The top student must be diligent

He expressed his gratitude sincerely: "I understand, thank you, teacher."

Xiao Yi nodded, and then said: "Okay, do you have any other questions?"

"No, no, if you have any more questions, I will come to you again." Liang Qiushi shook his head repeatedly.

"Well, remember to finish the paper early, and don't forget to submit the final draft before June."

"Sure!" Liang Qiushi gestured OK: "Don't worry, I have actually written the previous results a long time ago. If this extension is not completed, I will send you the previous results at that time, it doesn't matter."

Xiao Yi rolled his eyes: "Then you might as well send the previous results earlier."

Liang Qiushi waved his hand, said nothing, and left the office directly after saying goodbye.

Xiao Yi was the only one left again.

However, at this time, Xiao Yi recalled the process of guiding Liang Qiushi just now, and a thoughtful look appeared on his face.

"Diophantine approximation?"

"If Diophantine approximation is used to study the generation problem of invariant vectors...what will happen?"

He thought a little in his heart.

"Convert the problem into a solution problem of a Diophantine equation system?"

The inspiration in his mind began to jump up, and various thoughts began to collide with each other.

Finally, his intuition told him that this idea has great potential!

"If we fix a set of bases, represent the linear transformation ρ(g) with the matrix A_g, and represent the vector v with the coordinate vector (x_1,...,x_n), then the invariance condition ρ(g)v=v becomes a set of linear equations."

He picked up the draft paper next to him and wrote on it.

[A_g (x_1,...,x_n)^T =(x_1,...,x_n)^T,g∈G]

"If G is finitely generated, for example, generated by g_1,...,g_m, then the above conditions are equivalent to a Diophantine equation system..."

At this moment, Xiao Yi's eyes have gradually lit up.

"It seems that it really works!"

As long as it is converted into a Diophantine equation system, he can use the Diophantine approximation method to directly give a constructive solution!

And now, he is very close to this goal.

"Liang Qiushi, Liang Qiushi, I didn't expect you to be able to bring me some inspiration."

Xiao Yi sighed.

Although it is not a direct inspiration, it can only be said that when he was thinking about the theory of Diophantine equations just now, he accidentally touched on this inspiration.

Now, it has become a little easier for him to get such inspiration.

Even... it can no longer be called inspiration, but called... common sense?

"Okay, it's time to tackle this step next."

Xiao Yi raised his eyebrows slightly, and then continued to study the next step.

He could foresee that after this step was solved, the remaining obstacles would become quite simple.

...

Chapter 259 Request for the H-20 Project

As Xiao Yi expected, when he found the method of Diophantine approximation to solve this problem, the problem was solved by him like a broken bamboo.

"Now, I have found a unified method that can directly generate invariant vectors in each representation space."

After Xiao Yi successfully constructed this method on the draft paper, all the problems seemed to become clear in front of him.

As for the idea of ​​using Diophantine approximation, the method found is not only applicable to this problem.

It can be applied to various representation spaces and convert these problems into a Diophantine equation.

In other words, to a certain extent, he directly connected the relationship between Diophantine theory and representation theory more closely!

This can be regarded as an important breakthrough in the Langlands program!

After all, although the geometric Langlands conjecture has been proved, the Langlands conjecture has not been solved.

These are two completely different conjectures, just like the geometric Langlands program and the Langlands program, they are also different concepts and are independent of each other.

Maybe his achievement can help prove the Langlands conjecture?

At least his intuition tells him that it is possible.

However, whether to study the Langlands conjecture is not considered for the time being. If he has the energy to study the Langlands conjecture, he would rather try the "ultimate" conjecture, the Riemann conjecture.

As for now...

Let's first consider the problem behind the hail conjecture.

However, the problem behind it has indeed become simpler.

"Although it is simple, it is also a key point."

"Use this unified invariant vector generation method to realize the classification of periodic orbits..."

Xiao Yi narrowed his eyes slightly and thought about it.

"First, let's review the previous process. Since each invariant vector can be obtained by solving a Diophantine equation group, then we first assume that v=(v_1, v_2, ., v_n) is an invariant vector, then it satisfies such an equation group..."

[3v_i + 1 = v_j, if i is an odd number and H_i=(3i+1, i)

v_i/2 = v_k, if i is an even number and H_i=(i/2, i)]

"A periodic orbit of length 1 corresponds to a fixed point, that is, an invariant vector that satisfies v_i=i. Well, obviously, this condition is only satisfied when v_1=1."

"Then there is a periodic orbit of length 2..."

"Then there is a longer periodic orbit..."

Following Xiao Yi's thinking, the research on the problem has also reached a very in-depth level.

Until more than ten pages of draft paper have been used, Xiao Yi looked at the complete derivation process above and the result he finally got, and suddenly he was stunned.

At this time, he successfully constructed a tree structure of periodic orbits, and this tree structure is gradually recursive.

However, based on observation, he can clearly know that all possible hail sequence periodic orbits can be completely classified by a recursive tree structure.

Each layer of this structure corresponds to a periodic orbit of fixed length, and the branches of the tree correspond to the evolution from one periodic orbit to another longer periodic orbit.

That is to say, at this time, if he wants to prove the hail conjecture, he only needs to prove that: if the classification tree is infinite, then there must be a starting number whose corresponding hail sequence does not enter any periodic orbit, which will contradict the hail conjecture.

Therefore, in order to prove the hail conjecture, he only needs to prove that the classification tree is finite, and then he can complete the hail conjecture!

"So, this last question is derived?"

Xiao Yi was a little surprised.

Isn't it a bit...

simple?

Now, as long as he completes this last proposition, he can completely prove the hail conjecture.

Although the difficulty of this proposition seems to be very high, from his intuition, if he really wants to solve this problem, it will not be particularly troublesome.

However, thinking back to the time since he started studying the hail conjecture, he didn't seem to have encountered too many difficulties?

Until now, in just about a month, he has already studied this last step.

"This is probably... the horror of level 7."

Xiao Yi sighed in his heart.

After all, he has no shortage of inspiration. If he wants to solve the conjecture, isn't that just a matter of hustling?

For a moment, he even began to think, "The next question, should I study the Riemann hypothesis directly?"

It seems that there is a lot of potential!

If we really want to talk about difficulty, the hailstone conjecture is definitely not as difficult as the Riemann hypothesis. After all, the hailstone conjecture was born less than a hundred years ago, while the Riemann hypothesis has been born for more than a hundred years.

The most important thing is that there are definitely not as many mathematicians studying the hailstone conjecture as there are mathematicians studying the Riemann hypothesis.

However, the mathematical community still recognizes the difficulty of the hailstone conjecture. Otherwise, the great mathematician Erdős would not have lamented that the current mathematics is not ready to face such a problem.

For a moment, Xiao Yi's heart began to stir.

Of course, if you really want to study the Riemann hypothesis, you should wait until the current hailstone conjecture is completed.

Refocus your attention on the hailstone conjecture in front of you.

"Then let's continue."

However, at this moment, perhaps God didn't want him to solve this conjecture so calmly, his cell phone suddenly rang.

He saw the caller, it was Director Liu.

He raised his eyebrows slightly, why did Liu Cong call him now?

There shouldn't be anything important at this time, right?

Is it possible that they are planning to build a new nuclear fusion reactor and ask him to be a consultant?