The top student must be diligent.

Chapter 294: Riemann Hypothesis Lecture (VI)
Everyone at the scene was shocked that Xiao Yi was able to start proving so quickly.

It even seems that all the subsequent steps have already been basically conceived.

This is simply...

It's a bit too outrageous.

Especially the difficulty of this problem, in their opinion, it is completely impossible to start.

Perhaps this problem is not as difficult as the Riemann hypothesis or any other hypothesis.

But this is by no means a simple problem, especially since it now involves the new and far-reaching theory of generalized modular curves.

The complexity has increased by an unknown amount.

Although the generalized modular curve is quite important, its complexity can be seen by everyone present.

To prove an even more troublesome conclusion on such a complicated new theory...

Anyway, they felt that if they were asked to do it, they would at least think about it for a few days...right?

Well, maybe a week, or even a month.

Especially since this is just preliminary thinking.

After all, at the beginning, they didn't even know where to start.

It was not until now, after seeing Xiao Yi's method, that they understood a little bit.

But the key is, how did Xiao Yi manage to come up with it so quickly?
They are now completely puzzled.

Ultimately, it can only be attributed to one reason.

Because he is Xiao Yi.

That's the reason.

"…Now let us consider the Hecke characteristic H(M(E), s) of M(E). According to the construction of M(E), H(M(E), s) should contain L(E, s) as one of its factors. At the same time, the effect of σ on M(E) should induce some kind of automorphism of H(M(E), s)—"

[H(M(E), s)=ε(M)* p^(-s/2)* H(σ(M(E)), 1-s)]

"Then, combine the first and third steps, and we can get——"

【L(E， s)|H(M(E), s)】

"This means that the automorphism of L(E, s) can be elevated to the automorphism of H(M(E), s). In other words, the modular properties of E can be "embedded" into the modular properties of M(E)."

Having written this, Xiao Yi turned around and said to the mathematicians present with a smile: "At this point, I think the next steps have become very clear."

"So, from the matching of L-functions, we can infer the embedding of the curve. Specifically, if L(E, s) can be embedded into H(M(E), s), then there should be an isomorphism."

【φ: E→ M(E)】

“…so that φ induces an embedding from L(E, s) into H(M(E), s).”

Finally, after spending about half an hour, the entire blackboard was filled with the steps written by Xiao Yi, and it was just right, with no extra blank space.

Coupled with Xiao Yi's excellent blackboard writing, this blackboard is filled with a beauty that is unique to mathematics.

Xiao Yi put down the blackboard pen, clapped his hands, turned around, went to a small podium in front, picked up his water cup, took a sip, drank the last sip of water in it, then he put it back in place steadily, and then looked at the people present again.

At this moment, the entire venue was still silent.

Everyone was still immersed in the rigorous and accurate proof on the blackboard.

Although for the vast majority of the audience, the content of this blackboard is as incomprehensible as the Book of Heaven.

But it did not prevent them from finally understanding the feeling of "artwork" that Tao Terence and others had talked about.

This is truly a true work of art!
As for the mathematics masters in the front row, they were completely immersed in Xiao Yi's proof process and the current result.

Other ordinary audiences can now understand the feeling of what they are talking about in terms of artwork, let alone them?
Many partners of mathematicians feel that a mathematician’s true partner is mathematics, not themselves.

This is especially true for these top mathematicians.

Therefore, when faced with such a derivation process on a blackboard, they truly felt the wonderful beauty.

The beauty that is unique to mathematics.

And now, such beauty was once again brought to them by Xiao Yi.

It’s like the feeling when I first started reading the paper.

Their hearts were filled with such joy.

"It's so beautiful..."

They lamented.

Of course, in addition to its beauty, they were also amazed by the correctness of this proof.

Most of them might still be somewhat confused about Xiao Yi's answer to the previous question asked by Schultz.

However, since there is such a complete proof for this problem, they can easily understand it.

After all, Xiao Yi's proof was really easy to understand and they could understand it easily.

Of course, understanding the answer doesn’t mean I can do it.

It can only be said that Xiao Yi's proof process can be said to have taken "cleverness" to an extreme level.

As long as you understand where the clever part lies, then the whole process will be understandable.

But it is extremely difficult to find this clever spot at the very beginning.

Perhaps, Xiao Yi is the only one in the world who can discover such a coincidence.

As for others who want to solve this problem, they may only be able to prove it using other methods.

