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Chapter 103: Crazy Math Novice

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

Human thinking, especially mathematical thinking, requires continuity. When one is immersed in a certain state and is suddenly interrupted, it is difficult to get back to that state.

This is true.

Peter Schultz's thesis seemed to open the door to a new world for Qiao Yu.

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

He couldn't even find the right words to describe the feeling.

If I had to use a metaphor to describe this feeling, it would probably be a counterintuitive sense of geometry.

After reading the paper carefully, his brain seemed to be occupied by countless high-dimensional geometric spaces, but these spaces were not as smooth and continuous as the Euclidean space that we can see everywhere in our daily life, but were divided.

In particular, the rigid analytical space turned into a geometric object that was infinitely divided and layered in Qiao Yu's mind, and each dividing line was so precise and subtle.

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

It is these complex lines and geometric shapes that make the space no longer continuous, but instead present a discrete yet tight structure.

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

There is no smoothness or intuitiveness in this geometric world. It is constantly reconstructed by various expansion methods, extended infinitely, and finally constitutes a universe.

A universe interwoven with algebraic symbols and geometric figures. Every point in the universe contains infinite times of detailed divisions and infinite levels of details under a special law. These laws govern these points, lines, and surfaces, pushing the structure 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 concentrated state because of a deep feeling of fatigue transmitted from his brain, it was already twelve twenty-three in the morning.

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

However, considering that he had been on the bus for six hours today, it was probably normal for Qiao Yu to feel very tired.

So on the first night after coming to Beijing and studying at the university, Qiao Yu climbed into bed without even washing up. What's more annoying is that Qiao Yu, who always had a good sleep, had a strange dream that night. In the dream, he came to a strange maze world, which was composed of countless strange doors.

Every time Qiao Yu opened a door in his dream, he would 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 those in the real world.

It is possible to be here and there at the same time to test the probability?

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

It was not until Qiao Yu opened a door again that he suddenly felt a burst of light shining into his eyes, and then he subconsciously opened his eyes and woke up...

He forgot to draw the curtains last night, and the rising sun shone through the window into his eyes. He felt dizzy before he even got up.

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

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

Even if I didn't go to bed until one o'clock on the 30th, I would still get out of bed on time at seven forty the next day.

In this respect, Qiao Yu was different from other children since he was young.

Other children could never wake up, but Qiao Yu only needed to sleep for more than seven hours a day and take a ten or twenty minute nap at noon to be energetic all day long.

Although he seemed to have dreamed all night yesterday and could still remember those dreams clearly, Qiao Yu still felt in good spirits. He was not drowsy or listless due to the dreaming and simply got up from the bed.

Keeping in mind what Lao Xue had told him yesterday, Qiao Yu put on his clothes and dressed himself neatly before picking up the mouthwash cup and toothbrush on the windowsill. He then put the towel on his shoulders, walked out of the room and headed for the second floor around the corner.

Lao Xue took him there yesterday. The toilet and restroom are both on the second floor.

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

"Um, I'm..." Qiao Yu was about to introduce himself when the person on the other side suddenly realized something and asked, "Oh, you're the new student, right? Your name is, Qiao, um, Qiao what's that?"

"Qiao Yu, the 'yu' in metaphor!" Qiao Yu emphasized the word 'yu'.

When Lao Xue asked him about his identity yesterday, Qiao Yu was not defensive.

Today, this middle-aged man actually didn’t remember his name, which really broke his defense!

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. His mentor really had many students all over the world. This guy looked to be in his forties, but he still had to call his mentor Tian "teacher". This showed that he was of very high seniority.

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

"Oh, my name is Chen Zhuoyang, and I'm Director Tian's doctoral student. Because I'm responsible for some lectures and things like that, my office is upstairs. You can go up and sit down when you have time." Chen Zhuoyang said politely.

Although Qiao Yu looked very young and immature, the fact that the tutor had mentioned this to him specifically was enough to prove his importance to 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 ask hesitantly: "Um, Brother Chen, can I ask you a question, how old are you this year?"

"Me? 29? What's wrong?" After saying that, Chen Zhuoyang looked at Qiao Yu's astonished eyes, subconsciously touched his head, smiled self-deprecatingly and said, "Haha, I haven't even graduated yet, but my hair is almost falling out! Others say I look like I'm in my thirties, alas..."

