Tian Yanzhen knew that Qiao Yu was immersed in the paper and didn't want to disturb Qiao Yu, just like he didn't want to be disturbed by anything else or anyone when he was reading the paper.

Human thinking, especially mathematical thinking, needs coherence. When you are immersed in a certain state and are suddenly interrupted, it is difficult to get back to that state.

This is indeed the case.

Peter Schulz's paper seemed to open a door to a new world for Qiao Yu.

Those complex algebraic symbols and high-dimensional geometric structures have never been displayed in Qiao Yu's mind as three-dimensionally as they are today.

He couldn't even describe the feeling in precise words.

If Qiao Yu had to describe this feeling, it would probably be a counter-intuitive sense of geometry.

After carefully reading this paper, his brain seemed to be occupied by countless high-dimensional geometric spaces. However, these spaces are not as smooth and continuous as the Euclidean spaces that can be seen everywhere in our daily lives.

The divided space, especially the rigid analytical space, turned into geometric objects that were divided and layered infinitely in Qiao Yu's mind. Each dividing line was so precise and subtle.

Qiao Yu couldn't explain why those lines appeared there, but his subconscious told him that these lines should appear there.

It is these complicated lines and geometric figures that make the space no longer continuous, but present a discrete but tight structure.

These structures undergo complex and complex transformations in the P-progressive field.

There is no smoothness and intuition in this geometric world, but only constant reconstruction by various expansion methods, infinite extension, and finally forming a universe.

A universe intertwined with algebraic symbols and geometric figures. Every point in the universe contains infinite divisions and infinite levels of details under a special law. These laws govern these points, lines, and surfaces.

Pushing the construction of this cosmic geometry to perfection....

The only regret is that one night is not enough for Qiao Yu to fully understand even the first paper.

When Qiao Yu woke up from his focused state due to a deep sense of exhaustion transmitted from his brain, it was already 12:23 in the night.

He really felt very tired, even more tired than before when he studied Lao Xue's Diophantine equation until two or three in the morning.

However, considering that he had been riding for six hours today, Qiao Yu felt that it was probably normal to feel tired.

So on the first night when he came to the capital and was admitted to college, Qiao Yu climbed into bed without even washing up. What was even more annoying was that Qiao Yu, who always slept well, had a strange dream that night. In the dream, he came

We arrived in a strange labyrinth world, which is composed of countless strange doors.

Every time Qiao Yu opens a door in his dream, he will see a magnificent space composed of all kinds of strange geometric configurations. This world is so wonderful that even the physical rules followed by a photon are completely different from the real world.

Same. It could be here or there at the same time. Just try the odds?

No, maybe it can be here, it can be there, it can be everywhere.

Until Qiao Yu opened a door again, he suddenly felt a burst of light shining into his eyes, and then subconsciously opened his eyes and woke up...

He forgot to close the curtains last night, and the rising sun just hit his eyes through the window, and he felt dizzy before he even got up.

Qiao Yu quickly got up from the bed, picked up the mobile phone casually thrown beside the bed and took a look. Good guy, it was already eight twenty.

He had never woken up so late when he was in Star City.

Even if I didn't go to bed until 1 o'clock on the 30th day, I still got out of bed at 7:40 the next day.

In this regard, Qiao Yu was different from other children since he was a child.

Other children couldn't wake up, but as long as Qiao Yu slept for more than seven hours continuously every day and took a break for ten or twenty minutes at noon, he would be energetic enough for the whole day.

Although he seemed to have had a dream all night yesterday, and he still clearly remembers those dreams, Qiao Yu felt that he was in good spirits and was not sleepy because of too many dreams, so he simply climbed out of bed. Keeping in mind what Lao Xue said yesterday

explain,

Qiao Yu got dressed and dressed up neatly before picking up the mouthwash cup and toothbrush on the window sill, and then put the towel on his shoulders, then walked out of the room and walked to the second floor around the corner. Yesterday, Lao Xue took him

I've been there, the restrooms and restrooms are on the second floor.

