Zhang Shuwen hesitated for a moment, then stood up, walked to Qiao Yu's side, erased the last writing on the blackboard, and then started to explain on the spot.

"The Riemann-Roch theorem is a basic theorem in algebraic geometry, which is used to describe the dimensions of certain functions or forms on the algebraic curve. Specifically, the Riemann-Roch theorem applies to any divisor D on the algebraic curve X. The theorem states

The dimension of the function space L(D) associated with the divisor D on the algebraic curve.

Its specific statement is (D)=deg(D) 1-g (K-D). It has two parts that complement each other, describing the balance relationship between the divisor D and the remaining part K-D. ​​But there are special cases

, when the degree of D is large enough, (K—D) is zero, so in this case (D)=deg(D) 1—g, do you understand what this means?"

"The degree of D is large enough, and the relationship between dimension and degree is linear." Qiao Yu immediately replied. "Then when D is zero..."

"(0)=1—g (...., Professor Zhang, I understand what you mean.... So this part of the proof does not need to be so complicated, because the genus g(X) can be passed directly through Riemann—

Roch’s theorem shows that, hey, then this part of the proof will not be so troublesome... Let me think about it..."

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

"That is to say, when constructing the function..., dimQH1 (Cp is the dimension of the homology group after quantization, well, it depends on the genus g of the curve and the quantum calculation... This part can be obtained by calculating the canonical factor, H1C

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

Zhang Shuwen took a deep breath, keeping his expression calm, and then nodded.

"Great, then the next step is to prove... 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 is

will be limited, and Jacobian can further affect the number of rational points.

Genus is one of the core geometric invariants and cannot be simplified, so #C(K)sf(g,Jac(Cp)? Phew, no, looking at it this way, I feel that this method can really give the formula of constant C

Can you deduce it?" 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, weren't you really angry when you heard this? How dare you accept a 45-minute seminar when you don't have 100% confidence in proving something? I just saw the conference room and didn't

Chen Zhuoyang naturally couldn't say anything if he seemed to care.

On the stage, Professor Zhang snorted coldly and said: "It's still early, I believe you can prove it and even get a formula you want! But are those really useful?! At least you can

It only makes sense to simplify it to #C(K)sf(g)! It is okay to introduce Peter Schulz's theory. The mathematical proof process can be as complex and abstract as long as it is logic within the framework, but you have to

All complexity is limited to the intermediate steps of the proof!

The final result must be as simplified as possible! Otherwise, even if you prove the constant C and derive the result, and bring in so many set constants, think about how complicated the final formula will be? Others How to use it?

What real mathematics pursues is to complicate thinking and simplify results. Only concise 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 It’s not as simple as you think!”

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

Robert Green finally couldn't help but came over and asked, "Professor Tian, ​​what are Professor Zhang talking to that child?"

Just now Qiao Yu was using English when introducing his ideas, but when Zhang Shuwen came up to give Qiao Yu some pointers, he had already started using Chinese.

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

"Oh! God, are Zhang's requirements so strict? Doesn't he know that this child is only fifteen years old? At fifteen years old, he can really understand Schulz's theory and come up with such creative ideas. Zhang still doesn’t think it’s enough? Is he crazy? I even think this is indeed a research direction worth looking forward to 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 even too harsh.

"Yes, this is why I must hold this seminar. I also think this is a possibility worth looking forward to. However, it is still a little difficult for this child to complete this topic independently. So I am actually very grateful to Zhang Professor, at least he told Qiao Yu that subtraction in mathematics is sometimes much more difficult than proving the process itself."

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

"Although what Zhang said is very correct, there is no need to use such a harsh tone. This is unfair to a fifteen-year-old child." Robert Green still couldn't understand. After all, if Qiao Yu was his student, There's no way he could be so unkind.

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

"We in China have a saying that deep love leads to deep responsibility. If you are not particularly optimistic, given his character, you will not even bother to say nonsense to this child."

As he spoke, Tian Yanzhen couldn't help but laugh out loud, although it was very suppressed: "Haha, Professor Zhang is probably regretting in his heart that he has never met such a student, right? In fact, I also want to thank Professor Zhang. Originally, these words made me I also want to tell this kid, but I’m afraid it will dampen his enthusiasm.”

Robert Green shook his head. The thinking mode of the Chinese people is too weird. He is incapable of understanding. He does not point out the problem, but thanks others for pointing it out more severely? So if others point out the problem, it will not dampen the enthusiasm of a child. ?In his opinion, students like Qiao Yu should be encouraged more. After all, he is only fifteen years old. Only in this way can he stimulate his love and confidence in mathematics instead of criticizing.

What surprised Robert Green even more was that the child on the stage didn't show any frustration. He was still talking to Zhang Shuwen with a normal expression. Damn it, he couldn't understand Chinese.

..

Yes, Qiao Yu did not feel any frustration on the stage. Facing the serious-looking Zhang Shuwen, he just said seriously: "Thank you, Professor Zhang. I understand. I thought of the problem too simply! But I

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

Zhang Shuwen's face danced slightly, and he nodded and said: "Well, you are very talented in mathematics, but how to use this talent depends on whether you work hard and what kind of mathematical path you choose to take care of yourself." After saying that, he turned back.

Arrive at your position.

At this time, Tian Yanzhen had already put away his uncontrollable smile, and said calmly as usual: "If you have any other ideas, please tell me. Qiao Yu's idea is definitely not mature enough. Everyone can help."

