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Chapter 150 This is not paranoia, it is self-confidence

At UCLA, Tao Xuanzhi is hosting three mathematicians from MIT and Oxford University: Harvey Guth, James Maynard, and Yuantang Zhang.

It is obvious that the three guests are big names in the academic world.

In particular, James Maynard won the Fields Medal three years ago for his contributions to analytic number theory, especially his research on prime numbers.

In addition, Tao Xuanzhi is also one of the youngest Fields Medal winners ever. If it were not for Peter Schultz, he would still be the youngest Fields Medal winner.

So this meeting was of very high standard, with at least two Fields Medal winners present.

The reason why the four of them gathered together today was a paper recently published by Harvey Guth and James Maynard on the pre-release website: "New Progress in Estimation of Large Values ​​of Dirichlet Polynomials."

Tao Xuanzhi's evaluation is that this paper has made an important breakthrough in the field of analytic number theory and is a big step forward on the long road of proving the Riemann hypothesis.

Most importantly, Tao Xuanzhi believes that this is the first substantial breakthrough in the Riemann hypothesis in decades, and it also adds new tools and ideas to the study of the Riemann hypothesis.

This is undoubtedly a high evaluation, even though the paper is still under peer review. So we simply invited both authors to come.

Zhang Yuantang is because of his achievements and status in prime number research. There have always been small-scale discussions among mathematicians of this kind. After all, it is convenient for everyone to meet.

The four of them just went through a brainstorming session for more than three hours, mainly with Tao Xuanzhi and Zhang Yuantang raising some questions, and then the two authors made some explanations and even revisions.

For example, in Sections 7 and 8, page 63 cites an equation that did not exist before, a reference is missing before Lemma 12.3, a function suddenly appears but is not defined in the paper, and a step lacks a valid reason...

Well, some of the problems seem outrageous, but anyone who has used a computer to write a paper knows that some minor flaws are inevitable.

As long as they are not errors at the logical level, many errors are difficult to avoid. Especially for papers on number theory, which often need to be revised repeatedly, it is normal to miss a few formulas because of the author's state at that time.

This is also the reason why many paper reviewers will argue repeatedly with publishers, often based on the rigor of this scientific paradigm.

Especially in mathematics, generally speaking, pointing out that there is false evidence in the key steps of a paper is considered questioning. Pointing out such small problems is considered discussion.

It’s like when Wiles proved Fermat’s Last Theorem, the editorial department arranged six reviewers, who found a large number of problems during the review process. Fortunately, most of them were small problems that Wiles could clarify immediately. Of course, if they could not be clarified, they would be big problems.

The same was true of Perelman's proof of the Poincare conjecture. When it was first published, there were many problems, and some people felt that it was not explained clearly and needed a more detailed proof. This was also the reason why it caused some controversy later.

In short, theoretical mathematics is like this, which is one of the reasons why it requires extremely high talent.

Mathematics allows mathematicians to freely make various definitions without considering the laws of reality, and even to construct theories based on any starting point as long as the logic is self-consistent.

But at the same time, the proof process is extremely rigorous. A little logical conflict or omission may lead to the overthrow of the entire theory.

……

"Speaking of new tool frameworks, we have to mention Qiao Yu. By the way, everyone should know Qiao Yu, right?"

After talking about serious academic topics, Zhang Yuantang turned the topic to Qiao Yu.

The other three bosses nodded at the same time.

Needless to say, Tao Xuanzhi was able to obtain a Ph.D. from Princeton University at the age of 21. The value of this is very high. Anyone who has some understanding of Princeton University graduates knows how difficult it is.

The most important thing is that Tao Xuanzhi’s talent has never been based on a certain direction.

He has conducted many top-notch research in the fields of harmonic analysis, partial differential equations, combinatorial mathematics, analytic number theory, and representation theory, and has achieved master-level results in more than a dozen mathematical fields.

When he won the Fields Medal at the age of 31, he had already published more than important papers in major mathematics journals.

The most important of these is the proof of the Green-Tao theorem, which also provides a new path for the study of the twin prime conjecture.

In a sense, Tao Xuanzhi, Peter Schultz and Qiao Yu are all top-tier geniuses who have demonstrated mathematical talent at a very young age, so naturally they will attract attention.