After going up to the second floor and turning to another corner, Qiao Yu just walked into the bathroom. A middle-aged man walked out of the bathroom. The two met face to face, and the other person asked: "Huh? Who are you?"

"Um, I'm..." Qiao Yu was just about to introduce himself when the other person suddenly realized: "Oh, you are the teacher's new student, right? Your name is, Qiao, um, Qiao, what are you doing?"

"Qiao Yu, metaphor of metaphor!" Qiao Yu bit the word metaphor very hard.

Yesterday, when Lao Xue asked him what his identity was, Qiao Yu didn't get off guard.

Today this middle-aged man didn't even remember his name, which really made him lose his guard!

Sure enough, he is still a little Karami.

The middle-aged man said politely: "Ah, yes, yes, Qiao Yu! Hello, junior brother."

Qiao Yu felt a little emotional. As expected, his tutor was all over the world. This guy looked to be in his forties, and he still had to be called Teacher. His seniority was indeed very high.

Of course, Qiao Yu still asked politely: "Excuse me, are you..."

"Oh, my name is Chen Zhuoyang, and I am Director Tian's doctoral student. Because I have to be responsible for some lectures and the like, my office is upstairs. You can come and sit there when you are free." Chen Zhuoyang said politely. Although Qiao Yu

He looked very young and immature, but the instructor specifically mentioned it to him, which proved that he valued Qiao Yu, so Chen Zhuoyang behaved very politely.

However, these words set off a storm in Qiao Yu's heart. Looking at the other party's almost half-bald head and high hairline, he couldn't help but hesitantly asked: "Well, Senior Brother Chen, can I take the liberty to ask?

, how old are you this year?

"I'm? 29? Yesterday?" After saying that, Chen Zhuoyang looked at Qiao Yu's shocked eyes, subconsciously touched the top of his head, smiled to himself and said: "Haha, I haven't graduated yet, my hair is about to fall out! Others say I look like I’m in my thirties, ugh..."

"No, Senior Brother Chen, you are too modest. I believe you even if you say you are forty, really." Qiao Yu couldn't hold it back, and even forgot Qiao Chuan's reminder to not cause trouble when he went out. He was very honest. Said something.

"Well..." Chen Zhuoyang thought for a while and replied seriously: "Junior brother, you have to remember that it doesn't matter whether a man is handsome or not, the key is to be talented. And you haven't entered the threshold of mathematics yet, no I understand how difficult it is to learn mathematics well, but I’m pretty good at it.”

In fact, when Qiao Yu blurted out those words just now, he felt a little regretful. He originally wanted to apologize.

But as soon as Chen Zhuoyang's teaching came out, Qiao Yu couldn't help it anymore: "Well, Senior Brother Chen, I'm really not trying to make excuses. But the author of the paper I read yesterday was already a top professor at a well-known university at the age of 24. You're already 29, and you haven't even graduated with a doctorate yet. His talent is incomparable. I looked at his photos online and he has a lot of hair."

Chen Zhuoyang blinked and asked subconsciously: "Who are you talking about?"

Qiao Yu replied: "Peter Schulz."

Chen Zhuoyang stared at Qiao Yu silently, until Qiao Yu felt a little scared in his heart, and then he slowly said: "Junior brother, thank you for comparing me with the youngest Fields Medal winner! But I It is still recommended that you wait until you start writing your master's thesis.

Come talk to me about this issue again that day.

Although there is no shortage of resources for students who become teachers, you will know how strict the teacher's requirements are in the future. When you cry and come to beg your senior brother, well, I have decided to treat you like air. What did you say? It's no use.

After saying that, Chen Zhuoyang raised his head proudly and left.

The sound of "dong dong dong" going up the stairs reached his ears. Qiao Yu didn't even react yet. He was probably angry, right? Senior Brother Chen is so cute. He doesn't talk trash when he is angry. Even threats are cute. Extremely, and completely intimidating.

