, the entire venue seemed to be in silent mode.

Finally, Li Mu spoke.

"Let's recall here the Taniyama-Shimura theorem and how it was proved."

"If p is a prime number and E is an elliptic curve over a field of rational numbers, we can simplify the equation modulo p that defines E; In addition to the finite p-values, an elliptic curve is obtained on the finite field Fp with np elements. ”

"When my teacher Andrew Wiles proved it, he first considered using the Iwasawa theory to prove it, but after finding that this method did not work, he tried to use the Koliwakin-Fletcher method, but he encountered problems in a special type of Euler system."

"Eventually, he remembered why it would be better to try these two methods together, and with a single thought, my teacher completed the proof."

"Now, K-modulo theory has made K-theory related to modular forms, and all elliptic curves on rational number fields are modular, so we only need to communicate between K-theory and elliptic curves through the bridge of modular forms—"

"Success makes it very simple."

"And here, I must say that the combination between the Iwasawa theory and the Kolivakin-Fletcher method is also wonderful."

As he spoke, Li Mu turned around and continued to write on the blackboard.

And as he showed in a few steps, the eyes of the world-class mathematicians sitting in the first row immediately lit up.

"I see!"

"The Iwasawa Theory and the Kolivakin-Fletcher Method! He could have thought of such a thought! Then using Pontryagin's dual theorem, we Γ discrete groups formed by the roots of p-subunits in all complex fields......"

Faltins, who had been sitting upright, was now relaxed and leaned back 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 performance at this moment is no less than watching a super blockbuster with a score of 9.9, and he feels very happy.

And Deligne also shook his head at this time and sighed: "Unbelievable, unbelievable." ”

"Li Mu's knowledge reserve really gives people a bottomless feeling."

"Old, old."

At this time, Deligne had a very deep feeling.

As there are more and more branches of mathematics and the degree of refinement is getting deeper and deeper, these mathematical masters can basically only be said to be masters of mathematics who specialize in a certain direction, and no one can be omnipotent.

Even his teacher, the Mathematical Emperor Grothendieck, could not do it.

And those math problems are like enemies they have to challenge, and in the face of these enemies, they can only use the only math weapon they have in their hands to deal with them.

Therefore, they always fail, because to defeat these enemies, they often need to be proficient in more weapons in order to break through their flaws.

And Li Mu happens to be proficient in many directions and has a lot of weapons, so when he faces these enemies, he can often find the flaws of these enemies and defeat them.

For example, the hail conjecture and the twin prime conjecture in the past, and the Goldbach conjecture in the present.

Maybe......

Li Mu can also continue to find a successful path when studying physical problems, which is also the reason?

Deligne shook his head, his heart full of sighs.

It's just that suddenly he glanced at it, and he saw Wiles next to him, and he almost didn't smile.

And Wiles also noticed that Deligne looked over, and immediately said, "Did you hear that?" Li Mudu said that he used the Iwasawa theory and the Kolivakin-Fletcher method, which is the method I used back then, and you still question me as a teacher for not helping him. ”

"You can't talk nonsense about this kind of rumor in the future, otherwise I'll sue you for slander."

Deligne immediately said angrily: "The Iwasawa theory and the Koliwakin-Fletche method used by Li Mu are completely different from the ones you used back then, but he has made more modifications to the original method, and it is more perfect than your original combination." ”

Wiles spread his hands and said, "So this is my student!" How? Are you not convinced? ”

Deligne didn't want to pay attention to this guy even more.

Like a little kid, old naughty boy?

When this guy was still teaching at the Institute for Advanced Study in Princeton, he wasn't like this.

Of course, although he despised Wiles very much in his heart, Drinne was also very remorseful at this time.

Once, he also had an opportunity to accept Li Mu as his student, but he didn't cherish it, and he didn't regret it until today, if God gave him another chance to do it again-

He must rush ahead of Wiles and give Li Mu a precious gift.

At the beginning, he saw with his own eyes that Wiles gave the pen to Li Mu.