"Of course, Peter Schultz sent me an email and specifically mentioned Qiao Yu. He highly praised this young man. I have also read his papers. How should I put it... He is the most meticulous young mathematician I have ever met."

Tao Xuanzhi commented.

After hearing this evaluation, the other two mathematicians also nodded. In fact, this nod is meaningless and does not represent approval. It is more of a respect for the evaluation.

Zhang Yuantang smiled and said, "Not only is he rigorous, but more importantly, his ideas are very imaginative. Really, just like what I said just now, he proposed a brand new framework.

If he succeeds, he will not only be able to integrate the existing tools for studying number theory, but also perfectly combine number theory problems with geometry. Most importantly, after reading his ideas, I think he has a good chance of success. "

Zhang Yuantang's words immediately made the expressions of the three people become much more serious.

Whether it is integrating existing tools for studying number theory or completely combining number theory with geometry, this can be said to be a major breakthrough in mathematics.

Especially the latter.

There is no doubt that if Qiao Yu can really succeed, it will be a Fields Medal-level achievement.

Naturally, this also aroused the interest of the three big guys. The study of prime numbers is a number theory problem. If the theory proposed by Qiao Yu is really useful, it means that their research will have a new set of theoretical tools.

In particular, if geometric methods can be used without obstacles to solve number theory problems, it will be an important direction and one of the key areas in the development of modern mathematics.

After all, geometry is what provides many highly abstract and powerful tools for number theory.

"Um...can you tell me about this theory?" James Maynard asked cautiously.

After all, in the entire academic community, it is somewhat unreasonable to ask a third party about other people's research results that have not yet been officially published.

However, if it is just a general direction and does not involve proving the details, it does not matter.

So Zhang Yuantang nodded naturally.

He did not participate in Qiao Yu's subsequent work and did not know the details.

"Qiao Yu proposed a generalized modal number theory axiom system. Specifically, every natural number can be mapped into a modal space. This process is called modal mapping.

He defined the structure corresponding to the regular numbers. It includes the set of basic numbers, integers, fractions, and real numbers. It has the dependence of modal numbers and the self-referentiality of modal numbers. I will give you an example using arithmetic progressions..."

In this way, Zhang Yuantang spent more than twenty minutes explaining Qiao Yu's general idea.

A very general framework.

After listening to this, the three professors frowned and fell into deep thought.

There is no way, this is just a rough idea. It is still difficult to understand the content contained in it through simple explanation.

But everyone can understand the meaning of this.

"Wait, I can understand this kind of modal mapping. But since Qiao Yu is so ambitious, this framework must span a multi-dimensional modal framework, which raises a problem.

Many modal mappings are nonlinear and irreversible, which means that classical number theory methods cannot be directly applied within the framework. How can this problem be solved? "

After thinking for a moment, Tao Xuanzhi raised his own question.

Zhang Yuantang spread his hands and replied, "I don't really know the details of his handling. I can't ask in detail. But Qiao Yu should have a solution.

I remember he briefly explained that he constructed a supermodal operator matrix. Unlike traditional matrices, the elements in the matrix are not just arrays or linear operators.

Instead, it is a modal operator consisting of multiple mappings and self-referential relations. So each operator matrix has two dimensions, the normal dimension and the modal dimension.

The modal dimension can be used to represent the mapping of the matrix in different modal spaces. Even if this mapping is nonlinear and irreversible. "

Zhang Yuantang's answer was not that detailed. He knew that Qiao Yu had constructed such a matrix, but he really didn't know the more details.

That day, Qiao Yu put forward his idea. After a brief discussion, he left China and returned to the Western Hemisphere the next day.

It wasn't that he didn't want to stay for two more days, but mainly because Yanbei University didn't want to keep him any longer, so he naturally felt embarrassed to stay there all the time.

In fact, Tian Yanzhen had talked to him once about the possibility of returning to Yanbei University to teach, or getting a position at the Yanbei International Center for Mathematical Research, but Zhang Yuantang had never made up his mind.

"This idea... is very bold, and it seems to be effective. The overall integration of number theory and geometry... may even be more than just number theory and geometry, of course, if he can really succeed."

