Chapter 317: The Light of Inspiration From the NS Equation

Chapter 317 NS Equation - That Light of Inspiration

Hearing this, Xu Chuan looked at Cai Peng in surprise.

This former senior brother is really good to be able to get further study at Princeton in the Department of Mathematics of Nanjing University.

Although Nanjing University currently has him, a Fields Medal winner, the apparent strength of the Department of Mathematics has improved a lot. But to be honest, there is still a clear gap between the strength of the Department of Mathematics and those top universities with mathematics departments.

Not to mention Princeton and Paris Normal University, which are the top universities in the world, even Peking University, Shuimu, Fudan, and Nankai in China, Nanjing University cannot compare.

After all, the strength of a school or a department is not just based on the top one or two people. The foundation of these famous universities is still quite deep.

As for Nanjing University, apart from him, it can be said that there are almost no professors in the Department of Mathematics who are famous internationally.

Professor Zhou Hai's strength can be ranked in the upper middle level in China, but it is better to forget it in the international arena.

The fact that there are not many professors from Nanjing University who can give reports at the recent International Mathematics Conferences is a performance.

Before him, the mathematics department of Nanjing University was second-rate in China. In this environment, he was able to get a postgraduate degree in the mathematics department of Princeton by himself, which was very good.

And he remembered that Brother Cai’s talent in mathematics did not seem to be very good before. He did not expect that he would find the right path after the second proposal of his graduation thesis.

Fortune and misfortune are interdependent, this is true.

After thinking about it, Xu Chuan said, "I have not yet decided whether to take students next year."

"But since you have obtained a postgraduate degree in Princeton, it means that your talent and ability in mathematics have indeed improved a lot compared to before."

"Let me give you a test paper and you do it. If you can meet my standards and requirements, I can consider it."

To be honest, he did not think about accepting students this year.

But taking students is also what he should do. After all, he still holds the position of a full professor at Nanjing University. He should accept two or three students every year.

Of course, the most important thing is that Brother Cai Peng has obtained the qualification for postgraduate study in Princeton, and he has indeed met the threshold conditions for recruiting students.

If he does not meet the threshold, he will not accept.

Mathematics is a subject that requires both talent and hard work. It is impossible to just get by in this subject, especially under his guidance.

Hearing this, Cai Peng immediately swallowed his saliva nervously.

This big guy personally set the questions. Can he pass it?

The questions set by the big guys at the level of the Fields Medal are not difficult in his eyes, but for a small scum like him, I am afraid that he may not be able to solve them even if he racks his brains.

Before coming, he knew that the graduation thesis given by this big guy to the two students solved a world-class conjecture problem. Gu Bing and Amelia are still working hard for it.

This standard is the best among the best in Princeton.

Suddenly, he suddenly wanted to go to Princeton for further studies.

I am afraid.

This big guy is really awesome.

He had already killed his graduate thesis + doctoral study goals in his freshman year. If this falls into his hands, will he really have a chance to complete the graduation goal and further study?

After learning about Cai Peng's new direction of study and looking at his paper, Xu Chuan took out a pen and paper from the drawer, thought about it and started to write questions.

In about ten minutes, he wrote five questions on two pages of manuscript paper.

After handing it to Cai Peng, Xu Chuan smiled and said, "Five questions, give you three days. If you have the ability to finish it in advance, you can also hand it in to me in advance."

His minimum requirement for the five questions is to solve three.

Doctoral students can be said to have completed the accumulation of knowledge, and doctoral students who have further studies have entered the stage of creating knowledge.

For students at this stage, his requirements are not low.

If you are not talented enough, it is meaningless to force yourself to follow him, because it is difficult to keep up with his pace when explaining problems or teaching.

Cai Peng solemnly took the manuscript paper, nodded seriously, and said, "I will work hard!"

Xu Chuan smiled and said, "Don't have too much psychological pressure."

On the side, Zhou Hai said, "You go back and do the questions well first, I will talk to Professor Xu."

Cai Peng responded, said hello and left the office quickly.

Five questions in three days, with the level of this big guy, he still doesn't know if he can finish them. It seems that he has to stay up all night to work hard.

Cai Peng left, Zhou Hai picked up the teacup and drank a sip of warm tea, saying: "If you don't plan to teach students next year, there is no need to force it. It's also good to let him go to Princeton to study, which won't delay you or him."

