The top student must be diligent.

Chapter 171 The Upcoming International Congress of Mathematicians
In the 60-square-meter dormitory, in front of a large solid wood desk, Xiao Yi sat calmly looking at the draft papers on the table.

The contents of these draft papers are the conclusions he finally reached after such a long period of time.

The theorem he proposed is also a conjecture: [Let G be a compact, simple Lie group, and A be a Yang-Mills field defined on the four-dimensional sphere S^4. If there exists a non-zero Chern number c, then the lowest energy excited state of the Yang-Mills field A has a strictly positive mass gap. ]

Now, it has been thoroughly proven by him.

This officially means that the existence and mass gap problem of the Yang-Mills equation, one of the seven mathematical problems of the millennium and also the only two mathematical physics problems, has become the second problem to be solved among the seven problems.

He reached out and sorted out the draft papers on the desk one by one, and read them again from beginning to end.

The final proof originated from a sudden inspiration he got from the Hodge criterion conjecture that night almost two months ago, which allowed him to finally extract a key mathematical method from countless previous thoughts.

The most critical methods are Hodge decomposition and vertex algebra.

In the study of complex algebraic varieties, Hodge theory provides very powerful tools to analyze their geometric and topological properties.

The Hodge decomposition is one of the core concepts of Hodge theory, which decomposes the deram homology on a complex algebraic variety into (p, q)-type parts.

On the other hand, vertex algebra, as an important tool in quantum field theory and algebraic geometry, can be used to describe the algebraic structure in conformal field theory.

Combining the Hodge decomposition with the vertex algebra framework can provide new perspectives and tools for understanding the moduli space structure of complex algebraic varieties, and most importantly, it can generalize the Hodge structure to quantum field theory.

Xiao Yi directly named this mathematical method the Hodge-vertex algebra analytic method, and if it is extended to quantum field theory, it can probably be named quantum Hodge theory.

Whether it is this new mathematical method or the physical method of quantum Hodge theory, it can be called a very significant breakthrough.

The former can help the mathematical community to have a deeper understanding and research on algebraic geometry. Xiao Yi can even be sure that this method can help solve a large number of problems in algebraic geometry, such as the Hodge conjecture and Grothendieck's standard conjecture, the latter of which also includes the Hodge standard conjecture, in addition to motivation theory and so on.

The quantum Hodge theory may help the physics community to achieve a further understanding of quantum field theory, which may be regarded as a great breakthrough for the theoretical physics community, which has been stagnant for a long time.

You should know that in the field of quantum field theory, no new theories have been born for many years.

Most importantly, Xiao Yi used this new method to not only confirm the authenticity of the X-field, but also to make a prediction: the existence of the X-field must be accompanied by the appearance of an undiscovered particle, and this unknown particle was naturally named the X-particle by him.

A new particle that can be compared to the Higgs particle!

In general, this achievement can probably be regarded as the most valuable one among Xiao Yi's many achievements.
"It is indeed a conclusion of high value."

He recalled the voice that appeared in his mind when he completed the proof just now.

Indeed, a problem has to be at least as difficult as the Millennium Problem to be judged to be of high value.
After all, just to prove this problem, he came up with such an important method.

And now that the [Ruthless Combo] buff has been upgraded to level 5, he clearly feels that his understanding of everything in this world has been greatly improved again.

Even... maybe because the overall effect has been improved 20 times, he feels as if there has been a qualitative change.
Now he feels that he is terrifyingly strong.
Questions like the Riemann hypothesis and the Hodge hypothesis are all small matters!

Ahem... Well, it's not that strong.

But if he is really allowed to study these problems, there is a high probability that he will have a chance to achieve a breakthrough.

"But, this time it seems like an additional effect has been generated?"

"What's it called... Material Mastery?"

【Material Mastery: Whenever it comes to research related to materials, you can always see the key points.】

Hiss, this effect...

“Seems a bit awesome, huh?”

His brows couldn't help but move.

When it comes to material research, the academic community basically uses continuous trial and error to finally find the key points.

Therefore, it is very important to grasp the key and reduce the cost of trial and error.

The description of this additional effect is very interesting.

"Can you always see the key point?"

always?

Xiao Yi wondered if he could try to do a project related to materials when he looks back?

Well, let's wait until all the current matters are settled.

At least we have to deal with the International Congress of Mathematicians in a few months.

And speaking of the International Congress of Mathematicians…

By the way, what day is it today?
He opened the computer and took a look after a long time.

April 4……

Uh?
Where have you seen this date?
Suddenly he slapped his head.

