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Chapter 105: The young man is proud and confident

The first day of lectures is over.

After presenting the latest results, Robert Green began to explain the research process of him and his team.

For the professors and doctoral students who attended the lecture, the research process is obviously more important than the conclusion, which is the real dry goods. It involves the use of a series of mathematical tools, and these methods of use can often provide ideas for everyone's research.

This is the same idea that is exchanged in mathematics.

Mathematical tools themselves are neutral and widely applicable, and different researchers may use these tools in unique ways in their specific fields to create new ideas and methods.

Although no new mathematical tools were created in this process, in fact, those who can do this are already quite outstanding mathematicians.

As for those who create mathematical tools at every turn, such as Isaac Newton who invented calculus, John von Neumann who created game theory, David Hilbert who created Hilbert space, and Riemann, Gauss...

Still alive are Edward Witten,

The status of these people in the mathematics world is probably no different from the ancestors of the major sects in fantasy novels.

The reason why Peter Schultz is praised by Western mathematics giants and hailed as the greatest mathematician of the younger generation is that his work has opened up a series of new mathematical tools and methods.

In other words, he is still young now, and in a few decades, he will most likely be regarded as a founder of a school in the eyes of future generations.

If he could really solve a series of problems in the Langlands Program during his lifetime, then it might be controversial to say that his status surpassed that of his predecessors such as Newton, Hilbert, Gauss, and Grothendieck, but there is no doubt that he is on par with them.

Qiao Yu didn’t know whether others gained anything after listening to the lecture, but he certainly gained a lot.

The main benefit is a deeper understanding of Chabauty's method. Professor Robert has developed a local correction technique that has advanced the application of Chabauty, specifically using the Padé approximation and local height theory to more accurately control the distribution of rational points at different positions on the curve.

Another approach is to solve homogeneous indeterminate equations in the p-adic range and use the local geometric information in the p-adic analysis to restrict the possible rational points. This is particularly useful for curves with slightly lower genus but still complex geometric structures.

Qiao Yu also used similar techniques when writing his paper, but it was obviously not as smooth as the method developed by this professor and his team.

Anyway, after listening to the lecture, he was quite grateful for the journal that included his paper.

Sure enough, compared with others, the methods he used were quite LOW.

The only bright spot was probably that he made some minor improvements to the classic elliptic curve descent method when solving the problem.

Of course, he didn't know about this highlight at first, and it was Xue Song who told him.

There is no question-and-answer session in today's lecture. It is said that the more specific communication and question-and-answer sessions will be held on the third day of the lecture.

However, after the lecture, Professor Robert on the stage was still surrounded by several professors in the front row. Apparently, everyone was not satisfied with the last day of communication. Qiao Yu also wanted to join in the fun and get acquainted with everyone, especially to get close to Director Tian, ​​but Xue Song stopped him and said, "Let's go. We are busy."

"Ah? What are you busy with?" Qiao Yu was a little confused, then glanced at Director Tian not far away, but unfortunately Director Tian didn't look at him.

"Are you going to take this manuscript to the impromptu discussion meeting this afternoon? I need to re-edit your manuscript at noon, input it into the computer, and print out at least ten copies. Do you still think you have enough time? Are you not going to eat lunch?

Besides, you have to think about how to explain your ideas to the professors who are participating in the discussion. The afternoon discussion will last for 45 minutes, and you should speak for at least half an hour. Now that the discussion is over, what should you do in the afternoon? "Xue Song was busy reminding the teenager beside him.

I'm so envious.

This is Tian Yanzhen.

Even if a different tutor wants to promote his or her students in this way, he or she probably won’t have the courage to do so. Even if he or she has the courage, he or she won’t have the resources.

A 45-minute seminar is not something that can be held easily. First of all, everyone must be given the opportunity to attend.

For example, if he were to convene this seminar, I would have just told the senior professor that the content of this seminar is to discuss some mathematical ideas of my fifteen-year-old student. Although what he proposed has not been proven and is quite immature, I find it very interesting.

If they don't kick him out right away, they will probably ask him to leave politely and then put him on the blacklist and never have anything to do with him again.