Xu Chuan smiled and said: "Teaching students doesn't take up much energy."

"And to be honest, my requirements are not low. He can get the application of Princeton, which has reached the minimum standard. As for the five questions, if he can solve three, I will consider it. If not, then go to Princeton."

"If you can't keep up with the pace, you won't learn anything by following me to learn mathematics."

Zhou Hai nodded and said: "It depends on him. I can only help to build a bridge."

After a pause, he smiled again: "But speaking of it, my students who have some relationship with you seem to have opened their minds. Liu Jiaxin is the same. She used to study averagely, but she took off directly later."

Xu Chuan smiled and said, "Senior Jiaxin is different from Cai Peng. She is actually very talented, but working part-time to make money takes up too much of her time. After solving her life difficulties, her talent burst out. She helped me solve a modeling problem before."

"As for Cai Peng, I'm not sure. Maybe it's just like you said, after changing the field of study, he is like a fish in water."

Zhou Hai thought about it and smiled, "It may be true as you said. It seems that I have to pay more attention to those students who live in poverty in the future."

Xu Chuan smiled and said, "In fact, the school's scholarships and bonuses are enough for a college student, but Senior Jiaxin has a younger sister who is also studying, and her parents are gone, so it is difficult to make money."

The amount of scholarships and grants in 985-level universities is still very good, basically enough for poor students to complete their studies.

And if there are poor students in the school, basically as long as they are confirmed, they can get them, unlike primary schools, where there are all kinds of crooked ways. (PS: So college entrance examination candidates should work hard!)

After chatting with Zhou Hai for a while, Professor Zhou left, and Xu Chuan also picked up a math book from his desk.

He had a math class today. He had originally planned to let Gu Bing or Amelia take over, but he suddenly became interested, and since he had time, he simply went to the class himself.

In the classroom, the students who were waiting for class were stunned when they saw Xu Chuan, and then they burst into joy. This big guy actually came to class in person again, and he had stopped for a while before.

Xu Chuan didn't pay much attention to the reactions of these students. He has been teaching at Nanjing University for a while, and he is considered a semi-fixed NPC. The enthusiasm of these "players" has long passed.

After trying the headset, Xu Chuan said: "Students, today we will talk about Grothendieck's schematic language. Please turn to page 32 of the textbook. I hope you have already previewed it before."

As soon as these words came out, many students in the classroom were a little panicked.

"It's over. Today, the teacher is teaching in person. I must not have prepared enough! I only saw the Abel group layer on the topological space."

"Fuck! You at least prepared that much, I didn't even finish reading the algebraic clusters."

"It's over, this big guy teaches faster than turning pages, I'm dead this time, I'm going to fail the final algebra again."

"Someone come to save me!"

There was a cry of grief in the classroom.

After Xu Chuan took some classes, almost all the students of Nanjing University and even some students from other schools who came to take classes knew that the speed of this big guy's lectures was like riding a rocket.

If you don't pay attention, you will be "biu" directly into the sky.

In this big guy's class, picking up a pen and being confused for the whole class is not a story, this can really happen!

On the podium, Xu Chuan didn't care whether these people had prepared in advance. He opened the textbook and started the class.

".We have discussed Grothendieck's scheme language a long time ago, and we have also learned about the most basic definitions and properties of schemes in morphisms and condensed layers."

"Although we have given some concepts in the form of algebraic clusters, it is more convenient to discuss them in the language of schemes. For example, abstract algebraic clusters are actually finite schemes of integers on algebraic closed fields, such as Cartier divisors and reversible layers."

"."

In class, Xu Chuan first briefly reviewed the basic content, and then cut into the main text.

The formulas on the blackboard continued to appear and disappear as he explained and time passed, and the content of the class also extended from the initial Grothendieck scheme language to the use of schemes and cohomology to study curves.

In the large classroom, many students have begun to struggle and be confused, and some smarter ones have given up listening to the class, and took out their mobile phones from their pockets and started to stare at the podium to take pictures.

Even if you can't keep up with the pace of this big guy's class, you can't miss it.

Take it first, and then slowly replay it after returning to study.

After the first one, the second one followed. Soon, most of the students in the classroom started to hold up their phones to take pictures.

".Today's class ends here."

After writing the last equation on the blackboard, Xu Chuan threw the chalk in his hand onto the podium, turned around and said with a smile: "There is still a little time left. If you have any questions, you can ask them now."

