The first day of lectures is over.
After giving the latest results, Robert Green began to explain the research process of him and his team.
For professors and doctoral students who come to listen to the lecture, the research process is obviously more important than the conclusion. This is the real stuff. It involves the use of a series of mathematical tools, and these usage methods can often provide ideas for everyone's research. It is this kind of thinking that is communicated in mathematics.
Mathematical tools themselves are neutral and widely used. Different researchers may use these tools in unique ways in their specific fields to create new ideas and methods. Although no new mathematical tools are born in this process , but in fact, someone who can do this is already a very good mathematician.
As for those who constantly create mathematical tools, such as Isaac Newton who created calculus, John von Neumann who created game theory, David Hilbert who created Hilbert space, and Leigh, Gauss... Still alive such as Edward Witten,
The status of these people in the world of mathematics is probably no different from the ancestors of major sects in fantasy novels.
The reason why Peter Schulz is praised by Western mathematics tycoons and hailed as the greatest mathematician of the younger generation is precisely because his work has opened up a series of new mathematical tools and methods. That is to say, he is still He is young, and in a few decades, he will most likely be the grandmaster of the founding sect in the eyes of future generations.
If he can really solve a series of problems of the Langlands Program in his lifetime, then it may be controversial to say that his status surpasses predecessors such as Newton, Hilbert, Gauss, and Grothendieck, but there is no doubt that he is comparable. After listening to the lecture, Qiao Yu didn't know whether others had gained something, but he had indeed gained something.
The main gain is a deeper understanding of the Chabauty method. Professor Robert developed a local correction technique that promoted the application of Chabauty. Specifically, he used Padé approximation and local height theory to more accurately control different positions on the curve. The distribution of rational points.
The other is to solve the homogeneous indefinite equation in the p-adic range and use the local geometric information in the p-adic analysis to limit the possible rational points. It is especially suitable for slightly lower genus but still having complex geometric structures. Curve. Similar techniques were used by Qiao Yu when he wrote his paper. But it was obviously not as smooth as the method developed by the professor and his team.
In short, after listening to the lecture, he was quite grateful to the journal that included his paper. As expected, compared with others, the methods he used were quite low.
The only bright spot is probably that he made some small improvements to the classic elliptic curve descent method when solving the problem. Of course, he didn't know about this bright spot originally, but Xue Song told him about it.
There is no question and answer session in today's lecture. It is said that the more specific communication and question session will be placed 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. Obviously, everyone was not satisfied with the last day of communication. Qiao Yu originally wanted to join in the fun and become familiar with everyone. Especially when he was able to get close to Dao Tian, ​​but Xue Song stopped him: "Let's go, we are still 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 the impromptu seminar held in the afternoon with this manuscript? At noon, you will re-edit your manuscript, input it into the computer, and then print out at least ten copies. Do you still feel very free? You won't have lunch?
Besides, you still have to think about how to explain your ideas to the professors participating in the discussion. The afternoon discussion will be forty-five minutes, and you will have to talk for at least half an hour. Now that we have finished talking, what should we do in the afternoon?" Xue Song is busy. I had to pay attention to the teenage boy next to me. I was almost bursting with envy.
This is Tian Yanzhen.
Even if a different instructor wanted to praise his students so much, he most likely wouldn't have the courage. Even if he had the courage, he wouldn't have the resources. A forty-five-minute seminar can't be held without asking. First, everyone must let everyone know. I can give that face to participate.
For example, it was him who convened this seminar. I just told a 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 yet, it is not possible. It's quite immature, but I think it's interesting.
Instead of just kicking him out, they would probably invite him out politely, and then directly put him on the blacklist, and they would never have contact with each other forever. But Lao Tian is different. Even if he has slander in his heart, but There is a high probability that he will agree with a smile.
Some people even feel that it is an honor to receive an invitation. Alas.
It can only be said that people’s status and realm are different, and their decisions will be completely different. There is no pattern or pattern. It’s just that the more resources you have, the less you need to care about other people’s opinions, especially when doing something that does not violate any principles. This is true between people, and it is the same between countries.