And he didn't say anything, and even gave Wiles an assist.

I knew that this would happen today......

Regret not being at the beginning!

……

Of course, Li Mu's step also made other scholars realize what it means to think about genius.

看到这里的时候，他们都会不由自主的将自己代入到李牧的角度中，然后思考自己能否想到利用岩泽理论和科利瓦金-弗莱切方法结合，来解决这个问题，以及之后利用庞特里亚金对偶定理进行处理的思路，最终彻底实现K-模理论和椭圆曲线之间的统一。

In the end, 90% of people can only shake their heads, thinking that they must not have thought of such an idea.

And then there are 9% of people who are very decisive not to think about this kind of thing, they can't even do it, let alone think about how to deal with it next.

Of course, there are 1% of people who are more hard-mouthed and think they should be able to think of it, but this kind of people are also 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, here, we can easily get-"

"All elliptic equations on Q are K-modulus."

"That's it."

"We have successfully fused the elliptic curve, the K theory and the modular form to achieve the final unification."

He put his hands together and said in an annunciatory tone: "Putting aside the proof of Goldbach's conjecture later, at this point, I can confidently say that algebraic geometry and number theory have become more closely linked." ”

"Mr. Langlands' program is one step closer to its final realization."

As soon as the words fell, applause suddenly rang out, starting from the first row and ending at the end, everyone in the audience applauded.

The realization of Langlands' program is the common goal of all mathematicians, and Li Mu has achieved this step, which is already worthy of their warm applause.

Listening to the applause, Li Mu also smiled slightly, listening to the warm applause.

And until the applause gradually stopped, he continued: "In addition, I also make a prediction here, and the elliptic curve based on the K-mode theory plays a very important role in solving the Artin conjecture. ”

"If you are interested in solving the Artin conjecture, you may wish to try it with elliptic curves under the K-mode theory."

Hearing Li Mu's words, everyone present was stunned.

Artin's conjecture?

The Artin conjecture is also a very important issue in the Langlands program, because it directly corresponds to the function conjecture, which is one of the two parts of the Longlands program, that is, proving the Artin conjecture will help prove the functuality conjecture, and proving the funcerity conjecture is equivalent to half of the realization of the Longlands program.

For a while, many people thought about it, and finally their eyes lit up.

Truly!

The elliptic curve under the K-mode theory is indeed very helpful for solving the Artin conjecture.

The Artin conjecture speculates that a given integer a, which is neither a square number nor -1, is the original root modulus of an infinite number of prime p, and there is also an extended discussion of elliptic curves......

Many of the people present immediately made a decision, and after they went back, they would try to study the Atin conjecture.

Even if you can't prove it, if you achieve some results, you can still publish a paper in the first district.

After all, this is Artin's conjecture!

Li Mu on the stage took in the reactions of these listeners and smiled slightly, this is the meaning of solving a mathematical problem.

Because the theories and methods born in the process of solving one problem will help to 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.

"Now, it's time to settle the Goldbach conjecture once and for all – in fact, the next steps are pretty clear here."

"So, I'm not going to talk nonsense."

Li Mu wiped the blackboard that had been written all over the place, and then proceeded to the next steps like a bamboo.

The audience also followed the second paper they were looking through, followed Li Mu's proof, and continued to take notes.

Indeed, as Li Mu said, the next steps are very clear, he uses the elliptic curve under the K-mode to easily substitute the circle method into it, and then combines the sieve method.

Until 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 can substitute it into the original sieve function, and we can easily verify that the sieve function is greater than zero when λ=2."

"So far—"

Li Mu put down the blackboard marker in his hand, looked at the audience again, and announced crisply: "Obviously, we have successfully proved the Goldbach conjecture about even numbers. ”

“哥德巴赫寄出的那封信，在欧拉的手中未能完全启封，于是欧拉又将这封信，寄往了未来。”

"It has crossed the long river of time, and today, 280 years later, it has successfully reached the end."

"I'm honored to be the one who opened it."

"Thank you!"

(End of chapter)