Ultimate Scholar

【μ((C∩Br(x))\\E……】

【|u(y)u(z)|/d(y, z)……】

Li Mu on the stage continued to write the next steps, and didn't care about what happened off the stage.

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

For solving any mathematical problem, the train of thought and direction are the most important, and the wrong direction can only bring unnecessary waste.

Fortunately, he can often find the right direction.

This may also be regarded as the role of mathematical intuition.

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

The cycle repeats itself over and over again.

Because the audience at the scene held the original text of his thesis in their hands, there was no need to drag a lot of blackboards to record all the process.

Just 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 also 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 continues to reach a critical point, they will also light up from time to time, feeling wonderful for a certain step of Li Mu.

until an hour passed—

"...let us start by considering the case of the general limit space Mn j → X..."

"In subsection 6.28, by applying the results of the previous two subsections, 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)..."

"Then here..."

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

He smiled slightly and said, "After coming here, everyone should probably guess what I'm going to do next."

His words drew the attention of all the audience immediately.

What's next?

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

For those who understand, they immediately opened the first paper in their hands, which is the penultimate page 10 of "The Self-Consistent Properties of Elliptic Curves Under K-Module".

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

In the first row of seats, Faltings whispered.

This is the most critical step in the whole proof.

none of them.

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

Because it builds a bridge between two originally unrelated theories.

Li Mu, how did you do it?

Wiles on the side didn't speak either, and concentrated on Li Mu's proof.

His eyes narrowed slightly under the glasses.

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

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

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

Before Li Mu could say the next sentence, the entire venue seemed to be on silent mode.

Finally, Li Mu spoke.

"Please let us recall the Taniyama-Shimura theorem here, 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 defining E modulo p; except for a finite number of values ​​of p, we get an elliptic curve over a finite field Fp with np elements .”

"When my teacher Andrew Wiles proved it, he first considered using Iwasawa's theory to prove it, but after finding that this method didn't work, he tried to use the Kolivakin-Fletcher method again, but it failed. Another problem was encountered in a special Euler system."

"Until the end, he thought of how to combine these two methods to try, so a single thought made my teacher complete the proof."

"Now, the K-modulus theory has made K-theory contact the modular form, and all elliptic curves on the rational number field are modular, so we only need to connect the K-theory and the elliptic curve through the bridge of the modular form. To achieve communication-"

"Success, it becomes very simple."

"And here, I have to say that the combination of Iwasawa's theory and the Kolyvagin-Fletcher method is also brilliantly applied."

After speaking, Li Mu turned around and continued to write on the blackboard. \b

And following his demonstration in a few steps, the eyes of the world-class mathematicians sitting in the first row lit up immediately.

"I see!"

"Iwasawa's theory and the Kolivagin-Fletcher method! He can think of such an idea! Then using Pontryagin's duality theorem, Γ is dual to the discretization of all p-unit roots in the field of complex numbers group……"

Faltings' body, which was originally sitting upright, was also relaxed on the back of the seat at this moment, with a smile on his face.

As a very pure mathematician, his interest is nothing but mathematics, so seeing Li Mu's wonderful mathematics interpretation at this moment is no less than watching a super blockbuster with a rating of 9.9. happy mood.

And Deligne shook his head at this time and said with emotion: "Unbelievable, unbelievable."

"Li Mu's knowledge reserves really give 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 masters of mathematics can basically only be said to be masters of mathematics who specialize in a certain direction, and no one can do it. Almighty.

Even his teacher, the mathematics emperor Grothendieck, couldn't do it. \b

And those mathematical problems are like the enemies they want to challenge. Facing these enemies, they can only use the only mathematical weapon in their hands to deal with them.

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

However, Li Mu happened to be proficient in many directions and mastered many weapons, so when he faced these enemies, he was often able to find their weaknesses and defeat them.

Such as the hail conjecture and the twin prime number conjecture in the past, and the current Goldbach conjecture.

Maybe……

Is this also the reason why Li Mu can continue to find success paths when studying physics problems?