For example, if China suddenly has more than 20 aircraft carrier battle groups, thousands of fifth-generation fighter jets, and bombers, and then tells the world that everyone must watch the Spring Festival Gala this year, especially the rich friends who live in the west end of the village, hurry up and buy it. Broadcasting rights guarantee that even if the show is a mess, the ratings will skyrocket in an instant. It is truly unprecedented and unprecedented.
Qiao Yu didn't think so much, but felt that what Lao Xue said made sense.
So I didn't want to join in the fun anymore. I followed Lao Xue back to his room, turned on the computer and got busy. "What do you want to eat for lunch? I'll bring you the food." He saw Qiao Yu starting to work. , Xue Song asked. She felt more and more like a nanny, but fortunately, the doctoral student he brought will come to school in two days. "Well, just grab a packed lunch. I'm not picky about food, right?" , a little more meat." Qiao Yu said.
"Then I'll add two chicken drumsticks for you?" "Okay!"
Xue Song sighed and left. After a while, there was a knock on the door of the room. Qiao Yu said without raising his head: "Please come in." The door was pushed open and Tian Yanzhen walked in. Qiao Yu In the midst of his busy schedule, he turned his head and glanced, and quickly called out: "Director Tian."
"Well, are you making preparations?" "Yes!" "Let me take a look." "You sit down."
"Change it here. When you have not completed the proof, the wording should be more rigorous. Change it to, according to geometric intuition, it can be inferred that there is a constant C that depends on the geometric and arithmetic properties of the curve X, so that the number of rational points on the curve
N(X)≤C." "Oh."
"And here, your description is that the homology category QH(Cp) is an enhanced homology category... This does not emphasize the difference between it and the homology category in the general sense. I have thought about your idea carefully.
If you want to better analyze the local homology behavior of curves in p-ary complete space, you can introduce a quantized homology category. If you introduce quantization features at the homology level, you may be able to 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 the fields of topology and algebraic geometry, quantization refers to the process of discretization or upgrading a classical structure to a more complex structure. This process is usually non-commutative." Tian Yanzhen sees
Qiao Yu looked like he still didn't quite understand, so he picked up the paper and pen on the table and said, "We don't have much time. Let me explain it to you using geometric quantization in symplectic geometry as an example.
First we need to select a polarization in the phase space. You can think of it as determining a direction, or coordinates, in the classical phase space to simplify the complexity of the problem. Selecting a polarization can be seen as selecting a decomposition so that part of the coordinates are used
Describe quantum states, and 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 certain 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 classical phase space.
The symplectic structure of phase space and the result of polarization selection.”
As Tian Yanzhen spoke, he began to write down a specific example.
"You see, if the phase space of a single harmonic oscillator consists of position q and momentum p, forming a plane (q, p). The symplectic form can be written as w=dq^dp. We now want to quantize this plane into a
Hilbert space, first choose the polarization as д/др=..
Qiao Yu listened quietly to the instructor's explanation, and asked questions when he didn't understand anything. Ten minutes later, he suddenly understood again.
"Oh, I understand, my Q can represent quantized invariants, etc., let me think about it, I need a quantized homology category to decompose the homology group of the curve, so that I can explain the curve on the curve through quantization processing
The behavior of rational points in local quantum structures, right? Director Tian?”
"Um."
"Yes, yes, that's it. Let me use the pen. Well, it's in a quantum homology category..." Qiao Yu took the pen out of Director Tian's hand and quickly wrote what he said yesterday on the manuscript paper.
Wan Zhuo's first formula was completed. Tian Yanzhen looked at the series of formulas written by Qiao Yu and said without changing his expression: "Where is the proof process?"
"First of all, Q has been determined to be a quantum operator acting on the curvilinear 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 curvilinear homology. property. 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 greater the geometric complexity of the curve. High, the number of rational points is relatively small. By adding Q at this time, we can reach dimQH1(Cp)=f(g,Q). This is to make the changes in the local geometric structure more sensitive and further limit the rational points. number of.