The whole place was so quiet that the photographer was a little afraid to press the shutter, for fear of disturbing the moment when the truth was revealed.

It seemed that even they could feel the feeling of being enveloped by the truth.

Of course, such tranquility needs to be broken eventually.

And this breaker did not receive any glares.

"Well, that's my answer to that question."

Xiao Yi said.

"Professor Wiles, do you have any questions?"

Andrew Wiles stood up again.

At this moment, all that was left on his face was amazement and admiration.

"No, your answer... is much better than I expected."

"If I were you, I wouldn't have wandered in despair for so long..."

he sighed.

He has already put himself in Xiao Yi's shoes.

Xiao Yi today is like a refreshing version of his experience back then.

Xiao Yi just smiled and said, "But doesn't your experience of wandering add a bit more story to your life?"

“Failure is always present in life, but the joy of regaining success after a brief failure is even sweeter.”

Wiles was silent for a moment, then smiled and said, "Thank you for your guidance, God of mathematics!"

After he finished speaking, he sat back in his seat upright.

Xiao Yi nodded slightly.

The next moment, the whole audience burst into applause.

Not for anything else, but for this question and this answer.

Perhaps, it was also because of what Xiao Yi said.

"It was not until this time that I realized the gap between him and me could not be explained in a few simple words."

Terence Tao said while clapping.

“If I am the Mozart of mathematics, who is he?”

Feferman said with a grin: "Mathematical Jesus."

Tao Zhexuan compared the title of God of Mathematics and couldn't help but say, "Hmm... isn't this a loss of face?"

The people around him all laughed. …

"It seems that the crown of the Riemann hypothesis is finally going to be taken away by him."

Deligne sighed, and then added: "There can't be any more tricky questions, right? It shouldn't be that serious, right?"

Bombieri also smiled and said, "I don't think that will happen."

"However, after these two questions, I fully accept the crown, even if it is the Riemann hypothesis."

Deligne nodded: "Yes, I fully agree with that."

"Moreover, this also proves that we humans still have the ability to conquer mathematics."

"Although we organized the Prime Number Pioneer Project, which was considered as us bowing our heads to mathematics, in the end, Xiao Yi made us raise our heads again."

……

More audience members applauded.

Maybe he is a top mathematician, or maybe he is a student.

Or, a woman.

Luo Mingya sat in a corner, looking at Xiao Yi on the stage and clapping softly.

Originally, Xiao Yi wanted to arrange her in a position closer to the front, but in the end she said that she was not a mathematician and knew nothing about mathematics, so it would be better to give those good positions to those mathematicians who were really looking forward to this lecture. Then, in the end, she chose this position.

Although he was in the corner, he could still see Xiao Yi's various postures during the lecture.

Of course, this was also her first time to attend Xiao Yi's academic conference, and it was also her first time to see with her own eyes his true brilliance in mathematics and how high his attainments in mathematics were.

That was a style and achievement that awed the entire mathematics community. It was unprecedented and probably will never be matched in the future.

In addition to cheering for it in her heart, she was also deeply proud of it.

That was what attracted her most to him.

……

The applause lasted for a long time. After a while, Xiao Yi waved his hand, signaling everyone to stop.

"Okay, everyone, stop for a moment. Stop for a moment."

After hearing his words, the audience gradually stopped applauding.

"So, do you have any other questions?"

Xiao Yi asked again.

The whole place fell silent again.

Could there be more?

If there are still problems now, what kind of problems are they?

However, this time, after waiting for a full minute, no one raised their hand.

This also means that the mathematical community has completely recognized Xiao Yi's proof.

The Riemann hypothesis has truly become history!
Xiao Yi did not wait any longer, picked up the microphone, and said with a smile: "In this case, then I think... the Riemann hypothesis will end here and become a monument in the history of mathematics."

"From now on, the mathematical world will have one more theorem, Riemann's theorem, and many propositions established by it."

The applause rang out like thunder again, filling the entire lecture hall. Everyone began to cheer for this event and celebrate this historic moment today.

In the future, in the history of mathematics, it will also be recorded that the Riemann hypothesis was proved on November 2028, 11, and the prover was Xiao Yi.

This time the applause lasted even longer than before.

The applause just now was because of the wonderful questions and wonderful answers. The applause now is truly celebrating today's results.

The mathematicians in the front row all showed sentimental expressions.

This problem that they have been pursuing for who knows how long has finally been solved.

