Yuan Zhengxin was very satisfied with Qiao Yu's performance.

It can't even be described as very satisfied. The old man is very sorry. It's a pity that Qiao Yu is not his grandson! It would be great if this child was his grandson.

Especially at this time, when the sun was already setting, a ray of sunlight leaked in from the window and fell on the child's concentrated body, making Qiao Yu look like an angel covered with light. The more he looked at it, the more he fell in love with it.

How much do you like it?

Let's put it this way, the old man got up and went to the bathroom. When he came back, he made some noise, which alarmed Qiao Yu. Qiao Yu raised his head blankly, glanced at the place where the noise came from without focusing at all, and then immediately lowered his head and started. Continue to write and draw on the manuscript paper.

In fact, it had no effect, but the old man didn't even dare to drink water for the next time, fearing that if he had to go to the bathroom again, it would affect Qiao Yu's thinking. He even breathed a little softer. The old man looked at Qiao Yu from time to time. Yu glanced at him quietly, and finally the young man next to him stretched out his body.

But the old man's originally kind eyes became solemn because of this laziness. "How is the effect of studying here this afternoon?" the old man asked seriously.

Qiao Yu nodded vigorously, not caring about the old man's serious look, and replied loudly: "Grandpa Lecturer, that's very good! I'm not bragging to you, I feel like I'm going to read Needham's "Visualized Differential Geometry and Forms" "I've learned enough!" "Oh? You haven't asked me a single question, so I'm going to test you." The old man said seriously.

"No problem, you can take any test from this book except for the last bit!" Qiao Yu stood up confidently and handed the book to the old man beside him.

Yuan Zhengxin took the book from Qiao Yu, flipped through it casually, then put it aside and asked, "Needam believes that geometric intuition can affect the study and research of differential geometry. What do you think of his view?"

Qiao Yu replied loudly: "I think Needham's view is correct. First of all, geometric intuition is a concept emphasized in Needham's book, which refers to the ability to understand geometric objects and their properties through vision and spatial perception.

This intuition can help us form concrete spatial images when facing abstract mathematical concepts, making it easier to understand complex structures and relationships. Maintain focus on geometric intuition, especially when dealing with complex manifolds and surfaces. Through geometric intuition, researchers can better propose and verify conjectures, thereby making progress in the derivation and application of theories. Let me give you an example..."

Qiao Yu's voice, neither humble nor overbearing, was loud, clear, and ringing to the ears. The answers he gave were clear and logical, with opinions, analysis, and examples. Yuan Zhengxin probably understood why this child could get so many full marks, and why Zhang Shuwen went there After taking one look, I came to the conclusion that this child has amazing talent.

In this way, in the quiet small building full of ancient charm, two people, one old and one young, asked questions and answered questions, as if they had traveled through a hundred years, just like the way students faced their teachers in private schools in ancient times. The old man nodded slightly, finally showed a smile, and said: "The last question is, how do you understand the relationship between the tangent bundle and tangent space on the manifold, and discuss the impact of the geometric structure of the tangent bundle on the manifold."

Qiao Yu answered loudly without hesitation: "The tangent bundle is the set of all tangent spaces on the manifold, usually represented by TM, where each tangent space TpM corresponds to a point p on the manifold M. The tangent space is a vector space , and the tangent bundle is a structure that combines these tangent spaces.

The impact of the geometric structure of the tangent bundle on the manifold is that it allows us to define vector fields and differential forms. Through the study of the tangent bundle, we can analyze the local and global properties of the manifold. For example, smooth mapping, the topology of the manifold, and how Define geometric quantities on manifolds.

Oh, by the way, the tangent bundle also plays a key role in defining connections and calculating curvature, thereby helping us understand the curvature properties of manifolds..." The old man asked a total of six questions, covering the book Qiao Yu had read All aspects of the part.

Qiao Yu's answer left him unable to find a single fault. Really, every detail was almost perfect.

"Okay! Well, it's just past five o'clock, and there's still a while before dinner. Do you dare to challenge me on a question paper about differential geometry?" "No problem! Grandpa Shi, you want to test me, even if Come here!" Qiao Yu still looked confident and full of confidence.

"So confident? It seems I have to choose a more difficult one." After saying that, Yuan Zhengxin stood up, pulled out a folder from the bookcase behind him, selected a test paper from it and handed it over. Qiao Yu He took the paper and did not start working on it immediately. Instead, he read the paper from beginning to end, and then said with a sad face: "Master, it's okay to do the paper. Can I make a small request?" "Oh? What request?" The old man asked patiently.

