Section 319

After a brief thank you, Tao hung up the phone, quickly turned on the computer, found Xiao Yi's website in his favorite websites, and clicked on it.

Sure enough, after two years, a new paper finally appeared on this website.

Tao immediately downloaded the paper.

"Hail conjecture, hail conjecture..."

"How will Xiao Yi solve this conjecture?"

Tao's thirst for knowledge became quite abundant at this time.

As a big shot in the field of number theory, he is very interested in every conjecture in this field of number theory, otherwise, he would not have studied these problems.

He has studied the Goldbach conjecture and made some results, and he has also studied the twin prime conjecture and made some results.

The same is true for the hail conjecture. His original result is already quite close to the final proof.

However, most of the problems he has studied, no matter how hard he thinks about them later, can only be achieved to that extent.

As for why he doesn't spend more time on it?

That's because he knows very well that he can reach his limit.

But now, Xiao Yi can always go beyond his limit and achieve the ultimate goal.

For him, this feeling is quite powerless.

It's as if the other party is the complete version of himself, and he is just an incomplete version.

He once fell into such confusion.

But later he was relieved.

Why should he have such thoughts.

At least, Xiao Yi solved these problems, which is equivalent to satisfying his thirst for answers?

Although he didn't solve them himself...

He opened the paper and looked at the abstract first.

However, the simple abstract revealed something that was not simple at all.

"Representation theory? Is he crazy?"

When Tao Zhexuan saw the first sentence of the abstract, he couldn't help wiping his eyes.

Using representation theory to solve the hail conjecture?

Isn't this idea a bit...

Too outrageous?

He is now looking at Xiao Yi's paper, in fact, mainly to verify some of his previous thoughts.

Let's see if Xiao Yi's proof method is the same as the one he had thought of before. This way, it can also bring him some comfort.

[I was just a little bit away from thinking of it! ]

[What a pity, I also thought of one step before, but I was stumped when I got to the next step, what a pity. ]

And so on.

However, now it seems that Xiao Yi's method is completely beyond his imagination.

Representation theory...

Before, he never thought that he could use representation theory to prove the hail conjecture.

How did Xiao Yi think of it?

He couldn't help but have such a question in his heart.

But then, he smiled bitterly again.

It seems that I ask this question every time I read Xiao Yi's paper.

Okay, now it seems...

Maybe it is because Xiao Yi thought of using representation theory that he was able to prove it successfully?

However, as he continued to read below, he realized that his idea was only half right...

Maybe even less than half.

"It turns out to be like this!"

Just the first few steps have made him see the extraordinaryness of Xiao Yi's method.

He was still very confused about how Xiao Yi combined the representation theory method with the hailstone conjecture, but after seeing the first few steps, he understood it in his heart, and then he was amazed at this cross-disciplinary thinking.

Transforming a problem from number theory and dynamical systems into the language of representation theory requires a deep understanding of different fields and a unique mathematical perspective.

It is already difficult to even think of this step, let alone successfully integrating these steps into it.

So, it is not because Xiao Yi found a method of representation theory to study that he successfully proved the hailstone conjecture, but precisely because Xiao Yi chose the method of representation theory, representation theory can be used to prove the hailstone conjecture.

Otherwise, this method will most likely not be discovered for a long, long time, even until others successfully prove the hailstone conjecture with other methods.

Unless it is...

A long time after the Langlands Program is successfully implemented.

However, when he suddenly thought of the Langlands Program, Tao suddenly realized something, and then continued to read on. In the end, he found one of the most valuable conclusions in this paper, in addition to proving the hailstone conjecture.

That is to make a closer connection between Diophantine theory and representation theory.

And this achievement can naturally be regarded as a major breakthrough in the Langlands Program. Diophantine theory itself is an important field in number theory. It can achieve a deeper connection with representation theory, which is naturally a result of the Langlands Program.

"It's unbelievable..." Tao exclaimed, Xiao Yi could actually think of such an idea.

Such a result was achieved inadvertently.

Of course, it is normal to think about it. To be able to solve the hailstone conjecture, there must be some equally important results in it.

The value of the hail conjecture is never in the conclusion, but in the process.

The same is true for most problems in number theory.

If you continue to read further, there are more exciting ideas or methods, and each step is so brilliant.

Of course, it does not reduce the writing style of Xiao Yi's papers in those years.

One of the most important styles is that it is detailed and easy to understand. Xiao Yi's papers have not many simple steps, and they have always been a unique one in the mathematics world in terms of readability, so they are also highly praised by the academic community.

And until now, Xiao Yi's advantage is still maintained. Many students who have just officially entered the mathematics world often get recommendations from their teachers to let them read more of Xiao Yi's papers and learn his writing style.

Therefore, now there are already quite a few students who learn Xiao Yi's style when writing mathematics papers.

And not only these students, even now there are quite a few mathematicians who have begun to learn from Xiao Yi's writing style.

This is not to say that they want to learn well, but mainly because they found that learning Xiao Yi's writing style can help improve the citation rate of their papers. Many journals have also found this. Therefore, in order to improve the impact factor of the journal, editors will be more inclined to accept such papers when accepting articles.

To a certain extent, Xiao Yi has also brought changes to the mathematics community.

For this, Terence Tao has always praised.

Continuing to look down, Xiao Yi connected the invariant vector with the periodic orbit, and successfully discovered the one-to-one correspondence between the invariant vector in the representation space and the periodic orbit of the hail sequence. This ability to connect abstract algebraic concepts with specific number theory problems also made Terence Tao feel that he had gained something.

Until the end, the paper proved the finiteness of the classification tree, and the methods shown in the process also showed the depth of this paper.

"Another such a perfect paper."

Until the page on the computer turned to the side of the literature citation, Terence Tao finished reading all the contents of the text and made a final sigh.

Xiao Yi's papers always surprise him and never disappoint him.

"Perfect!"

After snapping his fingers, Tao Zhexuan began his daily routine after reading a heavyweight paper - writing comments on his blog.

Especially for such a wonderful paper, it is necessary for him to express his own views.

As for whether Xiao Yi's proof is correct...

Is there any need to say more?

Anyway, he couldn't find any mistakes in the whole paper because it was written in enough detail.

If he couldn't find any mistakes after reading it once, he didn't plan to read it again.

Anyway, he just assumed that Xiao Yi had successfully proved it.

As for whether this is a bit imprecise?

Well...

Considering that he is Xiao Yi, Tao Zhexuan feels that he has read it completely, which is already rigorous enough.

Entering the blog, he started typing.

[There is no doubt that we have received another good news today, that is, the hail conjecture has been successfully proved!

It has been almost 100 years since this problem was born. It is amazing that such a problem has puzzled our mathematics community for so many years.

Fortunately, Xiao successfully solved this problem when it was about to reach a century of history. It can be said that he saved the face of our mathematics community?

Ahem, let's get back to the topic. In short, I have read the paper once. Although it is not rigorous enough to conclude that Xiao has succeeded, I think the hail conjecture can be called the hail theorem, just like what I wrote in the first paragraph.