It’s the deadline for submitting report topics for the International Congress of Mathematicians!
He quickly logged into his email and saw the email from the conference organizing committee urging him to submit his manuscript.

So he used his hand speed, which had benefited from being single for 18 years, to quickly write the title and summary of his report and then sent it over.

After seeing the email was successfully sent, he breathed a sigh of relief.

He had been immersed in research during this period and had hardly paid attention to his email. He almost forgot the date.

He also knew that even if he was overdue, the conference organizing committee would definitely give him some time.

However, being able to submit on time is still very important, although he is almost submitting it on time, with only a few hours left before the deadline.

"Well, the next thing is to prepare for the report in a few months. Also, I have to relax. I've been exhausted during this period."

I stood up and stretched my body, moving it three circles to the left, three circles to the right, and twisting my neck and butt.

Suddenly, the mailbox on the computer interface suddenly notified that another letter had been received.

I took a look and saw that it was the email address of the International Congress of Mathematicians Organizing Committee that he had just replied to, and now he had sent another email back.

Automatic reply?
He opened it and took a look.

[Hello, Mr. Xiao Yi. First of all, thank you for submitting the report topic in time. In addition, I would like to ask, does submitting this topic mean that you have successfully proved the quality gap problem? ]

This shouldn't be an automatic reply, right?
Oh! It’s not an automatic reply.

Because there is a signature below, Chairman of the Organizing Committee of the 2022 International Congress of Mathematicians, Alexander Berezovsky.

Xiao Yi raised his eyebrows. Could it be that the other party thought that he had not yet solved the problem, and only sent a topic temporarily because he saw that the deadline was approaching?

Ok……

After a brief thought, he replied: [Regarding the quality gap problem, I have just come up with a wonderful proof method. You will know it when the lecture begins.]

He clicked to send the email, and finally he stopped paying attention to it, turned off the computer, and prepared to go to bed.

On the other side, Berezovsky and others gathered in front of the computer, waiting for Xiao Yi's reply.

As the notification sound of new email was heard on the computer, they hurriedly opened the new email.

Then they were all stunned.

So, the quality gap problem has really been proven?

For a moment, they looked at each other, and saw shock in each other's eyes.

Also, does this also mean that they became the first people in mathematics to know that the mass gap problem had been proved?

Berezovsky reacted quickly and ordered, "Quick! Publish the topic of Xiao Yi's report on the website."

"This year's conference will probably be the most eye-catching one in history!"...

the next day.

Max Planck Society, Institute for Mathematics, Bonn, Germany.

"It seems that today is the day when the International Congress of Mathematicians announces all the report details?"

In an office, Faltings asked Schultz.

"Really?" Schultz thought for a moment, then nodded: "It seems to be today."

So he turned on his computer and went to the official website of this year's International Congress of Mathematicians.

This year's International Congress of Mathematicians is held in St. Petersburg, Russia, not far from them, so of course they will participate.

Since you are going to participate, you can take a look at what these reports will talk about in advance and see if there are any that interest you, which can be regarded as advance preparation.

Professional mathematicians like them are naturally not like other ordinary mathematics scholars or enthusiasts who attend academic conferences purely for fun, and do not listen to any reports or miss any tea breaks.

They still hope to learn something from this meeting.

However, as soon as he opened the website, a huge piece of news on the homepage caught his attention.

[Plenary lecture: There is a strictly positive mass gap in Yang-Mills theory, speaker: Xiao Yi].

"Walter?!"

When Schultz saw the news, his eyes widened and he exclaimed in surprise.

Faltings, who was standing next to him, was startled by his exclamation and immediately said angrily, "Why are you shouting? You are so startled. Don't you know that I am already old?"

However, Schultz had no time to argue with Faltings about this matter. He pointed at the screen and said, "He he he he... Xiao Yi has decided to talk about the quality gap issue at the conference!"

"Huh? What?!"

After hearing what Schultz said, Faltings no longer cared about what happened just now and immediately came over. Then, the "big" news that was promoted on the front page immediately came into his sight.

So, he also showed an incredible expression: "He... has proved it? How did he prove it? What method did he use to prove it???"

These three questions in a row fully expressed the uneasiness in his heart at the moment.
"Who knows?" Schultz spread his hands.

How would he know?
He wants to know too!

It is almost a universally acknowledged fact that in order to prove the mass gap problem, new mathematical tools are definitely needed, and they must be new tools in algebraic geometry or topology. Of course, it may be both.

For mathematicians like them, they really want to know what kind of method can be used to prove the mass gap problem.

"I didn't ask you, don't be narcissistic."

Faltings certainly didn't think Schultz could answer it, so he urged, "Go in and take a look. The report topics are all there, and there should be a summary, right?"