James Maynard thought carefully for a while and then spoke.

The emotions are complicated. As I said before, if Qiao Yu succeeds, this will undoubtedly be another Fields Medal achievement.

These talented guys are always so unreasonable.

"No wonder Peter Schultz and Qiao Yu get along so well. They are on the same path," Tao Xuanzhi said with emotion.

This sentiment is very appropriate.

There is no doubt that Tao Xuanzhi and Peter Schultz are both outstanding talents of the younger generation. However, the reasons why they were recognized by the Fields Medal are completely different.

Peter Schultz built a completely new system, while Tao Xuanzhi solved an important mathematical problem. The two took different paths, and now it seems that Qiao Yu also wants to follow Peter Schultz's path.

"Not necessarily. As far as I know, Qiao Yu designed this framework in order to prove the twin prime conjecture. Perhaps after this framework is built, he will launch an attack on the twin prime conjecture.

In other words, he might be able to build a programmatic system that can provide guidance for mathematics while solving a series of number theory problems. Perhaps he can combine your two paths.

And it is very likely. After all, he has made a great contribution to the proof of the geometric Langlands conjecture. Really, I thought there might be a challenge, but I didn't expect the challenger to be so young. "

Zhang Yuantang expressed a different opinion.

He had talked with Qiao Yu face to face, and he knew better than anyone else how oppressive it was to discuss academic issues with Qiao Yu. He could quickly think of how to answer some of the superficial questions at the beginning.

But as the discussion got deeper and deeper, he found it really hard to cope with it. The most important thing was that Qiao Yu's questions always got to the heart of the matter, and even led him to think about something deeper.

So the deeper the discussion went, the more oppressive it felt. As for the next day, when he was about to respond to the challenge from the young people, this framework hit him directly in front of him, leaving him at a loss as to how to evaluate it.

So he was happy to let Tao Xuanzhi realize this as well.

"What you said makes me want to communicate with him. If he is also concerned about the prime number problem, I wonder if he will see our paper. What comments will he make?" James Maynard said with a smile.

For people like them, who have spent too much energy on the study of prime numbers, who wouldn't want to be the first to solve the problems that have puzzled people for hundreds of years?
"Yes, Professor Zhang, maybe you can help me contact Qiao Yu. I am very interested in his idea. If possible, maybe we can cooperate."

Tao Xuanzhi suddenly spoke.

Just now, based on what Zhang Yuantang said, he made some simple deductions in his mind and suddenly realized that Qiao Yu's idea was indeed likely to succeed.

He still didn't know how Qiao Yu solved some of the problems, but there was no doubt that this was a brand new mathematical idea.

A more unified mathematical expression makes the proof process of number theory clearer. It is no longer necessary to build a complex system for a specific problem and use different types of modal spaces to represent different problems.

Qiao Yu is an ambitious guy! He wants to build a grand unified theory of mathematics in his own way. Tao Xuanzhi even suspected that Qiao Yu wanted to replace the Langlands program. Yes, he wanted to use his modal space theory as a replacement.

This does not seem impossible, because although Qiao Yu's method is also abstract, it is not as difficult to understand as the Langlands program.

In particular, the geometrization of number theory problems can make some obscure number theory problems more intuitive in modal space.

"I can ask that kid, but even though he's only sixteen... well, he's not averse to communication, but he has his own way of choosing his partners."

Zhang Yuantang said with a strange look on his face.

In fact, ever since he learned about Qiao Yu's topic, he has been paying attention to the related progress. Of course, the result surprised him.

"Paranoia?" asked Harvey Guth, who had remained silent on the subject of Qiao Yu.

He knew the least about Qiao Yu, and had only heard about some of the things that happened at the World Congress of Algebraic Geometry, so he had not expressed any opinion just now.

"It's not paranoia, it's self-confidence to be precise. I think he probably thinks he can complete the project independently. So when he chooses collaborators, he likes to choose people who have a closer relationship with him, rather than people who can help the project."

Zhang Yuantang shook his head and corrected.

Well, this is understandable, and it can even be said that geniuses generally have this confidence.

Tao Xuanzhi also laughed and joked, "Indeed, if the main framework can be proven by itself, the rest are just minor verification work, and there is really no need for high-level collaborators.