But Lao Tian is different. Even if he has some complaints in his heart, he will most likely agree with a smile.

Some people even feel it is an honor to receive an invitation.

Hey……

It can only be said that people with different status and realms will make completely different decisions. There is no such thing as pattern or no pattern. The more resources you have, the less you need to care about other people's opinions, especially when doing things that do not violate any principles.

This is true between people and between countries.

For example, if China suddenly had more than 20 aircraft carrier battle groups, thousands of fifth-generation fighter jets and bombers, and then announced to the world that everyone must watch the gala together during the Spring Festival this year, especially the rich friends living in the west of the village, hurry up and buy the broadcasting rights.

I guarantee that even if the gala is filmed into a mess, the ratings will still shoot up instantly. It is truly unprecedented and will never be repeated.

……

Qiao Yu didn't think too much about it and felt that what Lao Xue said made sense.

So he didn't even think about joining in the fun, and followed Lao Xue back to his room, turned on his computer and started working.

"What do you want to eat for lunch? I'll go get it for you." Xue Song asked when he saw Qiao Yu start to work.

I feel more and more like a nanny, but fortunately, the doctoral students he is supervising will come to school in two days.

"Well, just take any box lunch. I'm not picky about food. Oh, and add more meat," said Qiao Yu.

"Then give you two chicken legs?"

"Okay!"

Xue Song curled his lips and left. After a while, there was a knock on the door. Qiao Yu said without even raising his head, "Come in."

The door was pushed open and Tian Yanzhen walked in. Qiao Yu turned his head and looked at her in his busy schedule, and hurriedly called out, "Director Tian."

"Well, are you preparing?"

"Yes!"

"Let me take a look."

"Sit down."

"Change it here. When you haven't completed the proof, the wording should be more rigorous. Change it to: Based on geometric intuition, it can be inferred that there is a constant C that depends on the geometric and arithmetic properties of the curve X, such that the number of rational points on the curve N(X) ≤ C."

"Oh."

"Also, your description of the homology category QH(Cp) as an enhanced homology category... does not emphasize its difference from the homology category in the general sense. I have carefully thought about your idea.

If you want to better analyze the local homology behavior of curves in p-advanced complete space, you can introduce a quantized homology category. If you introduce quantized features at the homology level, maybe you can capture subtle local changes in the geometric structure? "

"Ah? Quantization? But this has nothing to do with quantum physics, right?"

“I’m talking about the quantization of mathematics. In fields like topology and algebraic geometry, quantization refers to the process of discretizing or lifting classical structures to more complex ones, which are usually noncommutative.”

Seeing that Qiao Yu didn't quite understand, Tian Yanzhen picked up the paper and pen on the table and said, "We don't have much time, I will explain it to you using the example of geometric quantization in symplectic geometry.

First, we need to choose a polarization in the phase space, which can be understood as determining a direction or coordinate in the classical phase space to simplify the complexity of the problem. Choosing polarization can be seen as choosing a decomposition so that part of the coordinates are used to describe the quantum state, and the momentum becomes a differential operator acting on these quantum states.

Then, a Hilbert space is constructed through polarization conditions, which can be regarded as a kind of function space of the classical phase space. This function space contains all possible quantum states, that is, wave functions, and its structure depends on the symplectic structure of the classical phase space and the result of polarization selection. "

As Tian Yanzhen spoke, he had already started writing out a specific example.

"Look, if the phase space of a single oscillator consists of position q and momentum p, forming a plane (q, p). The symplectic form can be written as ω = dq ∧ dp. We now want to quantize this plane into a Hilbert space, first choosing the polarization as /p = 0..."

Qiao Yu listened quietly to the instructor's explanation and asked questions when he didn't understand something. Ten minutes later, he suddenly got it.

"Oh, I see. My Q can represent a quantized invariant. Wait, let me think about it. I need a quantized homology category to decompose the homology group of the curve. Then, I can explain the behavior of rational points on the curve in the local quantum structure through quantization, right? Director Tian?"