"No, Brother Chen, you are too modest. I would believe you even if you said you were forty, seriously." Qiao Yu really couldn't help it, and even forgot about Qiao Xi's reminder for him not to cause trouble when he went out, and said it very honestly.

"Hmm..." Chen Zhuoyang thought for a moment and replied seriously, "Junior brother, you have to remember that it doesn't matter whether a man is handsome or not. The key is that he has to be talented. And you haven't even entered the threshold of mathematics yet, so you don't know how difficult it is to learn mathematics well. I'm already pretty good."

In fact, when Qiao Yu blurted out those words just now, he felt a little regretful and had even thought about apologizing.

But as soon as Chen Zhuoyang said this, Qiao Yu couldn't help but say, "Well, Brother Chen, I'm not trying to argue. 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. Your talent is incomparable. I saw his photo online, and he had a lot of hair."

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

Qiao Yu replied: "Peter Schultz."

Chen Zhuoyang stared at Qiao Yu in silence until Qiao Yu felt a little uneasy. Then he slowly said, "Junior brother, thank you for comparing me with the youngest Fields Medal winner! But I still suggest that you wait until you start writing your master's thesis before discussing this issue with me.

Although being a teacher's student is not short of resources, you will know how strict the teacher's requirements are in the future. When you come crying and beg me, well... I have decided that I will treat you as air, and nothing you say will be of any use. "

After saying this, Chen Zhuoyang raised his head proudly and walked away.

The sound of "dong dong dong" coming up the stairs reached his ears, but Qiao Yu hadn't reacted yet. Maybe he was angry?
Senior Brother Chen is so cute. He never talks trash even when he’s angry. Even his threats are cute and have no intimidating effect at all.

The only pity is that we left too fast.

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

What a joke, a master's thesis, can't anyone write one with a brain? No matter how strict Director Tian is, he wouldn't ask him to solve a world-class mathematical problem for his master's thesis, right?

Mumbling these messy things in his mind, Qiao Yu quickly walked into the bathroom, tidied himself up, returned to the room to put his things back in place, then took his meal card and went out to have breakfast.

The research center is a long way from the cafeteria that Lao Xue took him to yesterday, but Qiao Yu thinks this is actually quite good. Walking back and forth is just the right way to exercise in the morning. And now it's winter vacation, so there aren't many cafeterias open. It should be more convenient after school starts.

I went to the cafeteria, bought two big buns and a cup of soy milk, and ate 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, but there were some proofs that I couldn’t understand yesterday, but when I looked at them again today, I felt I understood them thoroughly.

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

For example, when it comes to etale homology, the simplified computational framework provided by complete spaces. Yesterday, I couldn't figure out how this was done. After reading it today, I found that it is nothing more than mapping the homology classes in some complete spaces to the etale homology classes of classical geometric objects through the structure of the complete space.

This allows the homology class structure of algebraic varieties to be calculated more efficiently in a p-adic background.

After feeling that he had thoroughly understood these abstract things, Qiao Yu even wrote a related math problem on his manuscript paper based on the content he could not understand yesterday.

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

After finishing, Qiao Yu instantly felt very cute. He looked at the time and it was already half past ten in the morning. So he stood up and was about to go out to stretch his body. When he turned his head, he saw through the window that Director Tian and Lao Xue were walking towards the room together.

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

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

"Well, I came here specifically to see you. How is it? Are you used to it?" Tian Yanzhen asked when he entered the door.

"It's okay, I'm used to it. The water for washing my face in the morning was a little cold, but everything else is fine." Qiao Yu said truthfully.

"Don't you know to buy a thermos and get some boiling water? This kid..." Old Xue was speechless at Qiao Yu's complaints.

"Oh!" Qiao Yu responded honestly.

"Okay, it's normal that you're not used to it just now. You're so young, it's okay to wash your face with cold water, it will make you more energetic." Tian Yanzhen said as he walked to Qiao Yu's desk, glanced at the manuscript paper casually thrown next to the computer, and then picked it up with interest.

"Um, Director Tian, ​​this is a random question I came up with after reading a paper yesterday." Qiao Yu explained quickly.

This proves that he is very meritorious, as he started studying right after arriving at Yanbei University.

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

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

Qiao Yu answered, "It's the paper published by Peter Schultz in 2011 on the P-adic Hodge theory of rigid analytic space and complete space."