The only pity is that it went too fast.

He also wanted to tell this senior brother that if he couldn't even complete his master's thesis, he would definitely choose to drop out of school.

Are you kidding me, a master's thesis can be written if you have the brains? No matter how strict Director Tian is, the master's thesis will not require him to solve a world-class mathematical problem, right?

Muttering these messy things in his heart, Qiao Yu quickly walked into the bathroom, cleaned himself up, returned to the room, put the things back to their original places, then took the meal card and went out to have breakfast. It was more convenient.

He went to the canteen and bought two big buns and a cup of soy milk, and finished them on the way back. Then Qiao Yu opened yesterday's paper again.

I don’t know if it’s because my mind is clearer in the morning. There were some proof processes that I couldn’t understand yesterday, but I felt that I understood them after reading them again today.

I also roughly understand why some Chinese articles call Peter Schulz’s pioneering research a quasi-complete space. Because this theory can really be described as mysterious.

For example, when it comes to etale homology, the simplified calculation framework provided by the complete space. Yesterday, I couldn't understand how this was done. After reading it today, I found that it is nothing more than homology classes in certain complete spaces passing through the complete space.

The structure maps to the etale homology class of classical geometric objects.

In this way, the homology class structure of algebraic varieties can be calculated more efficiently in the p-adic background.

After feeling that he had completely understood these abstract things, Qiao Yu even wrote a related mathematical question on his own paper based on what he couldn't understand yesterday.

After finishing the question, Qiao Yu thought it was cool, so he picked up the pen again and answered the question based on his understanding of the theorems and lemmas given by Peter Schulz in the paper.

After finishing, Qiao Yu instantly felt cute. He looked at the time and saw that it was already half past ten in the morning. So he stood up and was about to go out to exercise. When he turned around, he saw Director Tian and Lao Xue through the window.

They were walking towards the room together. Coincidentally, the two of them were looking at him through the window, so Qiao Yu immediately reacted, ran to the door and opened it.

"Director Tian, ​​Teacher Xue, you are here, please come in quickly."

"Well, I came here just to see you. How is it? Are you still used to it?" Tian Yanzhen asked when he walked in.

"It's okay, I'm used to it. The water for washing my face in the morning is just a bit cold, but everything else is fine." Qiao Yu said truthfully. "Don't you know how to buy a kettle and get some boiling water? This kid..." Lao Xue said to Qiao Yu

Yu's complaints were speechless. "Oh!" Qiao Yu responded honestly.

Okay, it's normal that you haven't gotten used to it when you first came here. You're so young, it doesn't matter if you just wash your face with cold water, you'll be more energetic." Tian Yan said and walked to Qiao Yu's side, falling asleep and noticing the wrong paper he had left next to the computer.

Then they all picked it up with interest. "Well, Director Tian, ​​this is a random question I asked after reading a paper yesterday." Qiao Yu explained quickly.

This proves that he is very successful and started studying as soon as he arrived at Yanbei University.

"Yes, I know." Tian Yanzhen responded casually, and then carefully looked at the questions and the solution process on the manuscript paper.

Xue Song on the other side glanced at the paper on the computer, then turned to Qiao Yu with a confused expression: "What paper are you reading?"

Qiao Yu answered: "The paper published by Peter Schulz in 2011 on the P-radical Hodge theory of rigid analytic spaces and complete spaces."

Xue Song looked at Qiao Yu dumbfounded and said, "Didn't I tell you yesterday to download Professor Robert Greene's paper and study it? You will attend the professor's lecture tomorrow without any preparation? Look at Peter

·What’s Schulz’s paper for?”

"Ah?" Qiao Yu then remembered that Lao Xue had indeed mentioned it yesterday after he got the account for the library software.