Then restrict the rational points through the Jacobian. This is the method used by Professor Robert in today's lecture. We can change it and put it into the complete space. According to previous research, the higher the order of the Jacobian, it means that the curve can The number of rational points allocated may be smaller.
Finally, just construct this function. The first half of the right side of the function is the quantized homology group dimension, which depends on the genus g of the curve and the quantum operator Q. The second half reflects the geometric structure and rational points of the curve. Limitations. You are so awesome, Director Tian. You just gave me some pointers and I took a big step towards proving the existence of this constant C!”
Qiao Yu thanked him sincerely.
Tian Yanzhen was silent as he watched the proof process that Qiao Yu quickly wrote on the paper. He could feel his heartbeat speeding up.
"Bang bang bang..." Like a war drum being beaten.
What kind of comprehension speed is this? He thought 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 been exposed to.
After all, Qiao Yu has not received any systematic mathematics education, but as he talks, this guy suddenly clarifies a superficial idea yesterday to this point? And looking at the process, it seems that there is nothing wrong and it is quite rigorous. .It’s not a problem, but for a fifteen-year-old child, he really can’t ask for more!
"Have you ever had contact with Xin Jiejie before?" Tian Yanzhen suppressed the excitement in his heart and asked in as stable a tone as possible.
"No." Qiao Yu shook his head.
"Have you ever studied quantum physics specifically?" Tian Yanzhen asked again.
"No, I just know a little bit about the wave function, and that there is no certainty in the microscopic world, only probabilities. I haven't studied it specifically. I just read some popular science and understand the wave-particle duality." Qiao Yu shook his head again. Shaking his head. "Then you understand?"
"I understand. The principle is to make the curve contain quantum variables or quantum structures for micro-control and expand its operability. You have explained it so clearly. If you still don't understand, isn't that stupid?" said Wan Qiaoyu suddenly felt something was wrong. He reacted and asked cautiously: "Ah...could it be that my pushing process was wrong?"
Tian Yanzhen took a deep breath and shook his head. Suddenly he felt that some of his students who seemed quite smart before now seem to be a bit stupid. I will analyze them together with everyone in the manuscript for the afternoon seminar."
Qiao Yu nodded quickly and said, "I understand, Director Tian."
While the two were talking, the door was suddenly pushed open, and Professor Xue appeared at the door carrying two boxes of lunch. "Qiao, uh, Mr. Tian, ​​are you here too?"
"Well, at noon, I asked some school leaders to accompany Professor Robert to dinner. I was thinking about giving Qiao Yu his first seminar and explaining something to him." "Oh, you haven't eaten yet, have you? Why don’t you guys eat first?”
Tian Yanzhen hesitated for a moment, then nodded and said, "Okay, then I'll eat here. Don't run back and forth. I'll send a message to Xiao Li and ask him to bring you a piece." Originally he had no intention of eating here.
But there is no other way. Teaching students like Qiao Yu is really painful because it is very easy to gain a sense of accomplishment.
Especially 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 believes that no matter who has such a talent Students probably don’t mind communicating with each other for a while.
"No need to go to so much trouble. I just happened to ride to the cafeteria to eat and then come back."
"That's okay. You come back as soon as 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 a complicated mood and replied: "Okay. I'll do it as soon as possible." After saying that, he immediately turned and left. Tian Yanzhen opened the lunch box and said: "Hurry up and finish the meal. You still have a few propositions. I'll follow you. Please tell me briefly." "Okay, Director Tian."
Many times, when you really like someone, the behavior is similar.t.
".If The need for exponents related to the properties of modular homology groups greatly increases the complexity of the results." "Director Nada, how do you think this should be handled?"
"I think it is better to directly introduce Schulz'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 globalize local properties? But local information is usually associated with a specific prime number p, and different prime numbers may lead to different local behaviors. Is this more difficult?"
"But you proposed it, so you need to think about it. But as far as I know, in p-radical geometry, etale homology groups can be used to describe the local properties of algebraic curves. And as you just said, it is complete The sexual condition has ensured that we can deduce global results from local geometric structures, so this path is definitely feasible."