It’s a bit of a pity that the person who solved it is not me.

However, if I were really asked to do it, could I really solve it?

This question is destined to be a question that will never be answered - of course, that's not necessarily the case. They can also try hard to use other methods in the future to see if they can prove the Riemann hypothesis.

But unfortunately, even if they prove it in the future, they will not be the first to prove it.

After a few minutes of applause, it stopped again. Next, it was probably time for Xiao Yi to announce the end of this lecture.

Everyone is also looking forward to the final conclusion of this conference and the subsequent reaction from the whole world.

"Okay, then, let me thank you all for coming and attending this seminar for the last time. I hope that this seminar will leave a deep enough impression on you and that you will not be disappointed."

"Of course, before I wrap up, I have another discovery that I want to share with you all."

Everyone was stunned, and then the scene, which was originally slightly noisy, immediately became quiet. Everyone wanted to know what Xiao Yi had discovered?
"I believe you can see that a key idea in my proof is to connect different types of mathematical objects, such as elliptic curves, Riemann Zeta functions, Galois representations, etc., and use the interaction between them to gain new insights."

"Of course, this idea also exists in all aspects of mathematics, including when Professor Wiles proved Fermat's Last Theorem."

"So, after I finished the proof, I reviewed the whole process and realized that there seemed to be a formula that could connect the three mathematical objects."

At this point, Xiao Yi dragged over a small blackboard next to him and said while erasing the contents on it: "The three mathematical objects are automorphic representations, L-functions in number theory, and some natural objects in geometry."

"And now, we can give such a relationship. For every automorphic representation ρ, we can construct an L-function L(s, ρ) and a geometric natural object X(ρ) so that they satisfy the following relationship -"

【L(s,ρ)=L(s,X(ρ))】

"That is, the L-function of ρ is equal to some natural L-function of X(ρ)."

When the top mathematicians in the front row saw this relationship, they immediately narrowed their eyes and began to think carefully.

Soon, they realized how important this relationship was to the mathematical community.

"If this relationship holds, it will be a very powerful generalization of the Langlands Program."

"The Langlands program foresees that every automorphic representation should correspond to a geometric object and a number-theoretic L-function, and as for this relation, it can be further foreseen that there should be a direct equational relation between the geometric object and the L-function."

"That is to say, it can provide a unified perspective that links the three major branches of mathematics, algebra, geometry, and analysis. Through this relationship, we can transform problems in algebra into problems in geometry, such as transforming representation theory into the study of clusters, or problems in analysis, such as the properties of L-functions, and so on."

"I believe this will be a more unified relationship for mathematics."

"And I am now somewhat confident that this relationship can be established."

Afterwards, Xiao Yi showed the mathematicians present some simple deductions he had made on whether this relationship was valid.

Mainly some specific examples are given.

And those mathematicians who were able to understand fell into further deep thought.

If we look at the examples Xiao Yi gave, it seems that this new relationship is indeed an issue that is worthy of further consideration.

If it could really be proven...

This may not be another Riemann hypothesis.

Therefore, Xiao Yi proved a Riemann hypothesis and brought a new Riemann hypothesis to the mathematical community.

"In short, that's it. We can see that after some detailed deductions, we can connect some of the objects together, but how to extend this connection to the entire field still requires some specific research in the future. At present, it is probably still a bit complicated to truly prove this problem."

Xiao Yi continued speaking.

"So, I will just declare it as a new conjecture to the mathematics community for the time being, and hope that in the future the mathematics community will also be able to achieve a breakthrough on this issue."

Then he opened his hands and said, "Well, now, that's all I want to share."

“Mathematics is a completely different direction from other subjects. I hope that every friend present here can enjoy this difference in their future research and study of mathematics, and then bring more wonderful results to mathematics.”

"This is the end of my report. Thank you."

He opened his hands, bowed deeply, and turned to leave.

Those audience members who did not understand his new conjecture immediately applauded.

And those audience members who understood were awakened by the deafening applause a moment later, and then applauded again.

Everyone stood up and watched Xiao Yi leave. Even after his figure disappeared behind the door leading to the backstage, the applause did not stop. It was as if it was a concert, and everyone was looking forward to his return so that he could prove a conjecture for everyone.

Of course, such a thing is impossible to happen.

But no matter what, the Riemann hypothesis, engraved high on the crown of mathematics, eventually became a part of history amid such applause.

And mathematics continues to move forward, into the distant future.

(End of this chapter)