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Chapter 106 Math Isn’t That Easy…But It’s Not Hard Either!
Zhang Shuwen hesitated for a moment, then stood up, walked to Qiao Yu, erased the last word on the blackboard, and began his on-the-spot explanation.

"The Riemann-Roch theorem is a fundamental theorem in algebraic geometry that describes the dimension of certain functions or forms on an algebraic curve. Specifically, the Riemann-Roch theorem applies to any divisor D on an algebraic curve X, and states the dimension of the space of functions L(D) associated with the divisor D on the algebraic curve.

Its specific statement is (D) = deg (D) + 1g + (KD). It has two parts that complement each other and describe the balance between the divisor D and the remainder KD. But there is a special case, when the degree of D is large enough, (KD) is zero, so in this case (D) = deg (D) + 1g, do you understand what this means? "

"The degree of D is large enough, and the dimension and degree are in a linear relationship." Qiao Yu answered immediately.

"So when D is zero..."

"(0)=1g+(K)...Oh, Professor Zhang, I understand what you mean...So the proof of this part is not that complicated, because the genus g(X) can be directly derived through the Riemann-Roch theorem. Hmm, then the proof of this part is not that troublesome... Let me think about it..."

After saying that, Qiao Yu picked up the chalk and started writing on the other side of the blackboard.

"That is to say, when constructing the function... well, dimQH1(Cp is the dimension of the homology group after quantization, which depends on the genus g of the curve and the quantum operator Q... This part can be obtained by calculating the canonical factor to get the dimension of H1(Cp)...

So the dimension relationship after decomposition is directly dimQH1(Cp)=gf(Q). Professor Zhang, do you think this part of the derivation is correct?"

Zhang Shuwen took a deep breath, trying to keep his expression expressionless, and then nodded.

"Great, then the next step is to prove it... After deriving the dimension of the homology group, the larger the dimension of the quantized homology group, the higher the geometric complexity of the curve, and the number of rational points on the curve will be limited. In addition, the Jacobian can further affect the number of rational points...

Genus is one of the most core geometric invariants and cannot be simplified. So #C(K)≤f(g, Jac(Cp))? Oh, no, if you look at it this way, I feel like this method can really derive the formula for the constant C, right? "

Qiao Yu sighed subconsciously.

Really, Chen Zhuoyang in the audience was stunned when he heard Qiao Yu's words.

Although he was also shocked by Qiao Yu's understanding, wouldn't everyone be angry after hearing this?

How dare you accept a 45-minute seminar on something you are not % confident you can prove?

Seeing that no one in the meeting room seemed to care, Chen Zhuoyang naturally couldn't say anything.

Professor Zhang on the stage snorted coldly and said, "It's still early. I believe you can prove it and even get the formula you want! But are those really useful?! You have to at least simplify it to #C(K)≤f(g) for it to make sense!"
It is okay to introduce Peter Schultz’s theory. The mathematical proof process can be as complicated and abstract as long as it is logical within the framework, but you must limit all complexity to the intermediate steps of the proof!
The final result must be as simple as possible! Otherwise, even if you prove the constant C and derive the result, if you put so many set constants into it, think about how complicated the final formula will be? How can others use it?
Real mathematics pursues the complexity of thinking and the simplicity of results. Only simple results are truly useful and elegant mathematical tools! Too many constants or parameters will only increase the difficulty of understanding and calculation, and even if they are studied, they will be garbage! Mathematics is not as simple as you think! "

……

Zhang Shuwen's tone was extremely stern, but Tian Yanzhen looked very happy as she sat there.

Robert Green finally couldn't help but come over and ask, "Professor Tian, ​​what is Professor Zhang saying to that kid?"

Just now, Qiao Yu used English when introducing his ideas, but when Zhang Shuwen went up to give Qiao Yu instructions, he had already started using Chinese.

"He taught Qiao Yu not to get carried away, and warned the child that what he had proposed were just ideas, which were far from producing results, and that mathematical conclusions must be simplified." Tian Yanzhen explained with a smile.

"Oh my God, is Zhang so strict? Doesn't he know that this kid is only fifteen? At the age of fifteen, he can actually understand Schultz's theory and come up with such creative ideas. Zhang still thinks it's not enough? Is he crazy? I even think this is indeed a very promising research direction in the future."

