It's as if Peter Schulz is still racking his brains to train an AI that can understand mathematical theorems.

However, Qiao Yu feels that he has an advantage on this track. After all, he now owns a computer room, and his mathematics level is not bad. It seems that he is at least at the level of a potential Fields Medal winner.

In addition, his model will undertake a large number of professional tasks in the future, which means there will be a lot of multi-dimensional and complex data for it to train.

But as soon as Qiao Yu opened the paper and got into it, there was a knock on the door.

"Who is it?" Qiao Yu didn't open the door directly, but asked.

The sudden "dong dong dong" sound at night was really scary.

"I, Xue Song."

Lao Xue's voice drifted in from the corridor, and Qiao Yu quickly stood up and ran to open the door.

"Huh? Professor Xue, you haven't gone home yet at night?"

"Where do I have a home in the capital? I only have a dormitory, and when I go back to the dormitory I do research, so it's better to stay in the office."

"Forehead……"

"Well, I came to you mainly to discuss an issue. I guess you haven't slept yet and don't want to wait until tomorrow."

Only then did Qiao Yu notice that Xue Song had a stack of manuscripts in his hand. So he nodded and said, "Please come in."

No way, when a mathematics professor can't wait until tomorrow to discuss a certain mathematical problem, it means that he is really interested in this problem, so it is best not to let him down.

This was indeed the case. Following Qiao Yu into the small study, Xue Song handed Qiao Yu the manuscript in his hand before he even sat down.

"Look at this first. I discovered a very strange phenomenon today when I was studying the geometric mapping Ψ(Pussy)(s). In most cases, Ψ(s) is a smooth geometric object on s.

For example, manifolds or hypersurfaces. But near the zero point, there are obvious abnormalities in the properties of Ψ(s). This can probably be understood as the rupture of some geometric structure or the enhancement of symmetry."

Qiao Yu lowered his head and looked at the manuscript handed over by Xue Song. Of course, he also put into his mind what Xue Song said word for word.

He has always been very strong when it comes to multitasking.

According to the manuscript, Qiao Yu roughly knew the progress of Xue Song's research. The conclusion just now was discovered by Xue Song when constructing the symmetry group of modal space.

To sum up, the geometric map Ψ(s) is essentially controlled by a certain symmetry flow, and this flow can be explained by the role of Lie group G.

Then Xue Song came up with a very bold hypothesis: the modal geometric mapping Ψ(s) is composed of the orbit of a modal Lie group Gs.

It’s really interesting!

Because he had always communicated with Xue Song, Qiao Yu quickly read the entire manuscript.

After thinking for a moment, he said: "I think there are two possibilities. The first is that the modal Lie group Gs acts on the modal hyperspace H, so that Ψ(s) is the fixed point set of Gs.

Second, if Ψ(s) has enhanced symmetry at zero point s, does it mean that the group order of Gs has changed? Well, it is similar to the phase transition of enhanced symmetry?

Wait, let me think about it..."

Qiao Yu thought for a moment, then picked up the pen and added another formula to Xue Song's manuscript.

Xue Song came up to take a look, thought for a while, and said: "Directly describe the generator of the modal group as a Lie algebra?"

"Well, the geometric structure is controlled by the Laplacian operator of modal geometry." Qiao Yu nodded, and his brain began to enter a state of rapid thinking.

Xue Song frowned, pointed to the position of λ(s) in the formula, and asked: "What is this?"

"Eigenvalue."

Qiao Yu explained casually, and then suddenly said: "Have you discovered a problem? The zero point condition of the modal ζ function is equivalent to the resonance condition of the modal Lie algebra?"

Then I added another formula: "λ(s)=0 s"

He was still mumbling: "Look, s is the zero point of the modal ζ function."

Xue Song frowned and glanced at Qiao Yu. He felt a little strange. Were they discussing the same issue?

"Um...what are you talking about? Wait, modal zeta function? Did you make a modal mapping of the Riemann zeta function?"

"Um? Oh, I almost forgot. My latest paper hasn't been published yet. The latest issue of JAMS should be available in two days."

Qiao Yu slapped his forehead and explained quickly.

