Chapter 150 This is not paranoia, this is confidence
At the University of California, Los Angeles, Tao Xuanzhi is hosting three mathematicians from Massachusetts and Oxford University: Harvey Guth, James Maynard, and Zhang Yuantang.
Obviously the three guests are all bigwigs in academia.
Especially James Maynard, who just won the Fields Medal three years ago for his contributions to the field of analytic number theory, especially the study of prime numbers.
In addition, Tao Xuanzhi is also one of the youngest Fields Medal winners. If it were not for Peter Schulz, he is still the youngest Fields Medal winner.
Therefore, the meeting was of high standard, with at least two Fields Medal winners present.
The reason why the four of them got together today is because of a paper recently published by Harvey Guth and James Maynard on the pre-release website: "New Progress in Large-Value Estimation of Dirichlet Polynomials".
Tao Xuanzhi's evaluation is that this paper has made an important breakthrough in the field of analytic number theory and has taken a big step forward on the long road of proving the Riemann Hypothesis.
The most important thing is that Tao Xuanzhi believes that this is the first time in decades that a substantial breakthrough has been achieved on the Riemann Hypothesis. It also adds new tools and ideas to the research on the Riemann Hypothesis.
There is no doubt that this is a very high evaluation. Even though this paper is still in the peer review stage, I simply invited the two authors to come over.
Zhang Yuantang is because of his achievements and status in the study of prime numbers. There have always been small-scale discussions among mathematicians like this. After all, it is convenient for everyone to meet.
The four of them had just gone through more than three hours of brainstorming. Mainly, Tao Xuanzhi and Zhang Yuantang raised some questions, and then the two authors made some explanations and even modifications.
For example, in Sections 7 and 8, page 63 quotes an equation that did not exist before, a reference is missing before Lemma 12.3, a function that suddenly appears is not defined in the paper, and a certain step is missing
A valid reason…
Okay, some of the questions may seem outrageous, but anyone who has ever used a computer to write a paper will know that some small flaws are unavoidable.
As long as they are not logic-level errors, many errors are unavoidable. Especially papers on number theory often need to be revised repeatedly. At this time, due to the state of the paper's author at the time, it is actually normal to miss a few formulas.
This is also the reason why many paper reviewers will repeatedly argue with publishers, often based on the rigor of this scientific paradigm.
Especially in mathematics, generally pointing out that there is perjury in key steps of a paper is considered questioning. Pointing out such small problems is considered discussion.
Just like when Wiles proved Fermat's last theorem, the editorial department arranged for six reviewers. During the review period, a large number of problems were discovered. Fortunately, most of them were small problems that Wiles could clarify immediately. Of course, if he could not clarify them,
That's a big problem.
Perelman proved that the Poincaré conjecture was actually the same. When it was first released, some people felt that many issues were not explained clearly and more detailed proofs were needed. This was also the reason why it caused some controversy later.
In short, theoretical mathematics is like this, one of the reasons why the talent requirements are extremely high.
Mathematics allows mathematicians to freely make various definitions without considering the laws of reality, and even construct theories based on any starting point, as long as the logic is self-consistent.
But at the same time, it has extremely high requirements for the rigor of the proof process. A slight conflict or lack of logic may lead to the overthrow of the entire theory.
…
"When it comes to new tool frameworks, we have to mention Qiao Yu. By the way, everyone should know Qiao Yu, right?"
After chatting about serious academic topics, Zhang Yuantang brought the topic to Qiao Yu.
The other three big guys nodded at the same time.
Needless to say, Tao Xuanzhi was able to obtain a doctorate from Princeton University at the age of 21. The gold content in this is that anyone who has a certain understanding of Princeton University graduation knows how difficult it is.
The most important thing is that Tao Xuanzhi's talent is never based in one direction.
He has conducted many top-level research in the fields of harmonic analysis, partial differential equations, combinatorial mathematics, analytic number theory and representation theory. He has master-level performance in more than ten mathematical fields.
When he received the Fields Medal at the age of 31, he had published more than 80 influential papers in major mathematics journals.
The most important one is the proof of the Green-Tao theorem. This 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 first-rate geniuses who have already shown their mathematical talents at a very young age, so they will naturally pay attention to them.
"Of course, Peter Schulz sent me an email and specifically mentioned Qiao Yu. He admired this young man very much. I have also read his papers. How can I put it... He is the most thoughtful young man I have ever seen.
Mathematician."
Tao Xuanzhi commented.