"Um……"

"Yes, yes, that's right. Let me use the pen. Um, in a quantized homology category..." As he said that, Qiao Yu took the pen directly from Director Tian's hand and quickly completed the first formula he had thought of last night on the manuscript paper.

Tian Yanzhen looked at the formula written by Qiao Yu and said calmly, "Where is the proof?"

“First of all, Q has been determined to be a quantum operator acting on the curve homology group. Then the first step is to construct a quantum homology category. First, H is decomposed to construct a new quantum state, and then the quantum state dimension is used to describe the curve homology.

The second step is to find the relationship between the quantized homology group and the rational points. It is obvious here that the dimension of the homology group is directly related to the genus g of the curve. The larger the genus, the higher the geometric complexity of the curve, and the relatively small number of rational points.

At this time, adding Q in will give dimQH1(Cp)=f(g,Q). This is to make the changes in the local geometric structure more sensitive and further limit the number of rational points.

Then we restrict the rational points through Jacobian, which is the method used by Professor Robert in today's lecture. We can modify it and put it into the complete space. According to previous research, the higher the order of Jacobian, the fewer rational points can be assigned to the curve.

Finally, we just need to construct this function. The first half of the right side of the function is the dimension of the homology group after quantization, which depends on the genus g of the curve and the quantum operator Q. The second half reflects the geometric structure of the curve and the restrictions on rational points.

You are amazing, Director Tian. Just a few pointers have allowed me to take a big step towards proving the existence of the constant C!"

Qiao Yu expressed his gratitude sincerely.

Tian Yanzhen remained silent as he watched Qiao Yu quickly write the proof process on the manuscript paper.

He could feel his heartbeat speeding up.

"Bang, bang, bang..." Like the beating of war drums.

What kind of speed of comprehension is this? He thought that it would take at least half an hour to simply explain quantization to Qiao Yu, because it involved many complex mathematical concepts, many of which he was not sure whether Qiao Yu had ever come into contact with.

After all, Qiao Yu had not received any systematic mathematics education, but as he was talking, he suddenly clarified a rough idea he had yesterday to such an extent? And judging from the process, it seemed that there was nothing wrong with it, and it was quite rigorous.

It wasn't a problem, but for a fifteen year old, he really couldn't ask for more!
"Have you ever met Xin Ji before?" Suppressing the excitement in his heart, Tian Yanzhen asked in the most stable tone possible.

"No." Qiao Yu shook his head.

"Have you studied quantum physics specifically?" Tian Yanzhen asked again.

"No, I just know a little bit, such as wave functions and that there is no certainty in the microscopic world, only probabilities. I haven't done any special research, but I've read some popular science and learned about wave-particle duality and so on." Qiao Yu shook his head again.

"Then you understand?"

"I see. The principle is to allow the curve to contain quantum variables or quantum structures for micro-manipulation, to expand its operability. You have explained it so clearly. If you still don't understand, then you are stupid."

After saying that, Qiao Yu suddenly felt something was wrong. He reacted and asked cautiously, "Ah... Is it possible that the process of my pushing was wrong?"

Tian Yanzhen took a deep breath and shook his head. He suddenly felt that some of his students who seemed quite smart before now seemed a little stupid.

Then he slowly answered: "Math exams are divided into right and wrong, but there is no right or wrong in cutting-edge mathematical research. Judging whether the derivation process is correct depends only on whether you can justify yourself within the given theoretical framework without being picked out for any logical loopholes.

I haven't seen any logical loopholes in your derivation yet, but I can't represent the mathematical community. At least I only know a little bit about Peter Schultz's research, including the homology theory of P-adic geometry, but I'm not proficient in it. You can record all of this in the manuscript for the afternoon seminar and analyze it with everyone. "

Qiao Yu nodded quickly and said, "I understand, Director Tian."

As the two were talking, the door was suddenly pushed open and Professor Xue appeared at the door carrying two boxes of lunch.

"Qiao... um, Mr. Tian, ​​you are here too?"

"Well, I asked several school leaders to accompany Professor Robert to lunch at noon. I was thinking about explaining something to Qiao Yu since this is his first seminar."