After smiling sheepishly, he explained: "I forgot. Didn't you tell me about the deeds of Zheng Zhiqiang and Peter Schultz yesterday before leaving? Zheng Zhiqiang was a computer engineer, so I ignored him. , and then I downloaded Peter Schulz’s paper, and accidentally became fascinated by it.”

This answer left Xue Song speechless.

He told Qiao Yu yesterday that these two people seemed to be telling him not to be proud, but in the end, this guy went to study the complete space theory pioneered by Peter Schultz? This is not a typical case of not learning yet...

Xue Song was about to teach Qiao Yu a few words when Tian Yanzhen suddenly handed Qiao Yu's manuscript paper to Xue Song and said, "Professor Xue, take a look at it too." "Uh, okay." He took the paper from Tian Yanzhen. Xue Song didn't know what to say as soon as he glanced at the title of the manuscript paper.

Good guy...do you really understand this?

On the other side, Tian Yanzhen was already chatting with Qiao Yu: "Have I told you that the mathematical knowledge you are currently exposed to is not systematic?"

"Yes." Qiao Yu nodded honestly and replied blankly: "You told me in the dormitory when I was at CMO last time."

Well, so I worked with Professor Xue to help you develop a study plan, intending for you to make full use of these six months to give a rough overview of the entire algebraic geometry and number theory system.

Professor Xue also helped you develop a set of study and training courses, but I suddenly changed my mind just now. In the past six months, you can study on your own by reading papers in the library's database according to your interests. If there is anything you don't understand Yes, just ask Professor Xue or me. Professor Xue should be able to spare time to answer your questions at any time. I will spare half a day every week to discuss related issues with you.

If it is something beyond the research scope of Professor Xue and I, there are many professors in the research center, you can go and ask for advice. I will ask Professor Xue to send you a catalog later, which contains the contacts of all the professors in the research center and the School of Mathematics. methods and research directions.

I will say hello to you then, but when you ask other professors for advice, you must be more modest and respectful, understand? Everyone will answer your questions for free." Qiao Yu immediately nodded and assured: "Don't worry, Tian Director, I understand this, and to tell you the truth, I am actually the most polite."

something.

This not only allows you to cherish the opportunity to attend such lectures, but also pays the most basic respect to the specially invited lecture professor. It is also the most basic courtesy. Not only this time, but also other lectures that interest you in the future. You have to get used to doing this." "Understood, I will definitely remember this time, and I promise to read that professor's paper in the afternoon." Qiao Yu said immediately.

During the conversation between the two, Xue Song had already gone through Qiao Yu's question and the solution process. His mood was very complicated.

He had also read this early paper by Peter Schulz.

He was still at Princeton at that time.

To be honest, this thing was obviously too high-end for Xue Song at the time. Well, actually, the same is true for him now.

A complete space is a geometric object defined on a p-radical number body. The p-radical number itself is a relatively abstract and complex mathematical system, which is completely different from the intuitive real number or complex number system. What is even more frightening is that this theory It also combines several branches such as algebraic geometry and topology.

This is no longer a simple study of geometric objects, but also requires the ability to understand the algebraic structure of geometric objects and their behavior in different number fields.

If the above difficulties can be overcome, then the tilt theory involved in the complete space will really make people scratch their heads. By introducing a new algebraic geometry perspective, this idea combines a p-advanced complete body with characteristics The perfect bodies of p are related.

But this connection is very abstract and involves extremely abstract mathematical techniques such as algebraic closure and integral closure...

In other words, this theory cannot be understood with the help of existing mathematical frameworks. To understand Peter Schulz's ideas, one needs to establish a completely new mathematical system in the mind. Qiao Yu just downloaded a paper on this aspect yesterday. Just took a look and it seems that I can understand it today?!

Is this the advantage of a poor foundation? No, if the foundation is not good, you should not be able to understand this kind of paper at all.

To be honest, Xue Song found it difficult to understand how Qiao Yu understood the concepts created by Peter Schulz.

Xue Song actually heard what Tian Yanzhen said just now, but he couldn't raise any objections and just silently put Qiao Yu's manuscript paper back on the table.