Robert Greene said in confusion. Obviously from the perspective of this New York University professor, Zhang Shuwen was too strict and his requirements for Qiao Yu were too harsh.

"Yes, that's why I must hold this seminar. I also think this is a possibility worth looking forward to. But at present, it is still a little difficult for this child to complete this proposition independently. So I am actually very grateful to Professor Zhang. At least he told Qiao Yu that doing subtraction at the mathematical level is sometimes much more difficult than the proof process itself."

Tian Yanzhen helped Zhang Shuwen explain with a smile on her face.

"Although what Zhang said is very correct, there is no need to use such a harsh tone. It is not fair to a fifteen-year-old child."

Robert Greene still couldn't understand. After all, if Qiao Yu was his student, he would never be so ruthless.

Although he couldn't understand what Zhang Shuwen said, he could tell that his tone was even cold.

"We Chinese have a saying that goes, 'The deeper the love, the more severe the criticism. If he hadn't been particularly optimistic about the child, given his personality, he wouldn't even bother to say a word to him."

Tian Yanzhen couldn't help laughing out loud, even though he was very suppressed. "Haha, Professor Zhang must be feeling sorry that he didn't meet such a student, right? Actually, I want to thank Professor Zhang. I wanted to say these words to this kid this afternoon, but I was afraid that it would dampen his enthusiasm."

Robert Green shook his head. The Chinese people's way of thinking was too weird. He couldn't understand it. He didn't point out the problem himself, but thanked others for pointing it out more harshly? So if others pointed it out, it wouldn't dampen a child's enthusiasm?

In his opinion, students like Qiao Yu should be encouraged more, after all, he is only fifteen years old. This will stimulate his love and confidence in mathematics, rather than criticism.

What surprised Robert Greene even more was that the child on the stage did not show any frustration and was still talking to Zhang Shuwen as usual.

Damn it, he couldn't understand Chinese.

……

Yes, Qiao Yu on the stage did not show any frustration. Facing Zhang Shuwen who looked serious, he just said seriously: "Thank you, Professor Zhang. I understand. I thought the problem was too simple!

But I still insist that my idea is correct, and I will definitely prove this result. And without setting so many constants! "

Zhang Shuwen's expression eased a little, and he nodded and said, "Well, you are very talented in mathematics, but how to use this talent depends on your own efforts and what path of mathematics you choose. It's up to you."

After saying that, he turned around and returned to his seat.

At this time, Tian Yanzhen had already put away the smile that he had just been unable to suppress, and spoke calmly as usual: "Everyone, if you have any other ideas, please tell us. Qiao Yu's ideas are definitely not mature enough, so everyone should help this student of mine who is not very promising at the moment."

Then he turned his head and whispered to Robert Green beside him to explain his words.

The atmosphere in the conference room finally began to become lively.

Just as Tian Yanzhen said, Qiao Yu's ideas are extremely creative, but not mature. Most of the professors invited by Tian Yanzhen today have been immersed in the field of algebraic geometry for decades, so everyone has something to say.

Of course, this also includes Robert Greene who came all the way from across the ocean.

In fact, the NYU professor really liked Qiao Yu so much that at the end of the meeting, he took the initiative to say, "Qiao Yu, your name is pronounced like this, right? Maybe you can leave me a contact number so that I can introduce Peter Schultz to you in the future. I believe he would also like to meet you very much.

Do you know why? Because I think your ideas are very creative and valuable! Oh, by the way, remember to come to my lecture tomorrow. Believe me, there will be a lot of exciting content coming up, and you will definitely gain a lot! "

Robert Greene expressed his attitude through practical actions. In front of Zhang Shuwen, he encouraged Qiao Yu a lot.

"Thank you very much, Professor Robert. In fact, your lecture today has helped me a lot. I really enjoy reading your paper. All these ideas of mine popped out of my mind after reading your paper!"

Qiao Yu answered like this, which also made Professor Robert, who came from afar, smile.

When it comes to using processed truths to make people happy, Qiao Yu is even more skilled than his mathematical talent, of course, on the premise that he is willing to do so.

