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.
To be continued...