"Oh, you haven't eaten yet? How about you eat first?"

Tian Yanzhen hesitated for a moment, then nodded and said, "Okay, then I'll eat here. You don't have to run back and forth. I'll send a message to Xiao Li and ask him to bring you one."

He hadn't originally planned to eat here.

But there is no way. Teaching students like Qiao Yu is really addictive because it is easy to gain a sense of accomplishment.

In particular, this child's basic knowledge of mathematics is like a mixed abstract painting, but he can find his own context in this abstraction. This talent is probably no worse than Peter Schultz.

Tian Yan really believed that no matter who had such a student, they probably wouldn’t mind communicating with him for a while.

"No need to bother. I can just ride my bike to the cafeteria to eat and come back later."

"That's fine. Come back quickly after you finish eating and help Qiao Yu organize his things and record them into the computer. I just want to tell Qiao Yu something in advance." Tian Yanzhen nodded and said.

Xue Song nodded with mixed feelings and replied, "Okay. I'll do it as soon as possible." He turned around and left immediately. Tian Yanzhen opened the lunch box and said, "Hurry up and finish your meal. You still have a few questions. I'll briefly explain them to you."

"Okay, Director Tian."

Many times, when you really like someone from the bottom of your heart, the way you express yourself is actually the same.

……

"…if X is an algebraic curve of genus g, its moduli space Mg parameterizes all curves of genus g and imposes geometric constraints…"

"But have you ever thought that this would add another index that needs to be related to the properties of the modular form homology group, greatly increasing the complexity of the result?"

"Then Director Tian, ​​how do you think we should deal with this?"

"I think it would be better to directly introduce Schultz's p-adic Hodge theory and obtain further geometric constraints by analyzing the behavior of the curve in the p-adic number field Qp."

“You mean to make local properties global? But local information is usually associated with a specific prime number p, and different prime numbers may lead to different local behaviors, which is more difficult, right?”

"But you proposed this, so you have to think about it. But as far as I know, in p-adic geometry, the Etale homology group can be used to describe the local properties of algebraic curves. And as you just said, the completeness conditions have ensured that we can derive global results from local geometric structures, so this path is definitely feasible."

"Oh, I'll have to think about this more carefully...but I still need to introduce a constant, right?"

"You really need to think about it carefully. Qiao Yu, you must remember that the greatest gain for a mathematician's academic level in proving any mathematical conjecture is the proof method, not just the result..."

Xue Song, who had just had a quick meal, took over Qiao Yu's job of data entry, while the discussions and instructions between the master and apprentice kept drilling into his ears.

The mood became more complicated.

I envy Qiao Yu for being able to receive such careful guidance from a top academician. This is the treatment of a true disciple, and a particularly favored one.

Not to mention academician tutors, even ordinary tutors are unlikely to be so attentive and dedicated when guiding students.

Anyone who has worked as a graduate student for a big boss with a specific administrative position knows that big bosses basically won’t PUA students, but they also basically won’t care about your thesis, your studies, or whether you can graduate. Even when you graduate, they may not remember that they have a student like you...

Usually there is only one junior supervisor in charge of leading the group. Of course, the level of responsibility of the junior supervisor varies from person to person, so the level of masters trained by the big boss also varies greatly.

Qiao Yu is enjoying the special treatment of having an academician mentor who will take time out for tutoring even if he only has a box lunch at noon.

Of course, Xue Song also understood that this was all thanks to Qiao Yu's ability to accept this special treatment.

Discuss, discuss, there must be give and take in order to have a discussion.

The academician tutor gave a few words of advice, but the student ended up acting like a fool, not knowing what was going on and not knowing how to respond at all. There will definitely not be a next time.

Obviously, Qiao Yu accepted the offer, and then he really became a handyman.

Well, at least I’m still a director!

……

Tian Yan really left at 1:30 in the afternoon. In fact, he still had some basic theories and mathematical principles that he wanted to tell Qiao Yu all at once, but there was no way. He didn't have time to talk more as he had other things to do.

In the afternoon, there was the first seminar for his students, and he was somewhat worried about leaving the organizational work to others.

