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.
To be continued...