……

"I'm going to see the two professors off. You go upstairs to office 301 and wait for me. The door is not locked. Don't touch the things on the table, but you can read any book in the bookcase."

After the seminar, Tian Yanzhen gave Qiao Yu some instructions, then strode out of the meeting room and caught up with Robert Greene and Zhang Shuwen.

It’s hard to tell from his spirit that Director Tian is already nearly seventy years old.

Qiao Yu glanced at Xue Song and asked, "Teacher Xue, will you go with me?"

Xue Song shook his head and said, "Are you kidding? You can go to that place, but I dare not go in without an invitation. Just go and wait."

After saying that, Xue Song packed up his things and left.

Qiao Yu glanced at Brother Chen who was finishing up the work...

Forget it, he should go to the teacher’s office by himself.

Brother Chen is good in every way except that he is too serious.

Chen Zhuoyang finally breathed a sigh of relief when he felt Qiao Yu leave the office. He had to think carefully about how to get along with his junior brother in the future. He had to admit that when he first met his junior brother, he seemed to speak a little louder.

……

On the other side, outside the research center, Tian Yanzhen was seeing Zhang Shuwen off.

Robert Greene lived in a hotel just 400 meters away from the research center. He declined Tian Yanzhen's offer to take him to the hotel and walked out on his own. The main reason was that everyone was too smart, and the foreigner was very sensible, and he could see that the two Chinese colleagues had something to talk about.

"Do you want us to arrange a car to take you there?" Tian Yanzhen asked casually.

It's not that we have nothing to talk about. Although the two universities are very close, the campus is very large. It takes at least two or three kilometers to walk back.

"No need, it's fine to walk back, it's not that delicate." Zhang Shuwen replied listlessly.

"Well, shall I give you a ride?"

"No need."

"Just send me off to the school gate. I won't go out anymore." Tian Yanzhen insisted.

"Then let's go through the east gate." Zhang Shuwen said.

"Why don't you go hang out at the lake? It'll be busy next week after school starts."

"No, there will be a chance later."

"okay!"

After walking two steps silently, Tian Yanzhen suddenly said with a sigh: "Mr. Hamilton passed away last year."

Zhang Shuwen was silent for a moment, then said, "Well, respectable old man. You must be complaining in your heart, why it wasn't Simon Donaldson who died."

Tian Yanzhen smiled and said, "I'm not that mean. At our age, we've let go of many things. But then again, Donaldson is 69 this year, right?"

Zhang Shuwen nodded and said, "Yes, I remember he is one year older than you."

The two people's simple words are all about the history of the mathematical world.

Both Richard Hamilton and Simon Donaldson are internationally renowned mathematicians, especially the latter, a former Fields Medal winner, a fellow of the Royal Society of Britain and a dual member of the United States National Academy of Sciences, with many works.

However, there were indeed some minor conflicts between everyone back then.

From the perspective of ordinary people, it is probably a very important research result of Tian Yanzhen, but Donaldson's team thinks it is theirs. But the actual situation is more complicated, involving the recognition of the paper submitted to the pre-release website in advance.

For academic newcomers, well-known journals that still adhere to double-blind review may reject papers that have been published on pre-publication websites without hesitation, but it is different for these bigwigs.

The proof of an important proposition is often pre-published and reviewed by peers.

The situation at that time was that Tian Yanzhen was the first to publish the results, but Simon Donaldson believed that there were many problems in his proof process, so he published a new paper to solve the same problem, and insisted that Tian Yanzhen's paper was extensively revised after pre-publication and cited their process.

This happened many years ago. Apart from the two parties involved, few outsiders can get a full picture of what happened back then.

It was just that in that international debate, Simon Donaldson's team had an absolute advantage.

After all, he is a real Fields Medal winner, with a higher rank, a higher reputation in the international mathematics community, and more followers. Even Tian Yanzhen's mentor supports him.

Tian Yanzhen nodded and continued, "So we are old and need someone to take over. You have trained a lot of young people over the years. I have paid attention to them. Many of them are very good and have achieved good results. But it is also true that China's voice in the international mathematics community is still not loud enough."

Zhang Shuwen remained silent because this was indeed the fact.