Xue Song looked at Qiao Yu with a smile on his face and said, "Didn't I tell you yesterday to download Professor Robert Greene's paper and study it? You're going to listen to the professor's lecture tomorrow, but you didn't do any preparation? Why are you reading Peter Schultz's paper?"

"Ah?" Qiao Yu then remembered that Lao Xue did mention this after he had set up the library software account yesterday.

After smiling sheepishly, he explained, "I forgot. Before you left yesterday, you told me about the story of Zheng Zhiqiang and Peter Schultz. Zheng Zhiqiang was a computer scientist, so I ignored him and downloaded Peter Schultz's paper, and I accidentally got fascinated by it."

This answer left Xue Song speechless.

He told Qiao Yu about these two people yesterday and seemed to want him to not be so proud, but this guy ended up studying the complete space theory pioneered by Peter Schultz?
This is not typical yet...

Xue Song was about to teach Qiao Yu a lesson when Tian Yanzhen suddenly handed Qiao Yu's manuscript to Xue Song and said, "Professor Xue, take a look too."

"Um, okay." After taking the manuscript paper handed to him by Tian Yanzhen and just glancing at the title, Xue Song didn't know what to say.

Wow... do you really understand this?
On the other side, Tian Yanzhen had already started chatting with Qiao Yu: "Have I ever told you that the mathematical knowledge you have currently come into contact with is not systematic?"

"Yes." Qiao Yu nodded honestly and answered blankly: "Last time when I was in CMO, you told me in the dormitory."

"Well, so Professor Xue and I worked together to develop a study plan for you, so that you can make full use of this six months to make a general review of the entire algebraic geometry and number theory system.

Professor Xue has also developed a whole set of learning and training courses for you, but I just changed my mind. In the next six months, you can study by yourself by reading papers in the library's database according to your interests.

If you have any questions, just ask Professor Xue or me. Professor Xue should be able to take time to answer your questions at any time. I will take half a day every week to discuss relevant issues with you.

If it is beyond the research scope of Professor Xue and I, there are many professors in the research center who you can consult. I will ask Professor Xue to send you a directory with the contact information and research directions of all professors in the research center and the School of Mathematics.

I will help you say hello when the time comes, but when you ask other professors for advice, you must be humble and respectful, understand? Everyone is answering your questions for free."

Qiao Yu immediately nodded and assured, "Don't worry, Director Tian, ​​I understand this. To be honest, I am actually the most polite person."

Tian Yanzhen said again: "Just remember it. Don't read Schultz's paper this afternoon. Download two papers written by Professor Robert in the past five years and take a look at them. You should have come across some of the things in them, which can help you understand what he will talk about tomorrow.

This will not only teach you to cherish the opportunity to attend such lectures, but also show the most basic respect to the specially invited lecturer, and it is also the most basic courtesy. Not only this time, but also in other lectures that you are interested in, you should get used to doing this. "

"I understand. I will definitely remember this time and promise to read the professor's paper this afternoon." Qiao Yu said immediately.

While the two were talking, Xue Song had already roughly reviewed Qiao Yu's question and the solution process.

The mood is very complicated.

He also read this early paper by Peter Schultz.

He was still in Princeton at that time.

To be honest, this thing was obviously too high-end for Xue Song at that time. Well, in fact, it is still the same for him now.

Complete spaces are geometric objects defined on the p-adic number system. The p-adic number system itself is a relatively abstract and complex mathematical system, which is completely different from the intuitive real or complex number system.

What’s even more terrifying is that this theory also combines multiple 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 all the above difficulties can be overcome, then the tilt theory involved in the complete space is really puzzling. This idea introduces a new algebraic geometry perspective to associate a p-adic complete body with a perfect body of characteristic p.

But this relationship 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 the existing mathematical framework. To understand Peter Schultz's ideas, one needs to establish a completely new mathematical system in one's mind.

Qiao Yu downloaded a paper on this subject yesterday and took a quick look at it, but he seems to have understood it today?!
Is this an advantage of having a poor foundation? No, if you have a poor foundation, 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 Schultz.

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

This is probably a laissez-faire strategy. But the resources prepared for Qiao Yu are all top-notch, and it seems to be something that people look forward to seeing what results he can come up with by tinkering with it every day.

"Professor Xue, what do you think?" Tian Yanzhen asked. Xue Song shook his head, smiled bitterly, and said, "I think this arrangement is also good. It's only half a year anyway. Let's give him enough freedom first, and then see what his results are."