"Oh, then I need to think more about this...but I still need to introduce a constant, right?"
"You really have to think hard and carefully. Qiao Yu, you have to remember that when mathematics proves any mathematical conjecture, the biggest gain for a mathematician's academic level is the proof method, not just the result... Just have a meal
Xue Song from the fast food store took over from Qiao Yu and started doing the input work. The discussions and instructions between the master and the apprentice kept coming into his ears.
My mood became more complicated again.
I envy Qiao Yu to be given such careful guidance by a big-name academician. This is the kind of treatment he gets from a true disciple, and he is the kind of person who is particularly doted on. Not to mention academician tutors, even ordinary tutors are not very good at giving guidance to students.
Maybe so dedicated and dedicated.
Anyone who has worked as a graduate student for big bosses with specific administrative positions knows that big bosses basically don’t PUA students, but they also basically don’t care about your thesis, your studies, and whether you can graduate, and even when you graduate, they will
I may not even remember that there is a student like you...
Usually there is only one junior supervisor who is responsible for guiding people. Of course, the degree of responsibility of the junior supervisor varies from person to person, so the master's level of the master's degree led by the big boss also varies greatly. And Qiao Yu is enjoying the academician tutor even if he only has a box lunch for lunch.
, all have to take time for tutoring as a special treatment.
Of course, Xue Song also understood very well that this was due to Qiao Yu's ability to accept this special treatment. Discussion, discussion, only with back and forth can we have a discussion.
The academician tutor gave a few words of advice, but the student turned out to be like a fool. He looked confused and didn't know how to respond. Then there would be no next time. Obviously, Qiao Yu accepted this treatment, and then he really became a success.
Just a handyman.
Well, at least I’m still the little director!
Tian Yanzhen 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, but he had no choice. He had no time to talk more. He still had something to do. In the afternoon, he was his first student.
For a seminar, I was a little worried about leaving the organization 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 there were some things that needed to be modified. Anyway, there was still time. "I think you need a thunder machine here, preferably a thunder machine."
The printer was shot, please 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 often in the future." Xue Song said with certainty.
"Hey, 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?"
"It will be held at our Yanbei University, so I don't have to waste time running around."
"Hehe... Hurry up, the seminar is about to start in eighty 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.
Yan Zhen has something to say about this matter.
Although this situation can be said to be dominated by him.
44
On the other side, after Tian Yanzhen walked out of Qiao Yu's room, he took out his mobile phone and made a call. After the phone rang a few times, it was connected.
"Mr. Zhang, I heard you returned to China a few days ago?"
"Yes, I came back to do some errands." He said in a calm tone. "Are you in Huaqing now?"
"What's the matter? Professor Tian." The other person didn't answer directly.
"That's it. I have accepted a student. He should be a closed disciple." "Congratulations."
"He has a habit that looks a lot like you." "Oh?"
Come up with your own questions and solve them yourself."
"Well, this is a good habit. I have always emphasized that students should have this ability."
By the way, he is also similar to you. He was mainly self-taught before. So far, he has not systematically studied undergraduate mathematics tutorials." "Your students have not received systematic undergraduate education?"
"Yes, but he can independently understand Peter Schulz's paper. He also plans to use the complete space system built by Peter Schulz to solve the problem of accurately estimating the upper bound of rational number points on curves. I communicated with him
I came up with the idea and thought it was very feasible. I remember you also studied this issue when you were young.
The person on the other side was silent for a long time, and then he finally said, "Professor Tian, ​​are you calling me specifically to tell me a story?"
"At three o'clock in the afternoon, there will be a seminar on these ideas in the P2 conference room of Building 1 of the Research Center. Professor Robert from New York University will also participate. You will know if I am telling a story. By the way, he is
Born in 2009, fifteen years old this year, from Star City, China."
After waiting for a few seconds, there was no reply from the other party. Tian Yanzhen said calmly: "That's it for now. I personally hope you can come and see in person whether I'm lying. Goodbye." Then he hung up the phone directly.
.