Since the establishment of the Fields Medal, there has been only one Chinese mathematician who has won the Fields Medal, but the main requirement is that his academic experience was in Hong Kong Island and abroad, and he is of Chinese descent rather than Chinese nationality. Although there is only a difference of one letter, there are indeed many regrets hidden in it.

In any industry in this world, strength is respected. If one does not have significant achievements, one’s voice will naturally not be strong.

Tian Yanzhen sighed and continued, "The general direction of Huaxia mathematics research is still the development of scientific application. If we want to combine research with scientific and technological innovation and industrial implementation, applied mathematics will inevitably receive more support. Academic funds are limited, and the project funds for pure theoretical research will naturally be cut.

I am not complaining. After all, this is never something that can be decided by one or two people, but the limitations of this era. But in terms of theory, we are already backward, with little funding, and we do not pursue international cooperation. What do you think we should do? Not to mention that pure theoretical research in mathematics requires not only talent, but also a broad vision. "

Zhang Shuwen shook his head and said, "Mr. Tian, ​​I study number theory, and you study geometric analysis. Although I am very grateful for the help you provided me with your doctoral thesis, you should know that I don't want to get involved in any ideological disputes."

Tian Yanzhen nodded and said, "I don't mean to ask you to get involved in any disputes. You should also know Qiao Yu's talent. I just want to say that although he is my student, he is also a child trained by our Chinese people.

It would be fine if he remains unknown in the future, but if he does achieve something and encounters some bad things in the international mathematics community, I hope everyone can lend a hand. I don’t have much say in the first place, and if the child’s future is delayed because of these things, it would be meaningless.”

Zhang Shuwen smiled and said, "Mr. Tian, ​​if you don't have the confidence to train a child with such talent, you might as well not be his teacher. And I think he may be more interested in number theory in the future, so you can let me teach him."

Tian Yanzhen also laughed: "Haha, teacher is just a title. You know, his understanding of mathematics is not deep enough, and this kid has a wide range of interests. He came to school on the eighth day of the first lunar month. I gave him my document account to use. The original intention was to let him download Robert's paper to read it, so that he would not understand the lecture at all.

As a result, this guy downloaded five of Schultz's papers that night and started studying them. After reading them, he even gave himself an exercise on the theory of complete spaces. After I scolded him yesterday, he honestly read Robert's paper in the afternoon.

As a result, this incident happened today. So, if you want to teach him, I have no objection. Why don't you give him your contact information, and I will ask him to read your paper more and contact you directly if he doesn't understand anything, how about that? "

Zhang Shuwen frowned again: "What do you mean? He just came across Peter Schultz's research field on the eighth day of the first lunar month, and then he had those ideas and put them into practice? Professor Tian, ​​are you really serious?"

Tian Yanzhen suppressed his smile and answered seriously: "Yes, if you don't believe me, I have his ticket booking records and the thesis download records here, which can be opened to you. Of course, don't say you don't believe it, I was also shocked at the time. If it wasn't for this situation, I wouldn't call you today."

Zhang Shuwen said seriously: "Okay, let me take a look."

"Hmm?" Tian Yan was a little surprised. Good man, he was just being polite, but he really wanted to see it. But then again, this matter is indeed hard to believe. And the more knowledgeable you are, the harder it is to believe.

Tian Yanzhen even believed that if Peter Schultz knew that there was a fifteen-year-old child who could understand his proposition just by reading his paper for one night, he would probably find it unbelievable.

But the fact that they wanted to see the evidence so immediately showed a lack of trust between each other.

This guy can really do it.

But Tian Yanzhen still took out his phone. Fortunately, he didn't use anyone else to book the ticket for Qiao Yu...

"Take a look." Tian Yanzhen handed over the phone calmly: "This is the ticket booking information... Let me show you the background document download records. You don't think I would download five papers by Peter Schultz at seven o'clock in the evening, do you?"

After carefully reading the records, Zhang Shuwen fell silent again.

From the perspective of a normal mathematician, it is really impossible to understand why a fifteen-year-old child who has not received systematic education can understand Peter Schultz's paper in one night.