After Tian Yanzhen left, Qiao Yu obediently took over Xue Song's work, mainly because after the discussion just now, he felt that some places needed to be modified, and there was still time anyway.

"I think you need a printer, preferably a laser printer. Remember to mention it to Mr. Tian later." Xue Song suddenly said.

"Ah? Is it necessary?" Qiao Yu was stunned.

"It doesn't seem necessary now, but it will be needed frequently in the future." Xue Song said with certainty.

"Oh, by the way, I forgot to tell Director Tian about the first phase of IMO special training." Qiao Yu suddenly said in annoyance. "What happened to the special training?"

"Hold it at our Yanbei University, so I don't have to waste time running around."

"Heh... hurry up, the seminar will start in 80 minutes, and we need to finish it at least half an hour in advance."

Xue Song was too lazy to comment on this, because he felt that this reason might really convince Tian Yanzhen to speak up for this matter.

Although this situation can be said to be caused by him alone.

……

On the other side, Tian Yanzhen walked out of Qiao Yu's room, took out his mobile phone and made a call.

After the phone rang a few times, it was connected.

"Mr. Zhang, I heard that you returned to China a few days ago?"

"Yeah, I came back to do some things." He said in a light tone.

"Are you in Huaqing now?"

"Is there anything I can help you with? Professor Tian." The person on the other end did not answer directly.

"It's like this, I accepted a student, who should be my last disciple."

"congratulations."

"He has a habit that's a lot like you."

"Oh?"

“Make your own questions and solve them yourself.”

"Well, this is a good habit. I always emphasize that students should have this ability."

"By the way, he is quite similar to you in one respect. He mainly relied on self-study before, and so far, he has not systematically studied undergraduate mathematics courses."

“Your students haven’t received systematic undergraduate education yet?”

"Yes, but he was able to understand Peter Schultz's paper independently. He also had a wild idea to use the complete space system built by Peter Schultz to solve the problem of accurately estimating the upper bound of rational number points on a curve. I exchanged ideas with him and found it feasible. I remember that you also studied this problem when you were young."

The other party was silent for a long time, and then he spoke: "Professor Tian, ​​are you calling me just to tell me a story?"

"There will be a seminar about his ideas in the P1 conference room of Building 2 of the research center at 09 pm. Professor Robert from New York University will also attend. You will know if I am telling a story or not when you come and listen. By the way, he was born in , is fifteen years old this year, and is from Huaxia Star City."

After waiting for a few seconds, there was no reply from the other party, so Tian Yanzhen said calmly: "That's it for now. I personally hope you can come and see for yourself if I am talking big. Goodbye."

Then hung up the phone directly.

In the world of mathematics, there is a very interesting ranking method.

Fields Medal mathematician, one-hour presentation mathematician, forty-five-minute presentation mathematician, thirty-minute presentation mathematician, and others...

Both of them belong to the one-hour category, and the more powerful ones have mostly won the Fields Medal. So in fact, everyone has a similar status, and they are all the legendary touchstones of the Fields Medal, so there is no need to be too polite.

It’s enough as long as the notice has been received. Tian Yan really feels that if the other party does not come, it will be the other party’s loss, not his.

……

Opposite Yanbei, in an office at Huaqing University.

"Tian Yan's real phone number?"

"Ah."

"What does he want from you?"

“He accepted a student who was fifteen years old and had not systematically studied undergraduate mathematics courses. He was not only able to independently understand Peter Schultz’s paper, but also independently proposed a solution to the problem of the upper bound of the number of rational points on algebraic curves based on Schultz’s theoretical framework.”

After the words fell, the old man opposite was also stunned and asked in surprise: "Fifteen years old?"

Zhang Shuwen nodded.

"Do you believe?"

Zhang Shuwen shook his head and said, "But he said if you don't believe me, you can go and see for yourself. At 3 pm, that kid will host a seminar on this issue."

Silence...

"Go and have a look. If it's fake, expose him directly. Don't save face."

Zhang Shuwen: “…”

……

At 1:2 in the afternoon, Xue Song had already brought Qiao Yu to the P conference room in Building .