This is probably a herding strategy. But the resources prepared for Qiao Yu are all top-notch, so just watching him tinker with it every day and see what results he can produce seems to be something very exciting. " Professor Xue, what do you think?" Tian Yanzhen asked.

Xue Song shook his head, showed a wry smile, and said, "I think this arrangement is quite good. It only lasts for half a year. Let's give him enough freedom first, and then let him see the results." Tian Yanzhen nodded, looked at Qiao Yu and said : "That's it, do you have any questions?"

Qiao Yu quickly asked: "By the way, um, is there another Senior Brother Chen above this building?" Tian Yanzhen nodded and replied: "Yes, his office is arranged on the third floor. You guys met. ?」

Qiao Yu kept nodding and said: "Yes, yes, yes, I just want to ask what is the topic of Senior Brother Chen's doctoral research?"

Tian Yanzhen looked Qiao Yu up and down and smiled: "Why, it hurts to give advice to a fellow student? I used to help Professor Xue's master's student revise his thesis and feel that it does not reflect your level. Do you want to give advice to a doctor?" Qiao Yu? He repeatedly denied it and said, "No, no, I just want to hear how difficult it is for Senior Brother Chen to choose a topic, so that I can be mentally prepared for the topic he chooses for his doctorate in the future."

Tian Yanzhen shook his head and said, "Your situation is different from Senior Brother Chen. His topic selection has no reference for you."

But after finishing speaking, seeing Qiao Yu's eyes full of curiosity, he still revealed: "His research direction is geometric analysis on complex manifolds, specifically the existence of minimal surfaces on Keller manifolds. and stability. For example, how to construct a minimal surface through the variational method, the stability of the energy functional of the minimal surface, and the evolution of the minimal surface under the action of Ricci flow, etc. "

"Oh!" Qiao Yu nodded, looking stunned. "What? Have you studied these before?" Tian Yanzhen asked.

"No!" Qiao Yu shook his head and said, "I just think there is a reason why my senior brother has not graduated at the age of 29. When I watched the video before, a boss said that anything involving functional analysis is quite difficult.

of."

Xue Song next to him couldn't help it and said: "Functional analysis is difficult? No matter how difficult it is, it can't be more difficult than Schulz's complete space! As long as you can fully understand Hilbert space and Banach space, learn functional analysis

Analysis is a very simple matter."

Qiao Yu looked at Xue Song blankly and replied: "Ah? Is it because I saw that the p-adic analysis method involved in analyzing infinite dimensional space in the paper is using the techniques in functional analysis? There are many places in the paper

They all use P to enter Banach space."

Tian Yanzhen nodded and replied: "When analyzing the topological properties of p-adic numbers, function spaces and p-adic representation theory, the structure of Banach space does often appear. However, p-adic numbers are special and involve

Banach space is different from real number analysis, that is, what you said is that P enters Banach space.

Don’t get hung up on these things. You’ll understand when your knowledge is broad. In frontier research in mathematics, many mathematical tools overlap. This is why mathematicians need to have a comprehensive foundation. But I want to see if you can pass it.

To what extent have you achieved your own method? Are there any problems?"

Qiao Yu immediately shook his head and said, "No more."

"Then let's go first. Professor Xue will come to take you to Professor Robert's lecture tomorrow." After saying that, Tian Yanzhen left with Xue Song. Qiao Yu watched the two professors leave and stood up to move again.

I calmed down and read the paper again.

Director Tian said, just start reading Professor Robert’s papers in the afternoon.

There is still more than an hour before dinner, and Qiao Yu feels that he should have almost absorbed the nutrients in Peter Schulz's first paper...

The two mathematics tutors walked silently for a while under the warm sunshine. Xue Song couldn't help but ask: "Mr. Tian, ​​I think it's okay for Qiao Yu to let him study freely in the past six months, but what should we do with him after the school starts next year?"