Help me, a student who is not very successful yet.

Then he turned his head and explained his words in a low voice to Robert Green beside him. The atmosphere in the conference room finally began to become lively.

Just as Tian Yanzhen said, Qiao Yu's idea is very creative, but not mature. Most of the professors Tian Yanzhen invited today have been immersed in the field of algebraic geometry for decades. There are still things that everyone can say.

.Of course, including Robert Greene who came all the way from the other side of 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 information.

, I can introduce Peter Schulz to you if I have the opportunity in the future. I believe he also hopes to meet you very much.

Do you know why? Because I think your idea is very creative and very valuable! Oh, by the way, be sure to come to my lecture tomorrow. Believe me, there will be a lot of exciting content next, and you will definitely know it.

There are a lot of gains!" Robert Green expressed his attitude with practical actions, and encouraged Qiao Yu in front of Zhang Shuwen.

"Thank you very much, Professor Robert. In fact, your lecture today gave me a lot of help. I really enjoyed reading your paper. These ideas came to me after reading your paper.

!" Qiao Yu replied, which also made Professor Robert, who came from afar, smile.

When it comes to using processed truth to make people happy, Qiao Yu is even more expert than his mathematical talent, of course, if he is willing...

"I'm going to see off the two professors. You go upstairs and wait for me in Office 301. The door is unlocked. Don't touch the things on the table, but you can read the books in the bookcase as you like." After the seminar, Tian Yan

After Zhen confessed to Qiao Yu, he strode out of the conference room and caught up with Robert Green and Zhang Shuwen.

It was hard to tell from such a spirit that Director Tian was already nearly seventy years old.

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

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

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

Qiao Yu glanced at Senior Brother Chen who was doing the finishing work... Forget it, he would go to the teacher's office by himself.

Senior Brother Chen is good at everything, but he is too serious.

Chen Zhuoyang finally breathed a sigh of relief when he felt Qiao Yu leaving the office. He had to think carefully about how to get along with his junior brother in the future. I must admit that when he met his junior brother for the first time, he seemed to speak a little louder. .On the other side, outside the research center, Tian Yanzhen was seeing Zhang Shuwen off.

Robert Green lived in a hotel just over 400 meters away from the research center. He declined Tian Yanzhen's offer to send him to the hotel and left on his own. The main reason was that everyone was too smart and the foreigner was very knowledgeable. , it can be seen that the two Chinese colleagues have something to talk about.

"Would you like to arrange a car to take you there?" Tian Yanzhen asked casually.

There was nothing to talk about. Although the distance between the two universities was very close, the campus was very big. It would take two or three kilometers to walk back. "No, it's good to walk back. It's not that delicate." Zhang Shuwen replied in a wan tone. sentence.

"Well, I'll give it to you?" "No need."

"Let's see him off. I won't go out until we get to the school gate." Tian Yanzhen insisted. "Then let's go through the east gate." Zhang Shuwen said.

"Aren't you going to hang out by the lake? It will be lively after school starts next week." "No, there will be a chance in the future."

"Well, good!"

After taking two steps in silence, Tian Yanzhen suddenly said sadly, "Mr. Hamilton passed away last year."

Zhang Shuwen was silent for a moment and then said: "Well, respectable old gentleman. You must be complaining in your heart why Simon Donaldson is not the one who died."

Tian Yanzhen smiled and said: "I'm not that mean. At our age, many things have been put aside. But then again, Donaldson is 69 this year!" Zhang Shuwen nodded and said: " Yes, I remember he is one year older than you."

A few simple words from the two of them are the history of the mathematical world.

Both Richard Hamilton and Simon Donaldson are internationally renowned mathematicians, especially the latter, who is a former Fields Medal winner, an academician of the Royal British Academy, a double academician of the National Academy of Sciences of the United States, and has published numerous works. However, There were indeed some minor conflicts between everyone back then.

From ordinary people's perspective, it is probably a very important research result of Tian Yanzhen, but Donaldson's team thinks it is theirs. But the actual situation is actually more complicated, involving the issue of the recognition of papers submitted to pre-release websites in advance. For For new academics, well-known journals that still adhere to double-blind review may reject papers that have been published on the pre-publication website without hesitation, but it is different for these big names.

The proof of an important proposition is often pre-released 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 the proof process, so he published a new paper that solved the same problem, and insisted that Tian Yanzhen's paper was published after pre-publication.

They have made extensive revisions and quoted their process.

It happened many years ago, and apart from the two parties involved, very few outsiders can get a full picture of what exactly happened back then. It’s 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 many young people over the years. I have paid attention to them. Many of them are very powerful and have achieved good results. But China is

It is also true that the voice of the international mathematics community is still not loud enough.

Zhang Shuwen remained silent because this was indeed true.

Since the establishment of the Fields Medal, there has been only one mathematician from China who has won the Fields Medal, but his study experience was mainly in Hong Kong Island and abroad, and he is of Chinese descent rather than Chinese nationality. Although there is only a one-word difference, among them

There are indeed many regrets hidden there. In any industry in this world, strength is valued. If the results are not significant, the right to speak will naturally not be strong.

Tian Yanzhen sighed and continued: "The general direction of China's mathematics research is still towards scientific application. Research must be combined with technological innovation and industrial implementation. Applied mathematics will inevitably receive more support. Academic funds are limited.
To be continued...