After hearing this evaluation, the other two mathematicians also nodded. In fact, this nod was meaningless. It did not mean recognition, but more respect for the evaluation.
Zhang Yuantang smiled and said: "Not only is he rigorous, but more importantly, his ideas are very unconstrained. Really, as I said just now, he proposed a brand new framework.
If he can succeed, he will not only be able to integrate existing tools for studying number theory, but also perfectly combine number theory problems with geometry. The most important thing is, after reading his ideas, I think he has a good chance of succeeding.
"
Zhang Yuantang's words immediately made the three people's expressions 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 the world of mathematics.
Especially the latter.
There is no doubt that if Qiao Yu can really succeed, this will be a Fields Medal-level achievement.
Naturally, it also aroused the interest of the three big guys. The study of prime numbers is originally 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.
Especially if geometric methods can be used to solve number theory problems without any obstacles, this will be an important direction and one of the key areas for the development of modern mathematics.
After all, geometry is what provides many highly abstract and powerful tools for number theory.
"Um...is this theory convenient to talk about?" James Maynard asked cautiously.
After all, in the entire academic world, it is somewhat unreasonable to learn from a third party about other people's research results that have not been officially published.
But it doesn’t matter if it’s just a general direction and doesn’t involve proof details.
So Zhang Yuantang nodded naturally.
He was not involved in Qiao Yu's subsequent work, and he did not know the details.
"Qiao Yu proposed a system of axioms of generalized modal number theory. Specifically, every natural number can be mapped into a modal space. This process is called modal mapping.
He defined the structure corresponding to 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 an arithmetic sequence.
An example..."
In this way, Zhang Yuantang spent more than twenty minutes explaining Qiao Yu's general idea.
A very general framework.
After listening, the three professors frowned at the same time and fell into deep thought.
No way, this is just a rough idea, and it is still difficult to understand the content contained in it with a simple explanation.
But everyone can understand the meaning.
"Wait a minute, I can understand this kind of modal mapping. But since Qiao Yu's ambition is so great, this framework must span a multi-dimensional modal framework, and there is a problem.
There are many modal mappings that are nonlinear and irreversible, which means that classical number theory methods cannot be directly applied within the framework. How to solve this problem?"
After Tao Xuanzhi thought for a moment, he raised his question.
Zhang Yuantang spread his hands and replied: "I don't know much about the details of his handling. I can't ask carefully. 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 only arrays or linear operators.
It is a modal operator composed of multiple mappings and self-referential relationships. Therefore, each operator matrix has dual dimensions, ordinary dimensions and modal dimensions.
The modal dimension can be used to represent the mapping of matrices 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 anything more detailed.
Qiao Yu came up with his idea that day, and after everyone briefly discussed it, he left China the next day and returned to the Western Hemisphere.
It wasn't that he didn't want to stay for two more days, it was mainly because Yanbei University didn't want to keep him for more, so naturally he was embarrassed to stay there all the time.
In fact, Tian Yanzhen talked to him once about returning to Yanbei University to teach, or getting a position at Yanbei International Mathematics Research Center, but Zhang Yuantang never made up his mind.
"This idea... is very bold, and it seems to be effective. The integration of number theory and geometry... it may even be more than just number theory and geometry. Of course, I mean if he can really succeed."
James Maynard thought carefully for a while and then said.
The emotions are very complicated. As mentioned 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 hit it off so well, they walked the same path." Tao Xuazhi said with emotion.
This sentiment is very appropriate.
There is no doubt that Tao Xuanzhi and Peter Schulz are both amazing talents of the younger generation. However, the reasons why they were recognized by the Fields Medal are completely different.
Peter Schultz built a brand new system by relying on it, while Tao Xuanzhi relied on solving an important mathematical problem. The two took different paths. Now it seems that Qiao Yu also wants to follow Peter Schulz's path.
path.
"In fact, not necessarily. As far as I know, Qiao Yu designed this framework 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 may not only build a programmatic system that can guide the direction of mathematics, but also solve a series of number theory problems. Maybe he can combine your two approaches.
And it’s very possible. After all, he has made a great contribution to the proof of the geometric Langlands conjecture. Really, I thought I might face a challenge, but I didn’t expect the challenger to be so young.”
Zhang Yuantang expressed different opinions.
He had talked face-to-face with Qiao Yu, and he knew better than others the pressure he felt when discussing academic issues with Qiao Yu. He could quickly think of how to answer the first superficial questions.