?Or let him study with the elite class?"

Tian Yan really thought about it and said, "Don't make a decision in a hurry. Let's see what this child can do first. To be honest, I don't know how to teach him now. I just think I can try to let him do it on his own.

It might be more appropriate to take the initiative to explore the problem." Xue Song nodded. To be honest, he had no good way to teach Qiao Yu. Even when he saw the manuscript paper where Qiao Yu asked and answered questions, he began to doubt whether he could

He has the ability to be a good little guide for Qiao Yu.

It is no exaggeration to say that there are not many mathematicians in the world who can understand the content of Peter Schulz's research. After all, this is the most basic research in mathematics, aiming to build a connection between algebraic geometry, number theory and p analysis in multiple fields.

A bridge. Qiao Yu seemed to have an extremely amazing talent in comprehending these complex mathematical ideas. Not only had he never taught such students before, he had never met them before, so he suddenly felt a lot of pressure.

"Don't think so much. Speaking of Peter Schulz, there is another interesting thing. He gave his thesis to his supervisor Lambert. After reading it, Lambert told him that he could graduate with a Ph.D. So the real genius,

In fact, we don't need to worry too much." Tian Yanzhen added optimistically.

After hearing this, Xue Song took a deep breath and couldn't help but ask: "When you were teaching at Princeton, you came into contact with more students. Have you ever met anyone with Qiao Yu's talent?" Tian Yanzhen smiled.

He laughed and said, "You also studied at the Princeton School of Mathematics for eight years. You should know better than me what the situation of your classmates is like, right?"

Xue Song shook his head and replied: "There are really many geniuses, and I am one of the very ordinary ones, but there is really no genius that I admire from the bottom of my heart."

"That's because you are originally one of the geniuses." Tian Yanzhen said with emotion: "Students who can successfully graduate from Princeton are geniuses compared to ordinary people. Let alone graduate with a Ph.D., but mathematics and

Geniuses in the field of theoretical physics must be divided into three grades after all!"

One sentence made Xue Song completely lose interest in chatting.

It is really a hopeless field, where geniuses are divided into three, six or nine grades.

"If Qiao Yu is really the genius I thought he was, I have to thank you. If it weren't for that phone call, if I really missed it, I would regret it for the rest of my life." Tian Yanzhen looked at Xue Song, sincerely

Said. "You are serious!" Xue Song said politely.

"Okay, I'll go back first, and you can do your work. Oh, by the way, the joint training plan with Yu University has been drawn up. I have helped your students gain some rights and interests. If they are in joint training

If your achievements during this period meet the standards of Yanbei University, you can choose to get a diploma from Yujiang University or Yanbei University."

"Oh, thank you so much!"

"Xiao Xue, you're welcome!"

Watching Tian Yanzhen walk into the small building next to him, Xue Song stood there and thought for a moment, then smiled and took out his mobile phone, and walked out of the research center while editing the news. This news could probably be given to his group of doctoral students.

Is it infused with chicken blood?!

At noon, Qiao Yu went to the cafeteria alone for lunch. After coming back and taking a nap for ten minutes, Qiao Yu began to search for Professor Robert's papers in the backstage of the Yanbei University library and downloaded them. Listen to what you should hear, and you are also a student.

Essential advantages.

Especially what the instructor has repeatedly emphasized, and even related it to politeness and respect, is what you must listen to. As for the other things... you can actually have a choice.

People who can become important people will most likely not care about everything with students. Anyway, this is how Qiao Yu understands it. Just like Principal Zhang of Xingtie No. 1 Middle School.

As long as you follow his requirements and get results, Lao Zhang is really tolerant in other aspects.

Qiao Yu felt that even if he was so bored that he took down a few of Tieyizhong's signs, Lao Zhang would smile and ask the school's logistics department to make new ones, and then tell him that he would never make an example again. Dissertation retrieval at Yanbei University Library

The system searched for Robert Greene's name, and a bunch of papers suddenly appeared.