In the world of mathematics, there is a very interesting ranking method.
Fields medal mathematicians, one-hour lectures on mathematicians, 45-minute lectures on mathematicians, 30-minute lectures on mathematicians and other...
Both of them belong to the one-hour category, and most of the more powerful ones have won the Fields Medal. So in fact, everyone with similar status is the touchstone of the legendary Fields Medal, so there is no need to be too polite. Just let the notice arrive.
, Tian Yanzhen felt that if the other party didn't come, it would be the other party's loss, not his.
Opposite Yanbei, in a certain office at Huaqing University. "Tian Yanzhen's phone number?"
"Hmm." "What does he want from you?"
"He accepted a student who was fifteen years old and had never systematically studied undergraduate mathematics courses. Not only could he independently understand Peter Schulz's paper, but he also independently proposed a solution to the number of rational points on an algebraic curve based on Schulz's theoretical framework.
The solution to the world problem." After the words fell, the old man opposite was also stunned and said in surprise: "Fifteen years old?"
Zhang Shuwen nodded. "Do you believe it?"
Zhang Shuwen shook his head, and then said: "But he said if you don't believe it, you can go see it with your own eyes. At three o'clock in the afternoon, that kid will host a seminar on this issue." Silence...
"Go and have a look. If it's fake, just expose it without saving any face." Zhang Shuwen: ".."
At 2:30 in the afternoon, Xue Song had already brought Qiao Yu to the P2 conference room in Building 1.
Although generally speaking, the speaker should arrive last. However, Xue Song felt that Qiao Yu was already too arrogant and too indulgent. He would be too eccentric to recognize himself. Especially considering that Qiao Yu was only fifteen years old. He should still develop a good habit of humility.
The two of them had been in the conference room for less than five minutes before Chen Zhuoyang walked in. "Senior Brother Chen." This time Qiao Yu took the initiative to say hello.
Anyway, there was no one in the small conference room. Unlike in the morning, the conference room was already full of unfamiliar people when he arrived. Qiao Yu was embarrassed, but the current situation still supported him in showing his courtesy. Chen Zhuoyang After making a fuss, he didn't really want to deal with this guy Qiao Yu.
Especially when I learned that the protagonist of today's discussion was this junior fellow student, I was a little confused. He didn't even get this kind of treatment.
The most irritating thing is that this little talkative kid is holding a seminar and he has to be busy. Chen Zhuoyang asked: "Well, will you use PPT later?"
"Thank you, Senior Brother Chen, for your concern, but no, I don't have time to do it, and I don't know how to do it. I will just write on the blackboard later. Huh? There is no blackboard. Senior Brother Chen, how about you get me a blackboard?" Just put it there." Qiao Yu pointed enthusiastically to the open space in front.
Chen Zhuoyang rolled his eyes, he might as well make a PPT.
"Okay, I know where to put it. I'll ask the staff to bring one over later. Is the 90X120 one okay?" "Okay, please, Brother Chen. Oh, by the way, don't forget to ask for the pen."
Chen Zhuoyang sighed, turned around and walked away.
"Hey, Senior Brother Chen is good at everything else, but he is too serious. I can't see him smiling all day long." Qiao Yu commented. Xue Song was silent next to him, so he could understand Chen Zhuoyang.
If he were Qiao Yu's senior brother, he would probably be very 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. He assumed the responsibility of a senior brother and started to introduce Qiao Yu: "Qiao Yu, this is Professor Zhou Yang from the research center. Mainly studies low-dimensional topology and geometric groups.
"Hello, Professor Zhou."
"You are Qiao Yu. Mr. Tian values ​​​​you very much. Keep working hard." "Thank you, Professor Zhou, for your encouragement."
"This is Professor Cai Wenyuan. He mainly studies algebraic geometry and Lie groups..." "Hello, Professor Cai..."
"Professor Chu Weihua, a very high achievement in the field of algebraic topology..."