"Oh..." Xue Song nodded silently.

Fortunately, he is used to it.

Ever since Qiao Yu published his first "Annual Review of Mathematics", it seems that publishing a top issue was no longer a problem for this young man.

But he will soon be able to send it.

This topic has already accumulated a lot of content. This year, Xue Song is sure to publish one or two top journals, at least one of which is definitely qualified to be in the top four.

So I’m not envious.

"Wait a minute, Professor Xue, I discovered a very interesting situation, please wait while I do the math first."

Qiao Yu said something to Xue Song, then picked up a blank piece of manuscript paper on the table and started to calculate quickly.

Xue Song curiously came to the table with a stool and watched Qiao Yu's derivation process. With such a calculation, one frowned and looked at it. Unknowingly, Qiao Yu had scrawled and quickly deduced on seven or eight pieces of paper.

The entire desktop began to become visibly chaotic.

Finally, Qiao Yu stopped calculating and finally got a formula.

"Well, it's interesting. The topological invariant near the zero point actually exhibits strange behavior."

Qiao Yu muttered subconsciously.

"Well, it's like the ordinal number, or the Euler index?"

"The prepared description should be that the modal manifold degenerates into a low-dimensional symmetric subspace at zero points."

"You mean the classic Morse theory?"

"That's understandable. But Professor Xue, have you really not seen it yet?"

Qiao Yu turned his head, looked at Xue Song, and asked seriously.

"What do you see?" Xue Song frowned and looked at Qiao Yu in confusion. He began to hate the Riddler a little.

"Oh, by the way, you haven't read my paper yet. It's actually very simple, that... we seem to have proved the Riemann Hypothesis."

Xue Song laughed and said: "Haha... you just said that we proved the Riemann Hypothesis. Qiao Yu, you really know how to joke. Then should I show excitement now!"

Smiling, smiling, Xue Song suddenly stopped laughing because he found that Qiao Yu looked at him in a strange way, as if he was looking at a fool.

So Xue Song subconsciously suppressed his smile, pondered for two seconds, and asked unceremoniously: "Um... Qiao Yu, are you serious? Are you really done?"

"I'm very serious. I thought it would be a complicated process because there are many things that need to be verified, but I didn't expect it to be so simple.

The only regret is that we didn’t use any number theory methods. But I think the Riemann Hypothesis must have been proved. So is it true that Professor Xue hasn’t seen it yet?”

Xue Song stopped talking.

He raised his hand and picked up the manuscript paper on which Qiao Yu had just calculated, and then read it line by line, thinking for a long time, and even went through the Riemann function several times in his mind...

Finally, ten minutes later, Xue Song used all his strength and shook his head carefully.

"Although it doesn't match my identity, I really want to swear right now. Unfortunately, I haven't done any research on the Riemann Hypothesis, so I really don't see why you said you proved the Riemann Hypothesis!"

Really, after saying these words, Xue Song suddenly felt a sense of relief.

If you don't understand, you just don't understand. Although you may have passed by great wealth, you can't pretend to be like that.

Although for a moment just now, he really wanted to pretend like this.

After saying that, Xue Song took out his cell phone, turned on the camera, entered video recording mode, and pointed the lens at Qiao Yu: "Well, Qiao Yu, how about you write out the proof now?

Don’t worry, I won’t record your specific proof process, I will just keep some audio and video data. If I don’t have money to buy a house in the future, you can help me make a proof and I will take this video to auction.”

Qiao Yu suddenly laughed and said: "In fact, it is very simple at this step. Use group action to construct the modal operator, and through spectral decomposition, you will find that the eigenvalues ​​of the modal operator are symmetrical.

Haha, Professor Xue, you didn’t see it because you haven’t read the paper I submitted to JAMS. If you have read it, you should be able to easily analyze the modal operator by combining the properties of the modal ζ function.

The zero-mode condition of is equivalent to the Riemann Hypothesis.

Finally, by combining the modal group with the modal operator, this problem is solved. No need for those stupid methods, I verified it bit by bit, and all the non-trivial zero points are on the 1/2 line! I am very sure now
Chapter completed!