Schultz looked resentfully at Old Deng, who could probably be called his teacher.

You don’t even know it yourself, so why do you say I’m narcissistic?

But he still clicked on the link obediently.

The web page jumped and then entered the news link, which mainly talked about the details of Xiao Yi's report.

[Just yesterday, the conference organizing committee received news from Mr. Xiao Yi that he had successfully proved the existence of the Yang-Mills equation and the mass gap problem.

As we all know, the existence and mass gap problem of the Yang-Mills equation is one of the seven Millennium Problems proposed by the Clay Institute in 2000. It mainly involves the Yang-Mills theory in quantum field theory. As a rigorous mathematical physics problem, it also requires a lot of mathematical methods to solve.

If this problem can be solved, it will be a huge breakthrough for both mathematics and physics. Especially for mathematics, it may mean the emergence of some new mathematical methods, which may have a very important impact on the mathematical community.

In order to show our importance to this report, to allow Mr. Xiao Yi more time to give his report, and to avoid making you wait too long, we have decided to adjust the time of Mr. Xiao Yi's report to July 7, the first day of the conference, and the opening ceremony of the conference will be delayed to that evening.

If you want to view the theme and summary of this report, please click on the link: www.icm2022.org/……】

"Wow! It was specially arranged on the first day, even before the opening ceremony."

Looking at this news, Schultz was surprised.

"That's good. At least it's not on the last day. Otherwise, a lot of people would be dying of anxiety waiting for this report." Faltings said, "And this is the news they got from Xiao Yi. It should be basically confirmed now."

Schultz nodded. Xiao Yi had already confirmed it. Based on his past record, this wave was probably proven true.

Then he clicked on the following link: "Read the summary."

Then, they finally saw the summary of the report.

【For any compact simple gauge group G, there exists a nontrivial quantum Yang-Mills theory on R^4 with a mass gap Δ>0.

This is the description of the Yang-Mills existence and mass gap problem by A. Jaffe and E. Witten, and is also the official statement of the seven Millennium Problems of the Clay Institute. This problem is not only of great significance in theoretical physics, but also an important unsolved problem in mathematical logic and mathematical physics.

In this report, we will discuss how to prove the existence of Yang-Mills and the mass gap problem, and focus on a key new mathematical method: the Hodge-vertex algebra analytical method, as well as the significance of the conclusion in physics.

After reading this summary, both of them immediately grasped the most crucial thing.

Schultz: "Hodge-Vertex Algebra Analysis Method... This should be the new mathematical method he created, right?"

Faltings nodded slightly: "Based on the name, it mainly uses Hodge theory and vertex algebra?"

"Vertex algebra... nice idea."

The two looked at each other, as if they could see impatience in each other's eyes.

It’s like a gamer learns about a very interesting game and is eager to play it, but there are still three months before the game is officially released, which is very anxious.

Moreover, to some extent, their desire for this new method may be stronger than those game geeks' desire for new games.

After all, there is electronic impotence in the game, but their curiosity about mathematics will not be impotent.

……

What happened in this office at the Max Planck Institute for Mathematics also happened in many other mathematics-related schools or research institutions around the world.

In an instant, the report that had not yet begun became the focus of everyone's attention.

It is just like the Poincare conjecture back then, or even more popular. After all, when the Poincare conjecture was proved, the Internet platform was not as developed as it is now.
The analytical method of Hodge-vertex algebra mentioned by Xiao Yi in the abstract also spread quickly in the mathematics community.

Mathematicians almost all over the world are asking: What is the analytical method of Hodge-Vertex algebra?
But apparently, there is only one person in the world who knows the answer.

Anyone else who wants to know will have to wait until the report nearly three months later.

Of course, compared with the achievements in pure mathematics, the significance of the mass gap problem in physics is more attractive to ordinary people.

After all, keywords such as the four fundamental forces of the universe, a new particle, and cosmology are obviously much more attractive than algebraic geometry, projective clusters, and Hodge theory.

Otherwise, why do Nature and Science like such manuscripts?
Therefore, this news quickly spread from the mathematics community to the physics community, and for a time, it made theoretical physicists around the world even more excited.

They all wanted to know how Xiao Yi's proof could help them deepen their understanding of the strong interaction, whether they could use it to understand the X-field, and the possibility of unifying the strong and weak interactions.

As a result, the International Congress of Mathematicians received a large number of applications from physicists to attend the conference.

For a time, the conference three months later seemed to be a feast for both the mathematics and physics communities.

Time passed quickly.

The day of 2022 has officially entered July!
The International Congress of Mathematicians is about to be held!

(End of this chapter)