But as the discussion got deeper and deeper, it was really difficult for him to resist. The most important thing was that Qiao Yu's questions always got to the core of the problem, and even led him to think about something deeper.
So the deeper the discussion went, the more oppressive he 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 not sure how to evaluate it.
So he was happy to let Tao Xuanzhi realize this.
"What you said makes me want to communicate with him. If he is also concerned about the prime number problem, I don't know if he will see our paper and what his evaluation will be." James Maynard said with a smile.
For people like them, too much energy has been spent on the study of prime numbers. Who doesn't want to be the first to solve those 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.
He just made some simple deductions in his mind based on what Zhang Yuantang said, and suddenly found that Qiao Yu's idea was indeed possible to succeed.
He still didn't know how Qiao Yu solved some 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 ambitious! He wants to construct a unified theory of mathematics in his own way. Tao Xuanzhi even suspects that Qiao Yu wants to replace the Langlands program. Yes, just use his modal space theory to build one.
Replacement.
This does not seem impossible, because although Qiao Yu's method is also abstract, it is not as difficult to understand as Langlands' program.
In particular, the geometricization of number theory problems can make some obscure number theory problems more intuitive in the modal space.
"I can ask that kid, although he is only sixteen years old... How should I put it, he doesn't resist communication, but he has his own way of choosing collaborators."
Zhang Yuantang said with a strange expression.
In fact, ever since he learned about Qiao Yu's subject, he has been paying attention to the related progress. Of course, the result surprised him.
"Paranoid?" Harvey Guth, who had been silent on Qiao Yu's issue, asked.
He knew the least about Qiao Yu. He had only heard about some things that happened at the World Congress on Algebraic Geometry, so he had not expressed his opinion just now.
"It's not paranoia. To be precise, it should be self-confidence. I think he probably thinks he can complete this project on his own. So when he chooses collaborators, he likes to choose people who are closer to him than to the subject.
Helpful people."
Zhang Yuantang shook his head and corrected himself.
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 proved by itself, the rest will be detailed verification work, and it does not require skilled collaborators.
But I'm looking forward to what kind of results he can produce. Professor Zhang, you may make me unable to sleep well for a while, especially considering that someone can really solve many complex number theory problems at once."
Zhang Yuantang smiled and did not answer.
Not only him, but the other two people also felt a sense of urgency.
If someone really proves a series of difficult problems about prime numbers in an unprecedented way, this is not entirely good news for many mathematicians who have been studying prime numbers.
After all, no one wants to be a backdrop. If you don’t believe it, you can ask Sam and Frank.
"It's okay, let's ask first. I have never had any contact with Qiao Yu. It might be rude to send him an email rashly. Please, Professor Zhang."
Tao Xuanzhi thought for a while and said.
Zhang Yuantang smiled and nodded in agreement.
Discourtesy is just an excuse, these geniuses are proud.
…
Huaxia, Yanbei University.
At this time, Qiao Yu was indeed doing work that the professors on the other side of the ocean were concerned about.
He can ignore the verification work, but there are some tasks he needs to do first.
What Qiao Yu is doing at this time is to transform a series of problems he intends to solve using the modal space framework from classical expressions into modal space expressions.
For example, the classic expression of the twin prime conjecture is that there are infinite pairs of prime numbers (p, p+2), among which the prime numbers p and p+2 are both prime numbers.
Then the expression in multimodal space must 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 constraints.
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 shows periodicity and symmetry in the modal space.
To put it simply, a classic number theory problem is decomposed into three geometric problems.
If he can prove these three geometric problems in modal space, it means that he has completed the proof of the twin prime conjecture.
Of course, the premise is that his axiom system of generalized modal number theory can be widely recognized by the mathematical community, and it can be proved that this axiom system can indeed convert between geometry and number theory, and always maintains verifiability.
But then again, there are people who do the verification work, and he is the only one who does the transformation work himself.
After all, transforming the problem requires an extremely clear understanding of the axiom system and extremely high mathematical insight.
In the same way, 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 very smoothly.
Even the transformation of the Riemann Hypothesis is simpler than the Twin Prime Conjecture.
Moreover, in the classic interpretation, all zero points are distributed on a line. The distribution in the modal space is on a hyperplane.
Of course, the completion of the conversion does not mean that the problem will be solved immediately. There are still many things to be defined to achieve this step.
For example, geometric tools such as modal density, convolution, etc. In short, after geometricizing the problem and modalizing it, Qiao Yu will know what tools are needed to solve the problem, and then go to the framework to prove and transform them one by one.