With Chen Zhuoyang's introduction, the atmosphere in the conference room gradually became familiar. To be honest, the professors were quite unfamiliar with this child Qiao Yu. Everyone came to this seminar just for Tian Yanzhen's sake. As for Tian Yanzhen's face, everyone came to this seminar. Qiao Yu's idea is simpler. It doesn't matter if he's unfamiliar. Everyone will understand him later.
And with Senior Brother Chen as a tool beside him, Qiao Yu soon got to know all the professors who came to participate in the seminar. The mobile blackboard was also pushed into the conference room by the staff while everyone was chatting.
Soon, only three minutes before the start of the seminar, Tian Yanzhen, Robert, and another person Qiao Yu had never met walked into the conference room. Qiao Yu, who had sat back down in his seat, clearly felt The atmosphere in the conference room suddenly became a little stagnant...
It's not that Qiao Yu is sensitive, it's mainly that everyone showed it clearly.
We were still discussing in a low voice just now, but when the three people walked in, everyone suddenly stopped talking. It’s not like everyone suddenly became so disciplined just because foreign friends came, right?
Qiao Yu turned around and was about to express his thoughts to Xue Song. Suddenly he found that old Xue was staring at the three people who had just come in with a strange expression, so he asked in a low voice: "Teacher Xue, what's wrong with you?"
Xue Song turned around and said softly with an expressionless face: "Speak carefully later, don't waste the stage that Director Tian gave 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 participate in this seminar, so this was expected. Lao Xue once again emphasized not to waste Director Tian's opportunity. It could only be that this stranger had a great background. But he just looks like a thin and serious old man. Apart from some aura, there doesn't seem to be anything special about him.
Fortunately, there was no time for Qiao Yu to think wildly, and his mentor once again assumed the role of host.
"Thank you all very much for coming to this seminar today, and especially thank you to Professor Robert and Professor Zhang Shuwen for taking the time from their busy schedules to come and personally guide my student Qiao Yu on some of his wonderful ideas in mathematics.
The main issue discussed in today's seminar is about using tools such as complete space, modular form theory and P-radical geometry to study the upper bound on the number of rational points on the algebraic curve X. Okay, Qiao Yu, you can start speaking. ." After saying that, Director Tian sat back in his seat, and Qiao Yu stood up immediately without any stage fright.
"Thank you all teachers for coming to this seminar. Some of my immature thoughts have been printed out, which is the pile of manuscript paper placed on everyone's desk.
By the way, I would also like to express my special thanks to Professor Robert for inspiring me with today’s lecture, and my supervisor Professor Tian Yanzhen for his guidance. As Director Tian just said, I have recently read Professor Schultz and Professor Robert After writing my thesis, I suddenly came up with such a bold idea.”
As soon as Qiao Yu finished speaking, almost everyone picked up the report on the table. It was too simple. Just now, everyone came to the conference room a few minutes early and was busy chatting. No one picked it up and took a serious look at it. At last glance, Zhang Shuwen and Professor Robert, who were sitting next to Tian Yanzhen, had already picked up the simple book and started to look through it.
After Qiao Yu finished his opening remarks, he got to the point.
"My idea is to use the complete space theory built by Professor Peter Schulz, modular form theory, p-advanced geometry and quantized homology categories to derive the upper bound expression of rational points on the algebraic curve.
To do this, we first need to consider the geometric background of the curve Curve, Faltings theorem tells us that the number of rational points is limited.
But this is not enough, because we all hope to get a specific upper bound. According to a few sentences of analysis, the higher the genus, the complexity of the algebraic curve increases, which means that the number of rational points is relatively reduced. So my initial guess is: N ( Interesting methods to derive the results of constant C. However, these methods have not been proven yet, so I hope teachers can give me some inspiration. First, we introduce the modulus space. Let X be an algebraic curve with genus g. Its modulus space Mg parameterizes all curves with genus g.
Because the modular form is closely related to the modulus space, I understand it as certain functions defined on the modulus space, which provide geometric constraints on the complexity of the curve. In this way, let the level of the modular form be k, and we also assume that there is a constant A1, Such that: N(X)sC1(g, k)=A1-g-k^α..."