Although generally speaking, the speaker should arrive last, Xue Song felt that Qiao Yu was already too arrogant. If he was too indulgent, he would lose his self-recognition.

Especially considering that Qiao Yu is only fifteen years old, he should be allowed to develop the good habit of humility.

The two had been in the conference room for less than five minutes when Chen Zhuoyang walked in.

"Senior Brother Chen." This time Qiao Yu took the initiative to greet him.

Anyway, there was no one in the small conference room, unlike in the morning when he arrived, the conference room was already full of unfamiliar people. Qiao Yu felt embarrassed, but the current situation still allowed him to show his politeness.

Chen Zhuoyang curled his lips. He didn't want to pay any attention to this guy Qiao Yu.

Especially when I learned that the protagonist of today's discussion was this junior fellow apprentice, I was a little confused.

He doesn't even get this treatment.

The most infuriating thing is that he has to run around when this little brat with a loose mouth holds a seminar.

Chen Zhuoyang asked, "Well, are you going to use PPT later?"

"Thank you for your concern, Senior Brother Chen. But it's not necessary. I don't have time to do it, and I don't know how to do it. I'll just write it on the blackboard later. Eh? There's no blackboard. Senior Brother Chen, how about you get me a blackboard? Just put it there..."

Qiao Yu enthusiastically pointed to the open space in front.

Chen Zhuoyang rolled his eyes and said it would be better to make a PPT.

"Okay, I know where to put it. I'll ask the staff to bring one over later. Is a 90X120 one okay?"

"Okay, thank you, Senior Brother Chen. Oh, by the way, don't forget to ask for a pen."

Chen Zhuoyang curled his lips and turned away.

"Hey, Senior Brother Chen is good in everything except that he is too serious. You never see him smile all day long." Qiao Yu commented.

Xue Song stood by and said nothing. He could understand Chen Zhuoyang.

If he were Qiao Yu's senior brother, he would probably be annoyed to have such a junior brother.

Soon, several more professors walked into the conference room. Chen Zhuoyang had also returned to the conference room and took on the responsibility of a senior brother and began to introduce Qiao Yu: "Qiao Yu, this is Professor Zhou Yang from the research center. He mainly studies low-dimensional topology and geometric group theory..."

"Hello, Professor Zhou."

"You are Qiao Yu. Mr. Tian thinks highly of you. Keep up the hard work."

“Thank you Professor Zhou for your encouragement.”

"This is Professor Cai Wenyuan, who mainly studies algebraic geometry and Lie groups..."

"Hello, Professor Cai..."

"Professor Chu Weihua, your great achievements in algebraic topology..."

With Chen Zhuoyang's introduction, the atmosphere in the conference room gradually became familiar. To be honest, the professors were still quite unfamiliar with Qiao Yu, and they only came to attend this seminar for Tian Yanzhen's sake.

As for Qiao Yu, his idea is simpler. It doesn’t matter if he is unfamiliar with others, because everyone will understand him soon anyway.

And with Senior Brother Chen as a helper by his side, Qiao Yu soon got to know all the professors who came to attend the seminar.

While everyone was chatting, the mobile blackboard was pushed into the meeting room by the staff.

Soon, with only three minutes left before the seminar started, Tian Yanzhen, Robert, and another person Qiao Yu had never seen before walked into the conference room together.

Qiao Yu, who had already returned to his seat, clearly felt that the atmosphere in the meeting room suddenly became stagnant...

It’s not that Qiao Yu is too sensitive, it’s just that everyone showed it very obviously.

We were just discussing in low voices, but when three people walked in, everyone suddenly stopped talking.

It is not possible that everyone will suddenly become so disciplined just because foreign friends are coming, right?

Qiao Yu turned around and was about to express his thoughts to Xue Song when he suddenly discovered that Old Xue was staring at the three people who had just entered with a strange expression on his face.

So he asked in a low voice: "Teacher Xue, what's wrong with you?"

Xue Song turned around and said in a blank voice, "Speak carefully later, don't waste the stage that Director Tian has set up for you!"