Qiao Yu took the paper and said: "Look, some of the questions in this paper are not difficult in themselves, but it is too troublesome to use differential geometry methods. For example, the third question is to find the curvature and torsion of a parametric curve at point P. , if I use the method of differential geometry, I have to calculate the derivative first, and then solve it through the curvature and torsion formulas.

But if I use the algebraic method, the direct parameter substitution method, that is x=t, y=t^2, z=t^3. Through this relationship, the geometric properties of the curve can be directly derived, and then substituted into the geometric formula, saving calculations. The time of the cross product." The old man glanced at the question and nodded again and again, and asked kindly: "Well, it can indeed be solved like this, but why are you thinking of being lazy before you finish this paper?"

"Master, it's not that I'm lazy. It's mainly because I read a book all afternoon and I feel a little hungry. If I use differential geometry to do it all, it will take at least half an hour. I'll use some labor-saving methods and I'll finish it in twenty minutes. You can go back to eat!" Qiao Yu said, patting his belly.

Yuan Zhengxin was suddenly enlightened and immediately said: "Yes, yes, yes, this is my fault, Grandpa Shi. Hey, I forgot that you are only fifteen years old and it is the time when you are growing. I am not hungry, so I thought you are not hungry either. In this way, You don’t need to write a complete answer for a big question, just write an idea.” Unfortunately, there are only two people in the office, Qiao Yu and Yuan Zhengxin. If Mr. Yuan’s students see the self-blaming look of their teacher, they will probably be upset. Shocked and doubtful about life.

You must know that Mr. Yuan has always been famous for being strict. If it were someone else, let alone eating a little later, there is a high probability that he would not starve to death even if he didn't eat... "Grandpa, you are so kind. It only takes ten at most." Just five minutes will do!"

"Okay, you do the questions first. By the way, we'll eat here tonight. What do you want to eat?"

"Um? Well, let's eat in the cafeteria! Anyway, the food in Yanbei University's cafeteria is quite delicious. I just want to try the food in Qingbei cafeteria. And the food in the cafeteria is fast!" "That's not it. It may be worse than the ones at Yanbei Canteen. What kind of food do you usually like? When you do the questions, I will have them delivered."

"Master, I'm not picky about food. More meat would be great. By the way, mutton would be even better." "Okay!"

...

"Mr. Yuan, are you sure the meals will be packed directly to your office?" "Yes, remember to order more mutton, some of all kinds of mutton." "Well, okay, okay."

"Be quick!"

"Don't worry."

"How's it going? The food at Huaqing Canteen is also good, right?" "Yes, it's just as delicious as Yanbei Canteen."

"If it tastes good, eat more."

"Grandpa, you should eat too."

"I'm not hungry yet, please eat more. By the way, are you still used to it after coming to the capital?" After hearing this answer, Qiao Yu blinked. The old man didn't correct him, so that's fine.

"I'm quite used to it, but the heating in the room is too high, and I often wake up at night due to the heat." "Then it's better to leave a window open when sleeping to reduce stale air and allow fresh air. Overheating is really not good."

"Okay, grandpa. I'll try it tonight."

"Also, do you have enough living expenses in the capital? I heard from Teacher Lan that you still have to make money to support your family when you are in Star City?"

"That's enough! Grandpa, that's all from the old calendar! I won the gold medal in the Little Libaba International Mathematics Competition and was given a bonus of 200,000 yuan. I came first in IMO. The school secretly rewarded me 250,000 yuan. I gave it to The school bonus is left to my mother for living expenses, and I still have two hundred thousand. To be honest, I have never been so rich."

"Haha... you are very good, and the school is also very good. Speaking of which, Grandpa Shi, I also came through hard times."

"I know, I heard my grandpa say that it was very difficult for his generation, and it must be even more difficult for your generation. My grandpa was also a very good person, but it's a pity that he left when I was in the fifth grade." "Oh, God. It’s a holiday, what a pity. Putting aside these disappointing things, are you still reading Schulz’s paper?”

"Looking at it, Grandpa, I am studying his "Perfectoid Spaces" published in Publicationatiques deI'IHES in 2012. This paper makes the definition of perfect space clearer.