Under the stage, all the professors in the conference room have put away their previous relaxed attitude, and their expressions have become serious. The only ones whose expressions have not changed are probably Tian Yanzhen and Xue Song.
Chen Zhuoyang, who is sitting at the very back, can attest to this.
He was not interested in what Qiao Yu said, so he paid more attention to the expressions of the instructor and the two great men across from him.
It was obvious that Director Tian was very relaxed. He just watched Qiao Yu write on the blackboard quietly. One of the two big guys beside him frowned, and the other had already picked up a pen and started writing next to the manuscript. Painting... Chen Zhuoyang feels a little depressed...
Isn't it, everyone is serious? So it's not that Director Tian wants to push the younger brother, can everyone accept this kind of thing 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 was just thinking...
How could he play with his ideas like this? Chen Zhuoyang even felt that Director Tian was too anxious. After all, this junior was only fifteen years old! Although he was able to participate in CMO and get first place, it proved that he must have mastered high school knowledge, but college knowledge I don’t know if he has ever been exposed to it, but he knows nothing about scientific research!
He even thought that Qiao Yu could understand Peter Fields's paper because he was talking in his sleep. But now just looking at the expressions of the professors, that is not the case at all, because it can be seen that everyone is really starting to think... What the hell, is my junior brother really going to go against the will of heaven?
What made him even more desperate was that Qiao Yu on the stage not only didn't have any stage fright, but also became more and more excited as he talked, because many professors had begun to read his writing on the blackboard seriously. Wait, Professor Robert even took out his mobile phone to take a picture of his writing on the blackboard.
At this point, we can introduce the p-adic number field and Professor Schultz's homology theory. We know that for each prime number P, the properties of the etale homology group can constrain the local distribution of rational points on the curve.
Then according to Schulz's p-advanced Hodge theory, the following inequality can be derived: N(X)sC2(g,p)=A2·g2-log(p). There is a very important point here, Schulz's
A core feature of p-radical Hodge theory is its completeness.
So if the inequality we derived holds, we can start from the properties of the curve in the local domain and deduce the global geometric constraints. So we need to prove whether the inequality holds. To this end, under the guidance of Director Tian, ​​I thought of a way
, is to introduce a quantized homology category..."
During this half hour, Chen Zhuoyang felt like he was sitting on pins and needles.
Because there were only two students, Director Tian, ​​present in the entire conference room. One was talking in front, and the other could not understand what the junior brother was talking about... but the conference room was still eerily quiet, and there was not even any discussion.
Everyone was staring at Qiao Yu's writing on the blackboard.
Including those three math giants that most of the 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 am not yet able to handle the constants in the settings and provide a complete logical proof of specific tools, but I think this should be a new research direction, because once we launch
The result of constant C represents the upper bound on the number of rational points that can directly predict the correlation 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 conference room made him nervous again.
No, professors, aren't you going to say something?
We are all adults, so don’t look at your junior brother with that incredible expression, okay? He is only fifteen years old, and he should receive education about setbacks now! Everyone should severely criticize his ideas at this time! Chen Zhuoyang was thinking in his heart
He was thinking viciously, but when he saw Director Tian across from him raising his hand first and starting to applaud, he could only immediately raise his hand and applaud in cooperation...
"Pah, pah, pah.."
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 within a few dozen seconds, the applause stopped, and then Chen Zhuoyang finally heard the sound of nature.
"I have a question, Qiao Yu, why don't you just use the Riemann-Roch theorem in the third part?" Chen Zhuoyang glanced at Zhang Shuwen, who was facing him with a serious face. As expected, the professor is so powerful! "Ah? What is the Riemann-Roch theorem?"
Theorem?" Qiao Yu asked back, full of curiosity.
Everyone has different reactions.
For example, Qiao Yu stood there as if nothing had happened, but his nominal tutor, Professor Xue, felt very social and his face suddenly turned red.
As for the other professors, including Robert Greene, they were all at a loss. They probably couldn't understand that the little guy who just talked about algebraic curves for half an hour did not 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 in CMO and has not yet received a complete undergraduate education.
Chapter completed!