Tian Yanzhen nodded, looked at Qiao Yu and said, "That's settled, do you have any questions?"

Qiao Yu asked quickly, "By the way, is there another Senior Brother Chen in this building?"

Tian Yanzhen nodded and replied, "Yes, his office is on the third floor. Did you meet him?"

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

Tian Yanzhen looked Qiao Yu up and down and laughed: "Why, are you addicted to giving advice to your fellow students? You helped Professor Xue's master's students revise their papers before, but you felt that they did not reflect your level, so you want to give some advice to the doctoral students?"

Qiao Yu denied it repeatedly, "No, no, I just want to hear how difficult Brother Chen's topic selection is, so that I can be mentally prepared for the topic selection for my future doctoral program."

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

However, after he finished speaking, seeing Qiao Yu's eyes full of curiosity, he still revealed: "His research direction is geometric analysis on complex manifolds, specifically the existence and stability of minimal surfaces on Kähler manifolds.

For example, how to construct minimal surfaces through variational methods, the stability of energy functionals of minimal surfaces, and the evolution of minimal surfaces under the action of Ricci flow, etc. ”

"Oh!" Qiao Yu nodded, looking as if he suddenly realized something.

"What? Have you studied these too?" Tian Yanzhen asked.

“No!” Qiao Yu shook his head and said, “I just think there’s a reason why my senior hasn’t graduated at the age of 29. When I was watching a video before, a senior said that things involving functional analysis are very difficult.”

Xue Song next to him couldn't help but said, "Is functional analysis difficult? No matter how difficult it is, it can't be more difficult than Schultz's complete space! As long as you can fully understand Hilbert space and Banach space, learning functional analysis is a very simple matter."

Qiao Yu looked at Xue Song blankly and replied, "Ah? So the p-adic number analysis method involved in analyzing infinite-dimensional space in the paper is the technique used in functional analysis? P-adic Banach space is used in many places in the paper."

Tian Yanzhen nodded and replied: "When doing the analysis of the topological properties of p-adic numbers, function spaces and p-adic representation theory, the structure of Banach space does appear frequently. However, p-adic numbers are quite special, and the Banach space involved is different from the situation of real number analysis, which is what you call the p-adic Banach space.

Don't worry about it. You will understand when you have a broader knowledge. In cutting-edge mathematical research, many mathematical tools overlap. This is why mathematicians need a comprehensive foundation. But I want to see how far you can go in your own way. Any other questions?"

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

"Then we'll leave first. Professor Xue will come to take you to attend Professor Robert's lecture tomorrow." After saying that, Tian Yanzhen left with Xue Song.

Qiao Yu watched the two professors leave, then stood up, stretched his body, and started reading the paper again.

Director Tian said that we can start reading Professor Robert’s paper in the afternoon.

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

……

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

Tian Yanzhen thought for a moment and said, "Don't rush to make a decision. Let's see how far this child can go. To be honest, I don't really know how to teach him now. I just think it might be more appropriate to let him take the initiative to discover the problem himself."

Xue Song nodded. To be honest, he didn't have any good way to teach Qiao Yu. When he saw the manuscript paper where Qiao Yu asked and answered his own questions, he even began to doubt whether he had the ability to be a good guide for Qiao Yu.

It is no exaggeration to say that there are not many mathematicians in the world who can understand what Peter Schultz is doing. After all, this is the most basic research in mathematics, aiming to build a bridge between algebraic geometry, number theory and p-analysis.

Qiao Yu seemed to have an extremely amazing talent in understanding these complex mathematical ideas. Not only had he never taught such a student, he had never even met such a student before. He suddenly felt a great deal of pressure.

"Don't think too much. Speaking of Peter Schultz, there is an interesting story. He gave his thesis to his mentor Lambert. After reading it, Lambert told him that he could graduate with a doctorate. So we don't need to worry too much about true geniuses."

Tian Yanzhen added optimistically.

After hearing this, Xue Song let out a long sigh and couldn't help asking, "When you were teaching at Princeton, you came into contact with many students. Have you ever met anyone as talented as Qiao Yu?"

Tian Yanzhen smiled and said, "You also studied at Princeton's School of Mathematics for eight years. You should know more about your classmates than I do, right?"

Xue Song shook his head and replied, "There are really many geniuses. I am an ordinary one among them. But if I were to say who is the kind of genius that I really admire from the bottom of my heart, there really isn't one."