Qiao Yu understood the implication of this sentence and immediately turned his gaze to the person he didn't know yet.

Simple reasoning for him.

Director Tian had already made it clear that he would invite Professor Robert to attend the seminar, so this was something that was expected. Lao Xue once again emphasized not to waste the stage set up by Director Tian, ​​so this stranger could only be very important.

But he just looks like a skinny and serious old man. Apart from his aura, there seems to be nothing special about him...

Fortunately, Qiao Yu didn't have time to think too much, as his mentor once again took on the role of host.

“Thank you all for coming to today’s seminar, especially Professor Robert and Professor Zhang Shuwen for taking the time out of their busy schedules to personally guide my student Qiao Yu on his fantastic mathematical ideas.

The main issue discussed in today's seminar is about using tools such as complete space, modular form theory and P-adic geometry to study the upper bound of the number of rational points on the algebraic curve X. Okay, Qiao Yu, you can start your speech. "

After saying that, Director Tian returned to his seat. Qiao Yu was not at all nervous and stood up immediately.

“Thank you all for coming to this seminar. I have printed out some of my immature ideas. They are the stack of manuscript paper on your desk.

By the way, I would like to thank Professor Robert for his inspiration today, and my mentor Professor Tian Yanzhen for his guidance. As Professor Tian just said, after reading the papers of Professor Schultz and Professor Robert recently, I suddenly had such a bold idea. "

As soon as Qiao Yu finished speaking, almost everyone picked up the report on the table. It was too simple. Everyone had just arrived at the meeting room a few minutes early and were busy greeting each other. No one really picked up the report and took a serious look at it.

However, Zhang Shuwen and Professor Robert, who were sitting next to Tian Yanzhen, had already picked up the simple booklet and started flipping through it.

After Qiao Yu finished his opening remarks, he got straight to the point.

"My idea is to use the complete space theory built by Professor Peter Schultz, modular form theory, progressive geometry and quantized homology to derive the upper bound expression of rational points on algebraic curves.

To do this, we first need to consider the geometric background of the curve X, especially its genus g(X). Genus is an important topological invariant that represents the geometric complexity of the curve. For a curve with genus g>1, Faltings theorem tells us that the number of rational points is finite.

But this is not enough, because we all want to get a specific upper bound. According to geometric analysis, the higher the genus, the greater the complexity of the algebraic curve, which means that the number of rational points is relatively reduced. So my initial guess is: N(X)≤C(g).

Then I will use several assumptions to prove this result. Although I think this result is correct, I cannot prove the specific formula of constant C for the time being. But I have thought of several interesting ways to derive the result of constant C.

It's just that these methods have not been proven yet, so I hope that the teachers can give me some inspiration. First, we introduce the moduli space, let X be an algebraic curve with genus g, and its moduli space Mg parameterizes all curves with genus g.

Because modular forms are closely related to modular spaces, I understand them as certain functions defined on modular spaces that provide geometric constraints on the complexity of curves. Let the rank of modular forms be k, and let us assume that there is a constant A1 such that: N(X)≤C1(g, k)=A1gk^α..."

Below the stage, all the professors in the conference room had put away their previous relaxed attitudes and their expressions began to become serious.

The only two people whose expressions didn't change were probably Tian Yanzhen and Xue Song.

Chen Zhuoyang, who was sitting at the back, could testify to this.

He had little interest in what Qiao Yu said, so he paid more attention to the expressions of the instructor opposite and the two big shots.

It was obvious that Director Tian was in a relaxed state of mind, just quietly watching Qiao Yu writing on the blackboard. The two big guys beside him, one had a frown on his face, and the other had already picked up a pen and started writing and drawing next to the manuscript...

Chen Zhuoyang felt a little bit depressed...

No way, are you all serious? So it's not that Director Tian is trying to push the junior brother, but how can everyone agree with something that has not been proven?

Yes, when Chen Zhuoyang learned about this afternoon's seminar, he really felt that Director Tian just wanted his junior brother to get familiar with everyone. After all, Director Tian also said that Qiao Yu's ideas were just ideas...