Combined with the differential geometry I am looking at, I feel that on a complete p-adic, the structural ring of the complete space can be used for complex geometric analysis as long as it satisfies the characteristics of perfectoid. And his tower structure, too. People are shocked." "Yes, this is the way to read the paper, and connect it with what you have studied so far to do extended thinking. Schulz's work is groundbreaking. As far as I know, many teams are currently using his results. Do some basic research.

The parts involving geometry and number theory."

This hint was too obvious. After understanding it, Qiao Yu shook his head decisively and said: "Grandpa, I don't want to follow others to do the research they have already started. And my basics are still too poor now! Wait until I can make up for the basics." It’s a shortcoming, I’m going to start from scratch and make a result that you can be proud of!”

"Yes, yes, it's really not interesting to do it with others! My disciple is ambitious! Okay, you hurry up and make up the basics in the past six months. Grandpa is waiting for you to make achievements." "Grandpa, I'm full."

"Well, I have something else to do tonight, so I can only ask someone to drive you back." "No, grandpa, I can just walk back by myself."

"It's hard to walk so far just after dinner. Besides, it's inconvenient for me to drive to Yanbei to pick you up. Why don't you drop me off? You follow my arrangements." "Okay, grandpa."

"By the way, take this card and you can go in and out of Huaqing at any time in the future. There is still some balance in it. If you want to come to Huaqing to eat, just swipe this card. If you feel uncomfortable, the campus hospital in Huaqing is also You can use it directly." "Ah? Grandpa, isn't this bad?"

"You call me grandpa. In a hurry, I didn't prepare any elder gift for you. Just take it and don't refuse. I also have this book for you. You can take it back and read it at your leisure." Then thank you grandpa.

Qiao Yu took the book and the card, put the card casually into his pocket, and held the book in his hand and read it carefully.

"The Beauty of Mathematics" is an old book, but it can be seen that it is well preserved. When you open the title page, there is an autograph of the old man, which is so cool.

Qiao Yu put the book into his bag very solemnly.

"Come on, I'll take you downstairs to the car."

Yanbei University, Beijing International Mathematics Research Center, Chen Zhuoyang has a book spread out in front of him in his office, but his mind is obviously not on the book in front of him. From time to time he will raise his head and look out through the window. He is sitting This location just happens to be able to see the only way to this building. I originally chose this layout to observe whether the instructor would come over, but I didn't expect that I would use it to see my junior brother today.

Speaking of which, he was already in this state three hours ago. He went to the cafeteria for dinner at 5:30. When he came back, he went to the junior brother's room to take a look and found that the door was still locked and no one was inside. , then went back to the office to wait, and now it was almost half past six.

Junior brother, is he running to the other side and nowhere to go?

Forget it, Chen Zhuoyang feels like he can't stay in the office anymore. He has been restless. He might as well go back to his dormitory to calm down and read a book. After all, the academic pressure is still very high.

Suixi tidied up his desk, then picked up the books on the desk and walked out of the office. Not far from the entrance of the research center, he saw a black Audi car driving in this direction and then stopped steadily. At the intersection entering the research center, Chen Zhuoyang subconsciously glanced at the license plate. Well, it was not the car from the research center.

When he was about to walk over, he saw Qiao Yu getting out of the car and politely thanked the driver. "Qiao Yu?" Chen Zhuoyang called out.

"Hey, Senior Brother Chen? Are you going to go back to the dormitory?" Qiao Yu turned around and naturally walked to the senior brother's side.

The car behind him started directly and left behind the two of them. Well, Yanbei University has many gates. There is a gate in front that you can go around without turning around. "Ah, I just went out and you came back from the opposite side? That's Mr. Yuan." The car?" Chen Zhuoyang glanced at the car that had left and asked curiously.

"I don't know, my senior teacher just arranged a car for me, right? And I must come back, senior brother, I am a student at Yanbei University and I don't have a dormitory in Huaqing, so why don't I just sleep in my senior teacher's office?" "Seeing Qiao Yu acting as natural as if he had gone out for a walk and returned home, Chen Zhuoyang felt that his three views had been subverted again, and at the same time his curiosity was bursting. He naturally turned around and followed Qiao Yu back.

"Um, you were at the teacher in the afternoon, ah, no, what did you do with Mr. Yuan?"