Qiao Yu was startled. But he soon discovered that they were not all the same person. There seemed to be many people named Robert Green abroad.

Although I encountered a similar problem when searching for Peter Schulz, there was only one interference term, and that guy was still studying chemistry. The direction of his papers was completely different. But for Robert, many of his papers were mathematics-oriented.

Fortunately, Qiao Yu found that this paper retrieval system is actually very easy to use. It is not only rich in content, but also allows you to select the age. The advanced search page even supports searches for the author's affiliation. Qiao Yu remembered that Old Xue said that this professor was from New York University.

, which is much more convenient.

Soon, the serious Professor Robert's paper was downloaded.

I don’t know if it was because he first studied Peter Schulz’s paper that Qiao Yu’s brain opened up again. Qiao Yu actually felt that it was quite easy to understand the professor’s paper. Well, saying it was easy seemed a bit false.

Yes, but at least it’s not difficult.

For example, Qiao Yu really feels that those lemmas, preconditions of theorems, a series of concepts, and the proof process are easy to understand. It doesn't require too many brain cells to understand. But it's good to balance work and rest.

. Reading Peter Schulz’s paper yesterday was indeed too taxing on the brain. Today I should relax while reading a paper that is not so difficult to understand.

Although he was relaxed, it was already nine o'clock in the evening when Qiao Yu finished reading the two papers, and went to have dinner in the middle. After putting down the papers, Qiao Yu began to think habitually again, and suddenly an idea came to his mind.

.

To put it bluntly, the content of Professor Robert's research is the problem of accurately estimating the upper bound of the number of rational points for a given type of algebraic curve, especially a high-dimensional algebraic curve. This type of problem is actually closely related to the Diophantine equation. Finding the number of rational points

, and then study the distribution of these rational number points.

It is nothing more than that the geometric structure of high-dimensional algebraic varieties is often more complex, with more complex singularities, topological properties and different homology properties. These geometric properties all affect the distribution of rational points.

Therefore, there is actually only one research goal for this type of problem, which is to simplify the process of finding rational number points as much as possible, and to easily find the distribution of rational number points. This is equivalent to giving a high-order Diophantine equation and quickly determining whether there is a solution.

, and solve this type of equation.

Well, in short, this is how Qiao Yu understands it.

This is the cognition of a layman in mathematics. If Lao Xue were here at this time, after listening to Qiao Yu's thoughts, he would probably want to beat up this guy who doesn't know the heights of heaven and earth. The reason is also very simple. The research goal is simply too ridiculous.

.

Simplifying the process of finding rational points, but it is almost impossible to easily find the distribution of rational points on high-dimensional algebraic varieties. This is common sense in mathematics. Now everyone is doing nothing more than estimating rational points efficiently through geometric and algebraic tools.

Quantities and understanding their distribution through modern algebraic geometry tools.

As for quickly solving the Diophantine equation?

Even if it is determined that there is a solution to the solution of elliptic curves or more complex equations related to modular form, if he really wants to solve it, Lao Xue can only say haha.

Of course, these are not problems for Qiao Yu, a layman who doesn't have much respect for mathematics. In addition, he just studied Peter Schulz's mathematical ideas yesterday. A very bold idea suddenly came from Something popped up in Qiao Yu's mind and was out of control.

Why couldn't he try to use the theory created by Peter Schulz to solve this type of problem?

Regardless of whether it works or not, you can try to introduce a complete space into it. There are no suitable tools to deal with similar problems, but he can also create it himself.

Although this is a framework built by others, as long as the creation of tools within this framework and in compliance with the rules of this framework can solve the problem, it is definitely feasible. Then the problem before Qiao Yu now is very simple. How to introduce the problem of upper bound estimation of rational number points with algebraic curves into the framework of quasi-complete space theory? Qiao Yu, who was not afraid of tigers as a newborn, sat at the table and fell into deep thought.