But I am looking forward to what results he can come up with. Professor Zhang, you may keep me up at night for some time, especially when you think that someone can solve many complex number theory problems at once. "

Zhang Yuantang smiled and didn't answer.

Not only him, the other two also felt a sense of urgency.

If someone really proves a series of difficult problems about prime numbers in an unprecedented way, this would not be entirely good news for many mathematicians who have been studying prime numbers.

After all, no one wants to be a background board. If you don’t believe it, you can ask Sam and Frank.

"It's okay, just ask first. I haven't had any contact with Qiao Yu, and it might be rude to send him an email rashly. Please, Professor Zhang."

Tao Xuanzhi thought for a moment and said.

Zhang Yuantang smiled and nodded. Being rude is just an excuse. These geniuses are all proud.

……

Huaxia, Yanbei University.

At this time, Qiao Yu was indeed doing the work that professors on the other side of the ocean were concerned about.

He doesn't have to worry about the verification work, but he needs to do some work in advance.

What Qiao Yu was doing at this time was to transform a series of problems that he intended to solve using the modal space framework from classical expressions into expressions under the modal space.

For example, the classic statement of the twin prime conjecture is that there are infinitely many pairs of prime numbers (p, p+2), where both prime numbers p and p+2 are prime numbers.

Then the expression in multimodal space needs to be transformed into three questions.

1. In the modal space M, there are infinite pairs of modal points (r_p, r_p+2) such that the modal distance d_m(r_p, r_p+2) satisfies the fixed constraint.

2. The modal density function ρ_m(r) accumulates to infinity in the modal space region that satisfies the twin prime condition.

3. The distribution of twin prime pairs forms equally spaced points on the modal path Γ and exhibits periodicity and symmetry in the modal space.

Simply put, a classic number theory problem is decomposed into three geometric problems.

If he could prove all three geometric problems in modal space, it would mean that he had completed the proof of the twin prime conjecture.

Of course, the prerequisite is that his generalized modal number theory axiom system can be widely recognized by the mathematical community, and it can be proved that this axiom system can indeed be converted between geometry and number theory, and always remain verifiable.

But then again, someone else did the verification work, but he was the only one who did the conversion work himself.

After all, transforming the problem requires a very clear understanding of this axiomatic system and extremely high mathematical insight.

Similarly, the same steps are required to solve the Riemann hypothesis. First, convert the classical expression into a geometric expression under this framework, decompose the problem, and then prove it one by one.

This step actually went quite smoothly.

Even the transformation of the Riemann hypothesis is simpler than the twin prime conjecture.

And in the classical interpretation, all zeros are distributed on a line. But in modal space, they are distributed on a hyperplane.

Of course, the completion of the conversion does not mean that the problem can be solved immediately. There are still many things to be defined to achieve this step.

For example, geometric tools such as modal density and convolution. In short, after geometricizing and modalizing the problem, Qiao Yu knew what tools were needed to solve the problem, and then he could prove and transform them one by one according to the framework.

Qiao Yu did not think the same as the professors opposite him, or even as Director Tian and Mr. Yuan. He had no intention of building the entire theoretical framework first.

His plan is to build as needed.

What tools are needed to prove the upper bound conjecture? First derive the required tools in the form of theorems, and then prove the problem.

Then we will see what new tools are needed for the twin prime conjecture, and then proceed to the next stage of derivation, and then start proving it…

The advantage of doing this is naturally that you can publish the most articles, and others can't even say that you are writing a fake paper.

Whether adding new tools or solving new problems, they are the most popular content in the mathematics community. Even the Langlands Program is composed of many sub-conjectures.

This is actually the reason why Qiao Yu has no interest in evaluating the fund. After all, even if he gets the grant, the money is not in his personal account.

Instead, the funds will be transferred to the research center's account, and then a sub-account will be set up under it. When the money is needed, it will be transferred directly. Not to mention that the funds allocated to pure mathematical theory are generally not much.

It's mainly a reputation. But Qiao Yu doesn't think he's that anxious to gain fame. There's no need to rush to build a framework to benefit the mathematics community.