Qiao Yu didn't think what the professors opposite him thought, or even what Director Tian and Mr. Yuan thought. He had no intention of building the entire theoretical framework first.
His plan is to build it on demand.
To determine 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 look at what new tools are needed for the twin prime number conjecture, and then proceed to the next stage of derivation, and then start to prove...
The advantage of doing this is naturally that it can publish the most articles, and others can't even say that he is doing nothing.
Whether it is adding new tools or solving new problems, it is the favorite content of the mathematics world. Even the Langlands Program is also composed of many sub-conjectures.
This is actually the reason why Qiao Yu has no interest in appraising funds. After all, even if he gets the grant, the money is not in his personal account.
Instead, it will be transferred to the account of the research center, and then a sub-account will be divided below. When money is needed, it can be transferred directly. Not to mention that the funds allocated to pure mathematical theory are generally not much.
Mainly because of reputation. But Qiao Yu feels that he is not so anxious to seek fame. There is no need to be so anxious to build the framework and benefit the mathematics community.
After all, China's research progress in theoretical mathematics is far behind that of the West. After his new axiom system is fully contributed, there is a high probability that it will be the first to use it in some cutting-edge proposition proofs.
After completing these basic tasks, Qiao Yu stretched out and planned to ask other people about their work progress on WeChat.
Yesterday, he specially created a group chat and brought Qiao Xi, Xue Song and Chen Zhuoyang into a discussion group to facilitate his assignment of tasks.
Then he saw a new email notification appeared in his work mailbox, and it was Professor Zhang Yuantang's email, so he subconsciously clicked on it.
Even though he is somewhat famous in the mathematics community now, he doesn't actually exchange many emails on weekdays.
The main reason is that there is a lot of email communication within Professor Li’s research group at Huaqing.
As for other big bosses, they only occasionally send emails to discuss some issues. This is not only because everyone is busy, but also because Qiao Yu has not developed the habit of communicating by email.
"Qiao Yu:
We met in person. Today I was fortunate enough to be invited by Professor Xuanzhi to discuss his latest article "New Progress in Large-Value Estimation of Dirichlet Polynomials" with Professor Guth and Professor Maynard, and I felt that I gained a lot.
I remembered that when I was at Yanbei, you once said that you were very interested in the prime number problem, 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.
After discussing the article, I mentioned to three professors the axiom system of generalized modal number theory that you are trying to construct. Professor Tao Xuanzhi was very interested after hearing this.
In recent years, Professor Xuanzhi has also 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 search for properties similar to prime numbers in general sequence and functions.
And has made many achievements. For example, he developed the 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, reducing the distance between prime number pairs from 70 million to less than 600. Therefore, he hopes to establish cooperation with you and jointly discuss the geometricization of prime number problems.
If you are also interested, please let us know the convenient time or communication method.
Looking forward to seeing you, and wishing you Shunqi!
Zhang Yuantang."
After quickly scanning the letter, Qiao Yu subconsciously opened the web page 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 academia. He really doesn’t know many big figures in mathematics.
However, he knew that if he could invite Zhang Yuantang casually and ask Professor Zhang to write this letter to him specifically, he must be a big shot in the mathematics community.
This is indeed the case.
A quick search revealed two Fields Medalists. There is also one who, although he has not won the Fields Medal, seems to have a fairly low status in the world of mathematics.
The most important thing is that they are different from the Fields Medalist whom I met at the World Algebraic Geometry Conference last time. Both Tao Xuanzhi and James Maynard are still very young.
Tao Xuanzhi has just turned fifty this year, and James Maynard is even younger, still one year short of forty.
Qiao Yu could tell that Tao Xuanzhi was very active in the mathematics community. Last year, he teamed up with more than 60 mathematicians to develop questions and launched FrontierMath, a mathematical benchmark used to test the mathematical ability of artificial intelligence.
Simply put, FrontierMath is an original question bank. The benchmark contains hundreds of original and extremely challenging mathematical problems, covering the main branches of modern mathematics, such as number theory, real analysis, algebraic geometry, category theory, etc.
Then let the most advanced AI go to the question bank to answer the questions...
Seeing this, Qiao Yu suddenly realized that Yu Wei's sudden desire to do AI was indeed visionary.
Peter Schultz is engaged in AI mathematics research work with Microsoft, converting mathematical theorems into things that AI can understand, while Professor Tao Xuanzhi is engaged in AI mathematics test benchmark question bank.
