Li Mu's first report is over.

This report has a certain harvest for every participant.

For those students, they learned about Lee's space from Li Mu's report and how to use it simply, which is very meaningful for those who study geometry or are interested in studying fluid mechanics.

For mathematics scholars with a certain academic level, Li Mu's description of Li's space also gave them a deeper understanding, and maybe they could write a paper directly after they went back.

As for those big bulls, Li Mu's previous description certainly inspired them, but what surprised them the most was the new thing that Li Mu revealed in a few lines at the end.

Rewriting the Boltzmann equation?

On this point, the big bulls are all interested in it.

There are also many mathematicians who have studied the Boltzmann equation, just like Carlos Kenig, the chairman of the current IMU, whose main research direction is the Boltzmann equation, so at the end of Li Mu's report, he also showed great surprise.

He could fully see how shocking the changes in Li Mu's last few lines were to the Boltzmann equation.

Of course, there are many Fields Medal winners who have studied Boltzmann's equations, such as Pierre-Louis Lion, who belongs to the same Collège de France as Laurent Laphgue.

作为统计物理学中十分重要的一个方向，玻尔兹曼方程和纳维尔-斯托克斯方程一样，都深受诸多数学家们的重视。

一时之间，国际数学家大会的数学学者们，都开始讨论起了李牧的两天之后的那场菲尔兹奖讲座，那些大牛们更是无比的期待，当然，也主要是这些大牛们在热议着。

Because basically, only they can see the deep meaning from Li Mu's simple few lines.

It's just that the discussion of the big bulls naturally led others to follow the discussion.

So in this way, Li Mu's report two days later became the focus of everyone's attention.

……

"Hey, Leon, are you still sure you're not coming?"

Not long after the end of Li Mu's report, Laurent Laphguet gave it to Pierre-Louis Lyon.

Before coming here, Laurent Laphguet also asked Leon if he wanted to come, but Leon did not come to this International Congress of Mathematicians because of personal health reasons and because he did not believe that Li Mu had proved the problem of the Neville-Stokes equation in the millennium problem.

Leon on the other end of the line said, "Huh? What's going on, now that the conference has begun, there's no need for me to come. ”

Laurent Laverge laughed: "Believe me, you won't be disappointed, Li Mu's report will definitely make you feel worthwhile." ”

Leon was stunned for a moment, and asked suspiciously, "Li Mu?" I knew he had won this year's Fields Medal and became the youngest Fields Medal winner in history, so what did his report say? Isn't it his external source of internalization space? ”

"Huh...... No, if I'm not mistaken, his report should be over today, right? Leon said angrily, "You're making fun of me." ”

"Huh? Didn't you watch the live broadcast of Li Mu's report? Laphrogue asked.

"Not really."

"Hey, then I suggest you go and rewatch the recording now, especially the ending, go and go. ”

Laphrogue was now giving his friend Amway a video that he found interesting.

And Leon was a little puzzled, isn't it just a report, what's so special?

And it's the end......

And what about the ending?

It's just that in view of Laphrferg's enthusiastic appearance, he didn't want to let his old friend's enthusiasm down, so he said: "Okay, good, I'll watch it right away, I'll see it right away." ”

After hanging up the phone, Leon logged on to the official website of the 2022 International Congress of Mathematicians, and then entered the interface of Li Mu's report.

Because the report has ended, the full video of the report has been officially posted on the website, which can be downloaded and watched.

Leon quickly downloaded the video and opened it.

As the report began, listening to Li Mu's narration of the internalized space of the external force source in the video, Leon nodded from time to time.

In any case, the external source internalization space is indeed a very good new tool, and Leon is very much satisfied with it.

However, Li Mu's way of reporting is ...... It seems to be a little different from his previous reports, huh...... There is a feeling of being in class. ”

想到这里，利翁失笑地摇摇头，“大概是因为他开始成为老师了吧。 ”

He knows about Li Mu's number theory class, and even recommended Li Mu's number theory class video to his students.

Of course, the most important reason is that Li Mu's lecture method is very special, and Fields Medal winners like him will be amazed.

Shaking his head, he watched it very patiently, and even began to think about whether to let his students watch this video as well, which should help them better understand Lee's space.

And just like that, fifty minutes passed.

When he heard Li Mu in the video say that the main content of his report was the above, Leon came back to his senses.

"A very good level of report, the knowledge is easy to understand, and it is easy to understand, and it is easy to understand, I didn't expect this Li Mu to take a semester of classes, and the report is so good."