Deligne shook his head, full of sighs in his heart.

It's just that when he glanced out of the corner of his eye, he almost burst into laughter when he saw Wiles next to him.

And Wiles also noticed that Deligne looked over, and immediately said: "Did you hear that? Li Mu has already said that he used the Yanze theory and the Kolivakin-Fletcher method, which I used back then. You still doubt that my teacher has not brought him any help."

"You can't talk about such rumors in the future, otherwise I will sue you for defamation."

Deligne immediately said angrily: "The Yanze theory and the Kolivakin-Fletcher method used by Li Mu are completely different from what you used back then, okay? He has improved on your original method. More revisions, 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 talk to this guy anymore.

Like a child, old urchin?

This guy wasn't like this when he was teaching at the Institute for Advanced Study in Princeton.

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

In the past, he also had a chance to accept Li Mu as his student, but he didn't cherish it, until today he regretted it too much, if God gave him another chance to do it again——

He must give Li Mu a precious gift before Wiles.

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

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

I knew it would happen today...

I regret it!

...

Of course, this step by Li Mu also made other scholars realize what genius thinking is.

When they see this, they will involuntarily put themselves into the perspective of Li Mu, and then think about whether they can think of using the combination of Yanze theory and Kolivakin-Fletcher method to solve this problem, and then use Pang Triagin's idea of ​​dealing with the duality theorem finally completely realizes the unification between K-module theory and elliptic curve.

In the end, 90% of people can only shake their heads, thinking that they must have never thought of such an idea.

Then there are still 9% of people who don't think about this kind of thing decisively. 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, but these people are also insignificant.

On the podium, after Li Mu completed this step, the next steps became very clear.

After taking 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-modules."

"So far."

"We have successfully integrated elliptic curves, k-theory, and modular forms, and achieved the final unity."

He opened his hands and said in an announcing tone: "Let's not discuss the proof of Goldbach's conjecture for now. At this point, I can confidently say that the connection between algebraic geometry and number theory has become more tightened up."

"The program proposed by Mr. Langlands is one step closer to its final realization."

As soon as the words fell, applause suddenly sounded. 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 already deserves their warm applause.

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

And until the applause gradually subsided, he continued: "In addition, I am also making a prediction here. The elliptic curve based on the 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 use the elliptic curve under the K-module theory to try it out."

Hearing Li Mu's words, everyone present was taken aback.

Artin guess?

Artin's conjecture is also a very important problem in the Langlands program, because it directly corresponds to the functor conjecture, one of the two parts of the Langlands program, that is, proving the Artin conjecture will help to prove the functor sex conjecture, and proving the functor sex conjecture is equivalent to realizing half of the Langlands program.

For a while, many people began to think about it, and finally their eyes lit up.

really!

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

Artin conjectures that a given integer a that is neither a square number nor -1 is the original root modulus of infinitely many prime numbers p, and there are also extended discussions on elliptic curves, so think...

Many people present made a decision immediately, and tried to study Artin's conjecture after returning home.

Even if you can't prove it, if you get some results, you can publish a paper in the first district if you don't say much.

After all, this is Artin's conjecture!

Li Mu on the stage took a panoramic view of the audience's reactions and smiled slightly. This is the meaning of solving a math problem.

Because the theories and methods born in the process of solving a 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.

Afterwards, he also turned his head again and continued with the next steps.

"Then, the next step is to completely solve the Goldbach's conjecture—in fact, the next steps are very clear at this point."

"So, I will stop talking nonsense."

Li Mu wiped the blackboard that had been filled with writing, and proceeded to the next steps like a broken bamboo.

The audience off the court followed closely the second paper they read, followed Li Mu's proof, and continued to memorize 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 into it, 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 can substitute it into the original sieve function, and we 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 auditorium again, and announced neatly: "Obviously, we have successfully proved Goldbach's conjecture about even numbers."

"The letter sent by Goldbach was not completely unsealed in Euler's hands, so Euler sent this letter to the future."

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

"I am very honored to be its unsealed person."

"Thank you everyone!"