Hearing this, hands were raised in the classroom.

Xu Chuan picked one at random, and the other party stood up quickly.

"Professor, can you talk about the application of cohomology of Abel group layers on topological space?"

Hearing this question, Xu Chuan glanced at the student who stood up and smiled: "This is actually very easy to understand."

"As an application, first of all, it is proved that the arithmetic genus of the projective cluster X can be calculated by the cohomology group, and the arithmetic genus of the projective cluster depends on the projective embedding of X. Including flat morphisms and smooth morphisms, both of which can be processed by cohomology. In fact, cohomology can be used to better understand flatness and smoothness."

"Thank you, professor." The boy who asked the question thanked him with a thoughtful look and sat down. Obviously, Xu Chuan's answer brought him new inspiration.

Xu Chuan smiled and continued to ask students to answer questions.

After being picked to ask a question, a student stood up excitedly: "Professor, can you tell us about the staged proof of the NS equation?"

Xu Chuan was stunned for a moment, shook his head and smiled: "Why are you always interested in this?"

In his class, every time there is a question session, there are students asking this kind of questions, either interested in the Hodge conjecture or the NS equation.

Is this probably human nature? Knowledge and gossip, the public may be more willing to gossip?

Sensing that there seems to be hope, the classroom suddenly became lively.

"Professor, tell me!"

"This is also to broaden our horizons and let us understand the vastness of mathematics."

"Yes, let's take a look at the world of mathematics, professor"

After thinking about it, Xu Chuan looked at the time, 2:55 in the afternoon, and said: "Okay, since you want to listen, then I will briefly tell you."

As he said, he picked up the chalk from the table, erased the original formula on the blackboard, rewrote the lines of formulas, turned sideways and looked at the students in the classroom, and said:

"NS equations are special forms of diffusion-convection equations. They are all related to conservation laws and are the most The direct mathematical reflection of basic physical theorems. "

"In the process of studying the NS equation, I first started working with Professor Fefferman of Princeton."

"Fefferman used the Lie group with smooth differential manifold structure to make a smooth mapping, so that the Lie group G unitary table G played a continuous role on the Hilbert space, thus starting our journey to study the NS equation."

"Later."

In the classroom, Xu Chuan briefly explained the process of his study of the NS equation and wrote down some key step formulas on the blackboard with chalk.

For him, these things are just easy to come by.

For the students in the classroom, I am afraid that most of them don't understand it very well.

But it didn't matter. On the contrary, the students' eyes lit up. To them, it was just a story. They remembered as much as they could. When they went out for dinner or chat, they could brag about it: "I heard about Chuan Shen in the past."

On the podium, Xu Chuan finished the story with a smile: "For the NS equation, I have expanded it to the point where a smooth solution of the three-dimensional incompressible Navier-Stokes equation exists when a finite space is given and the initial value is infinitely smooth."

"Now there is only one last step, which is to spread the limitation of finite space to infinite space. This step may take a long time to study; of course, it may appear tomorrow."

As soon as he finished speaking, someone in the classroom shouted loudly: "Professor, if we want to solve the last step, do you have any ideas? Or should we say that Which direction to start from? "

Xu Chuan smiled and said, "I'm afraid this question is not what you should consider now. You should honestly lay a good foundation first."

After a pause, he continued, "But since you asked, I can still talk about it."

"In fact, for the NS equation, the solution may not be directly calculated. At least for now, the mathematical community may not be able to solve it."

"Because in a sense, its essence is the fluid micro-group hypothesis, that is, the fluid particles described by the NS equation are infinitesimal in space, but in fact they are infinite relative to molecules. "

After thinking for a while, Xu Chuan spoke again: "If you want to solve this last step, maybe you can start from the mathematics of micro-element fluids? Through the study of viscous fluids..."

On the podium, he stood there before he finished speaking, and his brows frowned unnaturally.

The four words "micro-element fluid" seemed like a light in the dark night, illuminating a way forward in his mind.

Without paying attention to the students who were already making some noise in the classroom, Xu Chuan quickly pulled out a piece of chalk from the podium and began to write on the blackboard.

PS: The second update of 8,000 words has been sent, and the third update is being written, which may be late (mainly because it is a bit brain-consuming to look up information, life is not easy, cat sighs.)

In addition: Thank you all for your monthly ticket support, thank you, thank you very much, and I will continue to update tomorrow!