"That's because you are one of the geniuses." Tian Yanzhen said with emotion: "Students who can successfully graduate from Princeton are geniuses compared to ordinary people. Not to mention those who can graduate with a doctorate. But geniuses in the fields of mathematics and theoretical physics are ultimately divided into different levels!"

This one sentence completely made Xue Song lose interest in chatting.

This is such a hopeless field, geniuses are divided into different levels...

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

"You're too kind!" Xue Song said politely.

"Okay, I'll go back first. You can do your thing. Oh, by the way, the joint training plan with Yujiang University has been drawn up, and I have also secured some rights for your students. If their achievements during the joint training period meet the standards of Yanbei University, they 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 door, Xue Song stood there thinking for a moment, then took out his mobile phone with a smile, and started editing a message while walking out of the research center.

This news should give his doctoral students some encouragement, right? !
……

At noon, Qiao Yu went to the cafeteria alone to have lunch. After coming back and taking a nap for ten minutes, Qiao Yu began to search for Professor Robert's paper in the background of Yanbei University Library and downloaded it.

The ability to listen to what needs to be heard is also a necessary quality for students.

Especially what the instructor has repeatedly emphasized, even linking it to politeness and respect, that is what you must listen to.

As for the rest... actually there are choices.

It is highly unlikely that someone who can become a big shot would haggle with his students over every little thing. At least that’s how Qiao Yu understood it.

For example, Principal Zhang of Xingtie No. 1 Middle School.

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

Qiao Yu felt that even if he was bored enough to take down a few of Tieyi Middle School's signs, Lao Zhang would smile and ask the school logistics department to make new ones, and then tell him that it would not happen again.

I searched for Robert Greene's name in the paper retrieval system of Yanbei University Library, and a bunch of papers appeared at once.

Qiao Yu was shocked, but soon found out that they were not all from the same person.

There seem to be a lot of people named Robert Greene abroad.

Although I encountered similar problems when searching for Peter Schultz, there was only one distractor, and that guy was studying chemistry. The direction of the paper was completely different.

But this guy Robert has written many papers on mathematics.

Fortunately, Qiao Yu discovered that this paper retrieval system is actually very easy to use. Not only is it rich in content, but you can also choose the year yourself. The advanced search page even supports searches of author units.

Qiao Yu remembered that Lao Xue said the professor was from New York University, which made things much more convenient.

Soon, Professor Robert's paper was downloaded.

I don’t know if it was because he studied Peter Schultz’s paper first that opened up Qiao Yu’s mind again, but Qiao Yu actually felt that the professor’s paper was quite easy to understand.

Well, it may sound a bit easy, but at least it’s not difficult.

For example, Qiao Yu really thinks that those lemmas, theorem preconditions, a series of concepts, and proof processes are easy to understand. You don't need to spend too many brain cells to understand them. But this combination of work and rest is pretty good.

Reading Peter Schultz's paper yesterday was really too taxing, so today I'll just relax by reading a paper that's 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 honestly. He then had dinner in between.

Putting down the paper, Qiao Yu began to think as usual, and suddenly an idea came to his mind.

To put it simply, what Professor Robert is studying is the problem of accurately predicting the upper bound of the number of rational points of a given type of algebraic curve, especially a high-dimensional algebraic curve. This type of problem is actually closely related to the Diophantine equation.

Find 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 homological properties. These geometric characteristics all affect the distribution of rational points.

Therefore, there is 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 being able to quickly determine whether a solution exists for a high-order Diophantine equation and solve this type of equation.

Well, that’s how Qiao Yu understood it anyway.

This is the understanding of a math layman. If Lao Xue were here at this moment, after listening to Qiao Yu's ideas, he would probably want to beat up this ignorant guy.

The reason is very simple. The research goal is simply too ridiculous.

Simplify 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 mathematical knowledge. Now what we do is to efficiently estimate the number of rational points through geometric and algebraic tools, and understand their distribution through modern algebraic geometry tools.

As for solving Diophantine equations quickly?

When it comes to solving elliptic curves or more complex equations related to modular forms, even if it is determined that there is a solution, if you really want to solve it, all Lao Xue can do is say haha.