After all, the mathematical knowledge involved is very diverse, including the most basic linear algebra, abstract algebra, real analysis and complex analysis; in terms of algebraic geometry, you need to study projective geometry, affine geometry, algebraic varieties and schemes, and homological algebra in depth;
There are also more advanced p-adic numbers and p-adic analysis, class field theory in algebraic number theory, Galois representation, the foundations of local and global fields, classical Hodge theory and p-adic Hodge theory, complete congruence space, homology and derived categories.

Oh, and there’s also category theory.

Of course, Zhang Shuwen also studied Peter Schultz's thesis, in which he used the language of category theory extensively. Concepts such as categories, functors, and natural transformations are indispensable for understanding his research.

"Of course, he only downloaded five articles and didn't read so many. I talked to him at noon, and he only read the 2011 article for the time being." Tian Yanzhen explained again.

After returning the phone to Tian Yanzhen, Zhang Shuwen did not say anything, but just walked forward driven by instinct. Soon, the east gate of Yanbei University was in front of him. Zhang Shuwen suddenly stopped, and Tian Yanzhen stopped as well.

"Compared with students like this, you as a mentor are a little lacking!" Zhang Shuwen said very pertinently.

Tian Yanzhen smiled but said nothing.

If the two of them switched places, he would have said the same thing after attending a similar seminar at Huaqing. Zhang Shuwen added: "Of course, I am also a little bit behind. But you are right about one thing, this child deserves to be carefully nurtured. I will send you my private email address later. If he has any questions in the field of number theory in the future, you can email me at any time to discuss. Of course, you can also leave him my private phone number."

"Okay!" Tian Yanzhen responded concisely.

There is no need to emphasize anything else. Mathematics also emphasizes a tradition of inheritance. This is not just the case in China, but all over the world. There are Eastern European schools, Soviet schools, Dutch schools, French schools, Swedish schools, American schools... There are all the schools that should be there. It can even be divided into more detailed categories.

Zhang Shuwen's statement showed that even though they were not teacher and student in name, they had already established a teacher-student relationship. Sometimes the teacher-student relationship is even more solid than the teacher-student relationship in name.

If students are really bullied in the future, how can teachers not step in to protect them?
Who doesn't have a protective mentality? As a teacher, I have the right to criticize my own talented students, but do you have the right to do so?

When they reached an agreement, Zhang Shuwen couldn't help but ask, "Will he have to pass on all the questions he asks me before sending them to me?"

Tian Yanzhen smiled and said confidently: "Not really, he is already my student."

"Goodbye." Zhang Shuwen said concisely.

"No thanks." Tian Yanzhen nodded.

After bidding farewell to Zhang Shuwen, Tian Yan turned around and walked towards the research center with genuine satisfaction. Qiao Yu was still waiting for him.

All the goals were achieved. Not only did he show off the talented student he discovered, but he also made the other party willing to tutor his own student for free, and by the way, he also safeguarded Qiao Yu's future.

……

Zhang Shuwen returned to Huaqing University and wanted to go back to the hotel to calm down, but he received a message from the old man on WeChat, asking if the seminar was over. After thinking for a moment, he simply went to the old man's office first.

There was no way out; he had to accept the old man's favor for many things he had done back then.

"came back."

"Ah."

"Are there really students like that?" Obviously, the old man was also very curious.

No matter how brilliant your life is, it cannot withstand the test of time.

Why is the Fields Medal only awarded to mathematicians under the age of 40? Because everyone knows that people are most energetic and their brains are most active before the age of 40. After the age of 40, even if the body is still healthy, the brain power begins to decline sharply.

So there are no great achievements before the age of forty, but after forty, it becomes very difficult.

This is especially true when you are 60 and gradually enter old age. After the passage of time, only experience remains. At this time, not only the brainpower is not enough to do scientific research, but also the energy is not enough. Perhaps you can only put your hope on the disciples who can inherit your legacy.

An excellent student will naturally make the big guys excited.

Zhang Shuwen certainly knew this mentality, because he was the same, so he chose to come back and resolve the doubts in person.

"Yes, amazing talent! I was thinking all the time what kind of talent could be able to read Schultz's paper in one night without much background knowledge before, and he was only fifteen years old."

The old man said decisively: "There is no such person. You have been deceived. I know my student well. He is very experienced in fraud."

Zhang Shuwen was also very moved. This inheritance from the master...