Nodding approvingly, Leon gave his rating.

But then he suddenly came back to his senses.

"Huh...... Laphrogue didn't say, is there anything I paid special attention to at the end? What to look out for? ”

Although the report was indeed very good, it was not enough to surprise him too much.

At this time, Li Mu in the video began to write a few lines of his preview on the blackboard.

Leon was immediately attracted to attention, and with the listing of the first line of formulas, Leon, whose main research direction is the Boltzmann equation, immediately recognized this line of formulas, which is related to the Boltzmann equation.

And with the next few lines of formulas listed in turn, as an expert in Boltzmann's equations, Leon couldn't take his eyes off it.

"Wait...... This is?! ”

stared closely at the blackboard in the video, but just as Li Mu was writing the key point, his hand stopped, and then turned his head to announce that the content of his preview was over, the report was over, and everyone was welcome to attend his Fields Medal lecture.

"Why did it stop!"

Seeing that it was over suddenly, Leon immediately became anxious.

Li Mu's few lines of formula are obviously to rewrite the Boltzmann equation, but it is incomplete, only revealing a corner, but it makes Leon scratch his liver a little anxiously.

He wanted to know how to rewrite it!

"Okay, okay, you do this preview, right?"

After sitting in front of the computer and being silent for a while, Leon took out his mobile phone on the spot and booked a ticket to St. Petersburg this afternoon.

He went to see the report himself, okay!

No way, he wanted to see a big change in the Boltzmann equations.

Just like the Neville-Stokes equation, it has gone through several changes to form the current shape.

Perhaps, this improvement by Li Mu will be able to bring great changes to the study of Boltzmann's equations.

……

The International Congress of Mathematicians continued to be held in full swing, and the two days passed quickly.

On July 10th, it arrived as scheduled.

The Fields Medal lectures were scheduled to begin at 10:50 a.m., which was also an hour, in Room 1 of the St. Petersburg International Conference Center.

Today's No. 1 conference hall is full of people.

There were even more people involved than Li Mu's last plenary report.

For no other reason, after the publicity in the past few days, his report, which may rewrite the Boltzmann equation, has become the focus of many relevant researchers.

And for all the people involved in the scene, this report is also more worthy of their attention.

Perhaps, this report, Li Mu will bring something new to the mathematical community?

No one wants to miss such an important moment.

……

“咦，老师，您怎么来了？ I remember you didn't come the other day? ”

In the second row, Cedric Villani saw Pierre-Louis Lyon and immediately greeted him with a smile.

Cedric Villani, winner of the 2010 Fields Medal, was also a student of Pierre-Louis Lyon.

As a student of Pierre-Louis Lyon, Cedric Villani's main research direction is also partial differential equations, and his greatest achievement is also in the Boltzmann equation.

However, Pierre-Louis Lyon did not seem to like the student who should have been his pride.

"Oh? Aren't you supposed to be sitting in the councilor's office and cheering for your ballot? How can you still have the leisure to attend a math conference? ”

Leon sneered.

It turned out that Cedric Villani went into politics seven years after winning the Fields Medal and basically stopped studying mathematics, so Leon was very disappointed in him.

Cedric Villani was probably used to the sarcasm of his mentor, and probably a little ashamed in his heart, so he didn't fight back, on the contrary, his face became depressed, and he said: "I lost the election, I lost the election." ”

Leon snorted, "You deserve it." ”

Cedric Villani sighed, but then said: "However, don't worry, teacher, I am now planning to start studying mathematics again, Li Mu's preview of the Boltzmann equation in the last report made me feel a huge shock, I didn't expect that the Boltzmann equation has a new angle to study——"

“我想和您澄清，当年我之所以开始投身政界，就是因为我已经对数学逐渐失去了兴趣，我感觉到玻尔兹曼方程存在着极限，但我想，李牧今天的报告，也许就会重新唤起我之前的兴趣。”

Listening to Cedric Villani's sincere words, Leon's brows moved, no matter what, he had always hated this student, and if he was really disheartened, he would definitely not bother to scold.

So, Leon finally said, "I hope you can really do what you say." ”

Of course, their little episode here did not attract the attention of the others, and Laurent Laffergue, who was probably sitting next to him, said to Cedric Villani: "I hope you don't disappoint your teacher again." ”

In the first row, today, next to Wiles, it was Faltins.