Of course, these were not a problem for Qiao Yu, a layman who did not have much awe of mathematics. In addition, he had just learned about Peter Schultz's mathematical ideas yesterday. Suddenly, a very bold idea popped up in Qiao Yu's mind, and he couldn't stop it.

Why couldn't he try to solve this kind of problem using the theory created by Peter Schultz?

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

Although this is a framework built by others, as long as we create tools within this framework and in accordance with the rules of this framework, as long as it can solve the problem, it will definitely be feasible.

Now the problem facing Qiao Yu is very simple: how to introduce the problem of estimating the upper bound of rational points on algebraic curves into the framework of the theory of quasi-complete spaces?
Qiao Yu, who was as fearless as a newborn calf, sat at the table and fell into deep thought.

A pen also began to scribble on the manuscript paper.

Ok……

This problem does not seem to be that simple, mainly due to the transformation of the problem.

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

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

But it doesn't matter, he is just a little Karami, he is just playing. It's free to try?

Soon, Qiao Yu wrote down the following words on the manuscript paper with great interest:

"Let X be a high-dimensional algebraic curve defined over the number field K, and X be a closed subset in a p-adic complete algebraic space. Then there exists a constant C that depends on the geometric properties of the curve X, such that the number of rational points on the curve satisfies: N(X)≤C."

Naturally, N(X) represents the number of rational points on the curve X.

It's just that Qiao Yu's intuition just now is 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 Schultz's theory to understand this proposition.

Now the first step he needs to take is to prove this proposition.

Because as long as it is proved that this constant C really exists, 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, the next step is to find the formula for the constant C and prove that the formula is correct.

And 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 approach.

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

Step one, open the refrigerator door, step two, put the elephant in, step three, close the refrigerator door.

The only problem is, he can’t seem to find a fridge the size of an elephant yet!
In particular, Qiao Yu suddenly discovered that even if the formula for constant C really exists, it will not only depend on the geometric properties of the curve, but may also depend on the characteristics of the number field K, the modular structure of the curve and even other algebraic geometry tools.

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

If it were a normal mathematician, he would probably give up at this point, but Qiao Yu was different. He was just a math novice and had already treated this challenge as a game.

Although I have no idea, but what if it succeeds?
And as I said, if you don’t have the tools, you can make it yourself.

Peter Schultz was only 21 years old back then, yet he was able to create such an awesome theoretical framework. There was no reason why he couldn't create a few usable mathematical tools at the age of , not to mention that the entire theoretical framework was provided by others, and he only needed to carry out secondary creation under the framework, which was obviously much easier.

After all, the rules are already there. He just needs to prove that his tool is correct through rigorous mathematical logic within the limitations of this framework.

So the next step can be further simplified. What kind of algebraic geometry tools can help him prove the existence of this constant C?

Qiao Yu thought for a long time with a frown on his face, and then confirmed again that first of all he needed a new homology category tool.

Then another line of handwriting appeared on the manuscript paper:
"The homology category QH(Cp) is an enhanced homology category, defined on the completion space of algebraic curves Cp. Its basic objects are the traditional homology classes H^i(Cp, Zp), but we need to treat them specially, through a new operator Q, which acts on the homology class, so that each object in the homology category has not only a topological structure, but also an additional invariant..."

Huh... Qiao Yu looked at this statement with great satisfaction. With this new homology category, the homology group of the curve can be decomposed more finely, which can greatly simplify the steps of proving the constant C. Perfect!

Sure enough, studying mathematics makes people happy!
Now a new problem arises: how to define this new operator Q? Qiao Yu feels stuck again...

MMPD, never mind! I can’t figure this out, so I’ll put it aside for now. Anyway, this tool is not enough to prove the constant C…

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

"The p-adic fuzzy measure μfuzzy(Cp) of an algebraic curve is a new measure function used to describe the fuzzy properties of an algebraic curve Cp in a p-adic geometric environment. It is defined as follows..."

　The 18st day check-in for the -word update is completed!
　　Thanks to book friends 20201229074741818, book friends 20241005192534569, and Rainbow x for their rewards and encouragement!
　　In addition: I found out from the book review section that there are actually friends who study mathematics reading this book. I would like to emphasize again that all the so-called new mathematical theories involved in the book are made up by the author. They have no reference value and no mathematical logic at all!

　　This is just a novel, just read it for fun, if you take it seriously, the author is crazy! If someone really develops similar new mathematical tools or theories, it is purely coincidental!

　　
　
(End of this chapter)