After all, China's research progress in theoretical mathematics is far behind that of the West. After his new axiomatic system is fully contributed, it is likely that others will be the first to use it to prove some cutting-edge propositions.

After finishing these basic tasks, Qiao Yu stretched himself and planned to ask others about their work progress on WeChat.

Yesterday, he specially created a group chat and added Qiao Xi, Xue Song and Chen Zhuoyang to the same discussion group to make it easier for him to assign tasks.

Then he saw a new email notification in his work mailbox, which was from Professor Zhang Yuantang, so he subconsciously clicked it.

Even though he is now somewhat famous in the mathematics community, he doesn't actually exchange many emails on a daily basis.

The main reason is that there are more email communications within Professor Li’s research group at Huaqing.

As for other big guys, they only send emails occasionally to discuss some issues. This is related to everyone being busy, but also to the fact that Qiao Yu has not yet developed the habit of communicating via email.

"Qiao Yu:
Today, I was fortunate to be invited by Professor Xuanzhi to discuss his latest article "New Progress in Estimation of Large Values ​​of Dirichlet Polynomials" with Professor Gus and Professor Maynard, and I felt that I gained a lot.

I remember that you said you were very interested in prime number problems when you were in Yanbei, so I recommended this article to you. The article has been published on the preprint platform arXiv, and the authors are James Maynard and Harvey Guth.

Besides discussing the article, I mentioned to the three professors the generalized modal number theory axiom system that you are trying to construct. Professor Tao Xuanzhi was very interested.

In recent years, Professor Xuanzhi has been trying to combine the analytic theory of prime numbers with the extreme value principle in combinatorial number theory to study the characteristic relationship between prime number distribution and modular form, and to find similar prime number properties in general number sequences and functions.

And achieved many achievements, such as the development of genetic sieve method to analyze the role of sieve method in complex sets, especially for constructing prime number sets with specific properties.

He is also committed to promoting the Polymath project, which has reduced the distance between prime number pairs from 7000 million to within 600. So he hopes to establish cooperation with you to jointly explore the geometric content of prime number problems.

If you are also interested, please let us know the convenient time or communication method.

Looking forward to your reply and best wishes!
Zhang Yuantang.

After quickly scanning the letter, Qiao Yu subconsciously opened the webpage and searched for the names Tao Xuanzhi, James Maynard, and Harvey Guth...

Yes, Qiao Yu is not only a layman in mathematics, but also a layman in the academic world. He really doesn't know many mathematics masters.

However, he knew that since he could easily invite Zhang Yuantang and even let Professor Zhang write this letter specifically for him, he must be a big shot in the mathematics community.

And indeed it is.

A quick search reveals two Fields Medal winners, and another who has never won a Fields Medal but seems to have a high status in the mathematics community.

The most important thing is that they are different from the Fields Medal winners I met at the World Algebraic Geometry Conference last time. Both Tao Xuanzhi and James Maynard are still very young.

Tao Xuanzhi just turned fifty this year, and James Maynard is even younger, with only one year left before he turns forty.

Qiao Yu could see that Tao Xuanzhi was very active in the mathematics community. Last year, he teamed up with more than 60 mathematicians to set questions and launched FrontierMath, a mathematics benchmark for testing the mathematical capabilities of artificial intelligence.

Simply put, FrontierMath is an original question bank. The benchmark contains hundreds of original and challenging math problems, covering the main branches of modern mathematics, such as number theory, real analysis, algebraic geometry, and category theory.

Then let the most advanced AI go to the question bank to answer the questions...

After seeing this Qiao Yu, I suddenly realized that Yu Wei’s sudden desire to do AI is indeed a visionary one.

Peter Schultz is doing AI mathematics research with Microsoft, converting mathematical theorems into something that AI can understand, while Professor Tao Xuanzhi is developing a benchmark question bank for AI mathematics tests.

After getting a general understanding of the life stories of these bigwigs, Qiao Yu casually logged into arXiv and downloaded the article "New Progress in Estimation of Large Values ​​of Dirichlet Polynomials" recommended by Professor Zhang to his computer.

It is worthwhile to read the latest papers of Fields Medal winners, not to mention that these Fields Medal winners are still concerned with the problem of prime numbers.