After roughly understanding the lives of these big guys, Qiao Yu casually logged into arXiv and downloaded the article "New Progress in Large Value Estimation of Dirichlet Polynomials" recommended by Professor Zhang to his computer.
It is still necessary to read the latest papers of Fields Medal winners. Not to mention that these Fields Medalists are still concerned about prime numbers.
To be honest, Qiao Yu was also afraid that someone would solve the twin prime conjecture and Riemann's hypothesis before him.
The former doesn't matter, the latter involves a bonus of 1.5 million US dollars.
At least for Qiao Yu now, he is still very interested in the 1.5 million U.S. dollars. It is worth tens of millions when converted into RMB. Not only is it given by Americans, but there is no need to pay taxes.
After all, the Fields Medal prize that a bunch of math people want to win is only 15,000 Canadian dollars, which is equivalent to less than 60,000 yuan in RMB.
Not to mention the Nobel Prize with a prize of 11 million Swedish kronor and the Turing Award with a prize of one million US dollars, they are not even comparable to the Wolf Prize.
The Wolf Prize is still worth $100,000!
This is probably what Qiao Yu is most dissatisfied with about the Fields Medal.
However, after downloading the paper, Qiao Yu thought for a while, and instead of reading the paper, he took a screenshot of Zhang Yuantang's email, opened the WeChat group, sent it directly, and then synchronized @ everyone.
"Comrades who are striving for the same goal, please take the time to read this email. Fields Medal mathematicians all want to cooperate with us, but I do not intend to cooperate with them.
Because I believe we can complete this project independently! So please work hard together! As long as we can achieve this result, even the Fields Medal bosses will be envious!
Dear comrades, please reply when you receive it!"
As the initiator of this topic, Qiao Yu feels that it is necessary to give everyone blood injections from time to time. Only in this way can everyone be urged to complete their respective tasks as soon as possible.
Although one person in the group was a professor who had given him a lot of advice and could even be said to have discovered him, one was his senior brother, and the other was his mother.
Qiao Yu has always believed that as long as he is not embarrassed, others will be embarrassed.
This seems to be the case.
Because the message was sent and I @everyone, but no one responded for a long time.
…
In the office, Xue Song is assigning tasks to his Ph.D. Since he has just arrived at Yanbei University, he has not been assigned an enrollment quota this year, but all the doctoral students from Yujiang University have followed him.
Feeling the phone vibrate twice, Xue Song picked it up and looked at it. For a moment, he didn't know what to say.
In his academic career for so many years, Xue Song has been a project leader and also worked on subjects with other people.
So much so that Qiao Yu's method of injecting chicken blood made him feel a little familiar and yet strange at the same time.
Normally when you work with professors, you don’t do this. On the contrary, tutors occasionally encourage you like this.
Especially the phrase "comrades who strive for the same goal" really made Xue Song find it difficult to comment.
But then again, Xue Song felt very excited that Tao Xuan was very interested in this topic.
But what did this ask him to reply to? He felt that he had lost all face when he received the reply, so he simply put down the phone as if he hadn't seen it.
He turned his attention to his students again.
"Junfei, work hard on this topic. To get a diploma from Yanbei University, the requirements will be higher. Qiao Yu's topic 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 from Qiao Yu, thought about it and then asked the instructor next to her.
"Teacher Yuan, is Professor Tao Xuanzhi famous?"
"Huh? Why did you mention him so suddenly?"
Qiao Xi handed the WeChat message to the instructor beside her.
Mr. Yuan took a look and smiled: "Haha, Xuanzhi is still very discerning."
Then he looked at Qiao Xi and explained: "Xuanzhi is one of the leaders of the new generation of mathematics. But you don't need to pay attention to this. Let that kid do it. It is only right that you lay the foundation first."
"Okay, teacher."
So Qiao Xi also casually threw her phone aside.
This guy... to say comrades, he probably wants someone to loosen his skin again.
…
In the middle of a meeting, Chen Zhuoyang felt a vibration. He took out his cell phone and glanced at it secretly. He quickly glanced at the letter that Qiao Yu had screenshotted, and he still felt a lot of emotion in his heart.
After reading what Qiao Yu sent, he really planned to reply to the one he received.
After all, compared to Xue Song and Qiao Xi, he really didn't think there was anything wrong with replying once.
But before his hand touched the phone keyboard, he heard someone calling his name.
"Chen Zhuoyang, why don't we start with you..."
unlucky……
Chapter completed!