Anyway, the old man in front of me and Professor Tian across from me both ended up having terrible relationships with their own mentors. The most regrettable thing is that these two were the most accomplished students of their respective mentors. If we look at them as a faction, in this inheritance process, it has almost become a tradition for the most outstanding disciples to turn against their mentors...

But he still shook his head and said affirmatively: "I can tell whether it is a scam or not. I believe that if you go there, you will come to the same conclusion. And even if it is a scam, the other party is just a 15-year-old child. You know, his mathematical derivation process is extremely rigorous.

Moreover, after I pointed out some mistakes and gave a brief explanation, he could quickly understand what I meant and immediately corrected and simplified the original derivation process. If you want to say that this is fake... then I can only say that this child is more talented! "

The old man was stunned, then asked: "What is the problem? Tell me how the child corrected it immediately."

Zhang Shuwen nodded.

This was also the reason why he did not reply directly to the message on WeChat, because he knew that when he truthfully affirmed the talent of that guy named Qiao Yu, this would definitely happen in the end.

So he wrote down Qiao Yu's conjectures and the proof process of the demonstration.

"…He did not use the Riemann-Roch theorem, but introduced a constant to deduce the result. I simply told him the content of the Riemann-Roch theorem, and he made the following changes…"

“Is this an on-site modification?”

"Yes, I will modify it on the spot. My explanation will not exceed three minutes."

The old man closed his eyes, thought for a moment, and said calmly, "There is probably only one possibility. His understanding of Schultz's theory is not at the technical level, but he has comprehended some of the essence of mathematics. He understands it, but he doesn't really understand it, or he doesn't fully understand it."

This sentence is very mysterious, but Zhang Shuwen understood it because the essence of mathematics is so mysterious.

However, Zhang Shuwen was not sure. He just nodded and said, "Maybe. We can't see what's in that kid's mind. But whether he understands it or not, it's enough to prove his mathematical talent. At least I can be sure that this kid's talent is at least not worse than Schultz's."

The old man nodded in approval, sighed deeply, opened his eyes wide with fighting spirit, and said loudly and firmly: "Shu Wen, how can Tian Yan really teach such a student? How can he teach well? He is not worthy of teaching! Don't you think so?"

Zhang Shuwen took a deep breath...

My goodness, how should he respond to this?

But he could also understand the old man. He had taught seventy or eighty doctoral students in his lifetime. It would have been fine if he had fallen out with the most successful student at the time, but until now he had not yet trained a student whose achievements surpassed his.

If it were him, he would probably never be willing to accept it in his life.

Of course, it would be fine if it stayed like this for the rest of my life, but in the world of mathematics, strength is ultimately a matter of strength.

If a student with whom you have the worst relationship wins the Fields Medal in the future and becomes a new generation leader in mathematics, how will history evaluate the merits and demerits of the two when the final verdict is reached?
Ordinary people would not consider these things because they are not qualified, but this old man is really qualified to consider them.

As for whether they are qualified to teach, they are already other people’s students. You can’t just snatch them away, right?

So Zhang Shuwen chose to remain silent. After all, this was a family affair of their lineage. After all, Qiao Yu could be considered the disciple of the old man in front of him, so he naturally would not make any comments.

"Didn't you ask for the child's contact information?" the old man asked again.

Very good, this question can be answered, but are you really trying to steal students?!

However, Zhang Shuwen could only shake his head and replied, "No, it's not convenient. But I left my personal email address to Tian Yanzhen, and I also promised that if Qiao Yu has any questions about number theory in the future, he can contact me at any time."

"Hey, you're confused! Then how long do you have to wait? Oh, where did he find this child?"

Zhang Shuwen thought about it and thought that this was probably not a secret, so he answered, "I found him among the CMOs. When I came back, I checked and found that he was ranked first among the CMOs this year and one of the only two candidates with full marks."

"Your name is Qiao Yu, right? Okay, I'll go ask."

"Then Mr. Zhengxin, I'll go back to the hotel first."

"Go ahead, take a good rest first."

……

On the other side of the street, at Yanbei University, Tian Yanzhen returned to the office.

As soon as he entered the room, he saw Qiao Yu curled up on the sofa, reading a book quietly. When he saw him come in, he jumped up immediately.