How can you play with your ideas like this? Chen Zhuoyang even thought that Director Tian was too anxious. After all, this junior brother was only fifteen years old! Although he was able to participate in the CMO and won the first place, it proved that he must have mastered the high school knowledge, but he didn't even know about college knowledge. He didn't know anything about scientific research!

He even thought that Qiao Yu was just dreaming when he could understand Peter Fields's paper. But now looking at the professors' expressions, it was not the case at all, because it could be seen that everyone had really started to think...

What the hell, is Junior Brother really going to defy the heavens?
What made him even more desperate was that Qiao Yu on the stage was not at all nervous, and he became more and more excited as he spoke, because many professors had already started to look at his blackboard seriously. Wait, Professor Robert even took out his cell phone to take a picture of what he wrote on the blackboard...

"…At this point we can introduce the -adic number field and Professor Schultz's homology theory. We know that for every prime number P, the properties of the etale homology group can constrain the local distribution of rational points on the curve.

Then according to Schultz's -Hodge theory, we can derive the following inequality: N(X)≤C2(g, p)=A2g2log(p). There is an important point here. A core feature of Schultz's -Hodge theory is its completeness.

So if the inequality we derived holds true, we can derive the global geometric constraints from the properties of the curve in the local domain, so we need to prove whether this inequality holds. To this end, under the guidance of Professor Tian, ​​I came up with a solution, which is to introduce a quantized homology category..."

During this half hour, Chen Zhuoyang felt like sitting on pins and needles.

Because there were only two students of Director Tian in the entire conference room. One was talking at the front, and the other could not understand what his junior was saying...

But the conference room was eerily quiet, without even a sound of discussion, and everyone was staring intently at Qiao Yu's writing on the blackboard.

Including the three mathematics giants who most professors in the conference room could only look up to.

Finally, Qiao Yu finished speaking...

“The above is my complete idea. The problem is that I cannot handle the constants in the setting yet, nor can I make a complete and logical proof of the specific tools. But I think this should be a new research direction, because once we derive the result of the constant C, it means that we can directly predict the upper bound of the number of rational points of the relevant curve.”

When Qiao Yu's voice finally dissipated in the air, Chen Zhuoyang finally breathed a sigh of relief and felt better.

But the quiet meeting room made him nervous again.

No, professors, you're not going to say anything?
We are all adults. Please stop looking at the little junior brother with that incredible expression. He is only fifteen years old. He should receive frustration education now! Everyone should criticize his ideas harshly at this time!
Chen Zhuoyang thought viciously in his heart, but when he saw Director Tian on the opposite side raise his hand first to start clapping, he could only immediately raise his hand and clap...

"Clap clap clap..."

The scattered applause seemed to make the professors react, and the conference room was immediately filled with applause.

Fortunately, there were not many people, and the applause stopped after a few tens of seconds. Then Chen Zhuoyang finally heard the heavenly sound.

"I have a question, Qiao Yu, why don't you just use the Riemann-Roch theorem in your third part?" Chen Zhuoyang glanced at Zhang Shuwen who had a serious face opposite him. It turned out that the professor was really awesome!
"Ah? What is the Riemann-Roch theorem?" Qiao Yu asked back with curiosity.

Everyone had different reactions.

For example, Qiao Yu, who was standing there, seemed to be nonchalant, but his nominal supervisor, Professor Xue, felt very socially awkward and his face turned red.

As for the other professors, including Robert Greene, they were all very confused. They probably couldn't understand why a kid who had just talked about algebraic curves for half an hour didn't know this important theorem in algebraic geometry and complex geometry.

Tian Yanzhen remained calm and explained in a gentle tone: "Professor Zhang, as I said before, Qiao Yu is only fifteen years old. He is a talent I discovered among the CMOs. He has not yet received a complete undergraduate education, so his knowledge of mathematics is relatively scattered. You can give him some guidance on the spot."

　I recommend this book to everyone. Just click the link below. It is a masterpiece of reversing cause and effect!
　　PS: The 20th day of updating 10,000 words has been completed!
　　Thanks to book friend 20240903120210936 for the reward and encouragement!
　　
　
(End of this chapter)