To be honest, Qiao Yu was also quite afraid that someone would solve the twin prime conjecture and the Riemann hypothesis before him.

The former doesn't matter, but the latter involves a prize of one and a half million US dollars.

At least for Qiao Yu now, he is still very interested in this 1.5 million US dollars. Converted into RMB, it is worth tens of millions. Not only is it given by Americans, but there is no tax on it.

After all, the Fields Medal, which a bunch of mathematicians are desperate to win, only has a prize of 1.5 Canadian dollars, which is less than yuan in RMB.

Not to mention the Nobel Prize with a prize of 1100 million Swedish kronor and the Turing Award with a prize of one million US dollars, it is not even comparable to the Wolf Prize.

The Wolf Prize is worth $100,000!

This is probably the thing that Qiao Yu is most dissatisfied with about the Fields Medal.

However, after downloading the paper, Qiao Yu thought about it and did not read the paper. Instead, he took a screenshot of Zhang Yuantang's email, then opened the WeChat group, sent it directly to the group, and then @ed everyone at the same time.

“Comrades who are striving for the same goal, please take some time to read this email. Fields Medal mathematicians want to cooperate with us, but I do not intend to cooperate with them.

Because I believe that we can complete this project independently! So please work hard together! As long as we can achieve this result, even the Fields Medal winners will be envious!

Comrades, please reply if you receive this! "

As the initiator of this project, Qiao Yu felt that it was necessary to give everyone a boost from time to time, so as to urge everyone to complete their tasks as soon as possible.

Although one of the people in the group was a professor who had given him many suggestions and could even be said to have discovered him, one was his senior, and another was his mother.

Qiao Yu has always believed that as long as he is not embarrassed, then others will be embarrassed.

And that seems to be the case.

Because the message was sent and everyone was @ed, but no one responded for a long time.

……

In the office, Xue Song was assigning tasks to his doctoral students. Because he had just arrived at Yanbei University, he had not yet been allocated a quota for enrollment this year, but all the doctoral students from Yujiang University had followed him.

Feeling his phone vibrate twice, Xue Song picked it up and took a look. For a moment he didn't know what to say.

During his many years of academic career, Xue Song has served as a project leader and also worked on projects with others.

So Qiao Yu's way of boosting his morale made him feel both familiar and strange.

Normally when working with professors, we don't do this. But our supervisors occasionally encourage us like this.

Especially the sentence "comrades working towards the same goal" really made Xue Song find it difficult to comment.

But then again, Xue Song was indeed excited that Tao Xuanzhi was very interested in this topic.

But what should he reply? If he replied "received", he would lose face, so he simply pretended he didn't see it and put down his phone.

He turned his attention back to his students.

"Junfei, work hard on this project. To get a diploma from Yanbei University, the requirements will be higher. Qiao Yu's project is a good breakthrough. Complete the verification work as soon as possible and submit it to me."

"Okay, boss."

……

Huaqing, Qiuzhai.

Qiao Xi glanced at the message sent by Qiao Yu, thought about it and then asked the mentor next to him.

"Teacher Yuan, is Professor Tao Xuanzhi very famous?"

"Huh? Why are you mentioning him all of a sudden?"

Qiao Xi handed the WeChat to the tutor beside him.

Old Yuan took a look and laughed: "Haha, Xuanzhi still has good vision."

Then he looked at Qiao Xi and explained, "Xuanzhi is one of the new generation of mathematical leaders. But you don't have to pay attention to these. Let that kid do it. It's better for you to lay a good foundation first."

"Okay, teacher."

So Qiao Xi also threw his phone aside.

This guy...he even said "comrades", maybe he wants someone to help him loosen up again.

……

Chen Zhuoyang, who was in a meeting, felt the vibration, took out his mobile phone and took a secret look at it, quickly browsing the letter that Qiao Yu had screenshotted, and felt a lot of emotion in his heart.

After reading what Qiao Yu sent, he really intended to reply with a received message.

After all, compared to Xue Song and Qiao Xi, he really didn't think there was anything wrong with replying with a "received".

However, before his hand touched the phone keyboard, he heard someone calling his name.

"Chen Zhuoyang, why don't I start with you..."

unlucky……

(End of this chapter)