Although Tian Yanzhen generally didn't like students who sat in an improper posture, he just smiled and said nothing.

After all, she is only fifteen years old. It is normal to have some minor flaws. As the saying goes, no one is perfect.

Well, to put it bluntly, people love the person and everything about him or her.

This is human nature. If it were someone you hate, the way you sit might send you to hell.

"Director Tian, ​​you're back." Qiao Yu took the initiative to greet him enthusiastically.

"Well, what book are you reading?"

"Uh..." Qiao Yu picked up the book again and looked at the cover before saying, "It's Needham's Visual Differential Geometry and Form."

"This book is a very good introductory reading on differential geometry. You can take it away later." Tian Yanzhen said with a smile.

"Thank you, Director Tian." Qiao Yu thanked him quickly, then looked around and said, "Would you like some water? I'll go get you some."

"No, no, just sit down."

"Oh, okay." After saying that, Qiao Yu sat down obediently opposite Tian Yanzhen.

He wouldn't be so presumptuous when there were people around.

"You weren't discouraged by what Professor Zhang said to you at the seminar today, were you?" Tian Yanzhen asked.

"No, I think Professor Zhang is right. If the result is too complicated, it is really of no use." Qiao Yu said seriously.

"That being said, it is not completely useless. Just like I told you this afternoon, the thoughts you made during the derivation are actually very valuable. Mathematical research has developed to this point, and not all propositions are invented to solve practical problems like calculus.

Just like some mathematical results, we may not see any use in the short term. But perhaps hundreds of years later, it will become a standard fact or an exercise. For mathematicians themselves, the result of the proof is like the exquisite box in the story of buying the casket and returning the pearl, and the proof method is the pearl inside.

For others, it is just the opposite. This is also the most attractive part of mathematics, so you don’t need to be discouraged at all, understand? "Tian Yanzhen patiently taught.

"Don't worry, Director Tian. I'm very stable. I'm just a newbie in mathematics. What am I afraid of? Besides, my goal is that my future achievements, whether for myself or others, will be that bead! Mathematics may not be that simple, but I don't think it's so difficult that it will make me feel desperate!"

Qiao Yu said with confidence.

A newborn calf should not be afraid of a tiger. Compared with the goal hidden in his heart, this is nothing!
If he couldn't even become a great mathematician who could call the shots in the mathematical world, Qiao Yu didn't think he would have a chance to realize his dream in the future.

After all, only by reaching the pinnacle in any industry can one become rich and powerful.

Qiao Yu's good attitude obviously made Tian Yanzhen feel very happy, and a smile appeared on his face involuntarily: "It's great that you have this attitude. In addition, you also felt today that basic knowledge of mathematics is also very important.

Although Professor Xue and I both think that you don’t need to set a specific course for you, you still need to make up for it. This is a list of books I drew up for you last night based on the catalog of books in the Yanbei University Library. You need to arrange your study time every day. You need to read these books as well as the thesis.”

After saying that, Tian Yanzhen picked up two printed book lists from the table and handed them to Qiao Yu.

"Thank you, Director Tian." Qiao Yu took the list and took a look. Wow, it was filled with two pages of densely printed books. They were all foreign works with English titles written directly, not translated versions. He couldn't help but mutter subconsciously, "So many?"

"You don't need to read all of them. This list of books has two stages. The books in the first stage are basic books. No matter whether you are interested in them or not, you must read them all. For the books in the second stage, you can choose to read in depth the areas you are interested in.

In this way, after a year at most, you should know which area of ​​mathematics you are more interested in, and you can't always be like this. Your age is the best time to concentrate on learning, and you can't waste even half a year. "

Having said that, Tian Yanzhen also became serious.

Qiao Yu rolled his eyes and immediately agreed, "Don't worry, Director Tian, ​​I will start reading these books tomorrow. But speaking of saving time, I have something to ask of you. Can you help me think of a solution?"

"What is there to ask for between us as master and disciple? As long as you think it will be beneficial to your studies, just ask for anything you need." Tian Yanzhen answered readily.

"Look, isn't the first phase of IMO special training going to start next month? I want to take more time to think about this proposition during this period. Do you think it is possible to hold the special training directly at Yanbei University so that I don't have to waste time running around?"