"Just read a book. After reading the book, Grandpa Master asked me some questions and made a practice paper. Then he saw that it was getting late, so he left me to have a light meal in Huaqing. Speaking of Huaqing Canteen The food here is also very delicious! Especially the boiled mutton. Oh, by the way, Grandpa Shi gave me a card before he left. He said that I could swipe my card to enter Huaqing in the future, and it could also be used to eat in the cafeteria. .Next time senior brother wants to go to Huaqing Canteen for dinner, tell me and I’ll treat you!”

"No, didn't you and Mr. Yuan talk about anything else?"

"We talked. Grandpa Shi asked me to study the Rosetta Stone problem and wanted to introduce me to a research team. I refused. After all, there is no point in working with others. We will do our own research in the future." "What? What kind of team?"

"It seems that it is a team of a professor from the Planck Institute of Mathematics. It mainly studies the Langlands Program of Geometry. There happens to be a professor from Huaqing University in the team. The master probably wants me to help others and accumulate experience. Experience or something, but I don’t think it’s necessary. Right, senior brother.”

Chen Zhuoyang looked at Qiao Yu with his mouth half open. He always felt that this might be made up, but it seemed that he couldn't make it up. Although there may be many people in this cutting-edge mathematics research team who know about it, there are definitely not many people.

Of course, Mr. Yuan definitely knew it very well, but he couldn't figure out why Mr. Yuan would specifically mention this to his junior brother without bringing Qiao Yu into the research group. If you add that Qiao Yu did express his disapproval of Schultz recently. If the paper has research, then it is more likely.

It is said that although Schulz himself did not participate in the study of the geometric Langlands Programme, his theory is an important prerequisite for solving this problem, so it seems that Mr. Yuan opposite him may really want to give his junior brother a hard hand. Old Yuan is confused!

Fortunately, my junior fellow apprentice is sober and refused. Otherwise, if he had joined such a world-class research group at the age of fifteen, would senior fellow apprentices like them still have to live?!

Of course, you can't say that. Chen Zhuoyang could only ignore his conscience and said with hatred: "Hey, little junior brother, such a good opportunity... just like you said, you can definitely go in and exercise. Learn more from those big guys, why did you refuse?"

Qiao Yu glanced sideways at Senior Brother Chen and said seriously: "Senior Brother Chen, you are so superficial! Our teacher Tian is also a big boss, and when he joins this kind of research group and signs the results, he is always at the end. If you can't even find it, who will remember you in the future?

If you have that time, why not quickly follow the teacher and learn more from Grandpa Shi, and then start a project on your own. If the research produces results, we will be the first authors. That would be so cool! So why are you anxious?

Besides, there are so many difficult problems in mathematics. The Millennium problems have been on the list for so many years and no one has been able to solve them. There are even more problems that are not on the list. Are you worried that there will be no problems in the future? So Ah, Senior Brother Chen, you need to enlarge the situation a little bit more, we need to accumulate more and more, and if we don’t make a fuss, it will be a blockbuster!”

Listening to Qiao Yu's old-fashioned words, Chen Zhuoyang couldn't help but think of the abbreviation of China's National Missile Defense System. This is so simple...

The problems of the millennium...

So, can normal people touch porcelain?

How many people have chosen this kind of proposition with great ambition, and then fell down on it for the rest of their lives, wasting their best years, failing to achieve any shit, just gambling on their lives...

He wanted to remind this junior fellow student that the senior fellow student across from him had just rejected the teacher's proposal and did not do research on the Riemann Hypothesis in the Millennium Problem, so he successfully won the Fields Medal. If it had been that year If Yuan Laozhen chose to pursue the Riemann Hypothesis, he might lose the Fields Medal.

.. never mind..

"By the way, senior brother, are you going back to the office again?"

Qiao Yu's voice brought Chen Zhuoyang back from his wandering thoughts. He looked at his junior brother beside him. He was a little embarrassed and just went to eat melon... He quickly followed Qiao Yu's words and said, "Well, yes, I just wrote half of it." I forgot my paper, I have to go back and get it."

"Speaking of thesis, I heard from the teacher that you are studying geometric analysis on complex manifolds?" Qiao Yu's tone suddenly became enthusiastic. Chen Zhuoyang looked at Qiao Yu warily, nodded, and replied: "Yes, what's the matter? ?」

"Brother, I have an idea. You see, everyone is studying the geometric Langlands conjecture. You should also make it more difficult for yourself and add Schultz's theory of nearly complete space. I have a feeling. , as long as you understand Schulz’s theory thoroughly, you will definitely be able to write new ideas in your doctoral thesis.”

Qiao Yu said seriously.
To be continued...