Faltins glanced at Wiles and said, "As a student of Li Mu, you don't know any news?" Can't reveal anything about it? ”

Hearing Faltins' words, Wiles looked over with some surprise and asked, "Huh?" The dignified Mr. Gerd Faltings would be interested in such a thing? ”

“…… Forget it, when I didn't ask. Faltins immediately stopped asking.

Indeed, according to his character, he should be disdainful of paying attention to such things.

Wiles' laughed suddenly.

"I'm interested, I'm interested, tell me about it." Tao Zhexuan, who was sitting on the other side of Faltins, came up and asked.

Wiles spread his hands: "I don't know, let's wait an hour later, Li Mu's report is over, and we'll know." ”

Tao Zhexuan rolled his eyes.

Wiles sat down innocently, but he was secretly happy in his heart, this feeling that no one else knew, only he knew.

In this way, the math gurus in the first few rows did not say a word, waiting for the start of the report.

There was no original report for them to read, so no one knew what Li Mu would say.

The clock passed quickly, and at 10:50, the time was up.

As Kenig, who served as the host, came to the stage, after a brief introduction, Li Mu followed the stage.

The same podium, the same hall.

The dress Li Mu was wearing was the same as two days ago.

But the only difference is the number of spectators at the scene and what Li Mu is going to talk about today.

Probably between these two, there is a relationship between the independent variable and the dependent variable.

"I'm meeting you again, and thank you for continuing to come to my presentation."

Walking onto the podium, Li Mu said with a smile.

"So, without nonsense, today's report officially begins."

"And the theme of today's report is—"

Li Mu turned around and looked at the PPT on the big screen behind him, and at the same time, he also pressed the button of the remote control in his hand, and the PPT page moved.

[Boltzmann equation and the internalization space of external force sources].

"In my last report, I mainly introduced the internalization space of external force sources."

"But when using external sources to internalize space, there will be a very obvious problem."

"Then the external force source internalization space can only combine the external force and the inner essence, but it cannot integrate the two."

"Before, it was like a bottle of water and oil, and no matter how much I shook it, they were always distinct."

"And that, in the end, will still affect our continued downward research."

"So, how do we solve this problem?"

"Let water and oil truly merge and completely separate from each other?"

Li Mu turned his head, came to the blackboard, and wrote down the first form of this report.

Then, the big bulls present immediately knew that this formula was a continuation of the formula he listed at the end of the report two days ago.

The first form immediately made them pause.

"I see!"

"That's what happened!"

"And so it was!"

Three different thoughts rang out in their hearts, but what they expressed together was the amazement in their hearts.

Li Mu on the stage began to narrate: "In an accident, I was surprised to find that the Boltzmann equation can be well combined with the internalized space of external force sources. ”

"Boltzmann's unique function can also completely make up for the shortcomings of the internal space of external force sources."

Li Mu smiled slightly: "Then, next, let me thoroughly deduce this process for everyone." ”

“要如何，才能够让玻尔兹曼方程，和外力源内化空间融合！ ”

So, the next thing he turned around again, picked up the pen, and began to write on the blackboard.

【（f/t）c=gI(g，Ω)[f(p'A，t)……】

With the listing of the formulas one by one, the big bulls in the audience watched it fascinated.

That's the art of math!

And the only people who can appreciate this kind of art are people like them.

Pierre-Louis Lyon had a look of great satisfaction.

The heart-scratching emotion that was hooked by Li Mu's preview of a few lines two days ago was completely released at this moment.

He saw that the Boltzmann equation still had very strong potential.

Laurent Laphgue, who was next to him, saw Leon's expression and said with a smile: "Why don't you regret coming over?" Well? ”

Leon waved his hand without turning his head and said, "Don't disturb me, I'm listening to the report." ”

Laurent Laffergue laughed, but it was clear that Leon's words had already answered his question sideways.

In the back row, Yun Wei and others also sat together.

Looking at Li Mu's description above, there was a deep amazement hidden in their eyes.

"Is there anything else that can stump him?"

Zhu Wen muttered, "Number theory, algebraic geometry, differentiation...... He even has such achievements as Lee's unfolding style. ”

Yun Wei was silent for a moment, and laughed at himself: "Maybe we have never been able to compare ourselves with him, after all, he wants to become...... Even beyond the likes of Grothendieck. ”

"Yes."

当然，对于在场的许多年轻数学天才们，他们此时都有着相同的感想。

It was a desperate gap, even so desperate that it was not despairing, but completely accepting.

……

(End of chapter)