A pen also started to scribble on the manuscript paper. Okay...

This problem seems not that simple, the main problem is the transformation of the problem.

After thinking for a long time, Qiao Yu came to the conclusion that if the upper bound estimation of rational number points can be transformed into a problem of homology and geometric properties on a complete geometric object, then it is logical to use the deep tools of p-adic geometry, such as complete Algebraic space, geometricalization of modular forms, and p-adic homology theory are used to analyze these rational number points.

I just don’t know if this transformation will make the problem more abstract and complex.

But it doesn't matter, he is just a little Kalami anyway, he is just playing. There is no money to try? So soon Qiao Yu wrote this paragraph on the manuscript paper with great interest:

"Suppose X is a high-dimensional algebraic curve defined on the number field K, and The number satisfies: N(X)≤C." Naturally, N(X) represents the number of rational points on the curve X.

It's just the intuition that just came out of Qiao Yu's brain that there must be such a constant C. The reason is very complicated. It is related to the geometric configuration of the curve in the complete space. You need to understand Peter Schulz's theory to understand it. This proposition. The first step he needs to do now is to prove this proposition.

Because as long as the existence of this constant C is proved, this conclusion will provide a solid theoretical basis for estimating the upper bound of the number of rational points on complex high-dimensional algebraic curves. After proving the first step, it is to find the formula of this constant C , and prove that this formula is correct.

Then - problem solved!

However, when Qiao Yu was preparing to prove this proposition with great ambition, he suddenly felt that the question he raised seemed a little difficult to start with.

He seems to be stuck in the cycle of steps it takes to put an elephant in the refrigerator.

The first step is to open the refrigerator door, the second step is to put the elephant in, and the third step is to close the refrigerator door. The only problem is that he seems to have not found a refrigerator as big as an elephant yet!

In particular, Qiao Yu suddenly discovered that even if this constant C formula really existed, it would not only depend on the geometric properties of the curve, but also may depend on the characteristics of the number field K, the modular structure of the curve and even other algebraic geometry tools. Because he

After racking his brains, Qiao Yu found that the existing algebraic geometry tools did not seem to support finding this C.

A normal math person would probably give up at this time, but Qiao Yu is different. He is just a newbie in math, and he has already regarded this challenge as a game. Although he has no clue, what if he succeeds?

And again, if you don’t have tools, you can make it yourself.

It makes no sense that Peter Schulz could create such a powerful theoretical framework when he was only 21 years old. It makes no sense that he couldn't create a few useful mathematical tools at the age of fifteen, let alone the entire theoretical framework.

It was provided by others, and he only needed to create it twice within the framework, which was significantly less difficult.

After all, the rules are already there. He only needs to prove that his tools are correct through rigorous mathematical logic within the constraints of this framework rule. So the next work can be further simplified. What kind of algebraic geometry

Tools can help him prove that this constant C exists.

Qiao Yu thought for a long time with a frown on his face, and then confirmed again. First, he needed a new homology category tool. So another row of writing appeared on the manuscript paper:

"The homology category QH(Cp) is an enhanced homology category, defined on the completion space of the algebraic curve Cp. Its basic object is the traditional homology class H^i(Cp,Zp), but we need to perform special processing on it,

Through a new operator Q, which acts on the homology class, each object in the homology category not only has a topological structure, but also has an additional invariant..."

Huh... Qiao Yu looked at this expression with great satisfaction. With this new homology category, the homology group of the curve can be decomposed more finely, and the steps to prove the constant C can be greatly simplified. Perfect! Sure enough, the research

Mathematics makes people happy!

So now a new question comes again, how to define this new operator Q, Qiao Yu feels stuck again.

MMPD, forget it! I can’t figure it out. Let’s put this aside for now. Anyway, to prove the constant C, this tool is not enough.

So Qiao Yu, who had gone completely crazy, started to create a second tool. Now he needed a new fuzzy measure function to approximate the constant C.
Chapter completed!