Qiao Yu immediately put forward the request he had long wanted.

To be honest, this is not an excessive request.

But Tian Yanzhen thought about it and asked, "Do you want to participate in IMO? In fact, given your current situation, it is not that important whether you participate in IMO or not."

Qiao Yu quickly shook his head and said, "No, Director Tian, ​​you said that you would let me participate in the IMO, and our Xingtie No. 1 Middle School has never produced an IMO gold medalist before, and I promised my alma mater that I must achieve this breakthrough."

Tian Yanzhen smiled and said, "Okay, this is indeed a serious matter. I will go and make a suggestion to the Mathematics Society. But let me make it clear first, I am only going to make a suggestion. As for whether it will work, I dare not guarantee it to you now. If the training location has been determined and cannot be changed, I can't do anything about it."

"It would be great if you could give me a suggestion. If not, there is nothing I can do. By the way, since you always have to give suggestions to the Mathematics Society, why not give me another suggestion? The special training in late March can be held in a school in Xingcheng.

This is not what I wanted to do. It was mainly because I promised Principal Zhang that after the second phase of special training was over, I would go back to my alma mater to give a mobilization speech to the senior high school graduates. My alma mater has been very good to me, and I can’t go back on my promise... Oh.”

I can only say that using the alma mater complex as an excuse is really too reasonable, so reasonable that Tian Yanzhen doesn't know how to refuse.

After all, it is a good thing if your students are grateful.

So Tian Yanzhen agreed without hesitation: "Okay. But I still say that, I can't promise you anything now. After all, it depends on whether there are any schools in Xingcheng that apply to host this event. The Mathematics Society has no affiliation with local schools."

"Thank you, Director Tian."

"Are you used to your life here? Do you need anything else?" Tian Yanzhen asked again.

"There's nothing else. In fact, I'm quite used to it, and the food in the cafeteria is also very appetizing... Oh, by the way, Teacher Xue asked me to tell you that I will definitely need a printer in the future. But I don't think it's necessary to use this for the time being?" Qiao Yu thought of Xue Song's instructions and said casually.

Tian Yanzhen said without hesitation: "This is really necessary. Professor Xue is a good person and very careful. Tell your senior brother Chen about this matter directly and tell him what I said. Ask him to go with you to transfer one and ask him to help you install and debug it."

"Uh? It's not good to trouble you, brother, with this matter, isn't it?"

"It's okay. We are all brothers. What's the problem? Forget it. I'll tell him later."

"Oh, I have nothing else to do then."

"Well, then go eat first. By the way, start reading the books on the list as soon as possible. I will check them every week."

"Don't worry about this, Director Tian. I know you are doing this for my own good, so you don't have to bother checking in your busy schedule. From now on, I promise to report my reading progress and insights of the week on WeChat before 10 a.m. every Monday. Forget about how you can criticize me!"

Look at this, a student who is extremely talented but still so humble and obedient, which mentor would not be confused?

Really, Tian Yan really tried hard to control his expression to avoid laughing too happily.

As a result, Xue Song, who introduced Qiao Yu to him, left a better impression in Tian Yanzhen's mind.

If he had followed his original idea and waited step by step until the IMO results came out before going for the interview, he might have been overtaken by the other party. Who could he turn to for justice?
"It's OK as long as you have this intention."

"Then I'll go eat first, Director Tian."

"Go ahead, eat more, and keep up with your studies on nutrition! If you run out of living expenses, tell me first."

"Okay, Director Tian! Don't worry, I'm never polite to my mentor."

Well, Qiao Yu said this sentence sincerely, and it means exactly what it says.

Tian Yanzhen looked at the proud Qiao Yu as he turned and left, and smiled.

Although Zhang Shuwen's personal email address had been sent to him, he was not in a hurry to give it to Qiao Yu today. After all, the child had not shown any special interest in number theory. He would wait until Qiao Yu found an interest.

　Thanks to Fenglin Wuhen, book friend 20210301106522696680, Zaiqulu_Delu Mengyu, and hikarusin for their rewards and encouragement!
　　PS: The 21st day of checking in for the -word update is completed!
　　
　
(End of this chapter)